· Contents 1 Introduction 1 1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.1.1 Conformal eld theory

National Research University Higher School of Economics

Faculty of Mathematics

Pavlo Gavrylenko

Isomonodromic deformations and quantum field theory

PhD thesis

SupervisorAndrei MarshakovDr.Sc., professor

Moscow – 2018

Contents

1 Introduction 11.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Conformal field theory . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Isomonodromic deformations . . . . . . . . . . . . . . . . . . . 71.1.3 Isomonodromy-CFT correspondence . . . . . . . . . . . . . . . 81.1.4 Twist fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1 List of the key results . . . . . . . . . . . . . . . . . . . . . . . 111.2.2 Organization of the thesis . . . . . . . . . . . . . . . . . . . . 13

2 Isomonodromic τ-functions and WN conformal blocks 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Isomonodromic deformations and moduli spaces of flat connections . 17

2.2.1 Schlesinger system . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Moduli spaces of flat connections . . . . . . . . . . . . . . . . 192.2.3 Pants decomposition of Mg

4 . . . . . . . . . . . . . . . . . . . 202.2.4 Pants decomposition for Mg

n . . . . . . . . . . . . . . . . . . . 212.3 Iterative solution of the Schlesinger system . . . . . . . . . . . . . . . 22

2.3.1 sl2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 sl3 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Remarks on W3 conformal blocks . . . . . . . . . . . . . . . . . . . . 312.4.1 General conformal block . . . . . . . . . . . . . . . . . . . . . 312.4.2 Degenerate field . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Free fermions, W-algebras and isomonodromic deformations 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Abelian U(1) theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Fermions and vertex operators . . . . . . . . . . . . . . . . . . 383.2.2 Matrix elements and Nekrasov functions . . . . . . . . . . . . 413.2.3 Riemann-Hilbert problem . . . . . . . . . . . . . . . . . . . . 433.2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Non-Abelian U(N) theory . . . . . . . . . . . . . . . . . . . . . . . . 453.3.1 Nekrasov functions . . . . . . . . . . . . . . . . . . . . . . . . 453.3.2 N -component free fermions . . . . . . . . . . . . . . . . . . . 463.3.3 Level one Kac-Moody and W-algebras . . . . . . . . . . . . . 473.3.4 Free fermions and representations of W-algebras . . . . . . . . 50

3.4 Vertex operators and Riemann-Hilbert problem . . . . . . . . . . . . 523.4.1 Vertex operators and monodromies . . . . . . . . . . . . . . . 523.4.2 Generalized Hirota relations . . . . . . . . . . . . . . . . . . . 563.4.3 Riemann-Hilbert problem: hypergeometric example . . . . . . 58

3.5 Isomonodromic tau-functions and Fredholm determinants . . . . . . 603.5.1 Isomonodromic tau-function . . . . . . . . . . . . . . . . . . . 603.5.2 Fredholm determinant . . . . . . . . . . . . . . . . . . . . . . 61

iii

CONTENTS

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Fredholm determinant and Nekrasov sum representations of isomon-odromic tau functions 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.1 Motivation and some results . . . . . . . . . . . . . . . . . . . 65

4.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.3 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . 70

4.1.4 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Tau functions as Fredholm determinants . . . . . . . . . . . . . . . . 74

4.2.1 Riemann-Hilbert setup . . . . . . . . . . . . . . . . . . . . . . 74

4.2.2 Auxiliary 3-point RHPs . . . . . . . . . . . . . . . . . . . . . 77

4.2.3 Plemelj operators . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2.4 Tau function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.5 Example: 4-point tau function . . . . . . . . . . . . . . . . . . 87

4.3 Fourier basis and combinatorics . . . . . . . . . . . . . . . . . . . . . 91

4.3.1 Structure of matrix elements . . . . . . . . . . . . . . . . . . . 91

4.3.2 Combinatorics of determinant expansion . . . . . . . . . . . . 94

4.4 Rank two case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4.1 Gauss and Cauchy in rank 2 . . . . . . . . . . . . . . . . . . . 99

4.4.2 Hypergeometric kernel . . . . . . . . . . . . . . . . . . . . . . 106

4.5 Relation to Nekrasov functions . . . . . . . . . . . . . . . . . . . . . 110

5 Exact conformal blocks for the W-algebras, twist fields and isomon-odromic deformations 121

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 Twist fields and branched covers . . . . . . . . . . . . . . . . . . . . 123

5.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2.2 Correlators with the current . . . . . . . . . . . . . . . . . . . 126

5.2.3 Stress-tensor and projective connection . . . . . . . . . . . . . 128

5.3 W-charges for the twist fields . . . . . . . . . . . . . . . . . . . . . . 129

5.3.1 Conformal dimensions for quasi-permutation operators . . . . 129

5.3.2 Quasipermutation matrices . . . . . . . . . . . . . . . . . . . . 130

5.3.3 W3 current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.3.4 Higher W-currents . . . . . . . . . . . . . . . . . . . . . . . . 132

5.4 Conformal blocks and τ -functions . . . . . . . . . . . . . . . . . . . . 134

5.4.1 Seiberg-Witten integrable system . . . . . . . . . . . . . . . . 134

5.4.2 Quadratic form of r-charges . . . . . . . . . . . . . . . . . . . 135

5.4.3 Bergman τ -function . . . . . . . . . . . . . . . . . . . . . . . . 138

5.5 Isomonodromic τ -function . . . . . . . . . . . . . . . . . . . . . . . . 140

5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.8 Diagram technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.9 W4(z) and the primary field . . . . . . . . . . . . . . . . . . . . . . . 147

5.10 Degenerate period matrix . . . . . . . . . . . . . . . . . . . . . . . . 148

iv

CONTENTS

6 Twist-field representations of W-algebras, exact conformal blocksand character identities 1516.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.3 W-algebras and KM algebras at level one . . . . . . . . . . . . . . . 153

6.3.1 Boson-fermion construction for GL(N) . . . . . . . . . . . . . 1536.3.2 Real fermions for D- and B- series . . . . . . . . . . . . . . . 155

6.4 Twist-field representations from twisted fermions . . . . . . . . . . . 1576.4.1 Fermions and W-algebras . . . . . . . . . . . . . . . . . . . . 1576.4.2 Twist fields and Cartan’s normalizers . . . . . . . . . . . . . . 1586.4.3 Twist fields and bosonization for gl(N) . . . . . . . . . . . . . 1616.4.4 Twist fields and bosonization for so(n) . . . . . . . . . . . . . 163

6.5 Characters for the twisted modules . . . . . . . . . . . . . . . . . . . 1646.5.1 gl(N) twist fields . . . . . . . . . . . . . . . . . . . . . . . . . 1656.5.2 so(2N) twist fields, K ′ = 0 . . . . . . . . . . . . . . . . . . . . 1666.5.3 so(2N) twist fields, K ′ > 0 . . . . . . . . . . . . . . . . . . . . 1666.5.4 so(2N + 1) twist fields . . . . . . . . . . . . . . . . . . . . . . 1676.5.5 Character identities . . . . . . . . . . . . . . . . . . . . . . . 1686.5.6 Twist representations and modules of W-algebras . . . . . . . 171

6.6 Characters from lattice algebras constructions . . . . . . . . . . . . . 1736.6.1 Twisted representation of g1 . . . . . . . . . . . . . . . . . . . 1736.6.2 Calculation of characters . . . . . . . . . . . . . . . . . . . . . 1756.6.3 Characters from principal specialization of the Weyl-Kac formula177

6.7 Exact conformal blocks of W (so(2N)) twist fields . . . . . . . . . . . 1826.7.1 Global construction . . . . . . . . . . . . . . . . . . . . . . . . 1826.7.2 Curve with holomorphic involution . . . . . . . . . . . . . . . 1846.7.3 Computation of conformal block . . . . . . . . . . . . . . . . . 1866.7.4 Relation between W (so(2N)) and W (gl(N)) blocks . . . . . . 188

6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.9 Identities for lattice Θ-functions . . . . . . . . . . . . . . . . . . . . 190

6.9.1 First identity for AN−1 and DN Θ-functions . . . . . . . . . . 1906.9.2 Product formula for AN−1 Θ-functions . . . . . . . . . . . . . 1926.9.3 An identity for DN and BN Θ-functions . . . . . . . . . . . . 193

6.10 Exotic bosonizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.10.1 NS ×R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.10.2 R×R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.10.3 l twisted charged fermions . . . . . . . . . . . . . . . . . . . . 1966.10.4 l charged fermions – standard bosonization . . . . . . . . . . . 198

Bibliography 199

v

1Introduction

In my thesis I present a correspondence between isomonodromic deformations ofhigher-rank Fuchsian linear systems and conformal field theory with higher-spin sym-metry, or W-symmetry. The correspondence that I describe is a generalization tohigher rank of the one found by Gamayun, Iorgov and Lisovyy in [GIL12]. This gener-alization is first found numerically and then proved in the free-fermionic framework byan explicit construction of the twist-fields that are at the same time monodromy fieldsand W-primary fields. Next I use this construction to give the Fredholm-determinantrepresentation of the general isomonodromic tau-function. The determinantal repre-sentation found in this way can also be proven without using any field theory by acareful analysis of derivatives of the determinant.

Another part of thesis deals with the special case that the monodromy groupis given by quasi-permutation matrices I present a construction of the W-primaryfields in terms of twisted bosons and give an expression of their conformal blocks interms of algebro-geometric objects associated with branched covers of the complexsphere. Such correlation functions are related to exact isomonodromic tau-functionsintroduced by Korotkin. I also present the interpretation of such fields in terms of freefermions and a computation of the characters of related W-algebra representations. Inthis part of the investigations also W-algebras of the orthogonal series are considered.

Basic concepts

In this section I try to give a self-contained overview of basic objects considered inthis thesis. My goal is to make this into an introduction for non-experts.

Conformal field theory

By a conformal field theory (CFT) is meant by default a two-dimensional quantumfield theory with conformal symmetry, i.e., the symmetry that preserves angles andmultiplies metrics by a scalar factor: gµν 7→ λgµν . A remarkable feature of thetwo-dimensional case is the fact that Lie algebra of local conformal transformationsbecomes infinite-dimensional and is generated by the holomorphic functions f : C→C, z 7→ f(z). In the infinitesimal form such transformations may be rewritten as

z 7→ z + ε(z) +O(ε2) (1.1)

1

1. Introduction

The Lie algebra of the corresponding vector fields ε(z)∂z has as a natural basis the`n = −zn+1∂z. In this basis its commutations relations acquire the following form:

[`n, `m] = (n−m)`n+m (1.2)

This Lie algebra is called Witt algebra, or Virasoro algebra with zero central extension.All local fields in a conformal theory transform under conformal transformations

in some non-trivial way. It happens though that one may always choose a basis in thespace of fields which is formed by elements that transform as differential forms of thekind φ∆,∆(z, z)dz∆dz∆. Such fields are called primary fields, and (∆, ∆) are calledtheir dimensions. Though in actual physical models ∆ is important, we will alwaysconsider only the holomorphic part. Infinitesimal transformation of the primary field,or, in other words, the action of the Lie derivative, is given by the formula

δε(z)φ∆(z) = (ε′(z)∆ + ε∂z)φ∆(z) (1.3)

By Noether’s theorem, any symmetry in a quantum theory gives rise to conservedcharges. Conformal transformations in CFT give rise to charges that are encoded bya single energy-momentum tensor T (z). The quantum version of Noether’s theorem isformed by the Ward identities that relate infinitesimal transformations of fields withthe action of conserved charges. In CFT they read as

δε(z)φ(z) =

˛

z

dw

2πiRT (w)φ(z) (1.4)

In this formula the integral goes around a small circle around w = z, and the symbolR means radial ordering of the operators

Rφ(z)ψ(w) = φ(z)ψ(w), |z| > |w|Rφ(z)ψ(w) = (−1)pψ ·pφψ(w)φ(z), |z| < |w|

(1.5)

Here pφ is fermionic parity. If we work in the path integral formulation we may justskip this notation: any product is already radially ordered.

A very important concept in CFT is the so-called operator product expansion, i.e.,the expansion of the radially ordered product of two fields in the neighbouring points:

A(z)B(w) =N∑

n=−∞

(AB)n(w)

(z − w)n+1(1.6)

Radial ordering will be usually omitted in all OPE expansions due to historical rea-sons, though it is important.

The singular part of the OPE contains all information about the commutationrelations between modes of the operators 1:

[An, B(w)] =1

2πi

˛

w

dzzn+∆−1A(z)B(w) (1.7)

1This can be easily deduced using properties of radial ordering and doing manipulations withcontour integrals

2

1.1. Basic concepts

where An = 12πi

¸zn+∆−1A(z)dz.

For example, one can write down an OPE of the primary field with the energy-momentum tensor:

T (z)φ∆(w) =∆φ∆(w)

(z − w)2+∂φ∆(w)

z − w+ reg. (1.8)

An analogous OPE for the energy-momentum tensor itself has the form

T (z)T (w) =c/2

(z − w)4+

2T (w)

(z − w)2+∂T (w)

z − w+ regular(= reg.) (1.9)

One can introduce the components of the Laurent expansion of the energy-momentumtensor

T (z) =∑n∈Z

Lnzn+2 (1.10)

and then rewrite the above OPEs in terms of these components:

[Ln, φ∆(z)] = (n+ 1)∆znφ∆(z) + zn+1φ∆(z) (1.11)

[Ln, Lm] =c

12(n3 − n) + (n−m)Ln+m (1.12)

The Lie algebra generated by the operators Ln is called the Virasoro algebra, and aswe see, it is a central extension of the algebra of vector fields by the element c calledcentral charge. The value of the central charge is an important characteristic of CFT.

Free bosonic CFT

One of not the most elementary, but very important examples of CFT is a free bosonictheory with N elementary fields φα(z, z) – Gaussian fields with the following OPEs:

φα(z)φβ(w) = −δαβ log |z − w|2 + reg. (1.13)

It is also useful to introduce derivatives of the φα, the so-called U(1) currents Jα(z) =i∂φα(z) with conformal dimension (1, 0). The commutation relation of the modesof such currents are given by [Jα,n, Jβ,k] = nδn+k,0δαβ. The Lie algebra with thesecommutation relations is called the Heisenberg algebra.

One can check that the energy-momentum tensor

T (z) =N∑α=1

: Jα(z)2 : (1.14)

actually generates the Virasoro algebra with c = N . In this way we can get a realiza-tion of the complicated Virasoro algebra in terms of a simpler Heisenberg.

