Geometric Quantization with Applications to Gromov … · 2013-09-05 · Geometric Quantization with Applications to Gromov-Witten Theory Emily Clader, ... 3.5. Semi-classical limit

Geometric Quantization with

Applications to Gromov-Witten

Theory

Emily Clader Nathan Priddis and

Mark Shoemaker

The notes that follow are expository and the authors make no claimto originality of any of the material appearing in them

Please send questions or comments to

Emily Clader ecladerumicheduNathan Priddis priddisnumicheduMark Shoemaker markashoegmailcom

Contents

Preface 1

Chapter 0 Preliminaries 501 Basics of symplectic geometry 502 Feynman diagrams 8

Chapter 1 Quantization of finite-dimensional symplecticvector spaces 15

11 The set-up 1512 Quantization of symplectomorphisms via the CCR 1913 Quantization via quadratic Hamiltonians 2314 Integral formulas 2515 Expressing integrals via Feynman diagrams 2716 Generalizations 32

Chapter 2 Interlude Basics of Gromov-Witten theory 3521 Definitions 3522 Basic Equations 3823 Axiomatization 41

Chapter 3 Quantization in the infinite-dimensional case 4331 The symplectic vector space 4332 Quantization via quadratic Hamiltonians 4533 Convergence 5134 Feynman diagrams and integral formulas revisited 5135 Semi-classical limit 54

Chapter 4 Applications of quantization to Gromov-Wittentheory 59

41 Basic equations via quantization 5942 Giventalrsquos conjecture 6243 Twisted theory 6444 Concluding remarks 67

Bibliography 69

Preface

The following notes were prepared for the ldquoIAS Program onGromov-Witten Theory and Quantizationrdquo held jointly by the De-partment of Mathematics and the Institute for Advanced Study atthe Hong Kong University of Science and Technology in July 2013Their primary purpose is to introduce the reader to the machineryof geometric quantization with the ultimate goal of computations inGromov-Witten theory

First appearing in this subject in the work of Alexander Giventaland his students quantization provides a powerful tool for study-ing Gromov-Witten-type theories in higher genus For example ifthe quantization of a symplectic transformation matches two totaldescendent potentials then the original symplectic transformationmatches the Lagrangian cones encoding their genus-zero theorieswe discuss this statement in detail in Section 35 Moreover accord-ing to Giventalrsquos Conjecture (see Section 42) the converse is truein the semisimple case Thus if one wishes to study a semisim-ple Gromov-Witten-type theory it is sufficient to find a symplectictransformation identifying its genus-zero theory with that of a finitecollection of points which is well-understood The quantization ofthis transformation will carry all of the information about the higher-genus theory in question

In addition quantization is an extremely useful combinatorialdevice for organizing information Some of the basic properties ofGromov-Witten theory for instance can be succinctly expressed interms of equations satisfied by quantized operators acting on thetotal descendent potential To give another example even if oneis concerned only with genus zero the combinatorics of expandingthe relation between two theories into a statement about their gen-erating functions can be unmanageable but when it is expressed viaquantization this unwieldy problem obtains a clean expression Therelationship between a twisted theory and its untwisted analoguediscussed in Section 43 is a key instance of this phenomenon

2 PREFACE

At present there are very few references on the subject of quan-tization as it is used in this mathematical context While a num-ber of texts exist (such as [4] [10] and [26]) these tend to focuson the quantization of finite-dimensional topologically non-trivialsymplectic manifolds Accordingly they are largely devoted to ex-plaining the structures of polarization and prequantization that onemust impose on a symplectic manifold before quantizing and lessconcerned with explicit computations These precursors to quan-tization are irrelevant in applications to Gromov-Witten theory asthe symplectic manifolds one must quantize are simply symplecticvector spaces However other technical issues arise from the factthat the symplectic vector spaces in Gromov-Witten theory are typ-ically infinite-dimensional We hope that these notes will fill a gapin the existing literature by focusing on computational formulas andaddressing the complications specific to Giventalrsquos set-up

The structure of the notes is as follows In Chapter 0 we give abrief overview of preliminary material on symplectic geometry andthe method of Feynman diagram expansion We then turn in Chapter1 to a discussion of quantization of finite-dimensional vector spacesWe obtain formulas for the quantizations of functions on such vectorspaces by three main methods direct computation via the canon-ical commutation relations (which may also be expressed in termsof quadratic Hamiltonians) Fourier-type integrals and Feynman di-agram expansion All three methods yield equivalent results butthis diversity of derivations is valuable in pointing to various gen-eralizations and applications We then include an interlude on basicGromov-Witten theory Though this material is not strictly necessaryuntil Chapter 4 it provides motivation and context for the materialthat follows

The third chapter of the book is devoted to quantization of infinite-dimensional vector spaces such as arise in applications to Gromov-Witten theory As in the finite-dimensional case formulas can beobtained via quadratic Hamiltonians Fourier integrals or Feynmandiagrams and for the most part the computations mimic those ofthe first part The major difference in the infinite-dimensional set-ting though is that issues of convergence arise which we make anattempt to discuss whenever they come up Finally in Chapter 4 wepresent several of the basic equations of Gromov-Witten theory in thelanguage of quantization and mention a few of the more significantappearances of quantization in the subject

PREFACE 3

Acknowledgements The authors are indebted to Huai-LiangChang Wei-Ping Li and Yongbin Ruan for organizing the work-shop at which these notes were presented Special thanks are due toYongbin Ruan for his constant guidance and support as well as toYefeng Shen for working through the material with us and assistingin the creation of these notes YP Lee and Xiang Tang both gaveextremely helpful talks at the RTG Workshop on Quantization orga-nized by the authors in December 2011 at the University of MichiganUseful comments on earlier drafts were provided by Pedro Acostaand Weiqiang He A number of existing texts were used in the au-thorsrsquo learning process we have attempted to include references tothe literature throughout these notes but we apologize in advancefor any omissions This work was supported in part by FRG grantDMS 1159265 RTG and NSF grant 1045119 RTG

CHAPTER 0

Preliminaries

Before we begin our study of quantization we will give a quickoverview of some of the prerequisite background material First wereview the basics of symplectic vector spaces and symplectic man-ifolds This material will be familiar to most of our mathematicalaudience but we collect it here for reference and to establish nota-tional conventions for more details see [8] or [25] Less likely to befamiliar to mathematicians is the material on Feynman diagrams sowe cover this topic in more detail The chapter concludes with thestatement of Feynmanrsquos theorem which will be used later to expresscertain integrals as combinatorial summations The reader who isalready experienced in the methods of Feynman diagram expansionis encouraged to skip directly to Chapter 1

01 Basics of symplectic geometry

011 Symplectic vector spaces A symplectic vector space (V ω)is a vector space V together with a nondegenerate skew-symmetricbilnear form ω We will often denote ω(vw) by 〈vw〉

One consequence of the existence of a nondegenerate bilinearform is that V is necessarily even-dimensional The standard exampleof a real symplectic vector space is R2n with the symplectic formdefined in a basis eα eβ1leαβlen by

langeα eβ

rang= 0

langeα eβ

rang= δαβ (1)

In other words 〈〉 is represented by the (skew-symmetric)matrix

(0 IminusI 0

In fact any finite-dimensional symplectic vector space admits a basisin which the symplectic form is expressed this way Such a basisis called a symplectic basis and the corresponding coordinates areknown as Darboux coordinates

6 0 PRELIMINARIES

If U sub V is a linear subspace of a symplectic vector space (V ω)then the symplectic orthogonal Uω of U is

Uω = v isin V | ω(u v) = 0 for all u isin UOne says U is isotropic if U sub Uω and Lagrangian if U = Uωwhich implies in particular that if V is finite-dimensional dim(U) =12dim(V)

A symplectic transformation between symplectic vector spaces(V ω) and (Vprime ωprime) is a linear map σ V rarr Vprime such that

ωprime(σ(v) σ(w)) = ω(vw)

In what follows we will mainly be concerned with the case V = Vprimeand it will be useful to express the symplectic condition on a linearendomorphism σ V rarr V in terms of matrix identities in Darbouxcoordinates Choose a symplectic basis for V and express σ in thisbasis via the matrix

(A BC D

Then σ is symplectic if and only if σT Jσ = J which in turn holds ifand only if the following three identities are satisfied

ATB = BTA (2)

CTD = DTC (3)

ATDminusBTC = I (4)

Using these facts one obtains a convenient expression for the inverse

σminus1 =

(DT minusBT

minusCT AT

An invertible matrix satisfying (2) - (4) is known as a symplecticmatrix and the group of symplectic matrices is denoted Sp(2nR)

012 Symplectic manifolds Symplectic vector spaces are thesimplest examples of the more general notion of a symplectic mani-fold which is a smooth manifold equipped with a closed nondegen-erate two-form called a symplectic form

In particular such a 2-form makes the tangent space TpM at anypoint p isin M into a symplectic vector space Just as every symplec-tic vector space is isomorphic to R2n with its standard symplecticstructure Darbouxrsquos Theorem states that every symplectic manifoldis locally isomorphic to R2n = (x1 xn y1 yn) with the sym-plectic form ω =

sumni=1 dxi and dyi

01 BASICS OF SYMPLECTIC GEOMETRY 7

Perhaps the most important example of a symplectic manifold isthe cotangent bundle Given any smooth manifold N (not necessarilysymplectic) there is a canonical symplectic struture on the total spaceof TlowastN To define the symplectic form let π TlowastN rarr N be theprojection map Then one can define a one-form λ on TlowastN by setting

λ|ξx = πlowast(ξx)

for any cotangent vector ξx isin TlowastxN sub TlowastN This is known as thetautological one-form The canonical symplectic form on TlowastN isω = minusdλ The choice of sign makes the canonical symplectic structureagree with the standard one in the case where N = Rn which sitsinside of TlowastN R2n as Spane1 en in the basis notation usedpreviously

In coordinates the tautological one-form and canonical two-formappear as follows Let q1 qn be local coordinates on N Thenthere are local coordinates on TlowastN in which a point in the fiberover (q1 qn) can be expressed as a local system of coordinates(q1 qn p1 pn) and in these coordinates

λ =sum

pαdqα

ω =sum

dqα and dpα

The definitions of isotropic and Lagrangian subspaces generalizeto symplectic manifolds as well A submanifold of a symplecticmanifold is isotropic if the restriction of the symplectic form to thesubmanifold is zero An isotropic submanifold is Lagrangian if itsdimension is as large as possiblemdashnamely half the dimension of theambient manifold

For example in the case of the symplectic manifold TlowastN with itscanonical symplectic form the fibers of the bundle are all Lagrangiansubmanifolds as is the zero section Furthermore if micro Nrarr TlowastN is aclosed one-form the graph of micro is a Lagrangian submanifold Moreprecisely for any one-form micro one can define

Xmicro = (x micro(x)) | x isin N sub TlowastN

and Xmicro is Lagrangian if and only if micro is closed In case N is simply-connected this is equivalent to the requirement that micro = d f forsome function f called a generating function of the Lagrangiansubmanifold Xmicro

8 0 PRELIMINARIES

Finally the notion of symplectic transformation generalizes inan obvious way Namely given symplectic manifolds (M ω1) and(N ω2) a symplectomorphism is a smooth map f Mrarr N such thatf lowastω2 = ω1

02 Feynman diagrams

The following material is drawn mainly from [12] other refer-ences on the subject of Feynman diagrams include Chapter 9 of [19]and Lecture 9 of [12]

Consider an integral of the form

~minusd2

eminusS(x)~dx (5)

where V is a d-dimensional vector space ~ is a formal parameter and

S(x) =1

2B(x x) +

mBm(x x) (6)

for a bilinear form B and m-multilinear forms Bm on V where gm areconstants The integral (5) can be understood as a formal series inthe parameters ~ and gm

021 Wickrsquos Theorem We begin by addressing the simpler ques-tion of computing integrals involving an exponential of a bilinearform without any of the other tensors

Let V be a vector space of dimension d over R and let B bea positive-definite bilinear form on V Wickrsquos theorem will relateintegrals of the form

l1(x) middot middot middot lN(x)eminusB(xx)2dx

in which l1 lN are linear forms on V to pairings on the set [2k] =1 2 2k By a pairing on [2k] we mean a partition of the set intok disjoint subsets each having two elements Let Π2k denote the setof pairings on [2k] The size of this set is

|Π2k| =(2k)

as the reader can check as an exerciseAn element σ isin Πk can be viewed as a special kind of permutation

on the set [2k] which sends each element to the other member of itspair Write [2k]σ for the set of orbits under this involution

02 FEYNMAN DIAGRAMS 9

Theorem 01 (Wickrsquos Theorem) Let l1 lN isin Vlowast If N is eventhen

l1(x) lN(x)eminusB(xx)

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

If N is odd the integral is zero

Proof First apply a change of variables such that B is of the formB(x x) = x2

1+ middot middot middot + x2

d The reader should check that this change of

variables changes both sides of the equation by a factor of det(P)where P is the change-of-basis matrix and thus the equality prior tothe change of variables is equivalent to the result after the changeFurthermore since both sides are multilinear in elements of Vlowast andsymmetric in x1 xd we may assume l1 = l2 = middot middot middot = lN = x1 Thetheorem is then reduced to computing

xN1 eminus(x2

1+middotmiddotmiddot+x2

This integral indeed vanishes when N is odd since the integrand isan odd function If N is even write N = 2k In case k = 0 the theoremholds by the well-known fact that

int infin

minusinfineminusx2

2 dx =radic

For k gt 0 we can use this same fact to integrate out the last d minus 1variables by which we see that the claim in the theorem is equivalentto int infin

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

To prove (7) we first make the substitution y = x2

2 Recall that the

gamma function is defined by

Γ(z) =

int infin

yzminus1eminusydy

and satisfies Γ(12) =radicπ as well as

Γ(z + k) = (z + k minus 1)(z + k minus 2) middot middot middot z middot Γ(z)

10 0 PRELIMINARIES

for integers k Thus we haveint infin

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

(2y)kminus12eminusy dy

= 2k+12Γ(k + 12)

= 2k+12(k minus 12)(k minus 3

2)Γ(1

=radic

2π(2k)

which proves the claim

022 Feynmanrsquos theorem Now let us return to the more generalintegral

Z = ~minusd2

eminusS(x)~dx

which for reasons we will mention at the end of the section issometimes called a partition function Recall that S has an expansionin terms of multilinear forms given by (6)

Because a pairing can be represented by a graph all of whosevertices are 1-valent Wickrsquos theorem can be seen as a method forexpressing certain integrals as summations over graphs in whicheach graph contributes an explicit combinatorial term The goal ofthis section is to give a similar graph-sum expression for the partitionfunction Z

Before we can state the theorem we require a bit of notation Ifn = (n0 n1 ) is a sequence of nonnegative integers all but finitelymany of which are zero let G(n) denote the set of isomorphism classesof graphs with ni vertices of valence i for each i ge 0 Note that thenotion of ldquographrdquo here is very broad they may be disconnected andself-edges and multiple edges are allowed If Γ is such a graph let

b(Γ) = |E(Γ)| minus |V(Γ)|where E(Γ) and V(Γ) denote the edge set and vertex set respectivelyAn automorphism of Γ is a permutation of the vertices and edgesthat preserves the graph structure and the set of automorphisms isdenoted Aut(Γ)