The simplest primary fields, or vertex operators of the constructed Virasoro, canbe given explicitly by the exponents

V~a(z) = : ei∑αaαφα(z)

: (1.15)

3

1. Introduction

One can check that the following OPE with the energy-momentum tensor holds forsuch fields:

T (z)V~a(w) =∆(~a)Va(w)

(z − w)2+∂V~a(w)

z − w+ reg. (1.16)

In this formula the conformal dimension is given by the formula ∆(~a) = 12

∑α

a2α.

In this way we can construct some examples of primary fields, but not an arbitraryone: in our case we have a serious problem, conservation of the U(1) charge. Namely,

colliding two fields with charges ~a and ~b we get another field with charge ~a+~b:

V~a(z)V~b(w) = (z − w)(~a,~b)V~a+~b(w) + . . . , (1.17)

whereas in the general CFT any fields can appear in this OPE. However, we willpresent an almost free-field generalization of this construction in Chapter 3, whichis not restricted by this charge-conservation condition.

The free-bosonic theory gives also an example of a theory with W-symmetry, thenon-linear higher spin symmetry. Generators of this symmetry are expressed viainitial bosonic fields as elementary symmetric polynomials (the energy-momentumtensor was a quadratic symmetric polynomial):

Wk(z) =N∑

α1<...<αk

: Jα1(z) . . . Jαk(z) : (1.18)

Clearly, there are only N such currents. It happens so that their commutators areactually non-linear functions of the initial generators. For example, in the N = 3 casethey look schematically like

T · T ∼ T

T ·W3 ∼ W3

W3 ·W3 ∼ T + (TT )

(1.19)

This algebra is very complicated in the general case, but nevertheless it can be studiedwith the help of various free-field techniques.

The field V~a(z) introduced above is also an example of a W-primary field since itsOPE with Wk(z) is given by the following formula:

Wk(z)V~a(w) =ek(~a)V~a(w)

(z − w)k+ less singular, (1.20)

where the ek are elementary symmetric polynomials. The main difference withthe usual conformal symmetry (1.8) is that, in general, coefficients near the lowerorders of this expansion are not given in terms of V~a(z). This causes one of the mainproblems of W-algebras: their vertex operators are not defined uniquely in the generalsituation. We propose some solution of this problem in Chapter 3.

4

1.1. Basic concepts

Free fermionic CFT

Another very important concept in two-dimensional physics is the boson-fermion cor-respondence, which relates free bosonic and free fermionic theories. The transforma-tion between these two theories can be given approximately (precise expressions arewritten in the main text) by the following formulas:

ψ∗α(z) ≈ : eiφα(z) :

ψα(z) ≈ : eiφα(z) :

Jα(z) = : ψ∗α(z)ψα(z) :

(1.21)

Here ψα(z) and ψ∗α(z) are N -component fermionic fields with the following OPEs andanticommutation relations:

ψ∗α(z)ψβ(w) =δαβz − w

+ reg.

ψ∗α,p, ψβ,q = δαβδp+q,0

(1.22)

The conformal dimensions of both ψ and ψ∗ are the same and equal to (12, 0).

From many points of view the fermionic description is much better. For example,instead of the complicated non-linear generators (1.20) the W-algebra has another setof nice fermionic operators which are just bilinear:

Wk(z) =N∑α=1

: ∂k−1ψ∗α(z)ψα(z) : (1.23)

Such a representation also gives us a better understanding of what is W-symmetry.Namely, its action on fermions is given by formula

Wk(z)ψ∗α(w) =∂k−1ψ∗α(w)

z − w+ reg. (1.24)

It may also be rewritten using (1.7) in terms of the modes Wk,n = 12πi

¸Wk(z)zk+n−1dz:[

Wk,n, ψ∗(w)

]= wn+k−1∂k−1ψ∗α(w) (1.25)

The above calculation demonstrates that the analogy between the vector fields−zn+1∂and the Virasoro generators Ln can be continued to an analogy between arbitrarydifferential operators zn+k−1∂k and W-generators Wk,n.

Another important concept in the free-fermionic theory are the group-like ele-ments: such operators act on the generators of the Clifford algebra ψα,n, ψ

∗α,n in a

linear way

O−1ψα,pO =∑β,q

Cα,n;β,qψβ,q (1.26)

Such operators were widely used before in the literature to construct solutions of inte-grable hierarchies, like KP, Toda, and their multi-component generalizations. Here weshow that they also appear in conformal theory: we find the general vertex operatorsfor the W-algebra in such a form.

5

1. Introduction

A remarkable property of a group-like element is the fact that any of its matrixelements can be expressed as a determinant of just two-particle ones. As an immediateconsequence of this property any correlation function of the group-like elements canbe expressed as some Fredholm determinant.

The AGT relation

There is an important object in conformal field theory, called the conformal block.For simplicity we consider the 4-point one:

F(∆0,∆t,∆1,∆∞; ∆0t; c|t) = 〈∆∞|φ∆1(1)P∆0tφ∆t(t)|∆0〉 (1.27)

To explain the meaning of this definition one has to recall that the symmetry algebraof the theory is the Virasoro algebra, and since it acts on the Hilbert space of thetheory, this Hilbert space decomposes into the sum of its highest-weight irreduciblerepresentations. In the general position they are Verma modules, i.e. modules withhighest weight |∆〉 such that

Lk>0|∆〉 = 0

L0|∆〉 = ∆|∆〉(1.28)

and the module itself is spanned by the vectors L−k1 . . . L−kn|∆〉 with k1 ≥ k2 ≥ . . . ≥kn.

Now one can say that projector P∆0t is a projector onto the Verma module withhighest weight (dimension) ∆, and any 4-point correlation function in conformal fieldtheory can be expanded over conformal blocks since its Hilbert space can be expandedinto Verma modules.

Conformal blocks itself are purely algebraic universal objects that can be computedjust from the commutation relations in the Virasoro algebra (1.12) and from thedefinition of the primary field (1.11). However, for the more complicated W-algebracase they can be computed algebraically only for the cases when two charges 2, ~at and~a1 have a very special form: ~a1 = (a1, b1, . . . , b1), ~at = (at, bt, . . . , bt). Fields with suchcharges are called semi-degenerate.

Virasoro conformal block is in general a concrete, but very complicated specialfunction, and until 2009 there were only two ways to compute it: by doing order-by-order computations in the Virasoro algebra or by using the Zamolodchikov recursionformula. The situation changed in 2009 when Alday, Gaiotto and Tachikawa proposedthe correspondence between 2D CFT and 4D N = 2 supersymmetric gauge theories.In this approach the conformal blocks become equal to the so-called Nekrasov in-stantonic partition functions. For our purposes the most important fact is that anycoefficient in the expansion of the instantonic partition function, and so of the confor-mal block, is given by an explicit combinatorial formula (we will write it for simplicity

2As we have seen in (1.20) for the free bosonic field, the action of the W-generators was expressedin terms of elementary symmetric polynomials ek(~a). It turns out that it is useful to use thisparametrization not only for the free case.

6

1.1. Basic concepts

only for c = 1) of the following kind:

F(a20, a

2t , a

21, a

2∞;σ2

0t; c = 1; |t) = (1− t)2a0at×

×∑Y+,Y−

t|Y1|+|Y2|∏s,s′=±

Zb(at + sa0 − s′σ0t|Ys, Ys′)Zb(a1 + sσ0t − s′a∞|Ys, Ys′)Zb((s− s′)σ0t|Ys, Ys′)

(1.29)

In this formula Y+, Y− are two Young diagrams, and Zb(ν|Y1, Y2) is some explicit fac-torized combinatorial expression depending on two Young diagrams and one complexnumber.

This formula for a conformal block was proved in 2010 by Alba, Fateev, Litvinovand Tarnopolsky. In their proof they presented such a basis that any matrix elementof the Virasoro vertex operator can be expressed in terms of Zb. What matters for usis that for c = 1 their basis is exactly the free-fermionic one. This is one more hintthat a fermionic description of conformal field theory is better than a bosonic one.

Isomonodromic deformations

There is a story from the beginning of 20th century when mathematicians started tostudy N ×N matrix linear systems with first-order singularities:

dΦ(z)

dz= Φ(z)

n∑k=1

Akz − zk

(1.30)

wheren∑k=1

Ak = 0. One may ask, at first, when such a system can be solved explicitly

in terms of some known special function. I present below an important list of someexamples which, however, does not cover everything:

• N = 2, n = 3. Always solvable in terms of hypergeometric function 2F1.

• n = 3, N – arbitrary, but the spectral type ofA1 is special: A1 ∼ diag(a1, b1, . . . , b1).Always solvable in terms of NFN−1. Here the analogy with the semi-degeneratefields is absolutely not accidental.

• n – arbitrary, but the monodromy group is a semidirect product of a permutationgroup and the diagonal matrices (quasi-permutation group). Always solvable interms of higher-genus theta-functions. This case corresponds at the CFT sideto the twist fields and is considered in Chapters 5, 6.

For n > 4 and for general A’s the Fuchsian system cannot be solved explicitly (thoughin Chapter 4 we give the formula that can give its explicit expansion in some regionof parameters). Instead of this it is reasonable to ask about the monodromy of sucha system. Namely, if we take some solution and continue it analytically around theloop γ encircling some singular point, we get another solution. Now any two solutionsof the system are connected by linear transformations, so we have

γ : Φ(z) 7→MγΦ(z) (1.31)

7

1. Introduction

The matrices Mγ ∈ GL(N) generate the monodromy group of the system, and analyticcontinuation around closed loops generates a map from π1(C\z1, . . . , zn) to GL(N)with the image coinciding with the monodromy group.

The problem of finding the monodromy group for a given system is also compli-cated. Instead of this one may look for such transformations of the systems thatpreserve the monodromy, the so-called isomonodromic transformations. It happensso that in general position we are able to move all singular points and to make somemodifications of the matrices Ak that preserve the monodromy: in this setting all ma-trices Ak become functions of z1, . . . , zn. Such a functional dependence is describedby a non-linear system of matrix equations, the Schlesinger equations :

∂Aj∂zk

=[Aj, Ak]

zj − zk∂Aj∂zj

= −∑k 6=j

[Aj, Ak]

zj − zk

(1.32)

There is also a non-trivial statement that can be verified explicitly that any solutionof the Schlesinger system gives some function of the zk, the tau-function, definedby its derivatives:

∂ log τ(z1, . . . , zn)

∂zk=∑j 6=k

trAkAjzk − zj (1.33)

This function is simpler than the fundamental solution itself. For example, for n = 3

the singular points it can be given explicitly by τ(z1, z2, z3) =3∏i<j

(zi− zj) trAiAj , while

the fundamental solution is still unknown in general. One of the first interesting casesis n = 4, N = 2: this tau-function solves Painleve VI equation and gives actuallyits general solution. This fact is one of the motivations to study isomonodromicdeformations: they give a convenient framework to study the equations from thePainleve family.

One of the achievements in the study of this tau-function for the Painleve VI casewas the work of Jimbo in 1982 where he obtained the first 3 terms of the tau-functionin terms of the monodromy. The next breakthrough in this direction was done byGamayun, Iorgov, Lisovyy and Teschner when they gave the general formula for theN = 2 tau-function, including arbitrary number of points, in terms of conformalblocks, which easily recovers the Jimbo formula. In the present thesis, see Chapters2-4, I present the generalization of their result for arbitrary N . In particular, I givein Chapter 4 a rigorous proof of this result without using any field theory.

Isomonodromy-CFT correspondence

Various parts of the correspondences between isomonodromic deformations, Painleveequations and quantum field theory (QFT) have been found in the late 70’s by Sato,Miwa and Jimbo. Archetypal formulas of such correspondence look like follows:

τ(x1, . . . , xn) = 〈O(x1) . . . O(xn)〉, Φ(y, y0) =y − y0

τ〈ψ∗(y)ψ(y0)O(x1) . . . O(xn)〉

(1.34)

8

1.1. Basic concepts

Here the O(x) are some disorder fields in some free field theory (like spin variable inthe Ising model), ψ(y), ψ∗(y0) are initial free fields, and Φ(y, y0) is a solution of somelinear problem. Such a correspondence was found for various massive and masslessbosonic and fermionic models. The only problem was that this correspondence wasfound 5 years before the creation of conformal field theory, otherwise this researchcould be related to CFT at that time.

There were several guesses that belong to Knizhnik and Moore that CFT is actuallyrelated to isomonodromic deformations, but they were not developed to get a finalexplicit answer. Such a development was done by Gamayun, Iorgov and Lisovyy in2012, when they gave the general solution of the Painleve VI equation as a linearcombination of c = 1 conformal blocks:

τ(t) =∑n∈Z

sn0tt(σ0t+n)2−θ2

0−θ2tCn(σ0t, θν)F(θ2

0, θ2t , θ

21, θ

2∞; (σ0t + n)2|t) (1.35)

Together with the AGT formula (1.29) this gave the general tau-function as an explicitseries. To explain this formula I give below the short dictionary of the correspondence:

Painleve VI CFT12

trA2ν = θ2

ν ∆ν = θ2ν

trM0Mt = 2 cos πσ0t ∆ = (σ0t + n)2

some function of trMµMν , trMν s0t

τ(t) 〈∆∞|φ∆1(1)φ∆t(t)|∆0〉[Φ(z)Φ(z0)−1]αβ

z−z01τ(t)〈∆∞|φ∆1(1)φ∆t(t)ψ

∗α(z)ψβ(w)|∆0〉

tr (n∑k=1

Akz−zk

)2 1τ(t)〈∆∞|φ∆1(1)φ∆t(t)T (z)|∆0〉

So the main rule is the following: dimensions (or higher W-charges) are symmetricfunctions of the eigenvalues of logarithms of the monodromy matrices.

Formula (1.35) was proved in several different ways, it was also generalized toarbitrary number of points with 2× 2 matrices.

In this thesis I present the same construction for the N × N case which relatesisomonodromic tau-function to a linear combination of conformal blocks of the W-algebra. In Chapter 2 we solve Schlesinger system numerically and conjecture thegeneral form of the tau-function, in Chapters 3, 4 we prove it using two differentapproaches. In Chapter 3 we construct explicitly W-primary fields as some fermionicgroup-like elements with given monodromy and then find the Fredholm-determinantformula for their correlator; in Chapter 4 we give the generalization of this formulato an arbitrary number of points and prove it.

Twist fields

The archetypal example of a twist field is Zamolodchikov’s construction of conformalfield with dimension ∆ = 1

16in c = 1 CFT. The first ingredient is the expression of

9

1. Introduction

the Virasoro algebra in terms of the half-integer Heisenberg algebra:[Jn+ 1

2, J−m− 1

2

]= (n+

1

2)δm,n

Ln =δn,016

+∑k∈Z+ 1

2

: JkJn−k :(1.36)

The usual bosonic representation (Fock module) is reducible, and it is expanded overthe infinite series of Verma modules with dimensions (1

4+n)2. This statement can be

obtained from the computation of characters with the help of the well-known Gaussformula:

q116∏∞

k=0(1− qk+ 12 )

=

∑n∈Z

q( 14

+n)2

∏∞k=1(1− qk)

(1.37)

The picture corresponding to this situation looks as follows: there is a bosonic fieldJ(z) which has monodromy around the origin J(e2πiz) = −J(z). This monodromyis actually related to the twist field O(0) sitting in the point z = 0. Its dimensionequals to 1

16.