We will associate a certain number FΓ to each graph known asthe Feynman amplitude The Feynman amplitude is defined by thefollowing procedure

(1) Put the m-tensor minusBm at each m-valent vertex of Γ

(2) For each edge e of Γ take the contraction of tensors attachedto the vertices of e using the bilinear form Bminus1 This willproduce a number FΓi

for each connected component Γi of Γ(3) If Γ =

⊔i Γi is the decomposition of Γ into connected compo-

nents define FΓ =prod

i FΓi

By convention we set the Feynman amplitude of the empty graph tobe 1

Theorem 02 (Feynmanrsquos Theorem) One has

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

where the outer summation is over all sequences of nonnegative integerswith almost all zero

Before we prove the theorem let us compute a few examples ofFeynman amplitudes to make the procedure clear

Example 03 Let Γ be the following graph

Given B V otimes V rarr R we have a corresponding bilinear form Bminus1 Vor otimesVor rarr R Moreover B1 isin Vor and so we can write the Feynmanamplitude of this graph as

FΓ = (Bminus1(minusB1minusB1))2

Example 04 Consider now the graph

Associated to B is a map B V rarr Vor and in this notation theFeynman amplitude of the graph can be expressed as

FΓ = minusB3(Bminus1

(minusB1)Bminus1

(minusB1))

Proof of Feynmanrsquos Theorem After a bit of combinatorial fid-dling this theorem actually follows directly from Wickrsquos theorem

First perform the change of variables y = xradic~ under which

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

Expanding the exponential as a series gives

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Each of the factors minusBi(y y) in this integral can be expressed as asum of products of i linear forms on V After unpacking the expresionin this way Zn becomes a sum of integrals of the formint

eminusB(yy)2(one linear form

)n1(product of two linear forms

)n2 middot middot middot

each with an appropriate coefficient so we will be able to applyWickrsquos theorem

Let N =sum

i i middot ni which is the number of linear forms in the aboveexpression for Zn We want to express the integral Zn using graphsTo this end we draw n0 vertices with no edges n1 vertices with 1 half-edge emanating from them n2 vertices with 2 half-edges n3 verticeswith 3 half-edges et cetera these are sometimes called ldquoflowersrdquo Weplace minusBi at the vertex of each i-valent flower A pairing σ isin ΠN canbe understood as a way of joining pairs of the half-edges to form fulledges which yields a graph Γ(σ) and applying Wickrsquos theorem weget a number F(σ) from each such pairing σ One can check that allof the pairings σ giving rise to a particular graph Γ = Γ(σ) combineto contribute the Feynman amplitude FΓ(σ) In particular by Wickrsquostheorem only even N give a nonzero result

At this point we have

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

where we have used the straightforward observation that the expo-nent

sumni(

i2minus 1) = N

2minus sum

ni on ~ is equal to the number of edgesminus the number of vertices of any of the graphs appearing in thesummand corresponding to n All that remains is to account for thefact that many pairings can yield the same graph and thus we willobtain a factor when we re-express the above as a summation overgraphs rather than over pairings

To compute the factor fix a graph Γ and consider the set P(Γ) ofpairings on [N] yielding the graph Γ Let H be the set of half-edges ofΓ which are attached as above to a collection of flowers Let G be thegroup of permutations of H that perserve flowers this is generatedby permutations of the edges within a single flower as well as swapsof two entire flowers with the same valence Using this it is easy tosee that

|G| =prod

(i)nini

The group G acts transitively on the set P(Γ) and the stabilizer ofthis action is equal to Aut(Γ) Thus the number of distinct pairingsyielding the graph Γ is prod

i(i)nini

|Aut(Γ)|

and the theorem follows

We conclude this preliminary chapter with a bit of ldquogenerating-functionologyrdquo that motivates the term ldquopartition functionrdquo for Z

Theorem 05 Let Z0 =(2π)d2

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

where the outer summation is over n as before and Gc(n) denotes the set ofisomorphism classes of connected graphs in G(n)

The proof of this theorem is a combinatorial exercise and will beomitted Note that if we begin from this theorem then the termsin Feynmanrsquos Theorem can be viewed as arising from ldquopartitioningrdquoa disconnected graph into its connected components This explainsthe terminology for Z in this particular situation

CHAPTER 1

Quantization of finite-dimensional symplectic vector

spaces

From a physical perspective geometric quantization arises froman attempt to make sense of the passage from a classical theory tothe corresponding quantum theory The state space in classical me-chanics is represented by a symplectic manifold and the observables(quantities like position and momentum) are given by smooth real-valued functions on that manifold Quantum mechanics on the otherhand has a Hilbert space as its state space and the observables aregiven by self-adjoint linear operators Thus quantization should as-sociate a Hilbert space to each symplectic manifold and a self-adjointlinear operator to each smooth real-valued function and this processshould be functorial with respect to symplectic diffeomorphisms

By considering certain axioms required of the quantization pro-cedure we show in Section 11 that the Hilbert space of states can beviewed as a certain space of functions Careful study of these axiomsleads in Section 12 to a representation of a quantized symplectictransformation as an explicit expression in terms of multiplicationand differentiation of the coordinates However as is explained inSection 14 it can also be expressed as a certain integral over theunderlying vector space This representation has two advantagesFirst the method of Feynman diagrams allows one to re-write it asa combinatorial summation as is explained in Section 15 Secondit can be generalized to the case in which the symplectic diffeomor-phism is nonlinear Though the nonlinear case will not be addressedin these notes we conclude this chapter with a few comments onnonlinear symplectomorphisms and other possible generalizationsof the material developed here

11 The set-up

The material of this section is standard in the physics literatureand can be found for example in [4] or [10]

16 1 QUANTIZATION OF FINITE-DIMENSIONAL SYMPLECTIC VECTOR SPACES

111 Quantization of the state space Let V be a real symplecticvector space of dimension 2n whose elements are considered to bethe classical states Roughly speaking elements of the associatedHilbert space of quantum states will be square-integrable functionson V It is a basic physical principle however that quantum statesshould depend on only half as many variables as the correspond-ing classical states there are n position coordinates and n momen-tum coordinates describing the classical state of a system whereas aquantum state is determined by either position or momentum aloneThus before quantizing V it is necessary to choose a polarizationa decomposition into half-dimensional subspaces Because thesetwo subspaces should be thought of as position and momentumwhich mathematically are the zero section and the fiber directionof the cotangent bundle to a manifold they should be Lagrangiansubspaces of V

The easiest way to specify a polarization in the context of vectorspaces is to choose a symplectic basis e = eα eβ1leαβlen Recall such abasis satisfies

langeα eβ

rang= 0

langeα eβ

rang= δαβ

where 〈middot middot〉 is the symplectic form on V The polarization may bespecified by fixing the subspace R = Spaneα1leαlen and viewing V asthe cotangent bundle TlowastR in such a way that 〈middot middot〉 is identified withthe canonical symplectic form1 An element of V will be written inthe basis e as sum

pαeα +sum

qβeβ

For the remainder of these notes we will suppress the summationsymbol in expressions like this adopting Einsteinrsquos convention thatwhen Greek letters appear both up and down they are automati-cally summed over all values of the index For example the abovesummation would be written simply as pαeα + qβeβ

Let V R2n be a symplectic vector space with symplectic basis e =eα eβ1leαβlen Then the quantization of (V e) is the Hilbert space He

of square-integrable functions on R which take values in C[[~ ~minus1]]Here ~ is considered as a formal parameter although physically itdenotes Plankrsquos constant

1Note that in this identification we have chosen for the fiber coordinates to bethe first n coordinates It is important to keep track of whether upper indices orlower indices appear first in the ordering of the basis to avoid sign errors

11 THE SET-UP 17

It is worth noting that while it is necessary to impose square-integrability in order to obtain a Hilbert space in practice one of-ten needs to consider formal functions on R that are not square-integrable The space of such formal functions from R to C[[~ ~minus1]]is called the Fock space

112 Quantization of the observables Observables in the clas-sical setting are smooth functions f isin Cinfin(V) and the result of ameasurement is the value taken by f on a point of V In the quantumframework observables are operators U on He and the result of ameasurement is an eigenvalue of U In order to ensure that theseeigenvalues are real we require that the operators be self-adjoint

There are a few other properties one would like the quantizationQ( f ) of observables f to satisfy We give one possible such list belowfollowing Section 3 of [4] Here a set of observables is called com-plete if any function that ldquoPoisson-commutesrdquo (that is has vanishingPoisson bracket) with every element of the set is a constant functionLikewise a set of operators is called complete if any operator thatcommutes with every one of them is the identity

The quantization procedure should satisfy

(1) Linearity Q(λ f + g) = λQ( f ) +Q(g) for all f g isin Cinfin(V) andall λ isin R

(2) Preservation of constants Q(1) = id(3) Commutation [Q( f )Q(g)] = ~Q( f g) where denotes

the Poisson bracket2

(4) Irreducibility If f1 fk is a complete set of observablesthen Q( f1) Q( fk) is a complete set of operators

However it is in general not possible to satisfy all four of theseproperties simultaneously In practice this forces one to restrict toquantizing only a certain complete subset of the observables or torelax the properties required We will address this in our particularcase of interest shortly

One complete set of observables on the state space V is givenby the coordinate functions pα qβαβ=1n and one can determine aquantization of these observables by unpacking conditions (1) - (4)above Indeed when f and g are coordinate functions condition (3)reduces to the canonical commutation relations (CCR ) where wewrite x for Q(x)

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

2This convention differs by a factor of i from what is taken in [4] but we chooseit to match with what appears in the Gromov-Witten theory literature

The algebra generated by elements qα and pβ subject to these commu-tation relations is known as the Heisenberg algebra Thus the abovecan be understood as requiring that the quantization of the coordi-nate functions defines a representation of the Heisenberg algebra BySchurrsquos Lemma condition (4) is equivalent to the requirement thatthis representation be irreducible The following definition providessuch a representation

The quantization of the coordinate functions3 is given by

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

forΨ isinHeIn fact the complete set of observables we will need to quantize

for our intended applications consists of the quadratic functions inthe Darboux coordinates To quantize these order the variables ofeach quadratic monomial with q-coordinates on the left and quantizeeach variable as above That is

Q(qαqβ) = qαqβ

Q(qαpβ) = ~qα part

partqβ

Q(pαpβ) = ~2 part2

partqαpartqβ

There is one important problem with this definition the com-mutation condition (3) holds only up to an additive constant whenapplied to quadratic functions More precisely

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

where C is the cocycle given by

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

and C( f g) = 0 for any other pair of quadratic monomials f andg This ambiguity is sometimes expressed by saying that the quan-tization procedure gives only a projective representation of the Liealgebra of quadratic functions in the Darboux coordinates

3To be precise these operators do not act on the entire quantum state spaceHe because elements of He may not be differentiable However this will not be anissue in our applications because quantized operators will always act on powerseries

12 QUANTIZATION OF SYMPLECTOMORPHISMS VIA THE CCR 19

113 Quantization of symplectomorphisms All that remainsis to address the issue of functoriality That is a symplectic diffeo-morphism T V rarr V should give rise to an operator UT He rarr HeIn fact we will do something slightly different we will associate to Tan operator UT He rarrHe In certain cases such as when T is upper-triangular there is a natural identification between He and He so ourprocedure does the required job More generally the need for suchan identification introduces some ambiguity into the functoriality ofquantization

Furthermore we will consider only the case in which T is linearand is the exponential of an infinitesimal symplectic transforma-tion which is simply a linear transformation whose exponential issymplectic For applications to Gromov-Witten theory later suchtransformation are the only ones we will need to quantize

The computation of UT is the content of the next three sections

12 Quantization of symplectomorphisms via the CCR

The following section closely follows the presentation given in[28]

Let T V rarr V be a linear symplectic isomorphism taking thebasis e to a new basis e = eα eβ via the transformation

eα = Aαβeβ + Cβαeβ

eα = Bβαeβ +Dβαeβ

Let pα and qβ be the corresponding coordinate functions for the basis

vectors eα and eβ Then pβ and qα are defined in the same way asabove The relation between the two sets of coordinate functions is

pα = Aβαpβ + Bαβq

β (9)

qα = Cαβpβ +Dαβ qβ

and the same equations give relations between their respective quan-tizations We will occasionally make use of the matrix notation

p = Ap + Bq

q = Cp +Dq

to abbreviate the aboveTo define the operator He rarrHe associated to the transformation

T observe that by inverting the relationship (9) and quantizing one

can view both pα qβ and pα qβ as representations of He The operator

UT will be defined by the requirement that

ˆqαUT = UTqα (10)

ˆpαUT = UTpα (11)

As the computation will show these equations uniquely specify UT

up to a multiplicative constantTo obtain an explicit formula for UT we restrict as mentioned

above to the case in which

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

then T is the derivative at t = 0 of the family of transformations Tt

The matrix(

a bc d

)satisfies the infinitesimal symplectic relations

bT = b cT = c aT = minusd (12)

which follow from the relations defining T by differentiationThis family of transformations yields a family of bases eαt et

αdefined by

eαt = A(t)αβeβ + C(t)βαeβ

etα = B(t)βαeβ +D(t)

βαeβ

with corresponding coordinate functions ptα q

αt Note that we obtain

the original transformation T as well as the original basis e andcoordinate functions pα qβ by setting t = 1

Using the fact that

Tminus1t = exp

(minusat minusbtminusct minusdt

(I minus ta minustbminustc I minus td

)+O(t2)

we obtain the relations

eα = eαt minus taαβeβt minus tcβαet

β +O(t2)

eα = etα minus tbβαe

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

βαpβ minus tbαβq

β +O(t2)

qαt = qα minus tcαβpβ minus tdαβqβ +O(t2)

and consequently

ptα = pα minus ta

βαpβ minus tbαβqβ +O(t2) (13)

qαt = qα minus tcαβpβ minus tdαβ qβ +O(t2)

Let Ut = UTt so that

ptαUt = Utpα (14)

qαt Ut = Utqα

Denote by u the infinitesimal variation of UT

∣∣∣∣∣t=0

By plugging (13) into the above equations and taking the derivativeat t = 0 we derive commutation relations satisfied by u

[pα u] = aβαpβ + bαβqβ

[qα u] = cαβpβ + dαβ qβ

These equations allow us to determine the infinitesimal variation ofU up to a constant

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

After identifying pα and qα with the operators ~ partpartqα

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

Expanding (14) with respect to t yields formulas for[pα

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

] Using these one can check that if Tt takes the

form Tt = exp(t(

a bc d

)) then Ut will take the form Ut = exp(tu) so in

particular UT = exp(u) Thus equation (15) gives a general formulafor UT

The formula simplifies significantly when T takes certain specialforms Let us look explicitly at some particularly simple cases