Another ingredient of the construction concerns the corresponding vertex operator:the field O(x) sitting in the arbitrary point and changing the sign of J(z) when it goesaround. The great discovery of Zamolodchikov was an exact formula for the conformalblock of such fields. For example, the 4-point block is given by simple formula:

F(1

16,

1

16,

1

16,

1

16; ∆; c = 1|t) =

(16t−1)∆eiπ∆τ(t)

(1− t) 18 θ3(0|τ(t))

(1.38)

where τ(t) is a period of the elliptic curve y2 = z(z − t)(z − 1).As far as we have the isomonodromy-CFT correspondence, we can use this con-

formal block in (1.35): this leads us to so-called Picard solution of Painleve VI. Fromthe point of view of the monodromy it corresponds to the quasi-permutations thatwere mentioned above: in this case one can find explicitly the general solution of then-point system.

In this thesis I present the generalization of Zamolodchikov’s construction to thecase of W-algebra. In contrast to the previous situation, here there is a richer collectionof twist fields that are labelled by the elements of the permutation group. Theypermute the bosonic currents leaving the W-generators untouched:

Jk(e2πiz)Os(0) = Js(k)(z)Os(0) (1.39)

In Chapter 5 we construct such fields and find the generalization of Zamolodchikov’sformula for their conformal blocks. We also show that using extended isomonodromy-CFT correspondence we can construct the tau-function from these conformal blocksand then identify it with the known tau-function found by Korotkin.

In Chapter 6 we find many generalizations of the character formula (1.37), wealso find a very close relation between the construction of the W-algebra twist fields

and the Lepowski-Wilson construction of the integrable representations of sl(N)1.We also relate this construction to the free-fermionic approach from Chapter 2. Incontrast to the previous considerations, here we also touch upon the W-algebras forthe orthogonal series and generalize all results related to twist-fields to this case.

10

1.2. Outline

Outline

Here I list the most important results of the thesis and then explain how the differentparts of the text are related to each other.

List of the key results

• Formula 2.53:

τ(t) =∑w∈Q

e(β,w)C(0t)w (θ0,θt,σ0t, µ0t, ν0t)C

(1∞)w (θ1,θ∞,σ0t, µ1t, ν1t)×

× t 12

(σ0t+w,σ0t+w)− 12

(θ0,θ0)− 12

(θt,θt)Bw(θi,σ0t, µ0t, ν0t, µ1∞, ν1∞; t)

This formula describes the conjectural form of the general N = 3, n = 4 tau-function.

• Formula 2.58:

C(0t)w (θ0, at,σ)C

(1∞)w (σ, a1,θ∞) =

∏ij G[1−at

N+(ei,θ0)−(ej ,σ+w)]G[1−a1

N+(ei,σ+w)+(ej ,θ∞)]∏

iG[1+(αi,σ+w)]

This formula gives the conjectural form of the structure constants for two semi-degenerate fields.

• Theorem 3.2: Vν(t) is a primary field of the conformal WN ⊕H algebra withthe highest weights uk(ν).

• Theorem 3.5:

Solution of the linear problem with n marked points is given by

(z − w)Kαβ(z, w) with

Kαβ(z, w) =〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)ψθ0

α (z)ψθ0β (w)|θ0〉

〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)|θ0〉(1.40)

whereas its isomonodromic tau-function is defined by

τ(t1, . . . , tn−2) = 〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)|θ0〉 (1.41)

• Formula 3.136: τ(t) = det (1 +Rt)

This formula expresses the 4-point isomonodromic tau-function as a Fredholmdeterminant with explicitly given kernel.

• Theorem 4.22: Fredholm determinant τ (a) giving the isomonodromic taufunction τJMU (a) can be written as a combinatorial series

τ (a) =∑

~Q1,... ~Qn−3∈QN

∑~Y1,...~Yn−3∈YN

n−2∏k=1

Z~Yk−1, ~Qk−1

~Yk, ~Qk

(T [k]

),

where Z~Yk−1, ~Qk−1

~Yk, ~Qk

(T [k]

)are expressed by (4.66), (4.63) in terms of matrix ele-

ments of 3-point Plemelj operators in the Fourier basis.

11

1. Introduction

• Theorem 4.32: This theorem describes the relation of our general Fredholmdeterminant and the particular hypergeometric one found before by Borodinand Deift.

• Theorem 5.1:Function

log τSW = 12

∑I,J

aITIJaJ +∑I

aIUI + 12Q(r)

solves the system (5.55), iff Q(r) solves the system ∂Q(r)∂qα

=∑

π(qiα)=qα

Res qiα(dΩ)2

dz

for α = 1, . . . , 2L, dΩ =∑α

dΩrα and other ingredients in the r.h.s. are given

by (5.16), (5.20) and the period matrix of C.

This theorem gives the solution for a Seiberg-Witten system.

• Formula 5.78: Q(r) =∑

qiα 6=qjβ

riαrjβ log Θ∗(A(qiα)−A(qjβ))−

∑qiα

(riα)2liα log d(z(q)−qα)1/liα

h2∗(q)

∣∣∣∣∣q=qiα

This formula gives the “r-charge contribution” to the exact conformal block.

• Formula 5.87: G0(q|a) = τB(q) exp

(12

∑IJ

aITIJ(q)aJ +∑I

aIUI(q, r) + 12Q(r)

)This is the general formula for the conformal block of twist fields (generalizationof Zamolodchikov’s formula).

• Theorem 6.2: The characters of the twisted representations are given by theformulas (6.85), (6.88), (6.95), (6.97).

• Theorem 6.3:If g1 ∼ g2 in G for different g1, g2 ∈ NG(h), then χg1(q) =χg2(q).

This theorem generalizes the Gauss identity from Zamolodchikov’s construction.

• Theorem 6.4: The conformal blocks (6.163) for generic W (o(2N)) twist fieldsare given by

G0(a, r, q) = τB(Σ|q)τ−1B (Σ|q)τSW (a, r, q)

where

∂qi log τB(Σ|q) =∑

π2N (ξ)=qi

Res tz(ξ)dξ, ∂qi log τB(Σ|q) =∑

πN (ζ)=qi

Res tz(ζ)dζ

i = 1, . . . , 2M

and

∂qi log τSW (a, r, q) =1

4

∑π2N (ξ)=qi

Res(dS)2

dz, i = 1, . . . , 2M

∂

∂aIlog τSW =

˛BI

dS, AI BJ = δIJ , I, J = 1, . . . , g−

12

1.2. Outline

Organization of the thesis

All the parts of this thesis are self-contained papers with their own introductions, sothey can be read independently. But nevertheless, there are some logical dependenciesbetween different parts. I show them on the following diagram:

Chapter5 Chapter6

Chapter2 Chapter3

Chapter4

Chapter 2 is devoted to the numerical solution of the Schlesinger system ofthe rank 3 and to the computation of corresponding isomonodromic tau-function.Its main task was to formulate and check the main conjectures for the higher-rankisomonodromy-CFT correspondence, which are then proved in the next chapters.

Chapter 3 deals with the free-fermionic construction of monodromy fields. Ax-iomatically such fields are defined by:

1) it is a fermionic group-like element

2) its two-particle matrix elements are expressed through the solution of 3-pointFuchsian system.

Then we prove that such fields are W-primaries, and at the same time their correlationfunction can be given as some Fredholm determinant. In this way we give the free-fermionic proof of the conjectures from Chapter 2. Also we get some integrablehierarchies which are related to such fields.

Chapter 4 is written in a pure mathematical language and absolutely rigorously,so it does not require any field-theory background. In this chapter we develop theframework in which the Fredholm determinant formula can be proved rigorously. Todo this first we cut the sphere with n punctures into n − 2 three-punctured sphere,and then introduce the spaces of functions on the obtained boundaries. Then weconstruct two projectors onto the space of functions that can be continued betweenthe different boundaries, PΣ and P⊕. After that we restrict these projectors on someother space H+: PΣ,+, P⊕,+. Thanks to this procedure they become non-degenerate.Then we define an infinite-dimensional determinant τ = detP−1

Σ,+P⊕,+. Next we provethat:

1) derivatives of this determinant coincide with the derivatives of isomonodromictau-function,

2) it is a Fredholm determinant, whose kernel in the 4-point case reproduces theone obtained in Chapter 3,

13

1. Introduction

3) the minor expansion of the determinant reproduces a combinatorial formulawhich in the known cases can be obtained from the isomonodromy-CFT + AGTcorrespondences. In this sense it gives one more independent proof of AGT forc = N .

Chapter 5 is devoted to the study of W-twist fields. The main techniques hereare the free-field conformal theory and the algebraic geometry of complex curves.From the field theory we get equations that are satisfied by the correlation functionsof twist fields, and then solve them in terms of period matrices, Abel maps and thetheta-function of some branched cover of the punctured sphere. This constructiongeneralizes the conformal block of Zamolodchikov.

Chapter 6 is also devoted to W-twist fields, but from the algebraic point of view.Here we consider the W-algebras for the orthogonal series, too. We start from the free-fermionic definition of W-algebras, their vertex operators in the spirit of Chapter 3,and then show that for quasi-permutation monodromy a lot of other bosonic construc-tions of these algebra can be obtained with the help of various exotic bosonizations.We also find a sequence of character identities that come from equivalences betweendifferent representations. In addition, we give a simple generalization of the exactconformal block from Chapter 5 to the orthogonal series.

References

The content of Chapters 2-6 is based on the following papers in order. Almost nochanges were done to avoid producing mistakes. Therefore some mathematical objectsare introduced several times, but any time the only properties that are needed for agiven chapter are introduced, so it would not confuse the reader.

• M. Bershtein, P. Gavrylenko, A. Marshakov, Twist-field representations of W-algebras, exact conformal blocks and character identities , [hep-th/1705.00957],Under review in Communications in Mathematical Physics

• P. Gavrylenko, O. Lisovyy, Fredholm determinant and Nekrasov sum represen-tations of isomonodromic tau functions, [math-ph/1608.00958], Submitted toCommunications in Mathematical Physics

• P. Gavrylenko, A. Marshakov, Free fermions, W-algebras and isomonodromicdeformations, Theor. Math. Phys. 2016, 187:2, 649–677, [hep-th/1605.04554]

• P. Gavrylenko, A. Marshakov, Exact conformal blocks for the W-algebras, twistfields and isomonodromic deformations, JHEP02(2016)181,[hep-th/1507.08794]

• P. Gavrylenko, Isomonodromic τ -functions and WN conformal blocks, JHEP09(2015)167,[hep-th/1505.00259]

14

2Isomonodromic τ -functions and WN

conformal blocks

Abstract

We study the solution of the Schlesinger system for the 4-point slN isomonodromyproblem and conjecture an expression for the isomonodromic τ -function in terms of2d conformal field theory beyond the known N = 2 Painleve VI case. We showthat this relation can be used as an alternative definition of conformal blocks for theWN algebra and argue that the infinite number of arbitrary constants arising in thealgebraic construction of WN conformal block can be expressed in terms of only afinite set of parameters of the monodromy data of rank N Fuchsian system with threeregular singular points. We check this definition explicitly for the known conformalblocks of the W3 algebra and demonstrate its consistency with the conjectured formof the structure constants.

Introduction

There are two topics in the mathematical physics that remained independent for a longtime: the theory of isomonodromic deformations, initiated by R. Fuchs, P. Painleveand L. Schlesinger in the beginning of 20th century (see [IN] and references therein),and the 2d conformal field theory (CFT) founded by A. Belavin, A. Polyakov andA. Zamolodchikov in 1984 [BPZ]. Both theories have wide range of applications.Conformal field theory describes perturbative string theory and second order phasetransitions in the 2d systems. The theory of isomonodromic deformations gives rise tonon-linear special functions such as Painleve transcendents, which appear in differentproblems of mathematical physics: for example, in the random matrix theory andgeneral relativity.

First relations between the theory of isomonodromic deformations and 2d quantumfield theory have been established in 1978-80 by M. Sato, M. Jimbo and T. Miwa[SMJ]. More recently, O. Gamayun, N. Iorgov and O. Lisovyy have discovered thatthe τ -function of the Painleve VI equation (related to the rank two Fuchsian systemwith four regular singular points on the Riemann sphere) can be expressed as a sum ofc = 1 conformal blocks, multiplied by certain ratios of the Barnes functions – a typicalexpansion of the correlation function in CFT [GIL12]. Their formula gives the general

15

2. Isomonodromic τ -functions and WN conformal blocks

solution of Painleve VI equation. This conjecture has already been proved in two ways:one proof is purely representation-theoretic and adapted initially for the 4-point τ -function [BShch] but can provide us with a collection of nontrivial bilinear relationsfor the n-point conformal blocks, whereas another one is based on the computationof monodromies of conformal blocks with degenerate fields and allows to consideran arbitrary number of regular singular points on the Riemann sphere [ILTe]. Thecorrespondence also extends to the irregular case: for instance, it gives exact solutionsof the Painleve V and III equations [GIL13], [ILT14], which are known to describecorrelation functions in certain massive field theories.

The present chapter is concerned with the extension of the isomonodromy-CFTcorrespondence to higher rank. Already in [GIL12] there was a suggestion that themonodromy preserving deformations of Fuchsian systems of rank N should be relatedto 2d CFT with central charge c = N − 1. One obvious and natural candidate forsuch a theory is the Toda CFT with WN algebra of extended conformal symmetry.We show that indeed the N × N isomonodromic problem corresponds to the WN

algebra, whose Virasoro part has central charge c = N − 1. These algebras werefirst introduced by A. Zamolodchikov in [ZamW], and their study was substantiallydeveloped in [FZ] (for the first nontrivial W3-case) and [FL] (for generic WN). Otherdevelopments in the theory of W -algebras are discussed in the review [BS].

The most condensed form of the commutation relations of W3 is given by theoperator product expansions (OPEs) of the energy-momentum tensor T (z) and theW -current W (z):

T (z)T (w) =c

2(z − w)4+

2T(z+w

2

)(z − w)2

+ reg. ,

T (z)W (w) =3W (w)

(z − w)2+∂W (w)

z − w+ reg. ,

W (z)W (w) =c

3(z − w)6+

2T(z+w

2

)(z − w)4

+

+1

(z − w)2

(32

22 + 5cΛ

(z + w

2

)+

1

20∂2T

(z + w

2

))+ reg.

(2.1)

where Λ(z) = (TT )(z)− 310∂2T (z).

The representation theory of this algebra is very similar to that of the Virasoroalgebra. In the generic case one has the Verma module with the highest vector |∆,w〉such that L0|∆,w〉 = ∆|∆,w〉, W0|∆,w〉 = w|∆,w〉. Hence the representation spaceis spanned by the vectors

L−m1L−m2 . . . L−mkW−n1W−n2 . . . |∆,w〉, m1 ≥ m2 ≥ . . . ≥ mk, n1 ≥ n2 ≥ . . . ≥ nk ,(2.2)

while the set of the highest weight vectors themselves corresponds to primary fields(vertex operators) of the 2d CFT. As in the Virasoro case, these fields can be deter-mined by their OPEs with higher-spin currents T (z) and W (z):

16

2.2. Isomonodromic deformations and moduli spaces of flat connections

T (z)φ(w) =∆φ(w)

(z − w)2+∂φ(w)

z − w+ reg.

W (z)φ(w) =wφ(w)

(z − w)3+

(W−1φ)(w)

(z − w)2+

(W−2φ)(w)

z − w+ reg.

(2.3)

However, the W-descendants such as (W−1φ) and (W−2φ) are not defined in general(this is to be contrasted with the Virasoro case where one has e.g. (L−1φ)(w) =∂φ(w)), which means that the 3-point functions involving such fields are not reallydefined. As a consequence, one cannot express the matrix elements

〈∆∞,w∞|φ(1)L−m1L−m2 . . . L−mkW−n1W−n2 . . . |∆0,w0〉

in terms of 〈∆∞,w∞|φ(1)|∆0,w0〉 only. It was shown in [BW] in an elegant way thatall such 3-point functions can be expressed in terms of an infinite number of unknownconstants

Ck = 〈∆∞,w∞|φ(1)W k−1|∆0,w0〉, k = 1, 2, . . . (2.4)

The problem is that having this infinite number of constants (which for the 4-pointconformal block actually becomes doubly infinite) one can adjust them as to obtainany function as a result. In this chapter we show that the isomonodromic approachcan fix this ambiguity in such a way that all these parameters become functions onthe moduli space of the flat connections on the sphere with 3 punctures. In the sl3case this space is 2-dimensional (we denote the corresponding coordinates by µ andν), so all Ck = Ck(µ, ν).

Note that for the WN algebra one would have the set of constants Ck1,...,kl withl = 1

2(N − 1)(N − 2) non-negative indices (e.g., this easily follows from analysis of

[BW]), which is half of the dimension of the moduli space of flat slN connections onthe 3-punctured sphere.

The chapter is organized as follows. In Section 2 we briefly discuss the origins ofthe Schlesinger system and the space of flat connections on the punctured Riemannsphere. Then we introduce a collection of convenient local coordinates on this space,which are related to pants decomposition of the sphere. In Section 3 an iterativealgorithm of the solution of the Schlesinger system is proposed. We then present a setof non-trivial properties of this solution, discovered experimentally, and put forwarda conjecture about isomonodromy-CFT correspondence in higher rank, which relatesWN conformal blocks to the isomonodromic tau function. In particular, for a collectionof known W3 conformal blocks we present the 3-point functions that can be used toconstruct the τ -function in the form of explicit expansion. In Section 4 we describethe problems of definition of the general W3 conformal block and discuss how theycan be addressed using the global analytic structure induced by crossing symmetry.We conclude with a brief discussion of open questions.

Isomonodromic deformations and moduli spaces of

flat connections

The main object of our study will be the Fuchsian linear system

17


d

dzΦ(z) =

n∑ν=1

Aνz − zν

Φ(z) = A(z)Φ(z) ,∑ν

Aν = 0 .

(2.5)

Here Aν are traceless matrices with distinct eigenvalues, Φ(z) is the matrix of Nindependent solutions of the system normalized as Φ(z0) = 1. It is obvious that uponanalytic continuation of the solutions along a contour γν encircling zν they transforminto some linear combination of themselves:

γν : Φ(z) 7→ Φ(z)Mν , (2.6)

where Mν ∈ GLN(C). The relation γn . . . γ1 = 1 from π1(CP 1\z1, . . . , zn, z0) im-poses the condition

M1 . . .Mn = 1 . (2.7)

The well-known Riemann-Hilbert problem is to find the correspondence

M1, . . . ,Mn → A1, . . . , An . (2.8)

It is easy to see that the conjugacy classes of Mν are

Mν ∼ exp (2πiAν) . (2.9)

The eigenvalues of Aν determine the asymptotics of the fundamental matrix solutionnear the singularities, so one can fix even this asymptotics and study the correspondingrefined Riemann-Hilbert problem. We will work only with traceless matrices Aν sincethe scalar part trivially decouples.

Schlesinger system

Since it is difficult to solve the generic Riemann-Hilbert problem exactly, one can firstask a simpler question: how to deform simultaneously the positions of the singularitieszν and matrices Aν but preserve the monodromies Mν . The answer follows from theinfinitesimal gauge transformation

Φ(z) 7→(

1 + εAν

z − zν

)Φ(z) ,

A(z) 7→ A(z) + εAν

(z − zν)2− ε[

Aνz − zν

, A(z)

],

(2.10)

that iszν 7→ zν + ε ,

Aµ 6=ν 7→ Aµ + ε[Aν , Aµ]

zν − zµ,

Aν 7→ Aν − ε∑µ 6=ν

[Aν , Aµ]

zν − zµ,

(2.11)

18


leading to the Schlesinger system of non-linear equations

∂Aµ∂zν

=[Aµ, Aν ]

zµ − zν,

∂Aν∂zν

= −∑µ6=ν

[Aµ, Aν ]

zµ − zν.

(2.12)

Note that one can fix zn =∞, then the corresponding matrix A∞ = −n−1∑ν=1

Aν will be

constant. A non-trivial statement is that the relations

∂

∂zµlog τ =

∑ν 6=µ

trAµAνzµ − zν (2.13)

are compatible and define the τ -function τ(z1, . . . , zn) of the Schlesinger system. It iseasy to see that the 3-point τ -function is given by a simple expression:

τ(z1, z2, z3) = const · (z1 − z2)∆3−∆1−∆2(z2 − z3)∆1−∆2−∆3(z1 − z3)∆2−∆1−∆3 ,

where ∆ν = 12

trA2ν . Let us now attempt to solve the Schlesinger system for the 4-point

case and compute the corresponding τ -function in the form of certain expansion.

Moduli spaces of flat connections

The main object of our interest is the τ -function. It depends on monodromy datawhich provide the full set of integrals of motion for the Schlesinger system. It will beuseful to start by introducing a convenient parametrizaton of this space.

One starts with n matrices Mν ∈ SLN , with fixed nondegenerate eigenvalues, i.e.there are n(N2−N) parameters. These matrices are constrained by one equation (2.7)and are considered up to an overall SLN conjugation, which decreases the number ofparameters by 2(N2 − 1). So the resulting number of parameters is

dimMslNn (θ1, . . . ,θn) = (n− 2)N2 − nN + 2 . (2.14)

Here θν ∈ h (h is the Cartan subalgebra) define the conjugacy classes: Mν ∼ e2πiθν .It is obvious that θν is equivalent to θν + hν , such that for all weights of the firstfundamental representation ei one has (ei,hν) ∈ Z. It means that hν ∈ ⊕ri=1Zα∨i ,where α∨i ∈ h are simple coroots (for the simply-laced case they coincide with theroots).

For the general Lie algebra this formula can be written as

dimMgn(θ1, . . . ,θn) = (n− 2) dim g− n · rank g . (2.15)

In particular, for n = 3 punctures on the sphere

dimMg3(θ1,θ2,θ3) = dim g− 3 · rank g .

This formula gives the number of non-simple roots of g. In the slN case it specializesto

dimMslN3 (θ1,θ2,θ3) = (N − 1)(N − 2) . (2.16)

19


This expression vanishes for sl2, which drastically simplifies the study of the corre-sponding isomonodromic problem. However already for sl3 this dimension is equal to2, i.e. it is nonvanishing. One way to simplify the problem is to set θ2 = ae1: in thiscase the orbit of the adjoint action

e2πiae1 7→ g−1e2πiae1g

has the dimension dimOae1 = dim g − dim stab(e1) = N2 − 1 − (N − 1)2 = 2N − 2.The total dimension is 2(N2 − N) + (2N − 2) − 2(N2 − 1) = 0. In this calculationthe first two terms correspond to the dimensions of orbits: two generic and one witha large stabilizer. The last term corresponds to one equation and one factorization.Hence

dimMslN3 (θ1, ae1,θ3) = 0 . (2.17)

This case is the best known on the side of W -algebras [FLitv07], [FLitv09], [FLitv12].In the mathematical framework, this situation corresponds to rigid local systems.

Pants decomposition of Mg4

We begin our consideration with an arbitrary Lie group G containing a Cartan torusH ⊂ G. The corresponding Lie algebras are g and h, respectively. At some point wewill switch to G = SLN(C) case.

The moduli space Mg4 is described by 4 matrices satisfying M1M2M3M4 = 1,

defined up to conjugation:

Mg4 = (M1,M2,M3,M4)/G . (2.18)

Let us introduce S = M1M2 and consider two triples

(M1,M2, S−1), (S,M3,M4) . (2.19)

Note that the products inside each of these triples are equal to the identity. Let usnow choose the submanifold with fixed eigenvalues of M1, . . . ,M4, S. One may alsouse the freedom of the adjoint action to diagonalize S

S = e2πiσ ,

where σ ∈ h. We thereby obtain a submanifold

Mg4(θ1,θ2;σ;θ3,θ4) = (M1,M2, e

−2πiσ), (e2πiσ,M3,M4)/H ⊂Mg4(θ1,θ2,θ3,θ4) ,

(2.20)where the remaining factorization is performed over the Cartan torus H ⊂ G. It isvery similar to what happens for Mg

3:

Mg3 = (M1,M2,M3)/G = (M1,M2, e

2πiθ3)/H , (2.21)

except that the conjugation is simultaneous for both triples. To relax this condition,let us define an extra Cartan torus acting on Mg

4:

h : (M1,M2, e−2πiσ), (e2πiσ,M3,M4) 7→ (M1,M2, e

−2πiσ), h−1(e2πiσ,M3,M4)h ,(2.22)

20


which looks like a relative twist of one part of the sphere with respect to another (inthe sl2 case it will be exactly the geodesic flow). Therefore one can say that

Mg4(θ1,θ2;σ;θ3,θ4)/H =Mg

3(θ1,θ2,−σ)×Mg3(σ,θ3,θ4) . (2.23)

The torus action is free, so locally it looks as a product (actually it is true evenglobally because the fibration (M1,M2,M3) 7→ (M1,M2,M3)/G is trivial: we cangive an algebraic parametrization for one representative from each conjugacy class).Therefore we have the equality for the open subsets (denoted by ≈):

Mg4(θ1,θ2;σ;θ3,θ4) ≈Mg

3(θ1,θ2,−σ)×H ×Mg3(σ,θ3,θ4) . (2.24)

The above considerations suggest the following choice of coordinates on Mg4:

• Gluing parameters σ: rank g items.

• Invariant functions on Mg3 ×M

g3 (for example, trM1M

−12 , trM−1

3 M4). Theyare invariant with respect to the action of “relative twists”: we have 2 dimMg

3

such functions.

• Relative twist parameters, which change under the twist (for example, trM2M−13 ,

trM−12 M3), rank g items. These coordinates will be denoted by β ∈ h.

This procedure is schematically depicted in Fig.4.6 for the sl3 case, where dimMsl33 =

2, dimMsl34 = 8. The coordinates on each copy of Msl3

3 are denoted by µ, ν. Theindices 1, 2, 3, 4 of the matrices are replaced by 0, t, 1,∞

Figure 2.1: Coordinates onMsl34 : eight = two σ’s + two β’s + µ0t + ν0t + µ1∞+ ν1∞

Pants decomposition for Mgn

Suppose that the coordinates onMgn−1 are chosen via the pants decomposition. Split

the matrices into two groups and define

Sn−3 = M1 . . .Mn−2 ,

Mgn = (M1, . . . ,Mn−2, S

−1n−3), (Sn−3,Mn−1,Mn)/G =

= (M1, . . . ,Mn−2, e−2πiσ), (e2πiσ,Mn−1,Mn)/H ≈Mg

n−1 ×H ×Mg3 .

(2.25)

Iteratively repeating this procedure, one is led to the following choice of coordinateson Mg

n:

21


• (n− 3) rank g gluing parameters σi,

• (n− 3) rank g relative twist parameters βi,

•n−2∑i=1

dimMg3(σi−1,θi+1,−σi) 3-point moduli of flat connections (here we identify

σ0 = θ1 and σn−2 = −θn).

Anticipating the result, let us mention that these coordinates are convenient fromthe CFT point of view: σi will parametrize intermediate charges in the conformalblock and βi will be the Fourier transformation parameters. This description wasshown to be valid in the sl2 case [GIL12], [ILTe] and was recently demonstrated tohold for slN case with dimMg

3 = 0 [GavIL]. From a more conceptual point of view,this decomposition illustrates that all extra parameters in the τ -function expansioncome from the 3-point functions.

Iterative solution of the Schlesinger system

In order to study the generic Schlesinger system, let us follow the approach proposedin the original paper of M. Jimbo [Jimbo] and in [SMJ, part 2].

Let us take the 4-point Schlesinger system and fix the singularities to be 0, t, 1,∞.The system becomes

t∂tA0 = [At, A0] ,

t∂tA1 =t

t− 1[At, A1] ,

∂tAt = −1

t[At, A0]− 1

t− 1[At, A1] .

(2.26)

Fixing the integral of motion A∞ = −A0 − At − A1, one obtains

t∂tA0 = [A0, A1 + A∞] ,

t∂tA1 = t(1− t)−1[A0 + A∞, A1] .(2.27)

The isomonodromic τ -function is defined by

∂t log τ =1

ttrAtA0 +

1

t− 1trAtA1 . (2.28)

Let us study the solution of the system (2.27) for the case when A1(t) is finite inthe limit t→ 0: A1(t) = A1(0) +O(tε>0). Under this assumption we have

t∂tA0(t) = [A0, A∞ + A1(0) +O(tε>0)] .

If the last term were absent, then the solution would be A0 = t−A∞−A1(0)A0tA∞+A1(0).