Example 11 Consider first the case where b = c = 0 In this casewe obtain

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

Let us verify this formula in a very easy case Suppose that a hasonly one nonzero entry and that this entry is off the diagonal so

that aβα = δ

jfor some fixed i j Assume furthermore that ψ is a

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)k(qi)k(q j)c jminusk

)cα(q j + a

iqi)c j

= ψ(ATq)

where we use that A = exp(a) = I + a in this case

Example 12 In the case where a = c = 0 or in other words T isupper-triangular the formula above directly implies

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

However in this case it is easy to check that B = b and we obtain

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

Example 13 Finally consider the case where a = b = 0 so that Tis lower-triangular Then

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

13 QUANTIZATION VIA QUADRATIC HAMILTONIANS 23

Remark 14 It is worth noting that this expression can be eval-uated using Feynman diagram techniques in which each diagramcorresponds to a term in the Taylor series expansion of the exponen-tial we will discuss this further in Section 154

In fact the above three examples let us compute UT for any sym-

plectic matrix T =

(A BC D

)= exp

(a bc d

)for which the lower-right

submatrix D is invertible4 To do so decompose T as

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

Each of the matrices on the right falls into one of the cases we cal-culated above Furthermore the quantization procedure satisfiesUT1T2

= UT1UT2

(up to a constant) since both sides satisfy (10) and(11) Thus the formula for the quantization UT of any such T is asfollows

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

or in matrix notation

(UTψ)(q) = exp(

2~(BDminus1q) middot q

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

One should be somewhat careful with this expression since thetwo exponentials have respectively infinitely many negative powersof ~ and infinitely many positive powers of ~ so a priori their compo-sition may have some powers of ~ whose coefficients are divergentseries Avoiding this issue requires one to apply each quantized op-erator toψ(Dminus1q) in turn verifying at each stage that the coefficient ofevery power of ~ converges A similar issue will arise when dealingwith powers of the variable z in the infinite-dimensional setting seeSection 33 of Chapter 3

13 Quantization via quadratic Hamiltonians

Before moving on to other expressions for the quantization UTlet us briefly observe that the formulas obtained in the previoussection can be described in a much simpler fashion by referring tothe terminology of Hamiltonian mechanics We have preferred the

4Throughout this text we will assume for convenience that D is invertibleHowever if this is not the case one can still obtain similar formulas by decompos-ing the matrix differently

longer derivation via the CCR because it more clearly captures theldquoobviousrdquo functoriality one would desire from the quantization pro-cedure but the Hamiltonian perspective is the one that is typicallytaken in discussions of quantization in Gromov-Witten theory (seefor example [6] or [16])

Let T = exp(F) be a symplectomorphism as above where

(a bc d

is an infinitesimal symplectic transformation Because the tangentspace to a symplectic vector space at any point is canonically identi-fied with the vector space itself we can view F V rarr V as a vectorfield on V If ω is the 2-form giving the symplectic structure the con-traction ιFω is a 1-form on V Since V is topologically contractiblewe can write ιFω = dhF for some function hF V rarr R This functionis referred to as the Hamiltonian of F Concretely it is described bythe formula

hF(v) =1

2〈Fv v〉

for v isin V where 〈〉 is the symplectic pairingBeing a classical observable the quantization of the function hF

Vrarr R has already been defined Define the quantization of F by

The quantization of the symplectomorphism T is then defined as

UT = exp(F)

It is an easy exercise to check that F agrees with the general for-mula given by (15) so the two definitions of UT coincide

One advantage of the Hamiltonian perspective is that it providesa straightforward way to understand the noncommutativity of thequantization procedure for infinitesimal symplectic transformationsRecall the quantization of quadratic observables obeys the commu-tation relation

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

for the cocycle C defined in (8) It is easy to check (for example byworking in Darboux coordinates) that the Hamiltonian hA associatedto an infinitesimal symplectic transformation satisfies

hA hB = h[AB]

14 INTEGRAL FORMULAS 25

Thus the commutation relation for infinitesimal symplectic transfor-mations is

[A B] = [AB] + C(hA hB)

For an explicit computation of this cocycle in an (infinite-dimensional)case of particular interest see Example 1341 of [6]

14 Integral formulas

The contents of this section are based on lectures given by XiangTang at the RTG Workshop on Quantization at the University ofMichigan in December 2011 In a more general setting the materialis discussed in [3]

Our goal is to obtain an alternative expression for UT of the form

(UTψ)(q) = λ

(φ(qpprime)minusqprimemiddotpprime)ψ(qprime)dqprimedpprime (16)

for a function φ R2n rarr R and a constant λ isin R[[~ ~minus1]] to be deter-mined Such operators since they generalize the Fourier transformof ψ are known as Fourier integral operators

The advantage of this alternate expression for UT is twofold Firstthey allow quantized operators to be expressed as sums over Feyn-man diagrams and this combinatorial expansion will be useful laterespecially in Section 35 Second the notion of a Fourier integral op-erator generalizes to the case when the symplectic diffeomorphism isnot necessarily linear as well as to the case of a Lagrangian subman-ifold of the cotangent bundle that is not the graph of any symplecto-morphism we will comment briefly on these more general settingsin Section 16 below

To define φ first let ΓT be the graph of T

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

HereR2n denotes the symplectic vector space obtained by equippingR2n with the opposite of the standard symplectic form so that thesymplectic form on the product is given by

〈(p q p q) (PQ P Q)〉 =sum

(minus piQ

i + Piqi + piQ

i minus Piqi)

Under this choice of symplectic form ΓT is a Lagrangian submanifoldof the product

There is an isomorphism of symplectic vector spaces

R2n timesR2n simminusrarr Tlowast(R2n)

(p q p q) 7rarr (q p p q)

where Tlowast(R2n) is equipped with the canonical symplectic form

〈(q p π ξ) (QPΠΞ)〉 =sum

(qiΠi + piΞ

i minusQiπi minus Piξi)

Thus one can view ΓT as a Lagrangian submanifold of the cotan-gent bundle Define φ = φ(q pprime) as the generating function for thissubmanifold Explicitly this says that

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

Let us restrict similarly to Section 12 to the case in which D isinvertible The relations defining ΓT can be rearranged to give

q = Dminus1q minusDminus1Cp

p = BDminus1q +DminusTp

Thereforeφ(q pprime) is defined by the system of partial differential equa-tions

partφ

partq= BDminus1q +DminusTpprime

partφ

partpprime= Dminus1q minusDminus1Cpprime

These are easily solved up to an additive constant one obtains

φ(q pprime) =1

2(BDminus1q) middot q + (Dminus1q) middot pprime minus 1

2(Dminus1Cpprime) middot pprime

The constant λ in the definition of UT is simply a normalizationfactor and is given by

as this will be necessary to make the integral formulas match thosecomputed in the previous section We thus obtain the followingdefinition for UT

(UTψ)(q) =1

( 12 (BDminus1q)middotq+(Dminus1q)middotpprimeminus 1

2 (Dminus1Cpprime)middotpprimeminusqprimemiddotpprime)ψ(qprime)dqprimedpprime

15 EXPRESSING INTEGRALS VIA FEYNMAN DIAGRAMS 27

We should verify that this formula agrees with the one obtainedin Section 12 This boils down to properties of the Fourier transformwhich we will define as

ψ(y) =

eminusixmiddotyψ(x)dx

Under this definition

(UTψ)(q) =e

12~ (BDminus1q)middotq

(Dminus1q)middotpprimeeminus12~ (Dminus1Cpprime)middotpprimeeminus

qprimemiddotpprimeψ(qprime)dqprimedpprime

= ine1

2~ (BDminus1q)middotqint

ei(Dminus1q)middotpprimeprimee~2 (Dminus1Cpprimeprime)middotpprimeprimeψ(pprimeprime)dpprimeprime

= e12~ (BDminus1q)middotq

eiQmiddotpprimeprimeprimeeminus~2 (Dminus1Cpprimeprimeprime)middotpprimeprimeprimeψ i(pprimeprimeprime)dpprimeprimeprime

eiQmiddotpprimeprimeprime(eminus ~2 (Dminus1C part

partqprime )middotpartpartqprime (ψ) i

)and(pprimeprimeprime)dpprimeprimeprime

= e12~ (BDminus1q)middotqe

minus ~2(DCT part

)middot partpartqψ(Dminus1q)

where we use the changes of variables 1~pprime = ipprimeprime ipprimeprime = pprimeprimeprime and

Dminus1q = iQ The integral formula therefore matches the one definedvia the CCR

15 Expressing integrals via Feynman diagrams

The formula given for UTψ in (17) bears a striking resemblace tothe type of integral computed by Feynmanrsquos theorem and in thissection we will make the connection precise

151 Genus-modified Feynmanrsquos Theorem In order to applyFeynmanrsquos theorem the entire integrand must be an exponential sowe will assume that

ψ(q) = e1~

Furthermore let us assume that f (q) is ~minus1 times a power series in ~so ψ is of the form

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

2~ (BDminus1q)middotq ~minusn

eminus1~

S(pprimeqprime)dpprimedqprime

S(pprime qprime) =(minus (Dminus1q) middot pprime + 1

2(Dminus1Cpprime) middot pprime + qprime middot pprime

)minus

~gFg(qprime)

Note that if we let y = (pprime qprime) then the bilinear leading term of S(pprime qprime)(in parentheses above) is equal to

2(Dminus1Cpprime)middotpprime+1

2(qprimeminusDminus1q)middotpprime+1

2pprime middot(qprimeminusDminus1q) =

B(y minusDminus1q y minusDminus1q)

where B(y1 y2) is the bilinear form given by the block matrix(Dminus1C I

and q = (0 q) isin R2nChanging variables we have

(UTψ)(q) = e12~ (BDminus1q)middotq ~minusn

eminus1~

( minusB(yy)2 minussumgge0 ~

gFg(qprime+Dminus1q))dy

Each of the terms minusFg(qprime +Dminus1q) can be decomposed into pieces thatare homogeneous in qprime

minusFg(qprime +Dminus1q) =sum

(minuspartm Fg

∣∣∣qprime=Dminus1q

middot (qprime)m)

where minuspartmFg|qprime=minusDminus1q middot (qprime)m is short-hand for the m-tensor

Bgm = minussum

m1 middot middot middotmn

partmFg

(partqprime1)m1 middot middot middot (partqprimen)mn

∣∣∣∣∣∣qprime=Dminus1q

(qprime1)m1 middot middot middot (qprimen)mn

in which the sum is over all n-tuples m = (m1 mn) isin Znge0 such

that |m| = m1 + middot middot middot + mn = m We consider this as an m-tensor in qprime

whose coefficients involve a formal parameter qThus we have expressed the quantized operator as

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

m Bgm(qprimeqprime))dy

This is essentially the setting in which Feynmanrsquos theorem appliesbut we must modify Feynmanrsquos theorem to allow for the presence ofpowers of ~ in the exponent This is straightforward but neverthelessinteresting as it introduces a striking interpretation of g as recordingthe ldquogenusrdquo of vertices in a graph

Theorem 15 Let V be a vector space of dimension d and let

S(x) =1

2B(x x) +

mBgm(x x)

in which each Bgm is an m-multilinear form and B00 = B01 = B02 = 0Consider the integral

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

where FΓ is the genus-modified Feynman amplitude defined below

Here Gprime(n) denotes the set of isomorphism classes of graphs withni vertices of valence i for each i ge 0 in which each vertex v is labeledwith a genus g(v) ge 0 GivenΓ isin Gprime(n) the genus-modified Feynmanamplitude is defined by the following procedure

(1) Put the m-tensor minusBgm at each m-valent vertex of genus g inΓ

i FΓi

Furthermore the Euler characteristic of Γ is defined as

χΓ = minussum

visinV(Γ)

g(v) + |V(Γ)| minus |E(Γ)|

Having established all the requisite notation the proof of thetheorem is actually easy

Proof of Theorem 15 Reiterate the proof of Feynmanrsquos theoremto obtain

Z =sum

n=(ngm)gmge0

~gngm+( m2 minus1)ngm

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

As before Wickrsquos theorem shows that the integral contributes thedesired summation over graphs modulo factors coming from over-counting A similar orbit-stabilizer argument shows that these fac-tors precisely cancel the factorials in the denominator The power of~ is

2minus 1

which is the sum of the genera of the vertices plus the number of edgesminus the number of vertices or in other words minusχΓ as required

It should be noted that for the proof of the theorem there is noparticular reason to think of g as recording the genus of a vertexmdashitis simply a label associated to the vertex that records which of them-tensors Bgm one attaches The convenience of the interpretation ofg as genus comes only from the fact that it simplifies the power of ~neatly

152 Feynman diagram formula for UT We are now ready togive an expression for UTψ in terms of Feynman diagrams In orderto apply Theorem 15 we must make one more assumption that F0

has no terms of homogeneous degree less than 3 in qprime Assuming thiswe obtain the following expression by directly applying the theorem

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

where FΓ(q) is the genus-modified Feynman amplitude given by plac-ing the m-tensor

partmFg

at each m-valent vertex of genus g in Γ and taking the contraction oftensors using the bilinear form

(Dminus1C I

)minus1

= minus(0 II Dminus1C

In fact since this bilinear form is only ever applied to vectors of theform (0 qprime) we are really only taking contraction of tensors using thebilinear form minusDminus1C on Rn

153 Connected graphs Recall from Theorem 05 that the loga-rithm of a Feynman diagram sum yields the sum over only connectedgraphs That is

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

where Gprimec denotes the set of isomorphism classes of connected genus-labeled graphs Thus writing

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

with Gprimec(g) collecting connected graphs of genus g we have

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

For those familiar with Gromov-Witten theory this expression shouldbe salientmdashwe will return to it in Chapter 3 of the book

154 The lower-triangular case At this point we can return toa remark made previously (Remark 14) regarding the computationof UTψ when T is lower-triangular In that case we may expressthe quantization formula obtained via the CCR as a graph sum in arather different way We explain how to do this below and show thatwe ultimately obtain the same graph sum as that which arises fromapplying Feynmanrsquos Theorem to the integral operator

(I 0C I

We showed in Section 12 that

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

Suppose that ψ(q) = esum

gge0 ~gminus1Fg(q) as above and expand both exponen-

tials in Taylor series Then (UTψ)(q) can be expressed as

iαβℓg

iαβ+sumℓg(gminus1)

prodiαβ

prodℓg

(minusCαβ

)iαβ ( partpartqα

partqβ

)iαβ prod

(Fg(q))ℓg (19)

Whenever a product of quadratic differential operators acts on aproduct of functions the result can be written as a sum over diagramsAs an easy example suppose one wishes to compute

partxparty( f gh)

for functions f g h in variables x and y The product rule gives nineterms each of which can be viewed as a way of attaching an edgelabeled

partpartx

partparty

to a collection of vertices labeled f g and h (We allow both ends ofan edge to be attached to the same vertex)

Applying this general principle to the expression (19) one can

write each of the productsprod

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

)iαβ Fg(q) as a sum over

graphs obtained by taking ℓg vertices of genus g for each g withvertices of genus g labeled Fg and attaching iαβ edges labeled

partpartx minusCαβ2

partparty

in all possible ways Each possibility gives a graph ˆGamma It is acombinatorial exercise to check that the contributions from all choicesof Γ combine to give a factor of

prodiαβmiddotprod ℓg

|Aut(Γ)|

Thus we have expressed (UTψ)(q) assum

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

where GiℓΓ(q) is obtained by way of the above procedure This isessentially the Feynman amplitude computed previously but thereis one difference GiℓΓ(q) is computed via edges whose two ends arelabeled whereas Feynman diagrams are unlabeled By summing upall possible labelings of the same unlabeled graph Γ we can rewritethis as

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

where GΓ is the Feynman amplitude computed by placing the m-tensor

partmFg

∣∣∣∣∣∣qprime=q

at each m-valent vertex of genus g in Γ and taking the contractionof tensors using the bilinear form minus 1

2(C + CT) = minusC Up to a multi-

plicative constant which we can ignore this matches the Feynmandiagram expansion obtained previously

16 Generalizations

As remarked in Section 14 one advantage of the integral formularepresentation of a quantized operator is that it generalizes in at leasttwo ways beyond the cases considered here

First if T is a symplectic diffeomorphism that is not necessarilylinear one can still define φ as the generating function of the graphof T and under this definition the integral in (16) still makes senseThus in principle integral formulas allow one to define the quan-tization of an arbitrary symplectic diffeomorphism As it turns out

16 GENERALIZATIONS 33

the formula in (16) is no longer quite right in this more general set-ting the constant λ should be allowed to be a function b = b(q pprime ~)determined by T and its derivatives Nevertheless an integral for-mula can still be obtained This is very important from a physicalperspective since the state space of classical mechanics is typically anontrivial symplectic manifold While one can reduce to the case ofR2n by working locally the quantization procedure should be func-torial with respect to arbitrary symplectic diffeomorphisms whichcan certainly be nonlinear even in local coordinates

A second possible direction for generalization is that the functionφ need not be the generating function of the graph of a symplecto-morphism at all Any Lagrangian submanifold L sub Tlowast(R2n) has agenerating function and taking φ to be the generating function ofthis submanifold (16) gives a formula for the quantization of L

CHAPTER 2

Interlude Basics of Gromov-Witten theory

In order to apply the formulas for UTψ to obtain results in Gromov-Witten theory it is necessary to quantize infinite-dimensional sym-plectic vector spaces Thus we devote Chapter 3 to the infinite-dimensional situation discussing how to adapt the finite-dimensionalformulas and how to avoid issues of convergence Before doing thishowever we pause to give a brief overview of the basics of Gromov-Witten theory Although this material will not be strictly necessaryuntil Chapter 4 we include it now to motivate our interest in theinfinite-dimensional case and the specific assumptions made in thenext chapter

21 Definitions

The material of this section can be found in any standard referenceon Gromov-Witten theory for example [7] or [19]

Let X be a projective variety Roughly speaking the Gromov-Witten invariants of X encode the number of curves passing througha prescribed collection of subvarieties In order to define these in-variants rigorously we will first need to define the moduli space

Mgn(X d) of stable maps

Definition 21 A genus-g n-pointed pre-stable curve is an al-gebraic curve C with h1(COC) = g and at worst nodal singulari-ties equipped with a choice of n distinct ordered marked pointsx1 xn isin C

Fix non-negative integers g and n and a cycle d isin H2(XZ)

Definition 22 A pre-stable map of genus g and degree d withn marked points is an algebraic map f C rarr X whose domain is agenus-g n-pointed pre-stable curve and for which flowast[C] = d Such amap is stable if it has only finitely many automorphisms as a pointedmap concretely this means that every irreducible component ofgenus zero has at least three special points (marked points or nodes)and every irreducible component of genus one has at least one specialpoint

36 2 INTERLUDE BASICS OF GROMOV-WITTEN THEORY

As eluded to in this definition there is a suitable notion of iso-morphism of stable maps Namely stable maps f C rarr X andf prime Cprime rarr X are isomorphic if there is an isomorphism of curvess Crarr Cprime which preserves the markings and satisfies f prime s = f

There is a moduli space Mgn(X d) whose points are in bijec-tion with isomorphism classes of stable maps of genus g and de-

gree d with n marked points To be more precise Mgn(X d) isonly a coarse moduli scheme but it can be given the structure ofa Deligne-Mumford stack and with this extra structure it is a finemoduli stack In general this moduli space is singular and may pos-sess components of different dimensions However there is alwaysan ldquoexpectedrdquo or ldquovirtualrdquo dimension denoted vdim and a class

[Mgn(X d)]vir isin H2vdim(Mgn(X d)) called the virtual fundamentalclass which plays the role of the fundamental class for the purposeof intersection theory The following theorem collects some of theimportant (and highly non-trivial) properties of this moduli space

Theorem 23 There exists a compact moduli spaceMgn(X d) of vir-tual dimension equal to

vdim = (dim(X) minus 3)(1 minus g) +

c1(TX) + n

It admits a virtual fundamental class [Mgn(X d)]vir isin H2vdim(Mgn(X d))

Expected dimension can be given a precise meaning in termsof deformation theory which we will omit In certain easy casesthough the moduli space is smooth and pure-dimensional and inthese cases the expected dimension is simply the ordinary dimensionand the virtual fundamental class is the ordinary fundamental classFor example this occurs when g = 0 and X is convex (such as thecase X = Pr) or when g = 0 and d = 0

Gromov-Witten invariants will be defined as integrals over themoduli space of stable maps The classes we will integrate will comefrom two places First there are evaluation maps

evi Mgn(X d)rarr X

for i = 1 n defined by sending (C x1 xn f ) to f (xi) In factthese set-theoretic maps are morphisms of schemes (or stacks) Sec-ond there are ψ classes To define these let Li be the line bundle on

Mgn(X d) whose fiber over a point (C x1 xn f ) is the cotangent

21 DEFINITIONS 37

line1 to the curve C at xi Then

ψi = c1(Li)

for i = 1 n

Definition 24 Fix cohomology classes γ1 γn isin Hlowast(X) and in-tegers a1 an isin Zge0 The corresponding Gromov-Witten invariant(or correlator) is

〈τa1(γ1) middot middot middot τan(γn)〉Xgnd =

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

1middot middot middot evlowastn(γn)ψan

While the enumerative significance of this integral is not imme-diately obvious there is an interpretation in terms of curve-counting

in simple cases Indeed suppose that Mgn(X d) is smooth and itsvirtual fundamental class is equal to its ordinary fundamental classSuppose further that the classes γi are Poincare dual to transversesubvarieties Yi sub X and that ai = 0 for all i Then the Gromov-Witteninvariant above is equal to the number of genus-g n-pointed curvesin X whose first marked point lies on Y1 whose second marked pointlies on Y2 et cetera Thus the invariant indeed represents (in somesense) a count of the number of curves passing thorugh prescribedsubvarieties

In order to encode these invariants in a notationally parsimoniousway we write

ai(z) = ai0 + ai

1z + ai2z2 + middot middot middot

for aijisin Hlowast(X) Then given a1 an isin Hlowast(X)[[z]] define

〈a1(ψ) an(ψ)〉Xgnd =int

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

middot middot middot

infinsum

evlowastn(anj )ψ

Equipped with this notation we can describe all of the genus-gGromov-Witten invariants of X in terms of a generating function

X(t(z)) =

n〈t(ψ) t(ψ)〉Xgnd

which is a formal function of t(z) isin Hlowast(X)[[z]] taking values in theNovikov ring C[[Q]] Introducing another parameter ~ to record the

1Of course this is only a heuristic definition as one cannot specify a line bundleby prescribing its fibers To be more precise one must consider the universal curve

π C rarrMgn(X d) This carries a relative cotangent line bundle ωπ Furthermore

there are sections si Mgn(X d) rarr C sending (C x1 xn f ) to xi isin C sub C We

define Li = slowastiωπ

genus we can combine all of the genus-g generating functions intoone

DX = exp

~gminus1F g

This is referred to as the total descendent potential of XIt is convenient to sum invariants with certain fixed insertions

over all ways of adding additional insertions as well as all choicesof degree Thus we define

〈〈a1(ψ) an(ψ)〉〉Xgm(τ) =sum

n〈a1(ψ) an(ψ) s s〉Xgn+md

for specified s isin Hlowast(X)

22 Basic Equations

In practice Gromov-Witten invariants are usually very difficultto compute by hand Instead calculations are typically carried outcombinatorially by beginning from a few easy cases and applyinga number of relations We state those relations in this section Thestatements of the relations can be found for example in [19] whiletheir expressions as differential equations are given in [23]

221 String Equation The string equation addresses invariantsin which one of the insertions is 1 isin H0(X) with noψ classes It states

〈τa1(γ1) middot middot middot τan(ψn) 1〉Xgn+1d = (20)

〈τa1(γ1) middot middot middot τaiminus1

(γiminus1) τaiminus1(γi) τai+1(γi+1) middot middot middot τan(γn)〉Xgnd

wheneverMgn(X d) is nonemptyThe proof of this equation relies on a result about pullbacks of

ψ classes under the forgetful morphism Mgn+1(X d) rarr Mgn(X d)that drops the last marked point Because this morphism involvescontracting irreducible components of C that becoming unstable afterthe forgetting operation ψ classes on the target do not pull back toψ classes on the source However there is an explicit comparisonresult and this is the key ingredient in the proof of (20) See [19] forthe details

It is a basic combinatorial fact that differentiation of the gener-ating function with respect to the variable ti

jcorresponds to adding

an additional insertion of τ j(φi) Starting from this one may ex-press the string equation in terms of a differential equation satisfied

22 BASIC EQUATIONS 39

by the Gromov-Witten generating function To see this fix a basisφ1 φk for Hlowast(X) and write

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

The ldquoexceptionalrdquo terms at the end arise from reindexing the summa-

tion because the moduli spacesMgn(X d) do not exist for (g n d) =(0 2 0) or (1 0 0) The first exceptional term is equal to

2~〈t0 t0〉X

where 〈〉X denotes the Poincare pairing on X because the ψ classes

are trivial on M03(X 0) The second exceptional term vanishes fordimension reasons

Taking the coefficient of ~minus1 we find thatF 0X

satisfies the followingdifferential equation

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Before we continue let us remark that the total-genus string equa-tion can be presented in the following alternative form

Qd~gminus1

(n minus 1)〈1 t(ψ) t(ψ)〉Xgnd =

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

This expression will be useful later

222 Dilaton Equation The dilaton equation addresses the sit-uation in which there is an insertion of 1 isin Hlowast(X) with a first powerof ψ attached to it

〈τa1(γ1) middot middot middot τanτ1(T0)〉Xgnd = (2g minus 2 + n)〈τa1

(γ1) τan(γn)〉Xgnd

Again it can be expressed as a differential equation on the generatingfunction In genus zero the equation is

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

The proof is similar to the above so we omit itA simple but extremely important device known as the dilaton

shift allows us to express this equation in a simpler form Define anew parameter q(z) = q0 + q1z + middot middot middot isin Hlowast(X)[[z]] by

q(z) = t(z) minus z (24)

so that qi = ti for i 1 and q1 = t1 minus 1 If we perform this change ofvariables then the dilaton equation says precisely that

1leilekjge0

partF 0X

partqij

= 2F 0X

or in other words that F 0X

(q(z)) is homogeneous of degree two

223 Topological Recursion Relations A more general equa-tion relating Gromov-Witten invariants to ones with lower powersof ψ is given by the topological recursion relations In genus zerothe relation is

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

〈〈τa1(γ1) φa〉〉X02〈〈φa τa2

(γ2) τa3(γ3)〉〉X03

where as above φa is a basis for Hlowast(X) and φa denotes the dualbasis under the Poincare pairing There are also topological recur-sion relations in higher genus (see [11] [14] and [15]) but we omitthem here as they are more complicated and not necessary for ourpurposes

23 AXIOMATIZATION 41

In terms of a differential equation the genus-zero topologicalrecursion relations are given by

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

Here we use gmicroν to denote the matrix for the Poincare pairing onHlowast(X) in the basis φα and gmicroν to denote the inverse matrix

224 Divisor Equation The divisor equation describes invari-ants in which one insertion lies in H2(X) (with no ψ classes) in termsof invariants with fewer insertions For ρ isin H2(X) it states

〈τa1(γ1) middot middot middot τan(γn) ρ〉Xgn+1d = 〈ρ d〉〈τa1

(γ1) middot middot middot τan(γn)〉Xgnd

(γiminus1) τaiminus1(γiρ) τai+1(γi+1) middot middot middot τan(γn)〉Xgnd

or equivalently

〈a1(ψ) anminus1(ψ) ρ〉Xgnd = 〈ρ d〉〈a1(ψ) anminus1(ψ)〉Xgnminus1d (26)

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

This equation too can be expressed as a differential equation on thegenerating function The resulting equation is not needed for thetime being but we will return to it in Chapter 4

23 Axiomatization

Axiomatic Gromov-Witten theory attempts to formalize the struc-tures which arise in a genus-zero Gromov-Witten theory One advan-tage of such a program is that any properties proved in the frameworkof axiomatic Gromov-Witten theory will necessarily hold for any ofthe variants of Gromov-Witten theory that share the same basic prop-erties such as the orbifold theory or FJRW theory See [23] for a moredetailed exposition of the subject of axiomatization

Let H be an arbitrary Q-vector space equipped with a distin-guished element 1 and a nondegenerate inner product ( ) Let

H = H((zminus1))

which is a symplectic vector space with symplectic form Ω definedby

Ω( f g) = Resz=0

(( f (minusz) g(z))

An arbitrary element ofH can be expressed assum

pkαφα(minusz)minus1minusk +

ℓge0

ℓφβz

ℓ (27)

in which φ1 φd is a basis for H with φ1 = 1 Define a subspaceH+ = H[[z]] of H which has coordinates qi

j Elements of H+ are

identified with t(z) isin H[[z]] via the dilaton shift

tij = qi

j + δi1δ j1

Definition 25 A genus-zero axiomatic theory is a pair (HG0)whereH is as above and G0 = G0(t) is a formal function of t(z) isin H[[z]]satisfying the differential equations (21) (23) and (25)

In the case where H = Hlowast(XΛ) equipped with the Poincare pair-ing and G0 = F 0

X one finds that the genus-zero Gromov-Witten the-

ory of X is an axiomatic theoryNote that we require an axiomatic theory to satisfy neither the

divisor equation (for example this fails in orbifold Gromov-Wittentheory) nor the WDVV equations both of which are extremely use-ful in ordinary Gromov-Witten theory While these properties arecomputationally desirable for a theory they are not necessary for thebasic axiomatic framework

CHAPTER 3

Quantization in the infinite-dimensional case

Axiomatization reduces the relevant structures of Gromov-Wittentheory to a special type of function on an infinite-dimensional sym-plectic vector space

H = H((zminus1)) (28)

As we will see in Chapter 4 the actions of quantized operators onthe quantization ofH have striking geometric interpretations in thecase where H = Hlowast(XΛ) for a projective variety X Because our ulti-mate goal is the application of quantization to the symplectic vectorspace (28) we will assume throughout this chapter that the infinite-dimensional symplectic vector space under consideration has thatform