Therefore it is natural to introduce

B = −A1(0)− A∞ = limt→0

(A0(t) + At(t)) ,

A0(t) = t−BA0(t)tB ,(2.29)

22

2.3. Iterative solution of the Schlesinger system

where A0(t) has a well-defined limit as t → 0. We see that in view of its definitionB describes the total monodromy around 0 and t in the limit t → 0. Since thedeformation is isomonodromic, this monodromy is constant and is given by M0Mt =M0t ∼ e2πiB. This allows to make the identification

B = σ . (2.30)

Our system then becomes

t∂tA0(t) = [A0(t), t−σ(A1(t)− A1(0))tσ] ,

t∂tA1 = t(1− t)−1[tσA0(t)t−σ + A∞, A1(t)] .(2.31)

Here we see an operator tadσ , which produces some fractional powers of t. It isconvenient to impose the condition that (σ,σ) 1, or at least that for all roots αone has |(σ, α)| < 1

2. This allows to organize the terms of the expansion in powers of

t according to their order of magnitude in the asymptotic behavior. If necessary, onecan perform an analytic continuation of the solution from the region with small σ.

We know that in the Lie algebra the operator tadσ acts by

tσEαt−σ = t(σ,α)Eα ,

tσHαt−σ = Hα ,

(2.32)

where α ∈ g∗ is a root and Eα, Hα are the elements of the Cartan-Weyl basis. Let usdefine a grading on the space of monomials

deg[tk+(σ,w)] = (k,w) ,

where w ∈ Qg is an element of the root lattice Qg =rankg⊕i=1

Zαi of g. It is useful to

define a filtrationQ0

g ⊂ Q1g ⊂ Q2

g ⊂ . . . ⊂ Qg (2.33)

on this root lattice, which is recursively constructed as follows: Q0g = 0, Q1

g is theset of all roots of g and 0, and

Qi+1g = x+ y|x ∈ Qi

g,y ∈ Q1g = Q1

g + . . .+Q1g .

Also define the double filtration Vn,m on the space of all fractional-power series:

tk+(σ,w) ∈ Vn,m ⇔ (k ≥ n) ∧ (w ∈ Qmg ) ,

Vn+1,m ⊂ Vn,m , Vn,m ⊂ Vn,m+1 .(2.34)

Each term of the filtration is generated by these monomials. This definition turns outto be useful because of the properties

t· : Vn,m → Vn+1,m ,

tadσ : Vn,m → Vn,m+1 ,

Vn1,m1 · Vn2,m2 → Vn1+n2,m1+m2 .

(2.35)

23


One can also see that the degrees present in Vn+1,m+k are larger then in Vn,m if σis sufficiently small (∀α ∈ Q1

g : |(σ, α)| < 1k). We also define a slightly ambiguous

notation Vn,w bytk+(σ,w) ∈ Vn,w ⇔ (k ≥ n) . (2.36)

Now we have all the ingredients that are necessary for the construction of aniterative solution of the system (2.31). Our initial data will be given by the triple ofmatrices σ, A0(0) and A1(0). Symbolically, the system (2.31) can be written as

A0(t) = F0(A0(t), A1(t)) ,

A1(t) = F1(A0(t), A1(t)) ,(2.37)

where “affine” bilinear (in the sense f(x, y) = axy + bx + cy + d) functions F0, F1

have the following properties:

F0 : Vn0,m0 × V0,0 → 0 ,

F0 : Vn0,m0 × Vn1,m1 → Vn0+n1,m0+m1+1 ⊂ Vn0+n1,∞ ,

F1 : Vn0,m0 × Vn1,m1 → Vn0+n1+1,m0+m1+1 + Vn1+1,m1 ⊂ Vn1+1,∞ .

(2.38)

Let us substitute into (2.37) the expressions

A0(t) = A0(0) +∞∑k=1

tkAk0(t) ,

A1(t) = A1(0) +∞∑k=1

tkAk1(t) ,

tkAk0(t), tkAk1(t) ∈ Vk,∞ .

(2.39)

From (2.38) we immediately see that (2.31) takes the form

Ak0(t) = fk0 (A<k0 (t), A≤k1 (t)) ,

Ak1(t) = fk1 (A<k0 (t), A<k1 (t)) .(2.40)

Because of the ≤ sign our strategy of solving will be to compute first Ak1(t), and thensubsequently determine Ak0(t). One can also write down explicit formulas for bilinearsfk1 and fk0 , which are immediate (though cumbersome) consequences of the system(2.31).

Now let us determine which powers (k,w) will be actually present in the solution.This will be done again iteratively, using only the properties (2.38):

• Taking A0(0) ∈ V0,0 and A1(0) ∈ V0,0, and computing F1, we get an element ofV1,1, therefore

A1 ∈ V0,0 + V1,1 + . . .

• Take A0(0) ∈ V0,0 and A1 ∈ V0,0 + V1,1 + . . ., then A0 ∈ V0,0 + V1,2 + . . .

• For A0 ∈ V0,0 + V1,2 + . . . and A1 ∈ V0,0 + V1,1 + . . . one finds A1 ∈ V0,0 + V1,1 +V2,3 + . . .

24


• Setting A0 ∈ V0,0 + V1,2 + . . . and A1 ∈ V0,0 + V1,1 + V2,3 + . . . yields A0 ∈V0,0 + V1,2 + V2,4 . . .

• . . .

Continuing this procedure one finds the following structure

A0(t) ∈∞∑k=0

Vk,2k ,

A1(t) ∈ V0,0 +∞∑k=1

Vk,2k−1 .

(2.41)

It is easy to check that these spaces are stable under the action of (F0, F1) describedby the rules (2.38). This is somewhat similar to the statement that the cone is stableunder the addition operation.

Indeed, let us try to find an element of A0(t) lying in Vk,2k+1. For this one wouldneed n0 + n1 ≤ k, m0 + m1 ≥ 2k, so m0 + m1 ≥ 2(n0 + n1). Since m1 ≤ 2n1 − 1 forn1 6= 0 (when F0 vanishes) and m0 ≤ 2n0, such an element cannot exist. Similarly,for A1, let us take an element lying in Vk,2k. One then needs to satisfy the constraintsn1 ≤ k − 1, m1 ≥ 2k (impossible) or n0 + n1 + 1 ≤ k and m0 + m1 + 1 ≥ 2k, whichimplies m0 +m1 ≥ 2n0 + 2n1 + 1. But m1 ≤ 2n1 and m0 ≤ 2n0, therefore one cannotget such an element neither.

Now let us compute the τ -function and try to understand in which elements ofthe filtration does it lie. Since we have

t∂t log τ(t) = − tr [t−σ(A1 + A∞)tσA0 + A20] + t(1− t)−1 tr [(A1 + A∞ + tσA0t

−σ)A1] ,(2.42)

naively it could be a term in V0,1. However, computing the constant part one finds

t∂t log τ(t) = tr (BA0 − A20) + . . . = tr (AtA0) + . . . =

=1

2tr (At + A0)2 − 1

2trA2

0 −1

2trA2

t + . . . =

=1

2(σ,σ)− 1

2(θ0,θ0)− 1

2(θt,θt) + . . . ,

(2.43)

where Aν ∼ θν . For convenience, let us introduce the notation

χ =1

2(σ,σ)− 1

2(θ0,θ0)− 1

2(θt,θt) (2.44)

The terms present in tr (t−σA1(t)tσA0(t)) that are closest to the boundary originate

from the constant part of A1(t). These terms belong to∞∑k=0

Vk,2k, therefore

log τ(t) ∈∞∑k=0

Vk,2k (2.45)

25


Figure 2.2: Filtration Q•sl2

Note that these estimates are too rough, since we have not taken into account thata number of the commutators actually vanish. The actual result turns out to be thesame for all three functions

log τ, A0, A1 ∈∞∑k=0

Vk,k ,

and it can be checked numerically. Moreover, it turns out that the expansion of theτ -function itself is even more restricted:

t−χτ(t) ∈∑w∈Qg

V 12

(w,w),w , (2.46)

which in fact provides an evidence for the 2d CFT description: different fractionalpowers come from t∆ for the different ∆’s, but the conformal dimension ∆ = 1

2(σ +

w,σ +w) is a quadratic function of w leading to the structure (2.46).

sl2 case

In this case we illustrate all procedures, definitions and statements using the exactsolution of [GIL12].

The Lie algebra sl2 is given by 3 generators Eα, E−α, Hα, such that

[Eα, E−α] = Hα ,

[Hα, E±α] = ±2E±α .(2.47)

The root lattice Qsl2 is shown in Fig.2.2. It is spanned by one root α. Q0sl2

is theempty square, Q1

sl2is the red rectangle, Q2

sl2is green and Q3

sl2is blue.

All monomials have the form tn+(σ,w) = tn+m(σ,α), and therefore can be depictedby the points of a two-dimensional lattice. Note that in our normalization (α, α) = 2.Several examples of the elements of this filtration are presented in Fig.2.3.Here the blue region represents V0,0, red corresponds to V1,1 and green is V3,4.

We can also show the “true” and “naive” lattice supports of the quantities A0(t),A1(t), log τ(t) and t−χτ(t). See Fig.2.4: green region is the “naive” support of A1(t),the blue region is the true support of A0(t), A1(t), log τ(t), which can be derivedexperimentally. Now one can use an exact formula for the tau function expansion[GIL12] (cf (2.50) below) to see that

τ(t) = tσ2−θ2

0−θ2t

∑k∈Z

t2σntn2

fn(t) , (2.48)

which in turn implies

tθ20+θ2

t−σ2

τ(t) ∈∞∑k=0

Vk2,k . (2.49)

26


Figure 2.3: Filtration V•,•

Figure 2.4: Support of the solutions

It looks like a miracle and means that a huge number of terms cancel out whenwe exponentiate, but this answer confirms the conjecture (2.46). This phenomenonis illustrated in Fig.2.5 in two ways. Upper bold numbers account for the degree inτ(t) (blue region), lower numbers correspond to the degree in log τ(t) (green region).Horizontal coordinate corresponds to the position in the sl2 root lattice.

00

11

11

42

42

93

93

Figure 2.5: Supports of τ(t) and log τ(t): circles correspond to the integral points ofthe x-axis, numbers inside show the y-coordinates of the cone and parabola.

27


Let us take the main formula from [GIL12]:

τ(t) =∑n∈Z

snC(0t)n (θ0, θt, σ0t)C

(1∞)n (θ1, θ∞, σ0t)t

(σ0t+n)2−θ20−θ2

tB(θi, σ0t + n; t) ,

(2.50)where B(. . . ; t) is the c = 1 Virasoro conformal block and

C(0t)n (θ0, θt, σ0t)C

(1∞)n (θ1, θ∞, σ0t) =

=

∏ε=±,ε′=±

G(1 + θt + εθ0 + ε′(σ0t + n))G(1 + θ1 + εθ∞ + ε′(σ0t + n))

G(1− 2σ0t)G(1 + 2σ0t).

(2.51)

Here (θν ,−θν) are the eigenvalues of the matrices Aν in the linear system (2.5),(e2πiσµν , e−2πiσµν ) are the eigenvalues of MµMν , s is the only variable depending on σ1t

(in a complicated way). The main properties of (2.50) and (2.51) can be summarizedas follows:

1. The support of τ(t) is as indicated in (2.49).

2. Relative twist parameter enters only via the factor sn in the structure constants.

3. The 3-point coefficients Cn factorize with respect to the pants decompositionparametrization.

We are now going to check these important properties in the sl3 case.

sl3 case

Figure 2.6: Filtration Q•sl3

Fig.2.6 illustrates the filtration on the sl3 root lattice. The red hexagon corre-sponds to Q1

sl3, Q2

sl3is shown in green and Q3

sl3is blue. It is difficult to visualize Vm,n,

28


since one would then need a 3d picture. One can however think of∞∑k=0

Vk,k as being a

cone with hexagonal section.Let us perform the numerical study of the 3 × 3 Schlesinger system. We first

determine which degrees (k,w) are present in log τ(t) and in τ(t) (Fig.2.7).

00

11

11

11

11

11

11

42

32

32

42

42

32

32

42

42

32

32

42

?3

73

73

73

73

?3

?3

73

73

73

73

?3

?3

7?3

7?3

7?3

7?3

?3

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?4

?5

?6

?7

Figure 2.7: Degrees present in t−χτ(t) and in log τ(t). Number χ is given by (2.44).

As above, the upper bold numbers correspond to degrees in t−χτ(t) and the lowerones to log τ(t). We mark with “?” sign those values which are obtained at the limitof machine precision or which are greater then 7 (so that they are not seen in thesolution up to the 7th order). Carefully analyzing this picture, one deduces that

log τ(t) ∈∞∑k=0

Vk,k ,

t−χτ(t) ∈∑w∈Qsl3

V 12

(w,w),w .(2.52)

It means that nonzero monomials of τ(t) fill a paraboloid, and not the naively expectedcone. In other words, a lot of nontrivial cancellations take place, which providesfurther evidence for the conjecture (2.46). We now list other nontrivial properties ofτ(t) revealed by our experimental study.

1. The expansion has the form

τ(t) =∑w∈Q

e(β,w)C(0t)w (θ0,θt,σ0t, µ0t, ν0t)C

(1∞)w (θ1,θ∞,σ0t, µ1t, ν1t)×

×t12

(σ0t+w,σ0t+w)− 12

(θ0,θ0)− 12

(θt,θt)Bw(θi,σ0t, µ0t, ν0t, µ1∞, ν1∞; t) .

(2.53)

29


2. The non-zero coefficients of the expansion start from t12

(w,w).

3. All the dependence on the relative twist parameters is hidden in β ∈ h, whichenters in a trivial way.

4. The dependence of structure constants on the 3-point monodromy parametersis factorized.

5. The first term in the expansion of conformal block has the form

B0 = 1 + [α + βC1(µ0t, ν0t) + γC1(µ1∞, ν1∞) + δC1(µ0t, ν0t)C1(µ1∞, ν1∞)]t+ . . .

This property is new, as compared to the N = 2 case, and we will see later thatit is very important.

All these facts tell us that almost all properties of sl2 case hold in the sl3 case.This leads us to

Main conjecture:

B0(θi,σ0t, µ0t, ν0t, µ1∞, ν1∞; t) is a conformal block of W3 algebra

The corresponding dimensions and W -charges are given by

∆ν =1

2(θν ,θν)

wν =

√3

2

∏i

(θν , ei) .(2.54)

The main advantage of the above definition of conformal block is that it dependsonly on 4 extra variables instead of a doubly-infinite set.