31 The symplectic vector space

Let H be a vector space of finite dimension equipped with anondegenerate inner product ( ) and let H be given by (28) Asexplained in Section 23 H is a symplectic vector space under thesymplectic form Ω defined by

Ω( f g) = Resz=0

(( f (minusz) g(z))

The subspacesH+ = H[z]

andHminus = zminus1H[[zminus1]]

are Lagrangian A choice of basis φ1 φd for H yields a symplec-tic basis for H in which the expression for an arbitray element inDarboux coordinates issum

ℓge0

ℓφβz

Here as before φα denotes the dual basis to φα under the pairing( ) We can identify H as a symplectic vector space with thecotangent bundle TlowastH+

44 3 QUANTIZATION IN THE INFINITE-DIMENSIONAL CASE

Suppose that T HrarrH is an endomorphism of the form

T =sum

Bmzm (29)

where Bm H rarr H are linear transformations Let Tlowast denote theendomorphism given by taking the adjoint Blowastm of each transformationBm with respect to the pairing Then the symplectic adjoint of T is

Tdagger(z) = Tlowast(minusz) =sum

Blowastm(minusz)m

As usual a symplectomorphism is an endomorphism T ofH thatsatisfies Ω(T f Tg) = Ω( f g) for any f g isin H When T has the form(29) one can check that this is equivalent to the condition

Tlowast(minusz)T(z) = Id

We will specifically be considering symplectomorphisms of theform

T = exp(A)

in which A also has the form (29) In this case we will require that Ais an infinitesimal symplectic transformation or in other words thatΩ(A f g) + Ω( f Ag) = 0 for any f g isin H Using the expression for Aas a power series as in (29) one finds that this condition is equivalentto Alowast(minusz) +A(z) = 0 which in turn implies that

Alowastm = (minus1)m+1Am (30)

There is another important restriction we must make the trans-formation A (and hence T) will be assumed to contain either onlynonnegative powers of z or only nonpositive powers of z Because of

the ordering of the coordinates pkα and qβ

ℓ a transformation

R =sum

with only nonnegative powers of z will be referred to as lower-triangular and a transformation

S =sum

with only nonpositive powers of z will be referred to as upper-triangular1 The reason for this is that if A had both positive and

1This naming convention disagrees with what is typically used in Gromov-Witten theory but we choose it for consistency with the finite-dimensional settingdiscussed in Chapter 1

negative powers of z then exponentiating A would yield a series inwhich a single power of z could have a nonzero contribution frominfinitely many terms and the result would not obviously be conver-gent There are still convergence issues to be addressed when onecomposes lower-triangular with upper-triangular operators but wedefer discussion of this to Section 33

Although in the finite-dimensional case we needed to start bychoosing a symplectic basis by expressing our symplectic vectorspace asH = H((zminus1)) we have already implicity chosen a polarizationH = H+ oplus Hminus Thus the quantization of the symplectic vectorspace H should be thought of as the Hilbert space H of square-integrable functions onH+ with values inC[[~ ~minus1]] As in the finite-dimensional case we will sometimes in practice allow H to containformal functions that are not square-integrable and the space of allsuch formal functions will be referred to as the Fock space

321 The quantization procedure Observables which classi-cally are functions onH are quantized the same way as in the finite-dimensional case by setting

qαk = qαk

pkα = ~part

partqαk

and quantizing an arbitrary analytic function by expanding it in aTaylor series and ordering the variables within each monomial in theform qα1

k1middot middot middot qαn

knpℓ1β1

middot middot middot pℓmβm

In order to quantize a symplectomorphism T = exp(A) of theform discussed in the previous section we will mimic the procedurediscussed in Section 13 Namely define a function hA onH by

hA( f ) =1

2Ω(A f f )

Since hA is a classical observable it can be quantized by the aboveformula We define the quantization of A via

and UT is defined by

UT = exp(A)

The next section is devoted to making this formula more explicit

322 Basic examples Before turning to the general upper-triangularand lower-triangular cases we will begin with two simple but crucialexamples These computations follow Example 1331 of [6]

Example 31 Suppose that the infinitesimal symplectic transfor-mation A is of the form

A = Amzm

where Am Hrarr H is a linear transformation and m gt 0To compute A one must first compute the Hamiltonian hA Let

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

pk1α(Amφα) zminus1minusk1 + (minusz)m

ℓ1ge0

ℓ1(Amφβ)(minusz)ℓ1

pk2αφα(minusz)minus1minusk2 +

ℓ2ge0

ℓ2φβz

Since only the zminus1 terms contribute to the residue the right-hand sideis equal to

mminus1sum

(minus1)kpkαpmminuskminus1β(Amφα φβ)+

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

(φαAmφβ) = (minus1)m+1(Amφα φβ)

Thus denoting (Am)αβ = (Amφα φβ) and (Am)αβ = (Amφα φβ) we can

hA( f ) =1

mminus1sum

(minus1)kpkαpmminuskminus1β(Am)αβ + (minus1)msum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

(minus1)k(Am)αβpart

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

Example 32 Similarly to the previous example let

A = Amzm

this time with m lt 0 Let (Am)αβ = (Amφα φβ) An analogous compu-tation shows that

minusmminus1sum

(minus1)k+1(Am)αβqαminusmminuskminus1q

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

Example 33 More generally let

A =sum

be an infinitesimal symplectic endomorphism Then the quadraticHamiltonian associated to A is

hA( f ) =1

(minus1)m+1(Aminuskminusmminus1)αβqαk qβm +

(minus1)m(Am)αβpk+mαqβ

Thus after quantization we obtain

(minus1)m(Am)αβqβ

partqαk+m

It is worth noting that some of the fairly complicated expressionsappearing in these formulas can be written more succinctly as dis-cussed at the end of Example 1331 of [6] Indeed the expression

partA =sum

(Am)αβqβ

partqαk+m

that appears in A for both m gt 0 and m lt 0 acts on q(z) isinH+ bysum

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

where [middot]+ denotes the power series truncation with only nonnegativepowers of z In other words if q =

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

By the same token consider the expressionsum

partqαk

partqβ

mminuskminus1

appearing in A for m gt 0 The quadratic differential operator partpartqα

partqβ

mminuskminus1

can be thought of as a bivector field onH+ and sinceH+ is a vector

space a bivector field can be identified with a tensor product of

two maps H+ rarr H+ Specifically partpartqα

partqβ

mminuskminus1

is the bivector field

corresponding to the constant map

φαzk+ otimes φβzmminuskminus1

minus isin H[z+] otimesH[zminus] H+ otimesH+Using the identity

mminus1sum

(minus1)kzk+zmminus1minuskminus =

zm+ + (minus1)mminus1zm

minusz+ + zminus

then it follows that

partqαk

partqβ

mminuskminus1

[A(z+) + Alowast(zminus)

z+ + zminus

where we use the pairing to identify (Am)αβφα otimes φβ isin H+ otimesH+ withAm isin End(H) Here the power series truncation is included to ensurethat (34) is trivially valid when m is negative

The modified expressions (33) and (34) can be useful in recog-nizing the appearance of a quantized operator in computations Forexample (33) will come up in Proposition 34 below and both (33)and (34) arise in the context of Gromov-Witten theory in Theorem164 of [6]

323 Formulas for the lower-triangular and upper-triangularcases Extending the above two examples carefully one obtains for-mulas for the quantization of any upper-triangular or lower-triangularsymplectomorphism The following results are quoted from [16] seealso [24] for an exposition

Proposition 34 Let S be an upper-triangular symplectomorphism ofH of the form S = exp(A) and write

S(z) = I + S1z + S2z2 + middot middot middot

Define a quadratic form WS onH+ by the equation

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

where qk = qαkφα and Wkℓ is defined by

kℓge0

zkwℓ=

Slowast(w)S(z) minus I

wminus1 + zminus1

Then the quantization of Sminus1 acts on the Fock space by

(USminus1Ψ)(q) = exp

(WS(q)

)Ψ([Sq]+)

for any functionΨ of q isinH+ Here as above [Sq]+ denotes the truncationof S(z)q to a power series in z

Proof Let A =sum

Amzm Introduce a real parameter t and denote

G(tq) = eminustAΨ(q)

Define a t-dependent analogue of WS via

Wt(q) =sum

kℓge0

(Wkℓ(t)qkqℓ) (35)

where sum

kℓge0

Wkℓ(t)

zkwℓ=

etmiddotAlowast(w)etmiddotA(z) minus 1

zminus1 +wminus1

Note that Wkℓ(t) =Wlowastℓk

(t)We will prove that

G(tq) = exp

(Wt(q)

([etAq

for all t The claim follows by setting t = 1To prove (36) let

g(tq) = log(G(tq))

and write Ψ = exp( f ) Then taking logarithms it suffices to show

g(tq) =Wt(q)

([etAq

) (37)

Notice that

dtG(tq) = minusA G(tq)

(minus1)ℓ(Aminuskminusℓminus1)αβqαk qβ

ℓG(tq) +

(minus1)ℓminus1(Aℓ)αβqβ

partqαk+ℓ

using Example (33) This implies that g(tq) satisfies the differentialequation

dtg(tq) =

(minus1)ℓAminuskminusℓminus1αβqαk qβ

partqαk

We will prove that the right-hand side of (37) satisfies the same dif-ferential equation

The definition of Wkℓ(t) implies that

dtWkℓ(t) =

ℓsum

ℓprime=0

AlowastℓprimeminusℓWkℓprime(t) +

kprime=0

Wkprimeℓ(t)Akprimeminusk + (minus1)kAminuskminusℓminus1

Therefore

dtWt(q) =

ℓprime

(AlowastℓprimeminusℓWkℓprime(t)qkqℓ

kprime

(Wkprimeℓ(t)Akprimeminuskqkqℓ

+ (minus1)k(Aminuskminusℓminus1qkqℓ)

ℓprime

(Wkℓprime(t)qkAℓprimeminusℓqℓ

)+ (minus1)k (Aminuskminusℓminus1qℓqk

ℓprime

(minus1)ℓprimeminusℓminus1(Aℓprimeminusℓ)

αβqβ

(Wkℓprime(t)qk φα

(minus1)k (Aminuskminusℓminus1qℓqk

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

Furthermore using Equation (33) it can be shown that

etAq]+

partqαk+ℓ

etAq]+

Thus both sides of (37) satisfy the same differential equationSince they agree when t = 0 and each monomial in q and ~ dependspolynomially on t it follows that the two sides of (37) are equal

The lower-triangular case has an analogous proposition but weomit the proof in this case

34 FEYNMAN DIAGRAMS AND INTEGRAL FORMULAS REVISITED 51

Proposition 35 Let R be a lower-triangular symplectomorphism ofH of the form R = exp(B) and denote

R(z) = I + R1z + R2z2 + middot middot middot Define a quadratic form VR onHminus by the equation

VR(p0(minusz)minus1 + p1(minusz)minus2 + p2(minusz)minus3 + middot middot middot ) =sum

kℓge0

(pkVkℓpℓ)

where Vkℓ is defined bysum

kℓge0

(minus1)k+ℓVkℓwkzℓ =

Rlowast(w)R(z) minus I

Then the quantization of R acts on the Fock space by

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

where VR(partq) is the differential operator obtained from VR(p) by replacing

pk by partpartqk

33 Convergence

In Chapter 1 we expressed the quantization of an arbitrary sym-plectic transformation by decomposing it into a product of upper-triangular and lower-triangular transformations each of whose quan-tizations was known In the infinite-dimensional setting howeversuch a decomposition is problematic because composing a seriescontaining infinitely many nonnegative powers of z with one con-taining infinitely many nonpositive powers will typically yield adivergent series This is why we have only defined quantization forupper-triangular or lower-triangular operators not products thereof

It is possible to avoid unwanted infinities if one is vigilant abouteach application of a quantized operator to an element of H Forexample while the symplectomorphism S R may not be defined it

is possible that (S R)Ψmakes sense for a givenΨ isinH if S(RΨ) hasa convergent contribution to each power of z This verification canbe quite complicated see Chapter 9 Section 3 of [24] for an example

34 Feynman diagrams and integral formulas revisited

In this section we will attempt to generalize the integral formu-las and their resulting Feynman diagram expansions computed inChapter 1 to the infinite-dimensional case This is only interesting in

the lower-triangular case as the Feynman amplitude of any graphwith at least one edge vanishes for upper-triangular operators

To start we must compute the analogues of the matrices A C andD that describe the transformation in Darboux coordinates Supposethat

R =sum

is a lower-triangular symplectomorphism If

ekα = φα(minusz)minus1minusk eℓα = φαzℓ

then it is easily check that

ekα = R middot ekα =sum

kprimege0

(minus1)kminuskprime (Rkminuskprime )αγekprimeγ +

ℓprimege0

(minus1)minus1minusk(Rℓprime+k+1)αγeℓprimeγ

eℓβ = R(eℓβ) =sum

ℓprimege0

(Rlowastℓprimeminusℓ)γ

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

Then the relations among these coordinates are

pkα =sum

kprimege0

(minus1)kprimeminusk(Rkprimeminusk)γαpkprimeγ

kprimege0

(minus1)minus1minuskprime(Rℓ+kprime+1)γβpkprimeγ +sum

ℓprimege0

(Rlowastℓminusℓprime)βγqγ

ℓprime

That is if we define matrices A C and D by

(A lowast)(kprimeγ)

(kα)= (minus1)kprimeminusk(Rkprimeminusk)

C(ℓβ)(kprimeγ) = (minus1)minus1minuskprime(Rℓ+kprime+1)γβ

D(ℓβ)

(ℓprimeγ)= (Rlowastℓminusℓprime)

then the coordinates are related by

p = A p

q = C p +Dq

These matrices have rows and columns indexed by

(k α) isin Zge0 times 1 dbut the entries vanish when k is sufficiently large

As in Section 14 1 the integral formula will be expressed in termsof a function φ Hrarr R defined by

φ(q pprime) = (Dminus1q) middot pprime minus 1

2(Dminus1

C pprime) middot pprime

It is not necessary to invert D as one has Dminus1 = A T in this caseThus the above gives an explicit formula for φ

Equipped with this we would like to define

(URψ)(q) = λ

where λ is an appropriate normalization constant The problem with

this though is that the domain of the variables q = qβ

ℓand pprime = pprime

over which we integrate is an infinite-dimensional vector space Wehave not specified a measure on this space so it is not clear that (39)makes sense

Our strategy for making sense of (39) will be to define it by itsFeynman diagram expansion as given by (18) Modulo factors of2π which are irrelevant because UR is defined only up to a realmultiplicative constant the answer is

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

where as before Gprime(n) is the set of isomorphism classes of genus-labeled Feynman diagrams and FΓ(q) is the genus-modified Feyn-man amplitude given by placing the m-tensor

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

∣∣∣∣∣∣∣∣∣∣s=Dminus1q

middotprod

ℓ)mℓβ

at each m-valent vertex of genus g and taking the contraction oftensors using the bilinear form minusDminus1C Here as before Fg is definedby the expansion

ψ(q) = esum

gge0 ~gminus1Fg(q)

Note that det(Dminus1C ) is well-defined because although these matriceshave indices ranging over an infinite indexing set they are zerooutside of a finite range

35 Semi-classical limit

The most important feature of quantization for our purposes thatit relates higher genus information to genus-zero information Wewill return to this principle in the next chapter in the specific contextof Gromov-Witten theory Before we do so however let us give aprecise statement of this idea in the abstract setting of symplecticvector spaces

LetH be a symplectic vector space finite- or infinite-dimensionalFix a polarizationH =H+ oplusHminus and suppose that for each g ge 0

XH+ rarr R F g

YH+ rarr R

are functions on H+ Package each of these two collections into thetotal descendent potentials

DX = exp

~gminus1F g

DY = exp

~gminus1F g

Let LX and LY be the Lagrangian subspaces of H that under theidentification of H with TlowastH+ coincide with the graphs of dF 0

respectively That is

LX = (p q) | p = dqF 0X subH

and similarly for LY

Theorem 36 Let T be a symplectic transformation such that

UTDX = DY

T(LX) = LY

(The passage from DX to LX is sometimes referred to as a semi-classicallimit)

Proof We will prove this statement in the finite-dimensional set-ting but all of our arguments should carry over with only notationalmodifications to the infinite-dimensional case To further simplify itsuffices to prove the claim when T is of one of the three basic typesconsidered in Section 12

Case 1 T =

(I B0 I

) Using the explicit formula for UT obtained

in Chapter 1 the assumption UTDX = DY in this case can be written

35 SEMI-CLASSICAL LIMIT 55

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

Taking logarithms of both sides picking out the coefficient of ~minus1and taking derivatives with respect to q we find

dqF 0Y = Bq + dqF 0

X (40)

Now choose a point (p q) isin LX so that dqF 0X= p Explicitly the

point in question is pαeα + qαeα so its image under T is

pαeα + qαeα = (p + Bq)αeα + q

using the expressions for eα and eα in terms of the e basis obtained inChapter 1 Thus the statement that T(p q) isin LY is equivalent to

dqF 0Y = p + Bq

Since p = dqF 0X

by assumption this is precisely equation (40) Thisproves that T(LX) sub LY and the reverse inclusion follows from theanalogous claim applied to Tminus1

Case 2 T =

(A 00 AminusT

) This case is very similar to the previous

one so we omit the proof

Case 3 T =

(I 0C I

) Consider the Feynman diagram expres-

sion for UTDX obtained in Section 15 Up to a constant factor theassumption UTDX = DY in this case becomes

~gminus1F g

~minusχΓ

|Aut(Γ)|FXΓ (q) (41)

where FXΓ

(q) is the Feynman amplitude given by placing the m-tensor

partmF Xg

at each m-valent vertex of genus g in Γ and taking the contraction oftensors using the bilinear form minusC

Recall that if one takes the logarithm of a sum over Feynmandiagrams the result is a sum over connected graphs only and in this

case minusχΓ = g minus 1 Thus if one takes the logarithm of both sides of(41) and picks out the coefficient of ~minus1 the result is

F 0Y (q) =

Γ connected genus 0

FΓ(q)

|Aut(Γ)|

Let us give a more explicit formulation of the definition of FΓ(q)For convenience we adopt the notation of Gromov-Witten theoryand write

F X0 (q) = 〈〉 + 〈eα〉Xqα +

2〈eα eβ〉Xqαqβ + middot middot middot = 〈〈〉〉X(q)

where 〈〈φ1 φn〉〉X(q) =sum

kge01k〈φ1 φn q q〉X (k copies of q)

and the brackets are defined by the above expansion Then deriva-tives of F 0

Xare given by adding insertions to the double bracket It

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

Here V(Γ) and E(Γ) denote the vertex sets and edge sets of Γ respec-tively while H(v) denotes the set of half-edges associated to a vertexv The summation is over all ways to assign an index ih isin 1 dto each half-edge h where d is equal to the dimension ofH+ For anedge e we write e = (a b) if a and b are the two half-edges comprisinge

Thus we have re-expressed the relationship between F Y0 and F X

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

Now to prove the claim choose a point (p q) isin LY We will provethat Tminus1(p q) isin LX Applying the same reasoning used in Case 1 to

the inverse matrix Tminus1 =

(I 0minusC I

) we find that

Tminus1(p q) = pαeα + (minusCp + q)αeα

Therefore the claim is equivalent to

dqF Y0 = dminusCdqF Y

0+qF X

From the above one finds that the ith component of the vectordqF 0

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

On the other hand the same equation shows that the ith componentof the vector dminusCdqF Y

0+qF 0

Xis equal to

〈〈ei〉〉X(minusCdqF Y0 + q)

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

(minusCiaib)(minusC)kjek + qℓeℓ

Γ1Γn

n|Aut(Γ1)| middot middot middot |Aut(Γn)|sum

ihw1isinV(Γ1)wnisinV(Γn)

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

(minusCiaib)(minusC)kc jc〈ei ek1

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

(minusCiaib)(minusC)kc jc〈〈ei ek1 ekn〉〉(q)

Upon inspection this is equal to the sum of all ways of starting with adistinguished vertex (where ei is located) and adding n spokes labeledk1 kn then attaching n graphs Γ1 Γn to this vertex via half-edges labeled j1 jn This procedures yields all possible graphswith a distinguished vertex labeled by eimdashthe same summation thatappears in the expression for dqF Y

0 Each total graph appears inmultiple ways corresponding to different ways of partitioning it intosubgraphs labeled Γ1 Γn However it is a combinatorial exercise

to verify that with this over-counting the automorphism factor infront of each graph Γ is precisely 1

|Aut(Γ)|

Thus we find that dminusCdqF0Y+qF X0 = dqF Y

0 as required

CHAPTER 4

Applications of quantization to Gromov-Witten theory

In this final chapter we will return to the situation in whichH = Hlowast(XΛ)((zminus1)) for X a projective variety In Section 41 weshow that many of the basic equations of Gromov-Witten theory canbe expressed quite succinctly as equations satisfied by the action ofa quantized operator on the total descendent potential More strik-ingly according to Giventalrsquos conjecture there is a converse in certainspecial cases to the semi-classical limit statement explained in Section35 we discuss this in Section 42 below In Section 43 we briefly out-line the machinery of twisted Gromov-Witten theory developed byCoates and Givental This is a key example of the way quantizationcan package complicated combinatorics into a manageable formula

Ultimately these notes only scratch the surface of the applicabilityof the quantization machinery to Gromov-Witten theory There aremany other interesting directions in this vein so we conclude thebook with a brief overview of some of the other places in whichquantization arises The interested reader can find much more in theliterature

41 Basic equations via quantization

Here with give a simple application of quantization as a way torephrase some of the axioms of Gromov-Witten theory This sectionclosely follows Examples 1332 and 1333 of [6]

411 String equation Recall from (22) that the string equationcan be expressed as follows

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

60 4 APPLICATIONS OF QUANTIZATION TO GROMOV-WITTEN THEORY

Applying the dilaton shift (24) this is equivalent to

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

From Example 32 with m = minus1 one finds

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

2~〈q0 q0〉 minus part1z

Thus applying (33) it follows that the string equation is equivalentto

zDX = 0

412 Divisor equation In a similar fashion the divisor equationcan be expressed in terms of a quantized operator Summing overg n d and separating the exceptional terms equation (26) can bestated assum

Qd~gminus1

(n minus 1)〈t(ψ) t(ψ) ρ〉Xgnd =

Qd~gminus1

n(ρ d)〈t(ψ) t(ψ)〉gnd

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

The left-hand side and the second summation on the right-handside combine to give

minussum

Qd~gminus1

(n minus 1)

lang[ρ(t(ψ) minus ψ)

t(ψ) t(ψ)

After the dilaton shift this is equal to partρz(sum~gminus1F X

g )As for the first summation let τ1 τr be a choice of basis for

H2(XZ) which yields a set of generators Q1 Qr for the Novikovring Write

ρ =rsum

ρiτi

41 BASIC EQUATIONS VIA QUANTIZATION 61

in the dual basis τi for τi Then the first summation on the right-hand side of (42) is equal to

ρiQipart

partQi

~gminus1F Xg

The first exceptional term is computed as before

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

The second exceptional term is more complicated in this case Werequire the fact that

M11(X 0) X timesM11

and that under this identification

[M11(X d)]vir = e(TX timesLminus11 ) = e(TX) minus ψ1cDminus1(TX)

where L1 is the cotangent line bundle (whose first Chern class is ψ1)and D = dim(X) Thus

〈ρ〉X110 =int

e(TX) middot ρ minusint

cDminus1(TX) middot ρ = minus 1

cDminus1(TX) middot ρ

Putting all of these pieces together we can express the divisorequation as

minus 1

2~(q0ρ q0) minus partρz

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

or in other words as

)middot DX =

ρiQipart

partQi

minus 1

cDminus1(TX) middot ρDX

It should be noted that the left-hand side of this equality makes sensebecause multiplication by ρ is a self-adjoint linear transformation onHlowast(X) under the Poincare pairing and hence multiplication by ρz isan infinitesimal symplectic transformation

42 Giventalrsquos conjecture

The material in this section can be found in [17] and [23]Recall from Section 23 that an axiomatic genus zero theory is a

symplectic vector spaceH = H((zminus1)) together with a formal functionG0(t) satisfying the differential equations corresponding to the stringequation dilaton equation and topological recursion relations ingenus zero

The symplectic (or twisted) loop group is defined as the set M(z)of End(H)-valued formal Laurent series in zminus1 satisfying the symplec-tic condition Mlowast(minusz)M(z) = I There is an action of this group on thecollection of axiomatic genus zero theories To describe the action itis helpful first to reformulate the definition of an axiomatic theory ina more geometric though perhaps less transparent way

Associated to an axiomatic genus zero theory is a Lagrangiansubspace

L = (pq) | p = dqG0 subHwhere (pq) are the Darboux coordinates on H defined by (27) andwe are identifyingH TlowastH+ by way of this polarization Note herethat G0(t) is identified with a function of q isinH+ via the dilaton shiftand that this is the same Lagrangian subspace discussed in Section35

According to Theorem 1 of [18] a function G0(t) satisfies the requi-site differential equations if and only if the corresponding Lagrangiansubspace L is a Lagrangian cone with the vertex at the origin satisfy-ing

L cap TfL = zTfLfor each f isin L

The symplectic loop group can be shown to preserve these prop-erties Thus an element T of the symplectic loop group acts onthe collection of axiomatic theories by sending the theory with La-grangian cone L to the theory with Lagrangian cone T(L)

There is one other equivalent formulation of the definition ofaxiomatic genus-zero theories in terms of abstract Frobenius mani-folds Roughly speaking a Frobenius manifold is a manifold equippedwith a product on each tangent space that gives the tangent spacesthe algebraic structure of Frobenius algebras A Frobenius manifoldis called semisimple if on a dense open subset of the manifold thesealgebras are semisimple This yields a notion of semisimplicity foraxiomatic genus zero theories Given this we can formulate thestatement of the symplectic loop group action more precisely

42 GIVENTALrsquoS CONJECTURE 63

Theorem 41 [18] The symplectic loop group acts on the collection ofaxiomatic genus-zero theories Furthermore the action is transitive on thesemisimple theories of a fixed rank N

Here the rank of a theory is the rank of HThe genus-zero Gromov-Witten theory of a collection of N points

gives a semisimple axiomatic theory of rank N which we denoteby HN The theorem implies that any semisimple axiomatic genus-zero theory T = (HG0) can be obtained from HN by the action ofan element of the twisted loop group Via the process of Birkhofffactorization such a transformation can be expressed as S R inwhich S is upper-triangular and R is lower-triangular

Definition 42 The axiomatic τ-function of an axiomatic theoryT is defined by

τTG = S(RDN)

where S R is the element of the symplectic loop group taking thetheory HN of N points to T andDN is the total descendent potentialfor the Gromov-Witten theory of N points

If T is in fact the genus-zero Gromov-Witten theory of a space Xthen we have two competing definitions of the higher-genus poten-tial DX and τT

G Giventalrsquos conjecture is the statement that these two

Conjecture 43 (Giventalrsquos conjecture [17]) If T is the semisimpleaxiomatic theory corresponding to the genus-zero Gromov-Witten theory ofa projective variety X then τT

In other words the conjecture posits that in the semisimple caseif an element of the symplectic loop group matches two genus zerotheories then its quantization matches their total descendent poten-tials Because the action of the symplectic loop group is transitiveon semisimple theories this amounts to a classificaiton of all higher-genus theories for which the genus-zero theory is semisimple

Givental proved his conjecture in case X admits a torus actionand the total descendent potentials are taken to be the equivariantGromov-Witten potentials In 2005 Teleman announced a proof ofthe conjecture in general

Theorem 44 [29] Giventalrsquos conjecture holds for any semisimple ax-iomatic theory

One important application of Giventalrsquos conjecture is the proof ofthe Virasoro conjecture in the semisimple case The conjecture states

Conjecture 45 For any projective manifold X there exist ldquoVirasoro

operatorsrdquo LXmmgeminus1 satisfying the relations

[LXm L

Xn ] = (m minus n)LX

m+n (43)

such that

LXmDX = 0

for all m ge minus1

In the case where X is a collection of N points the conjecture holds

by setting LXm equal to the quantization of

Lm = minuszminus12Dm+1zminus12

)z = z2 d

The resulting operators Lmmgeminus1 are the same as N copies of thoseused in Wittenrsquos conjecture [30] and the relations (43) indeed holdfor these operators

Thus by Wittenrsquos conjecture the Virasoro conjecture holds forany semisimple Gromov-Witten theory by setting

LXm = S(RLmRminus1)Sminus1

for the transformation S R taking the theory of N points to the

Gromov-Witten theory of X1

43 Twisted theory

The following is due to Coates and Givental we refer the readerto the exposition presented in [6]

Let X be a projective variety equipped with a holomorphic vector

bundle E Then E induces a K-class onMgn(X d)

Egnd = πlowast flowastE isin K0(Mgn(X d))

Mgn(X d)

1In fact one must check that LXm defined this way agrees with the Virasoro

operators of the conjecture but this can be done

43 TWISTED THEORY 65

is the univeral family over the moduli space Consider an invertiblemultiplicative characteristic class

c K0(Mgn(X d))rarr Hlowast(Mgn(X d))

Any such class can be written in terms of Chern characters

c(middot) = exp

skchk(middot)

for some parameters skA twisted Gromov-Witten invariant is defined as

〈τ1(γ1) middot middot middotτn(γn) c(Egnd)〉Xgnd =int

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

These fit into a twisted genus-g potential F g

cE and a twisted totaldescendent potential DcE in just the way that the usual Gromov-Witten invariants do

There is also a Lagrangian coneLcE associated to the twisted the-ory but a bit of work is necessary in order to define it The reason forthis is that the Poincare pairing on H(XΛ) should be given by three-point correlators As a result when we replace Gromov-Witten in-variants by their twisted versions we must modify the Poincare pair-ing and hence the symplectic structure on H accordingly Denotethis modified symplectic vector space byHcE There is a symplecticisomorphism