It is easy to check this definition for the case when W3-block can be definedalgebraically. This becomes possible when θt = ate1 and θ1 = a1e1, where e1 isthe weight of the first fundamental representation. The best way to present thisconformal block is to use Nekrasov formulas [Nek] which can be applied to conformalfield theory in view of the extended AGT [AGT] correspondence, first established in[Wyll], [MirMor]. The most convenient (for c = 2) expression for the conformal blockcan be found in [FLitv12]:

Bw(θ∞, a1,σ, at,θ0; t) = B(θ∞, a1,σ +w, at,θ0; t)

B(θ∞, a1,σ, at,θ0; t) = (1− t)13ata1

∑~Y

t|~Y |Zbif (−θ∞, a1,σ|~0, ~Y )×

× 1

Zbif (σ, 0,σ|~Y , ~Y )Zbif (σ, at,θ0|~Y ,~0) ,

(2.55)

where

Zbif (θ, a,θ′|~ν, ~ν ′) =

3∏i,j=1

∏s∈ν′i

(−Eν′i,νj(i(θ, ej)− i(θ

′, ei)|s)− ia

3

)×

×∏t∈νj

(Eνj ,ν′i(i(θ

′, ei)− i(θ, ej)|t)− ia

3

),

(2.56)

30

2.4. Remarks on W3 conformal blocks

and the quantities E are defined by

Eλ,µ(x|s) = x− ilµ(s)− iaλ(s)− i . (2.57)

It yields exactly the same result as our computations using iterative solution of theSchlesinger system.

We have also conjectured in this case and checked experimentally a formula forthe structure constants, which is a straightforward generalization of (2.51):

C(0t)w (θ0, at,σ)C(1∞)

w (σ, a1,θ∞) =

=

∏ij G[1− at

N+ (ei,θ0)− (ej,σ +w)]G[1− a1

N+ (ei,σ +w) + (ej,θ∞)]∏

i

G[1 + (αi,σ +w)].

(2.58)

Here ei denote the weights of the first fundamental representation and αi are all rootsof slN (in our case N = 3). This formula was recently proved [GavIL] for generalN . One can also observe a similarity between this formula and Toda 3-point functioncomputed in [FLitv07].

Remarks on W3 conformal blocks

General conformal block

Here we consider for simplicity the c = 2 case, but the generalization to arbitrary c isstraightforward. First we explain how the WN conformal block is defined algebraically.For that let us compute the following expression:

B(θ∞,θ1,σ,θt,θ0; t) = 〈−θ∞|φθ1(1)Pσφθt(t)|θ0〉 , (2.59)

where |θ0〉 and 〈−θ∞| are the highest-weight vectors with the charges given by (2.54),Pσ is the projector onto the whole Verma module (2.2) with given highest weight. Thisconformal block can be computed by inserting the resolution of the identity in theVerma module. One can take, for instance, the naive basis (2.2), or (if we do notnecessarily want to preserve the L0 grading) the basis from [BW], or (if we wish toadd the Heisenberg algebra) the AGT basis from [FLitv12], [BBFLT]. Let us call the

vectors of this basis |σ, ~Y 〉 and suppose that

L0|σ〉 = (∆(σ) + |~Y |)|σ, ~Y 〉 .

Their scalar products will be Kσ(~Y , ~Y ′) = 〈σ, ~Y |σ, ~Y ′〉. This allows to express con-formal block as

B(θ∞,θ1,σ,θt,θ0; t) = tχ∑~Y ,~Y ′

t|~Y |〈−θ∞|φθ1(1)|σ, ~Y 〉K−1(~Y , ~Y ′)〈σ, ~Y ′|φθt(1)|θ0〉 =

= tχ∑~Y

t|~Y |Q(~Y )Q(~Y ) ,

(2.60)

31


where Q(~Y ) =∑~Y ′

K−1(~Y , ~Y ′)〈σ, ~Y ′|φθt(1)|θ0〉 and Q(~Y ) = 〈θ∞|φθ1(1)|σ, ~Y 〉 and χ

is given by (2.44). The claim of [BW] is that Q(~Y ) and Q(~Y )

Q(~Y ) = Q(~Y |C1, . . . , C|~Y |) = γ0(~Y ) +

|~Y |∑k=1

γk(~Y )Ck ,

Q(~Y ) = Q(~Y |C1, . . . , C|~Y |) = γ0(~Y ) +

|~Y |∑k=1

γk(~Y )Ck ,

(2.61)

are “triangular” “affine” linear functions of infinitely many arbitrary parametersCk, Ck defined by

Ck = 〈−θ∞|φθ1(1)W k−1|σ〉, Ck = 〈σ|W k

1 φθt(1)|θ0〉 . (2.62)

Degenerate field

Let us consider the case θt = e1 (the weight of the first fundamental representation).The fusion rules for such fields are known to be given by

[e1]⊗

[θ] =⊕k

[θ + ek] . (2.63)

Let us also shift θ0 7→ θ0 − en, multiply the conformal block by t23 = t(e1,e1), and

define the quantity

Φnk(t) = t(e1,e1)B(θ∞,θ1,θ0 + ek − en, e1,θ0 − en; t) =

= t(θ0,ek)+(1−δkn)∑~Y

t|~Y |Q(~Y , C1, . . . , C|~Y |)q(

~Y ) , (2.64)

where q(~Y ) do not contain any free parameters [BW].Now denote the degenerate field φe1(t) by ψ(t) and consider the correlation func-

tion

t(e1,e1)〈−θ∞|φθ1(1)ψ(t)|θ0 − ek〉 .

In the region t → 0 (s-channel) it can be expanded in the basis of conformal blockswritten above. But if we set t → 1 or t → ∞ (t- and u-channel), then we will havethe following OPEs

ψ(t)φθ1(1) =∑k

Cθ1+eke1,θ1

· (t− 1)(θ1,ek) (φθ1+ek(1) + descendants) ,

t(e1,e1)〈−θ∞|ψ(t) =∑k

Cθ∞+ekθ∞,e1

· t−(θ∞,ek) (〈−θ∞ − ek|+ descendants) .(2.65)

These formulas suggest that the space of conformal blocks involving ψ(t) is 3-dimensionaland near each point we have a basis with asymptotics prescribed by θν . It is clear that

32

2.4. Remarks on W3 conformal blocks

upon analytic continuation of Ψ1k(t) around 0, 1,∞ one gets some linear combinationsof the basis elements

γ0 : Φ1k(t) 7→∑k′

Φ1k′(t)(M0)k′k ,

γ1 : Φ1k(t) 7→∑k′

Φ1k′(t)(M1)k′k ,

γ∞ : Φ1k(t) 7→∑k′

Φ1k′(t)(M∞)k′k .

(2.66)

In our case M0 = diag(e2πi(θ0,e1), e2πi(θ0,e2), e2πi(θ0,e3)). That these formulas must holdcan be expected on general grounds (crossing symmetry) and from the fact that thespace is 3-dimensional. However, looking at the formula (2.64), the freedom in choice

of Ck can give us Φ1k = t(θ0,e1)+(1−δk1)∞∑k=0

fltl with arbitrary fn’s. It means that W -

algebra itself does not account for the global structure of conformal blocks and thisinformation should be introduced as an extra input.

Now suppose that we have some globally-defined multivalued functions Φ1k. Thenwe have three monodromiesM0,M1,M∞ and one can solve the refined 3-point Riemann-Hilbert problem. Suppose that its solution is given by the matrix F (t) such that

d

dtF (t) =

(A0

t+

A1

t− 1

)F (t) , (2.67)

A0 = diag ((θ0, e1), (θ0, e2), (θ0, e3)) and F (t) is normalized in such a way that F (t) =tA0(1 + O(t)). Next let us compute Ri(t) =

∑k

Φ1k(t)(F (t)−1)ki. This vector has the

trivial monodromies around all singular points, it is regular there and R(0) = (1, 0, 0),so that R(t) = (1, 0, 0). It means that

Φ1k(t) = F1k(t) . (2.68)

This formula allows us to fix all constants Ck. This is done in the following way:we solve the 3-point Riemann-Hilbert problem, take F11(t) and read the coefficients ofconformal block from its series. These coefficients are triangular linear combinations(2.61) of Ck (i.e., kth term of the conformal block expansion involves only Cj≤k).This construction thus expresses Ck via the moduli (µ, ν) of flat connections on the3-punctured sphere.

Ck = Ck(µ, ν) . (2.69)

All constants are expressed in terms of only two parameters. If we now recall the 5thexperimental property of the τ -function, its origin can be easily understood: the firstterm of the conformal block (with the structure constants fixed above) depends onlyon C1(µ, ν) and C1(µ, ν) and this dependence is at most bilinear.

Verlinde loop operators

Here we can slightly modify our point of view: now all possible vertex operatorsdefined by (2.3) and (2.4) have to be considered simultaneously. They form some ∞-dimensional vector space, which can be identified with the space of 3-point conformal

33


blocks (and which was one-dimensional in the Virasoro case). One can define theaction of the Verlinde loop operators on this space in the same way as it was done in[CGTe]. This action is given by some operators V (γ) depending on the loop γ.

If we now look at the results of [ILTe] then we realize that (2.50) can be definedalternatively as the common eigenvector of all possible Verlinde loops. One can actin the same way for the case of 3-point conformal blocks

V (γ) · 〈Y |φθ1,µ,ν(1)|Y ′〉 = Mγ(µ, ν) · 〈Y |φθ1,µ,ν(1)|Y ′〉 (2.70)

This procedure defines the basis of the “right” vertex operators φθ1,µ,ν(1) characterizedby some Ck(µ, ν). It looks more natural in this approach that τ -function constructedfrom such operators should solve the Riemann-Hilbert problem.

The question about interpretation in c 6= N − 1 case is still open: the problem iscaused by non-commutativity of the algebra of V (γ). Moreover, even in the minimal

model-like cases c = N − 1 − N(N2 − 1) (k−1)2

kfor integer k, when the algebra is

commutative again, the relation to the isomonodromic deformations becomes unclear(see “concluding remarks” in [BShch] for discussion of the Virasoro case).

Conclusions

We have discovered several important properties of the isomonodromic τ -functionsin higher rank, which can be interpreted as signatures of the isomonodromy-CFTcorrespondence for the WN case. This allows to give a definition of the general WN

conformal block, depending only on a finite number of parameters. It is also possibleto prove [GavIL] that the algebraic way to define well-known conformal blocks forsemi-degenerate fields agrees with the above definition.

We have also considered a particular conformal block with degenerate field andshown that its global structure is not fixed algebraically. The requirement of thecorrect global behavior of this object yields an expression for the whole infinite seriesof constants in theW3 conformal block in terms of the solution of the 3-point Riemann-Hilbert problem.

These expressions can be written in terms of coordinates on the moduli space offlat connections on sphere with 3 punctures. This is expected to be universal andwork for any conformal block (not only for those with degenerate fields). We havechecked experimentally some properties, which support this conjecture.

Finally, let us list some remaining open problems:

• One needs to check that the procedure of fixing Ck is self-consistent.

• If the constants Ck can be fixed in such a way, we may try to prove that theτ -function can be given as a sum of the general WN conformal blocks.

• A constructive solution of the 3-point Riemann-Hilbert problem is still missing.

• It would be interesting to understand the meaning of Zbif (θt,σ,θ0;µ, ν|~Y , ~Y ′)in the context of isomonodromy-CFT correspondence. It can be done for thecase ~Y ′ = ~0 and it is interesting what happens for the arbitrary Young diagrams.

34

2.5. Conclusions

• There as an approach to the definition of conformal blocks of the light fields inthe limit c→∞ [FR]. In that case explicit integral expression for the conformalblock was derived. All the information about the 3-point functions enters thisdefinition via several functions of one variable. The open problem is to obtainthe monodromy properties of such conformal blocks and to identify the choiceof 3-point functions that gives the conformal blocks arising in our approach.

• It is also important to understand the meaning of the results [BMPTY] aboutpartition functions of TN theories without lagrangian description (which arebelieved to be the counterparts of the general WN 3-point functions) from theisomonodromic point of view.

35

3Free fermions, W-algebras and

isomonodromic deformations

Abstract

We consider the theory of multicomponent free massless fermions in two dimensionsand use it for construction of representations of W-algebras at integer Virasoro centralcharges. We define the vertex operators in this theory in terms of solutions of thecorresponding isomonodromy problem. We use this construction to get some newinsights on tau-functions of the multicomponent Toda type hierarchies for the class ofsolutions, given by the isomonodromy vertex operators and get useful representationfor the tau-function of isomonodromic deformations.

Introduction

The aim of the chapter is to present briefly the main free-fermionic constructions thatappear in the study of correspondence between the problem of isomonodromic defor-mations and two-dimensional conformal field theories – for some class of the theorieswith extended conformal symmetry. An interest to the two-dimensional conformalfield theories (CFT) with extended nonlinear symmetries, generated by the higherspin holomorphic currents, has been initiated by pioneering work [ZamW]. Thesetheories with so called W-symmetry possess many features of ordinary CFT, includ-ing the free field representation [FZ, FL], which becomes especially simple for thecase of integer Virasoro central charges. However, even in this relatively simple caseit turns already to be impossible to construct in generic situation the W-conformalblocks [BW], which are the main ingredients of the conformal bootstrap definition ofthe physical correlation functions [BPZ].

This interest has been seriously supported already in our century by rather non-trivial correspondence between two-dimensional CFT and four-dimensional super-symmetric gauge theory [LMN, NO, AGT], where the conformal blocks have to becompared with the Nekrasov instanton partition functions [Nek, NP] producing inthe quasiclassical limit the Seiberg-Witten prepotentials [SW]. This correspondencealso meets serious problems beyond SU(2)/Virasoro level: both on four-dimensionalgauge theory and two-dimensional CFT sides. These difficulties can be attackedusing different approaches, for example in [GMtw] we have demonstrated how the

37

3. Free fermions, W-algebras and isomonodromic deformations

exact conformal blocks for the twist fields [ZamAT87, ZamAT86, ApiZam] in theo-ries with W-symmetry can be computed, using the technique developed previously in[KriW, Mtau, GMqui].

Here we present another approach to the study of the CFT vertex operators in thetheories with extended conformal symmetry, based on their free-fermionic construc-tion. It is clear, that it should work (at least) in the cases of integral central charges,where it is intimately related with the recently discovered there CFT/isomonodromycorrespondence [GIL12, Gav]. We are going to discuss the operator content of thesetheories with nontrivial monodromy properties, and then turn to the problem of com-putation of the matrix elements of generic monodromy operators. Finally, we aregoing to relate these matrix elements with the tau-functions of two different classesof problems – the tau-functions of the multicomponent classical integrable hierarchiesof Toda type, and the tau-functions of the isomonodromic deformations.

Abelian U(1) theory

Fermions and vertex operators

Introduce the standard two-dimensional holomorphic fermionic fields with the actionS = 1

π

´Σd2zψ∂ψ, so that

ψ(z)ψ(z′) =1

z − z′+ . . . (3.1)

orψr, ψs = δr+s,0, r, s ∈ Z + 1

2,

ψ(z) =∑r∈Z+ 1

2

ψrzr+1/2

, ψ(z) =∑s∈Z+ 1

2

ψszs+1/2

(3.2)

with the half-integer mode expansion. The bosonization formulas read

ψ(z) =: eiφ(z) := e−∑n<0

Jnnz−n

e−∑n>0

Jnnz−n

eQzJ0 ,

ψ(z) =: e−iφ(z) := e

∑n<0

Jnnz−n

e

∑n>0

Jnnz−n

e−Qz−J0 ,

(3.3)

where

J(z) =: ψ(z)ψ(z) := i∂φ(z) =∑n∈Z

Jnzn+1

,

[Jn, Jm] = nδn+m,0, n,m ∈ Z, [Jn, Q] = δn0 ,

(3.4)

where normal ordering means, that all negative modes stand to the left of all positive,and all Q to the left of J0.

Consider now generic vertex operators for the bosonic fields

Vν(z) =: eiνφ(z) := e−ν

∑n<0

Jnnz−n

e−ν

∑n>0

Jnnz−n

eνQzνJ0 ≡ V −ν (z)V +ν (z)eνQzνJ0 (3.5)

38

3.2. Abelian U(1) theory

which satisfy the obvious exchange relations, following from the Campbell-Hausdorffformula

V +α (z)V −β (w) =

(1− w

z

)αβV −β (w)V +

α (z) ,

Vα(z)Vβ(w) =( zw

)αβ (1− w

z

)αβ (1− z

w

)−αβVβ(w)Vα(z) .

(3.6)

One can also write

Vα(z)Vβ(w) = (z − w)αβ : Vα(z)Vβ(w) : . (3.7)

Since vertex operators contain the factor eνQ, they shift the vacuum charge

Vν(z) : Hσ → Hσ+ν (3.8)

when acting onto a sector in full Hilbert space

H =⊕σ

Hσ(3.9)

corresponding to the definite value of this charge. Notice that we do not impose anyspecial constraints to the (real) values of the vacuum charges σ ∈ R.

The Hilbert space Hσ is constructed by the action of the negative bosonic gener-ators

J−n1 . . . J−nk |σ〉 (3.10)

on the vacuum vector J0|σ〉 = σ|σ〉, and these states can be labeled by the Youngdiagrams with the row lengths n1, . . . , nk.

One can also construct the action of the fermionic operators on this vector space.Then the bosonization formulas (3.3) will generally produce the fractional powers inholomorphic coordinate z due to the factors zJ0 , while e±Q just shift the vacuumcharge by ±1. It means that one can define the (multiple) action of the modes of theoperators

ψσ(z) =∑r

ψσrzr+1/2+σ

, ψσ(z) =∑r

ψσrzr+1/2−σ (3.11)

in the direct sum of the Hilbert spaces

Hσ =⊕n∈Z

Hσn (3.12)

naturally labeled by some fractional σ ∈ R/Z.Basis in the each spaceHσ

n can be given by the vectors generated by the zero-chargeexpressions of the fermionic modes. As in bosonic representation, these vectors canbe labeled by the Young diagrams

|Y, σ〉 =∏i

ψσ−piψσ−qi |σ〉 (3.13)

where now pi and qi are the Frobenius coordinates of the Young diagram. In ourconvention they are half-integer, and can be easily read of the following picture:

39


@@@

i.e. one has to cut the diagram by the main diagonal and just take the areas of therows and columns starting from the diagonal cells. For example, the Young diagramfrom the picture has pi = 9

2, 5

2, 3

2 and qi = 9

2, 5

2, 1

2.

The states in the dual toHσ module can be obtained by the Hermitian conjugation

〈σ, Y | = 〈σ|∏i

ψσqiψσpi. (3.14)

Our main aim in what follows is to compute the matrix elements of the operatorVν(1) = Vν between the arbitrary fermionic states

Z(ν|Y ′, Y ) = 〈θ + ν, Y ′|Vν(1)|Y, θ〉 . (3.15)

The most straightforward way is to use explicit bosonic representation (3.5) of thevertex operator

Z(ν|Y ′, Y ) = 〈σ + ν|∏j

ψq′jψp′jV−ν V

+ν e

νQ∏i

ψ−piψ−qi|σ〉 =

= 〈0|∏j

V −−νψq′jV−ν · V −−νψp′jV

−ν

∏i

V +ν ψ−piV

+−ν · V +

ν ψ−qiV+−ν |0〉 =

= 〈0|∏j

(V −ν )−1ψq′jV−ν · (V −ν )−1ψp′jV

−ν

∏i

V +ν ψ−pi(V

+ν )−1 · V +

ν ψ−qi(V+ν )−1|0〉 .

(3.16)

It is easy to understand from (3.3) and (3.5) that the consequent triple products ofoperators in this formula can be considered as certain adjoint action, or just conjuga-tions of the fermions, which turn under such action just into the linear combinationsof themselves. At the level of generating functions it looks like

V +ν ψ(z)(V +

ν )−1 = (1− z)νψ(z) , V +ν ψ(z)(V +

ν )−1 = (1− z)−νψ(z) ,

(V −ν )−1ψ(z)V −ν =

(1− 1

z

)νψ(z) , (V −ν )−1ψ(z)V −ν =

(1− 1

z

)−νψ(z) ,

(3.17)

or, more generally

Vν(w)−1ψσ+ν(z)Vν(w) =( zw

)νexp

(ν∑n∈Z

′ 1

n

zn

wn

)ψσ(z) ,

Vν(w)−1ψσ+ν(z)Vν(w) =( zw

)−νexp

(−ν∑n∈Z

′ 1

n

zn

wn

)ψσ(z) ,

(3.18)

where the formal series in the r.h.s. can be rewritten with the help of the Fouriertransformation as

exp

(ν∑n∈Z

′ zn

n

)=

sin πν

π

∑k∈Z

zk

k + ν. (3.19)

40


This is a particular case of transformations from GL(∞), realized by∑ars : ψ−rψs :∈ gl(∞), ars →∞, |r − s| → ∞ , (3.20)

moreover, corresponding to the situation, when ars = ar−s (a well known example ofsuch transformation is generated by the currents Jn =

∑r : ψrψn−r : from (3.4)). It is

true in the most general case: if one computes any matrix elements of such operator,they always can be expressed in terms of those with only two extra fermion insertions,i.e. we do not need an explicit form of the operator Vν = V −ν V

+ν – just the only fact

of the adjoint action, and we are going to use this property in more complicated nonAbelian situation below.

In particular, one can compute (3.16) first using the Wick theorem

Z(ν|Y ′, Y ) = det

(〈σ + ν|ψq′jψp′jVν |σ〉〈σ + ν|ψq′jVνψ−qi |σ〉−〈σ + ν|ψp′jVνψ−pi |σ〉〈σ + ν|Vνψ−piψ−qi |σ〉

)= detGν (3.21)

and then to apply (3.17) to the matrix elements in (3.21).

Matrix elements and Nekrasov functions

The two-fermion matrix elements of the matrix G = Gν (its rows are labeled byxa = q′j ∪ −pi, whereas columns are labeled by yb = p′j ∪ −qi, here wedenote by p and q some positive half-integer numbers) are expressed as

G(q′, p′) = 〈0|ψq′ψp′V −ν |0〉 =

q′− 12∑

m=0

(ν)mm!

(−ν)p′+q′−m(p′ + q′ −m)!

,

G(−p,−q) = 〈0|V +ν ψ−pψ−q|0〉 =

q− 12∑

n=0

(−ν)nn!

(ν)p+q−n(p+ q − n)!

,

G(−p, p′) = −〈0|ψp′V −ν V +ν ψ−p|0〉 = −

p′− 12∑

m=0

(−ν)mm!

(ν)m+p−p′

(m+ p− p′)!,

G(q′,−q) = 〈0|ψq′V −ν V +ν ψ−q|0〉 =

q− 12∑

n=0

(−ν)nn!

(ν)n+q′−q

(n+ q′ − q)!.

(3.22)

These expressions are easily computed, using adjoint action (3.17) for the components

V +ν ψ−p(V

+ν )−1 =

∞∑m=0

(ν)mm!

ψ−p+m, V +ν ψ−q(V

+ν )−1 =

∞∑m=0

(−ν)mm!

ψ−q+m

(V −ν )−1ψqV−ν =

∞∑m=0

(ν)mm!

ψq−m, (V −ν )−1ψpV−ν =

∞∑m=0

(−ν)mm!

ψp−m

(3.23)

41


with (ν)m = ν(ν + 1) . . . (ν +m− 1), (ν)0 = 1, and there are explicit formulas for thesums in the r.h.s. of (3.22)

b∑m=0

(ν)mm!

(−ν)a−m(a−m)!

=(ν)b+1(−ν)a−bνab!(a− b− 1)!

b∑m=0

(−ν)mm!

(ν)a+m

(a+m)!= − (−ν)b+1(ν)a+b+1

ν(a+ ν)b!(a+ b)!

(3.24)

which can be easily proven by induction. It allows to rewrite matrix elements (3.22)in the factorized form

G(q′, p′) =1

ν(p′ + q′)

(ν)q′+ 12(−ν)p′+ 1

2

(q′ − 12)!(p′ − 1

2)!,

G(−p,−q) = − 1

ν(p+ q)

(ν)p+ 12(−ν)q+ 1

2

(p− 12)!(q − 1

2)!,

G(−p, p′) =1

ν(p− p′ + ν)

(ν)p+ 12(−ν)p′+ 1

2

(p− 12)!(q′ − 1

2)!,

G(q′,−q) = − 1

ν(q′ − q + ν)

(ν)q′+ 12(−ν)q+ 1

2

(q − 12)!(q′ − 1

2)!.

(3.25)

The determinant from (3.21) can be therefore written as

deta,b

G(xa, yb) =∏j

(−ν)p′j+ 12(ν)q′j+ 1

2

ν(p′j − 12)!(q′j − 1

2)!

∏i

(ν)pi+ 12(−ν)qi+ 1

2

ν(pi − 12)!(qi − 1

2)!· deta,b

G(xa, yb) (3.26)

where now for two new sets xa = q′j ∪ −pi − ν, yb = −p′j ∪ qi − ν

G(xa, yb) =sgn(xayb)

xa − yb, (3.27)

and the corresponding determinant can be computed using the Cauchy determinantformula

deta,b

1

xa − yb=

∏a<b(xa − xb)

∏a>b(ya − yb)∏

ab(xa − yb),

so one gets finally

Z(ν|Y ′, Y ) = ±∏j

(−ν)p′j+ 12(ν)q′j+ 1

2

ν(p′j − 12)!(q′j − 1

2)!

∏i

(ν)pi+ 12(−ν)qi+ 1

2

ν(pi − 12)!(qi − 1

2)!×

×∏

i>j(p′i − p′j)

∏i<j(pi − pj)

∏i>j(q

′i − q′j)

∏i<j(qi − qj)

∏ij(q

′i + pj + ν)

∏ij(p

′i + qj − ν)∏

ij(p′i + q′j)

∏ij(pi + qj)

∏ij(q

′i − qj + ν)

∏ij(pi − p′j + ν)

(3.28)It is easy to see that this expression has the structure

Z(ν|Y ′, Y ) = ± Zb(ν|Y ′, Y )

Z120 (Y ′)Z

120 (Y )

(3.29)

42


where

Z120 (Y ) =

∏i

(pi − 1

2

)!(qi − 1

2

)!

∏ij(pi + qj)∏

i<j(qi − qj)∏

i<j(pi − pj), (3.30)

while

Zb(ν|Y ′, Y ) =∏i

ν−1(−ν)p′i+ 12(ν)q′i+ 1

2

∏j

ν−1(−ν)qj+ 12(ν)pj+ 1

2×

×∏

ij(q′i + pj + ν)

∏ij(p

′i + qj − ν)∏

ij(q′i − qj + ν)

∏ij(p

′i − pj − ν)

.

(3.31)

In this normalization one can check that

Zb(ν|Y ′, Y ) = ±∏t∈Y

(1 + aY (t) + lY ′(t) + ν)∏s∈Y ′

(1 + aY ′(s) + lY (s)− ν) (3.32)

is exactly the Nekrasov bi-fundamental function of the U(1) gauge theory at c = 1 orε1 + ε2 = 0. Notice also that

Zb(0|Y, Y ) = Z120 (Y )Z

120 (Y ) =

∏s∈Y

(1 + aY (s) + lY (s))2 = ZV (Y )−1(3.33)

is Nekrasov function for the pure U(1) gauge theory, which corresponds to the Plancherelmeasure on partitions [LMN].

Riemann-Hilbert problem

The following simple observation is extremely important for our generalizations below.Consider the correlator

〈θ|Vν(1)ψσ(z)ψσ(w)|σ〉 = δθ,σ+νzσw−σ(1− z)ν(1− w)−ν

z − w(3.34)

which is easily computed using bosonization rules (3.3). One finds then, that

(z − w)〈θ|Vν(1)ψσ(z)ψσ(w)|σ〉 = φ(z)φ(w)−1 (3.35)

is expressed actually through the solutions of a simple linear system

dφ(z)

dz= φ(z)

(σ

z+

ν

z − 1

)(3.36)

It means that this linear system can be used to define all two-fermion matrix elements,e.g. in the region 1 > |z| > |w|

〈θ|Vν(1)ψσ(z)ψσ(w)|σ〉 =∑p,q

1

zp+12−σwq+

12

+σ〈θ|Vν(1)ψσpψ

σq |σ〉 (3.37)

and together with the Wick theorem it defines all matrix elements, or just the vertexoperator Vν , uniquely – up to a numeric factor. In its turn the linear system itselfis determined by the monodromy properties (here very simple) of φ(z) at z = 0and z = 1 (and related to them monodromy at z = ∞). Hence, the problem ofcomputation of the two-fermion correlation functions can be reformulated in terms ofa Riemann-Hilbert problem.

43


Remarks

• Formulas (3.28), (3.31) give a very explicit representation for the matrix elementand bi-fundamental Nekrasov function in terms of the Frobenius coordinatesof the corresponding Young diagrams (this representation, for example, is farmore adapted for practical computation, than the formulas (3.32)). However,it is sometimes not easy to see directly, that these formulas possess some niceproperties: for example satisfy the “sum rules” like∑

Y

t|Y |Zb(α1|∅, Y )Zb(α2|Y, ∅)

Z0(Y )=∑Y

t|Y |Z(α1|∅, Y )Z(α2|Y, ∅) =

=∑Y

t|Y |〈0|eiα1φ(1)|Y, 0〉〈Y, 0|eiα2φ(1)|0〉 = (1− t)α1α2

(3.38)

where the r.h.s. immediately follows from resolution of unity and the correlatorof two exponentials

〈0|eiα1φ(1)eiα2φ(t)|0〉 = (1− t)α1α2 (3.39)

which is instructive to compare with the computation from [KKMST, Koz].