HcE rarrH

x 7rarrradic

We define the Lagrangian cone LcE of the twisted theory by

LcE =radic

c(E) middot (pq) | p = dqF 0cE subH

where we use the usual dilaton shift to identifyF 0cE(t) with a function

of q isin (HcE)+The quantum Riemann-Roch theorem of Coates-Givental gives

an expression forDcE in terms of a quantized operator acting on theuntwisted Gromov-Witten descendent potentialDX of X

Theorem 46 [6] The twisted descendent potential is related to theuntwisted descendent potential by

ℓgt0

sℓminus1

chℓ(E)cDminus1(TX)

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

(2m)(chℓ(E)z2mminus1)and

ℓgt0

sℓminus1(chℓ(E)z)and

Here B2m are the Bernoulli numbers

The basic idea of this theorem is to write

c(Egnd) = exp

skchk(Rπlowast flowastE)

(πlowast(ch( f lowastE)Tdor(Tπ))

using the Grothendieck-Riemann-Roch formula A geometric theo-rem expresses Tdor(Tπ) in terms of ψ classes on various strata of themoduli space and the rest of the proof of Theorem 46 is a difficultcombinatorial exercise in keeping track of these contributions

Taking a semi-classical limit and applying the result discussed inSection 35 of the previous chapter we obtain

Corollary 47 The Lagrangian cone LcE satisfies

LcE = exp

0leℓleD

s2mminus1+ℓB2m

(2m)chℓ(E)z2mminus1

This theorem and its corollary are extremely useful even whenone is only concerned with the genus zero statement For example itis used in the proof of the Landau-GinzburgCalabi-Yau correspon-dence in [5] In that context the invariants under considerationknown as FJRW invariants are given by twisted Gromov-Witten in-variants only in genus zero Thus Theorem 46 actually says nothingabout higher-genus FJRW invariants Nevertheless an attempt todirectly apply Grothendieck-Riemann-Roch in genus zero to obtaina relationship between FJRW invariants and untwisted invariantsis combinatorially unmanageable thus the higher-genus statementwhile not directly applicable can be viewed as a clever device forkeeping track of the combinatorics of the Grothendieck-Riemann-Roch computation

44 CONCLUDING REMARKS 67

44 Concluding remarks

There are a number of other places in which quantization provesuseful for Gromov-Witten theory For example it was shown in[22] [20] [13] that relations in the so-called tautological ring an

important subring of Hlowast(Mgn(X β)) are invariant under the actionof the symplectic loop group This was used to give one proof ofGiventalrsquos Conjecture in genus g le 2 (see [9] and [31]) and can alsobe used to derive tautological relations (see [21] [2] and [1]) Werefer the interested reader to [23] for a summary of these and otherapplications of quantization with more complete references

Bibliography

[1] D Arcara and Y-P Lee Tautological equations in M31 via invariance conjec-tures arXiv preprint math0503184 2005

[2] D Arcara and Y-P Lee Tautological equations in genus 2 via invarianceconstraints Bulletin-Institute of Mathematics Academia Sinica 2(1)1 2007

[3] S Bates and A Weinstein Lectures Notes on the Geometry of Quantization Amer-ican Mathematical Society 1997

[4] M Blau Symplectic geometry and geometric quantization Found athttpwwwictptriesteit˜mblaulecturesGQpsgz

[5] A Chiodo and Y Ruan LandaundashGinzburgCalabindashYau correspondence forquintic three-folds via symplectic transformations Inventiones mathematicae182(1)117ndash165 2010

[6] T Coates Riemann-Roch Theorems in Gromov-Witten Theory PhD the-sis University of California at Berkeley 1997 Also available athttpmathharvardedu tomcthesispdf

[7] D A Cox and S Katz Mirror Symmetry and Algebraic Geometry AmericanMathematical Society 1999

[8] A C Da Silva Lectures on Symplectic Geometry Springer Verlag 2001[9] B Dubrovin and Y Zhang Frobenius manifolds and Virasoro constraints

Selecta Mathematica 5(4)423ndash466 1999[10] A Echeverrıa-Enrıquez M C M Lecanda N R Roy and C Victoria-Monge

Mathematical foundations of geometric quantization Extracta mathematicae13(2)135ndash238 1998

[11] T Eguchi and C Xiong Quantum cohomology at higher genus Adv TheorMath Phys 2(hep-th9801010)219ndash229 1998

[12] P Etingof Mathematical ideas and notions of quantum field theory Found athttpwww-mathmitedu etingoflectps 2002

[13] C Faber S Shadrin and D Zvonkine Tautological relations and the r-spinWitten conjecture arXiv preprint math0612510 2006

[14] E Getzler Intersection theory on M14 and elliptic Gromov-Witten invariantsJournal of the American Mathematical Society 10(4)973ndash998 1997

[15] E Getzler Topological recursion relations in genus 2 In Integrable Systems andAlgebraic Geometry pages 73ndash106 World Sci Publishing 1998

[16] A Givental Gromov-Witten invariants and quantization of quadratic hamil-tonians Mosc Math J 1(4)551ndash568 2001

[17] A Givental Semisimple frobenius structures at higher genus Internationalmathematics research notices 2001(23)1265ndash1286 2001

[18] A Givental Symplectic geometry of Frobenius structures In Frobenius mani-folds pages 91ndash112 Springer 2004

70 BIBLIOGRAPHY

[19] K Hori S Katz R Pandharipande R Thomas and R Vakil Mirror SymmetryClay Mathematics Institute 2003

[20] Y-P Lee Wittenrsquos conjecture Virasoro conjecture and invariance of tautolog-ical equations arXiv preprint mathAG0311100 2003

[21] Y-P Lee Invariance of tautological equations I conjectures and applicationsJournal of the European Mathematical Society 10(2)399ndash413 2008

[22] Y-P Lee Invariance of tautological equations II Gromov-Witten theory Jour-nal of the American Mathematical Society 22(2)331ndash352 2009

[23] Y-P Lee Notes on axiomatic GromovndashWitten theory and applications InAlgebraic geometrymdashSeattle 2005 pages 309ndash323 Proc Sympos Pure MathVol 80 Part 1 Amer Math Soc 2009

[24] Y-P Lee and R Pandharipande Frobenius manifoldsGromov-Witten theory and Virasoro constraints Found athttpwwwmathethzch rahulPart1ps 2004

[25] D McDuff and D Salamon Introduction to Symplectic Topology Oxford Uni-versity Press 1998

[26] W G Ritter Geometric quantization arXiv preprint math-ph0208008 2002[27] Y Ruan Analytic continuation of Gromov-Witten theory Unpublished notes

2010[28] A Schwarz and X Tang Quantization and holomorphic anomaly Journal of

High Energy Physics 2007(03)062 2007[29] C Teleman The structure of 2d semi-simple field theories Inventiones mathe-

maticae 188(3)525ndash588 2012[30] E Witten Two-dimensional gravity and intersection theory on moduli space

Surveys in Diff Geom 1(243)74 1991[31] L Xiaobo and T Gang Quantum product on the big phase space and the

Virasoro conjecture Advances in Mathematics 169(2)313ndash375 2002

Preface

Chapter 0 Preliminaries


02 Feynman diagrams

Chapter 1 Quantization of finite-dimensional symplectic vector spaces

11 The set-up





16 Generalizations

Chapter 2 Interlude Basics of Gromov-Witten theory

21 Definitions

22 Basic Equations

23 Axiomatization

Chapter 3 Quantization in the infinite-dimensional case



33 Convergence



Chapter 4 Applications of quantization to Gromov-Witten theory


42 Giventals conjecture

43 Twisted theory


Bibliography

The notes that follow are expository and the authors make no claimto originality of any of the material appearing in them

Please send questions or comments to

Emily Clader ecladerumicheduNathan Priddis priddisnumicheduMark Shoemaker markashoegmailcom

Contents

Preface 1

Bibliography 69

Preface

2 PREFACE

PREFACE 3

CHAPTER 0

Preliminaries

langeα eβ

rang= 0

langeα eβ

rang= δαβ (1)

(0 IminusI 0

6 0 PRELIMINARIES

(A BC D

ATB = BTA (2)

CTD = DTC (3)

ATDminusBTC = I (4)

σminus1 =

(DT minusBT

minusCT AT

sumni=1 dxi and dyi

λ =sum

pαdqα

ω =sum

dqα and dpα

8 0 PRELIMINARIES

02 Feynman diagrams

~minusd2

eminusS(x)~dx (5)

S(x) =1

2B(x x) +

mBm(x x) (6)

|Π2k| =(2k)

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

xN1 eminus(x2

int infin

minusinfineminusx2

2 dx =radic

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

2 Recall that the

Γ(z) =

int infin

yzminus1eminusydy

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

Contents

Preface 1

Bibliography 69

Preface

2 PREFACE

PREFACE 3

CHAPTER 0

Preliminaries

langeα eβ

rang= 0

langeα eβ

rang= δαβ (1)

(0 IminusI 0

6 0 PRELIMINARIES

(A BC D

ATB = BTA (2)

CTD = DTC (3)

ATDminusBTC = I (4)

σminus1 =

(DT minusBT

minusCT AT

sumni=1 dxi and dyi

λ =sum

pαdqα

ω =sum

dqα and dpα

8 0 PRELIMINARIES

02 Feynman diagrams

~minusd2

eminusS(x)~dx (5)

S(x) =1

2B(x x) +

mBm(x x) (6)

|Π2k| =(2k)

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

xN1 eminus(x2

int infin

minusinfineminusx2

2 dx =radic

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

2 Recall that the

Γ(z) =

int infin

yzminus1eminusydy

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

Preface

2 PREFACE

PREFACE 3

CHAPTER 0

Preliminaries

langeα eβ

rang= 0

langeα eβ

rang= δαβ (1)

(0 IminusI 0

6 0 PRELIMINARIES

(A BC D

ATB = BTA (2)

CTD = DTC (3)

ATDminusBTC = I (4)

σminus1 =

(DT minusBT

minusCT AT

sumni=1 dxi and dyi

λ =sum

pαdqα

ω =sum

dqα and dpα

8 0 PRELIMINARIES

02 Feynman diagrams

~minusd2

eminusS(x)~dx (5)

S(x) =1

2B(x x) +

mBm(x x) (6)

|Π2k| =(2k)

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

xN1 eminus(x2

int infin

minusinfineminusx2

2 dx =radic

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

2 Recall that the

Γ(z) =

int infin

yzminus1eminusydy

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

2 PREFACE

PREFACE 3

CHAPTER 0

Preliminaries

langeα eβ

rang= 0

langeα eβ

rang= δαβ (1)

(0 IminusI 0

6 0 PRELIMINARIES

(A BC D

ATB = BTA (2)

CTD = DTC (3)

ATDminusBTC = I (4)

σminus1 =

(DT minusBT

minusCT AT

sumni=1 dxi and dyi

λ =sum

pαdqα

ω =sum

dqα and dpα

8 0 PRELIMINARIES

02 Feynman diagrams

~minusd2

eminusS(x)~dx (5)

S(x) =1

2B(x x) +

mBm(x x) (6)

|Π2k| =(2k)

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

xN1 eminus(x2

int infin

minusinfineminusx2

2 dx =radic

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

2 Recall that the

Γ(z) =

int infin

yzminus1eminusydy

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

PREFACE 3

CHAPTER 0

Preliminaries

langeα eβ

rang= 0

langeα eβ

rang= δαβ (1)

(0 IminusI 0

6 0 PRELIMINARIES

(A BC D

ATB = BTA (2)

CTD = DTC (3)

ATDminusBTC = I (4)

σminus1 =

(DT minusBT

minusCT AT

sumni=1 dxi and dyi

λ =sum

pαdqα

ω =sum

dqα and dpα

8 0 PRELIMINARIES

02 Feynman diagrams

~minusd2

eminusS(x)~dx (5)

S(x) =1

2B(x x) +

mBm(x x) (6)

|Π2k| =(2k)

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

xN1 eminus(x2

int infin

minusinfineminusx2

2 dx =radic

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

2 Recall that the

Γ(z) =

int infin

yzminus1eminusydy

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

CHAPTER 0

Preliminaries

langeα eβ

rang= 0

langeα eβ

rang= δαβ (1)

(0 IminusI 0

6 0 PRELIMINARIES

(A BC D

ATB = BTA (2)

CTD = DTC (3)

ATDminusBTC = I (4)

σminus1 =

(DT minusBT

minusCT AT

sumni=1 dxi and dyi

λ =sum

pαdqα

ω =sum

dqα and dpα

8 0 PRELIMINARIES

02 Feynman diagrams

~minusd2

eminusS(x)~dx (5)

S(x) =1

2B(x x) +

mBm(x x) (6)

|Π2k| =(2k)

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

xN1 eminus(x2

int infin

minusinfineminusx2

2 dx =radic

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

2 Recall that the

Γ(z) =

int infin

yzminus1eminusydy

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

6 0 PRELIMINARIES

(A BC D

ATB = BTA (2)

CTD = DTC (3)

ATDminusBTC = I (4)

σminus1 =

(DT minusBT

minusCT AT

sumni=1 dxi and dyi

λ =sum

pαdqα

ω =sum

dqα and dpα

8 0 PRELIMINARIES

02 Feynman diagrams

~minusd2

eminusS(x)~dx (5)

S(x) =1

2B(x x) +

mBm(x x) (6)

|Π2k| =(2k)

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

xN1 eminus(x2

int infin

minusinfineminusx2

2 dx =radic

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

2 Recall that the

Γ(z) =

int infin

yzminus1eminusydy

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

λ =sum

pαdqα

ω =sum

dqα and dpα

8 0 PRELIMINARIES

02 Feynman diagrams

~minusd2

eminusS(x)~dx (5)

S(x) =1

2B(x x) +

mBm(x x) (6)

|Π2k| =(2k)

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

xN1 eminus(x2

int infin

minusinfineminusx2

2 dx =radic

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

2 Recall that the

Γ(z) =

int infin

yzminus1eminusydy

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

8 0 PRELIMINARIES

02 Feynman diagrams

~minusd2

eminusS(x)~dx (5)

S(x) =1

2B(x x) +

mBm(x x) (6)

|Π2k| =(2k)

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

xN1 eminus(x2

int infin

minusinfineminusx2

2 dx =radic

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

2 Recall that the

Γ(z) =

int infin

yzminus1eminusydy

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

2 dx =(2π)d2

radicdet B

σisinΠN

iisin[N]σ

Bminus1(li lσ(i))

xN1 eminus(x2

int infin

minusinfineminusx2

2 dx =radic

minusinfinx2keminus

minusx2

2 dx =radic

2π(2k)

2kk (7)

2 Recall that the

Γ(z) =

int infin

yzminus1eminusydy

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

10 0 PRELIMINARIES

minusinfinx2ke

minusx2

2 dx = 2

int infin

x2keminusx2

int infin

= 2k+12Γ(k + 12)

2)Γ(1

=radic

2π(2k)

Z = ~minusd2

eminusS(x)~dx

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

i FΓi

Z =(2π)d2

radicdet(B)

n=(n0n1)

infinprod

ΓisinG(n)

~b(Γ)

|Aut(Γ)|FΓ

(minusB1)Bminus1

(minusB1))

eminusB(yy)2esum

mge0 gm(minus~

m2 minus1

Bm(yy)m )dy

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

12 0 PRELIMINARIES

eminusB(yy)2prod

(i)nini

(minus ~ i

2minus1Bi(y y))ni

n=(n0n1)

(i)nini~ni(

i2minus1)

eminusB(yy)2prod

(minus Bi(y y)

Denote

eminusB(yy)2prod

(minusBi(y y))nidy

Let N =sum

Z =(2π)d2

radicdet(B)

(i)nini

σisinΠN

~b(Γ(σ)FΓ(σ)

sumni(

i2minus 1) = N

2minus sum

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

|G| =prod

(i)nini

i(i)nini

|Aut(Γ)|

det(B) Then one has

log(ZZ0) =sum

n=(n0n1)

infinprod

ΓisinGc(n)

~b(Γ)

|Aut(Γ)|FΓ

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

CHAPTER 1

spaces

11 The set-up

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

langeα eβ

rang= 0

langeα eβ

rang= δαβ

pαeα +sum

qβeβ

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

11 THE SET-UP 17

[pα pβ] = [qα qβ] = 0 [pα qβ] = ~δ

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

qαΨ = qαΨ

pαΨ = ~partΨ

partqα

Q(qαqβ) = qαqβ

partqβ

partqαpartqβ

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

C(pαpβ qαqβ) =

1 α β

2 α = β(8)

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

β (9)

p = Ap + Bq

q = Cp +Dq

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

(A BC D

)= exp

(a bc d

In other words if

(A(t) B(t)C(t) D(t)

)= exp

(ta tbtc td

The matrix(

a bc d

αdefined by

βαeβ

Using the fact that

Tminus1t = exp

)+O(t2)

β +O(t2)

βt minus td

βαetβ +O(t2)

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

This implies

ptα = pα minus ta

β +O(t2)

and consequently

ptα = pα minus ta

ptαUt = Utpα (14)

qαt Ut = Utqα

∣∣∣∣∣t=0

u = minus 1

2~cαβpαpβ +

~aβαqαpβ +

2~bαβqαqβ + C

and qα respec-

tively we obtain

u = minus~2

cαβpart

partqαpart

partqβ+ a

βαqα

partqβ+

2~bαβq

αqβ + C (15)

dt)kUt

)∣∣∣∣t=0

and[qα

dt)kUt

)∣∣∣∣t=0

form Tt = exp(t(

a bc d

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

(UTψ)(q) = exp

(aβαqα

partqβ

)ψ(q) = ψ(ATq)

that aβα = δ

monomial

ψ(q) =prod

(qα)cα

(UTψ)(q) = exp

iqi part

partq j

)prod(qα

c jsum

iqi part

partq j

)k prod(qα

c jsum

(k)(c j minus k)

)cα(q j + a

iqi)c j

= ψ(ATq)

(UTψ)(q) = exp(

2~bαβq

αqβ)ψ(q)

(UTψ)(q) = exp(

2~Bαβq

αqβ)ψ(q)

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

plectic matrix T =

(A BC D

)= exp

(a bc d

(A BC D

(I BDminus1

)middot(

I 0DCT I

)middot(

DminusT 00 D

= UT1UT2

(UTψ)(q) = exp(

2~(BDminus1)αβq

αqβ)

(minus~

2(DCT)αβ

partqαpart

partqβ

)ψ((DminusT)αβq)

(UTψ)(q) = exp(

(minus~

(DCT part

)middot partpartq

)ψ(Dminus1q)

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

(a bc d

hF(v) =1

2〈Fv v〉

UT = exp(F)

[Q( f )Q(g)] = ~Q( f g) + ~2C( f g)

hA hB = h[AB]

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

[A B] = [AB] + C(hA hB)

(UTψ)(q) = λ

(p q p q)

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

sub R2n timesR2n

(minus piQ

i + Piqi + piQ

i minus Piqi)

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

(qiΠi + piΞ

(q p p q))

∣∣∣∣∣∣pα = A

βαpβ + Bαβq

(q pprime

partφ

partqpartφ

partpprime

partφ

φ(q pprime) =1

(UTψ)(q) =1

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

ψ(y) =

(UTψ)(q) =e

= ine1

minus ~2(DCT part

ψ(q) = e1~

ψ(q) = esum

gge0 ~gminus1Fg(q)

(UTψ)(q) = e1

eminus1~

)minus

~gFg(qprime)

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

eminus1~

(minuspartm Fg

middot (qprime)m)

Bgm = minussum

partmFg

(UTψ)(q) = e1

eminus1~

(minusB(yy)2 +

sumgmge0

S(x) =1

2B(x x) +

mBgm(x x)

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

Z = ~minusd2

eminusS(x)~dx

Z =(2π)d2

radicdet(B)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ

i FΓi

χΓ = minussum

visinV(Γ)

Z =sum

n=(ngm)gmge0

(m)ngmngm

eminusB(yy)2prod

(minusBgm)ngmdy

2minus 1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q) (18)

partmFg

(Dminus1C I

)minus1

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

ΓisinGprimec

~minusχΓ

|Aut(Γ)|FΓ(q)

F g(q) =sum

ΓisinGprimec(g)

|Aut(Γ)|FΓ(q)

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

(UTψ)(q) =(2π)ne

radicdet(Dminus1C)

~gminus1F g(q)

(I 0C I

(UTψ)(q) = exp

(minus~

2Cαβ part

partqαpart

partqβ

)ψ(q)

iαβℓg

prodiαβ

prodℓg

(minusCαβ

partqβ

)iαβ prod

(Fg(q))ℓg (19)

partxparty( f gh)

partpartx

partparty

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

(minusCαβ

2)iαβ

prod(partpartqα

partpartqβ

partparty

|Aut(Γ)|

iαβℓgΓ

~minusχΓ

|Aut(Γ)|GiℓΓ(q)

(UTψ)(q) =sum

~minusχΓ

|Aut(Γ)|GΓ

partmFg

16 Generalizations

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

CHAPTER 2

21 Definitions

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

c1(TX) + n

evi Mgn(X d)rarr X

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

21 DEFINITIONS 37

ψi = c1(Li)

for i = 1 n

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

ai(z) = ai0 + ai

[Mgn(Xd)]vir

infinsum

evlowast1(a1j )ψ

infinsum

evlowastn(anj )ψ

X(t(z)) =

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

DX = exp

~gminus1F g

22 Basic Equations

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

t j = tijφi

partt10

~gminus1F g

Qd~gminus1

langt(ψ) t(ψ) 1

rangXgn+1d

Qd~gminus1

(n minus 1)

langt(ψ) t(ψ)

tij+1τ j(φi)

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

parttij

~gminus1F g

2~〈t(ψ) t(ψ) 1〉X030 + 〈1〉110

2~〈t0 t0〉X

partF 0X

partt10

2〈t0 t0〉X +

partF 0X

parttij

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉X (22)

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

partF 0X

partt11

1leilekjge0

partF 0X

parttij

minus 2F 0X (23)

1leilekjge0

partF 0X

partqij

= 2F 0X

〈〈τa1+1(γ1) τa2(γ2)τa3

(γ3)〉〉X03(τ)

(γ2) τa3(γ3)〉〉X03

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

part3F 0X

partti1j1partti2

j2partti3

microν

part2F 0X

partti1j1parttmicro

gmicroνpartF 0

partti2j2partti3

j3parttν

or equivalently

nminus1sum

langa1(ψ)

[ρai(ψ)

anminus1(ψ)

gnminus1d

23 Axiomatization

H = H((zminus1))

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

ℓge0

ℓφβz

ℓ (27)

tij = qi

j + δi1δ j1

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

CHAPTER 3

H = H((zminus1)) (28)

Ω( f g) = Resz=0

(( f (minusz) g(z))

ℓge0

ℓφβz

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

T =sum

Bmzm (29)

Blowastm(minusz)m

T = exp(A)

R =sum

S =sum

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

qαk = qαk

pkα = ~part

partqαk

knpℓ1β1

hA( f ) =1

2Ω(A f f )

UT = exp(A)

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

A = Amzm

f (z) =sum

ℓge0

ℓφβz

ℓ isinH

hA( f ) =1

2Ω(A f f )

2Resz=0

(minusz)m

ℓ1ge0

ℓ2ge0

ℓ2φβz

mminus1sum

(minus1)mpm+kαqβ

k(Amφ

α φβ)

minus 1

kpm+kα(Amφβ φ

By (30) we have

hA( f ) =1

mminus1sum

pm+kαqβ

k(Am)αβ

This implies that

mminus1sum

partqαk

partqβ

mminuskminus1

+ (minus1)msum

(Am)αβqβ

partqαm+k

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

A = Amzm

minusmminus1sum

k+ (minus1)m

kgeminusm

(Am)αβqβ

partqαk+m

A =sum

hA( f ) =1

partqαk+m

partA =sum

(Am)αβqβ

partqαk+m

(Am)αβqβ

partqαk+m

ℓφγzℓ

(Am)αβqβ

kφαzk+m

sumℓ qi

ℓφiz

ℓ then

partAq = [Aq]+ (33)

partqαk

partqβ

mminuskminus1

partqβ

mminuskminus1

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

partqβ

mminuskminus1

mminus1sum

minusz+ + zminus

partqαk

partqβ

mminuskminus1

z+ + zminus

WS(q) =sum

kℓge0

(Wkℓqkqℓ)

kℓge0

zkwℓ=

wminus1 + zminus1

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

(WS(q)

)Ψ([Sq]+)

Proof Let A =sum

Wt(q) =sum

kℓge0

where sum

kℓge0

Wkℓ(t)

zkwℓ=

zminus1 +wminus1

G(tq) = exp

(Wt(q)

([etAq

g(tq) = log(G(tq))

g(tq) =Wt(q)

([etAq

) (37)

Notice that

ℓG(tq) +

partqαk+ℓ

dtg(tq) =

partqαk

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

dtWkℓ(t) =

ℓsum

ℓprime=0

kprime=0

Therefore

dtWt(q) =

ℓprime

kprime

ℓprime

αβqβ

kℓℓprime

αβqβ

(Wkℓprime(t)qk

partqαℓprime

qℓprime

partαk+ℓ

(Wt(q)

etAq]+

partqαk+ℓ

etAq]+

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

kℓge0

(pkVkℓpℓ)

kℓge0

(URΨ)(q) =

(~VR(partq)

](Rminus1q)

pk by partpartqk

33 Convergence

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

R =sum

kprimege0

ℓprimege0

βeℓprimeγ

Let pkα and qβ

ℓbe defined by

pkαekα +sum

ℓge0

ℓeℓβ =

pkαekα +sum

ℓge0

ℓeℓβ

pkα =sum

kprimege0

ℓprimege0

ℓprime

D(ℓβ)

p = A p

q = C p +Dq

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

2(Dminus1

(URψ)(q) = λ

(URψ)(q) =1radic

det(Dminus1C )

n=(n0n1)

ΓisinGprime(n)

~minusχΓ

|Aut(Γ)|FΓ(q)

msummℓβ=m

mprodmℓβ

partmFg(s)prod

(partsℓβ)mℓβ

middotprod

ℓ)mℓβ

ψ(q) = esum

gge0 ~gminus1Fg(q)

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

XH+ rarr R F g

YH+ rarr R

DX = exp

~gminus1F g

DY = exp

~gminus1F g

UTDX = DY

T(LX) = LY

Case 1 T =

(I B0 I

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

~gminus1F g

2~Bαβq

αqβ)

~gminus1F g

dqF 0Y = Bq + dqF 0

X (40)

dqF 0Y = p + Bq

Since p = dqF 0X

Case 2 T =

(A 00 AminusT

Case 3 T =

(I 0C I

~gminus1F g

~minusχΓ

where FXΓ

partmF Xg

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

F 0Y (q) =

FΓ(q)

|Aut(Γ)|

F X0 (q) = 〈〉 + 〈eα〉Xqα +

follows that

FXΓ (q) =

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

F Y0 (q) =

|Aut(Γ)|sum

visinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

(I 0minusC I

) we find that

0+qF X

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

Yis equal to

|Aut(Γ)|sum

ihwisinV(Γ)

langlangei

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)isinE(Γ)

(minusCiaib)

0+qF 0

Xis equal to

=〈〈ei〉〉Xsum

|Aut(Γ)|sum

ihwisinV(Γ) jk

langlange j

hisinH(w)

rangrangX

(q)prod

vwisinV(Γ)

langlang prod

hisinH(v)

rangrangX

e=(ab)isinE(Γ)

Γ1Γn

j1 jnk1 knℓ1ℓm

langlange jc

hisinH(wc)

rangrangX

vw1 wn

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

ekn qℓ1eℓ1

qℓmeℓm〉m

Γ1Γn

j1 jnk1 kn

langlange jc

hisinH(wc)

rangrangX

langlang prod

hisinH(v)

rangrangX

(q)prod

e=(ab)

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

|Aut(Γ)|

0 as required

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

CHAPTER 4

Qd~gminus1

(n minus 1)

lang[t(ψ)

t(ψ) t(ψ)

2~〈t0 t0〉

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

minus 1

2~(q0 q0) minus

Qd~gminus1

(n minus 1)

lang[q(ψ)

t(ψ) t(ψ)

z= minus 1

2~〈q0 q0〉 minus

qαk+1

partqαk

= minus 1

zDX = 0

Qd~gminus1

(n minus 1)

lang[ρt(ψ)

t(ψ) t(ψ)

2~〈t(ψ) t(ψ) ρ〉030 + 〈ρ〉110 (42)

minussum

Qd~gminus1

(n minus 1)

t(ψ) t(ψ)

ρ =rsum

ρiτi

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

ρiQipart

partQi

~gminus1F Xg

〈t(ψ) t(ψ) ρ〉030 = (q0ρ q0)

M11(X 0) X timesM11

〈ρ〉X110 =int

minus 1

~gminus1F Xg

ρiQipart

partQi

minus 1

~gminus1F Xg

)middot DX =

ρiQipart

partQi

minus 1

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

τTG = S(RDN)

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

[LXm L

m+n (43)

such that

LXmDX = 0

for all m ge minus1

)z = z2 d

43 Twisted theory

Mgn(X d)

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

c(middot) = exp

skchk(middot)

[Mgn(Xd)]vir

evlowast1(γ1)ψa1

n c(Egnd)

HcE rarrH

x 7rarrradic

LcE =radic

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

ℓgt0

sℓminus1

e(X) and

s jch j(E)

mgt0ℓge0

s2mminus1+ℓB2m

ℓgt0

c(Egnd) = exp

LcE = exp

0leℓleD

s2mminus1+ℓB2m

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

70 BIBLIOGRAPHY

Preface


02 Feynman diagrams


11 The set-up





16 Generalizations


21 Definitions

22 Basic Equations

23 Axiomatization




33 Convergence






43 Twisted theory


Bibliography

Documents

Geometric Quantization with Applications to Gromov … · 2013-09-05 · Geometric Quantization with Applications to Gromov-Witten Theory Emily Clader, ... 3.5. Semi-classical limit