• One can also easily extract some useful information from particular cases of(3.34), which include a nice identity (cf. with [NO, BAW])

(1− z)ν(1− w)−ν

z − w=

1

z − w+

∞∑a,b=0

(−ν)a+1(ν)b+1

(a+ b+ 1)νa!b!zawb (3.40)

containing some part of the matrix elements from (3.25).

• According to (3.7)

ψ(z + t/2)ψ(z − t/2) =1

t: exp

(ˆ z+t/2

z−t/2J(ξ)dξ

): (3.41)

Expansion into the powers of t gives the infinite series of the currents of W1+∞algebra

: ψ(z + t/2)ψ(z − t/2) :=1

t:

(exp

(ˆ z+t/2

z−t/2J(ξ)dξ

)− 1

):=

=∑k>0

tk−1

(k − 1)!Uk(z)

(3.42)

where explicitly

U1(z) = J(z), U2 = 12

: J(z)2 :, U3 =1

3

(: J(z)3 : +

1

4∂2J(z)

), . . .

(3.43)and one implies bosonic normal ordering for the bosons and fermionic for thefermions. These formulas have been used many times (see e.g. [Pogr, OP, LMN,NO, Mint]) to relate the generators of the W1+∞ algebra with the fermionicbilinear operators, and we just recall them in order to generalize below to muchless trivial non Abelian case.

44

3.3. Non-Abelian U(N) theory

Non-Abelian U(N) theory

Nekrasov functions

Consider now more general case of Nekrasov functions, corresponding to the U(N)non-Abelian theory. They can be expressed in terms of U(1) functions (3.31), (3.32)by the following product formula

Zb(θ′, ν,θ|Y ′,Y ) =

N∏α,β=1

Zb(ν − θ′α + θβ|Y ′α, Yβ) (3.44)

For the diagonal elements Z0(θ|Y ) = Zb(θ, 0,θ|Y ,Y ) one gets

Z0(θ|Y ) =N∏

α,β=1

Zb(−θα + θβ|Yα, Yβ) = ±∏i<j

Z2b (−θα + θβ|Yα, Yβ) ·

∏α

Z0(Yα)

(3.45)or, after taking the square root, just

Z120 (θ|Y ) =

∏α<β

Zb(−θα + θβ|Yα, Yβ) ·∏α

Z120 (Yα) (3.46)

Now for simplicity it is better to replace θ′α − ν 7→ θ′α or θα + ν 7→ θα, then ν simplydisappears from (3.44). Consider now the normalized matrix element

Z(θ′,θ|Y ′,Y ) =Zb(θ

′,θ|Y ′,Y )

Z120 (θ′|Y ′)Z

120 (θ|Y )

=

=

N∏α,β=1

Zb(−θ′α + θβ|Y ′α, Yβ)∏α<β Zb(−θα + θβ|Yα, Yβ)

∏α

Z120 (Yα) ·

∏α<β Zb(−θ′α + θ′β|Y ′α, Y ′β) ·

∏α

Z120 (Y ′α)

(3.47)Using representation (3.28), (3.31) for the U(1) functions in terms of the Frobenius co-ordinates, one finds that the ratio of products of the elementary Cauchy determinantsfrom there is actually combined into more sophisticated unique Cauchy determinant

Z(θ′,θ|Y ′,Y ) = detIJ

1

xI − yJ×

×∏i,α

f1,α(θ′,θ, p′α,i)f2,α(θ′,θ, q′α,i)f1,α(θ,θ′, pα,i)f2,α(θ,θ′, qα,i)(3.48)

with two multi-sets of variables entering the determinant of the form

xI = −q′α,i − θ′α ∪ pα,i − θαyI = p′α,i − θ′α ∪ −qα,i − θα

(3.49)

45


up to quite nontrivial diagonal part, which can be still read from (3.28) and (3.47),giving the following factors for (3.48)

f1,α(θ,θ′, pα,i) =1

(pα,i − 12)!

∏β

(θ′β − θα)pα,i+ 12√

θ′β − θα

∏β 6=α

√θβ − θα

(θβ − θα)pα,i+ 12

f2,α(θ,θ′, qα,i) =1

(qα,i − 12)!

∏β

(θα − θ′β)qα,i+ 12√

θα − θ′β

∏β 6=α

√θα − θβ

(θα − θβ)qα,i+ 12

(3.50)

Existence of the determinant formula (3.48) is very important, since it actually impliesthat Nekrasov functions Z(θ′,θ|Y ′,Y ) can be identified with the matrix elements ofsome vertex operator, characterized as in the Abelian U(1) case by its adjoint action,which is still a linear transformation but now of the N -component fermions. Weare going indeed to introduce this vertex operator below using the theory of (N -component) free fermions, generalizing the Abelian case considered above. In generalsituation this operator is characterized by solution to auxiliary linear problem onsphere with three marked points, while explicit formulas of this section just correspondto particular case of the hypergeometric-type solutions.

N-component free fermions

Hence, consider the generalization of the free-fermionic construction from U(1) to thenon-Abelian U(N) case. First, introduce the algebra

ψα,r, ψβ,s = 0 , ψα,r, ψβ,s = 0 ,

ψα,r, ψβ,s = δα,βδr+s,0

r, s ∈ Z + 12, α, β = 1, . . . , N

(3.51)

of the canonical anticommutation relations for the components of the fermionic fieldswith free first-order action S = 1

π

∑Nα=1

´Σd2zψα∂ψα, so that (3.51) are equivalent to

the operator product expansions

ψα(z)ψβ(w) =δαβz − w

+ Jαβ(w) +O(z − w)

ψα(z)ψβ(w) = reg. ψα(z)ψβ(w) = reg.

(3.52)

Similarly to (3.3) it is also possible and useful to introduce the bosonization formulasfor these fermionic fields

ψα(z) = exp

(−∑n<0

Jα,nnzn

)exp

(−∑n>0

Jα,nnzn

)eQαzJα,0εα(J0)

ψα(z) = exp

(∑n<0

Jα,nnzn

)exp

(∑n>0

Jα,nnzn

)e−Qαz−Jα,0εα(J0)

(3.53)

Here Jα,n form the Heisenberg algebra

[Jα,n, Jβ,m] = nδαβδm+n,0, [J0,α, Qβ] = δαβ (3.54)

46


and εα(J0) =∏α−1

β=1(−1)J0,β , we may also note that εα(x + y) = εα(x)εβ(y). Theseextra sign factors do the same as the Jordan-Wigner transformation: they convertcommuting objects into the anticommuting ones.

A standard representation of this algebra Hσ is constructed from the vacuumvector |σ〉, with the charges J0|σ〉 = σ|σ〉 and killed by all positive modes

ψσα,r>0|σ〉 = 0 , ψσα,r>0|σ〉 = 0 . (3.55)

Basis vectors of this representation can be given by

|pα,i, qα,i,σ〉 =N∏α=1

|pα,i|∏i=1

ψα,−pα,i

|qα,j |∏j=1

ψα,−qα,j

|σ〉 (3.56)

The letters pα,i and qα,i, at least in the case when #pα = #qα, should be interpreted asFrobenius coordinates of the N -tuple of the Young diagrams. It will be also convenientin what follows to use the vacuum-shifting operators P n

α

P 0α = 1, P n<0

α = ψσα,n+ 1

2ψσα,n+ 3

2. . . ψσ

α,− 12|σ〉 ,

P n>0α = ψσ

α,−n+ 12ψσα,−n+ 3

2. . . ψσ

α,− 12|σ〉

(3.57)

and the corresponding states

|n,σ〉 =N∏α=1

P nαα |σ〉 . (3.58)

in particular for the vectors n = ±1β with components nα = ±δαβ.

Level one Kac-Moody and W-algebras

Consider the W-algebras for g = sl(N) series, possibly extended to gl(N) where weshall call it WN ⊕ H. Their generators in current representation can be identifiedwith the symmetric functions of the normally ordered currents J(z) ∈ h ⊂ g with thevalues in Cartan subalgebra, or equivalently, up to a coefficient, as certain “Casimir

elements” in the universal enveloping U(sl(N)1). The Virasoro central charge at levelk = 1 is

c =k dim g

k + CV=N2 − 1

1 +N= N − 1 (3.59)

When embedded to U(gl(N)1) this current algebra has nice representation in termsof the multi-component free holomorphic fermionic fields

Jαβ(z) =: ψα(z)ψβ(z) :, α, β = 1, . . . , N (3.60)

The WN -algebra can be defined in terms of invariant Casimir polynomials of thecurrents, commuting with the screening charges Qαβ =

¸Jαβ(z) (it is enough to

require commutativity only with those, corresponding to the positive simple roots).

47


Then the W-generators turn to be just the symmetric polynomials of the diagonalCartan currents Jα = Jαα(z) ∈ h, i.e.

Wn(z) =∑

α1<α2<...<αn

: Jα1(z)Jα2(z) . . . Jαn(z) : , n = 1, . . . , N (3.61)

One can consider the representations of U(gl(N)1) and WN ⊕ H in Hσ. For thispurpose it is convenient to introduce the generating functions

ψσα (z) =∑r∈Z+ 1

2

ψσα,r

zr+12

+σα, ψσα (z) =

∑r∈Z+ 1

2

ψσα,r

zr+12−σα

. (3.62)

where shifts of the powers of the coordinate z come naturally, e.g. from the bosoniza-tion formulas (3.53). For these fields instead of (3.52) one gets

ψσα (z)ψσβ (w) = δαβzσαw−σα

z − w+ : ψσα (z)ψσβ (w) : =

=δαβz − w

+ δαβσαw

+ : ψσα (w)ψσβ (w) : +O(z − w) ,

(3.63)

then it is clear that the modes of Jσαβ(w) from (3.52) acquire in this representationthe form

Jσαβ,n = δαβδn,0σα +∑p∈Z+ 1

2

: ψσα,n−pψσβ,p :

(3.64)

As in the U(1) case (see (3.42)) in the fermionic realization of WN ⊕H, then one canchoose the set of generators in a form of the fermionic bilinears:

∑α

ψσα (z +t

2)ψσα (z − t

2) =

N

t+∞∑k=1

tk−1

(k − 1)!Uσk (z) . (3.65)

The l.h.s. of this formula gives

ψσα (z +t

2)ψσα (z − t

2) =

1

t

(1 + t

2z

1− t2z

)σα+

+1

t

∑m∈Z

1

zm

tz

(1 + t2z

)m+1

∑p∈Z+ 1

2

(1 + t

2z

1− t2z

)p+ 12

+σα

: ψσm−p,αψσα,p : .

(3.66)

48


Introducing two collections of polynomials 1

(1 + x

2

1− x2

)p= 1 +

∞∑k=0

uk(p)xk

(k − 1)!,

x

(1 + x2)m+1

(1 + x

2

1− x2

)p+ 12

=∞∑k=1

vk,m(p)xk

(k − 1)!,

(3.68)

the generators in the r.h.s. of (3.65) explicitly become

Uσk,m =∑α

(δm,0uk(σα) +

∑p∈Z

vk,m(p+ σα) : ψσα,m−pψσp,α :

). (3.69)

This set of generators of WN ⊕ H contains commuting zero modes Uσk,0 which wereshown to play an important role in the study of the extended Seiberg-Witten theoryand AGT correspondence [LMN, MN, Mint, FLitv12]. It is also important to noticethat commutation relations between these generators are linear, the only place whenthe non-linearity appears are the relations between these generators.

Using the bosonization rules (3.53) one can rewrite these generators in the conven-tional form. To perform explicit splitting of this algebra into WN ⊕H it is convenientto redefine Jα(z) 7→ Jα(z)+j(z), where the new currents already satisfy the condition∑Jα = 0 and the operator product expansions (OPE)

j(z)j(w) =1N

(z − w)2+ reg. Jα(z)Jβ(w) =

δαβ − 1N

(z − w)2+ reg. (3.70)

Now we take the bilinear expression

∑α

ψα(z +t

2)ψα(z − t

2) =

∑α

: eiϕ(z+ t2

)+iφα(z+ t2

) :: e−iϕ(z− t2

)−iφα(z− t2

) : =

=1

t: eiϕ(z+ t

2)−iϕ(z− t

2) :∑α

: eiφα(z+ t2

)−iφα(z− t2

) :(3.71)

with j(z) = i∂ϕ, Jα(z) = i∂φα(z) and expand it into the powers of t. Comparing

1One can also notice at the level of the generating functions (3.68) that

vk,0(p) = uk(p+1

2)− uk(p− 1

2) .

First polynomials are given explicitly by

u1(p) = p, u2(p) =p2

2, u3(p) =

p3

3+p

6,

u4(p) =p4

4+p2

2, u5(p) =

p5

5+ p3 +

3p

10, . . .

(3.67)

49


with (3.65) we get the following formulas:

U1(z) = Nj(z), U2(z) = T (z) +N

2: j2(z) :,

U3(z) = W3(z) + 2NT (z)j(z) +N

3

(: j3(z) : +

1

4∂2j(z)

),

U4(z) = −W4(z) +1

2(TT )(z) + 3W3(z)j(z) + 3 : j2(z) : T (z)+

+N

4

(: j4(z) : + : j(z)∂2j(z) :

), U5(z) = . . .

(3.72)

where T (z) = −W2(z) is the stress-energy tensor, and (AB)(z) is the “interacting”normal ordering

(AB)(z) =

˛z

dw

w − zA(w)B(z)

One find therefore, that one basis is related with the other by some complicated,though explicit and triangular transformation. Here we can see that generators Uk(z)are actually dependent, namely, if N = 3, then W4(z) = 0 and U4(z) becomes somenon-linear expression of the lower generators.

It is also easy to see that for the states (3.58)

Jσα,0|n,σ〉 = (σα + nα)|n,σ〉 , Uσk,0|n,σ〉 = uk(σ + n)|n,σ〉 ,Uσk,m>0|n,σ〉 = 0.

(3.73)

It is sometimes useful to decompose the whole Hilbert space into the sectors Hσ =⊕n∈ZN

Hσn with fixed h ∈ gl(N) charges and also into the sectors Hσl =⊕∑nα=l

Hσn with

fixed overall u(1) = gl(1) charge. Summarizing all these facts we can formulate thefollowing

Theorem 3.1. Spaces Hσl are representations of gl(N)1, and for general σ spacesHσn are the Verma modules of WN ⊕H algebra with the highest weight vectors |σ,n〉and with basis vectors |Y ,n,σ〉, ∀Y .

Proof is extremely simple: gl(N)1 generators have zero fermionic gl1-charge, WN⊕H generators have zero charges with respect to the whole Cartan subalgebra h, so thespaces Hσl and Hσn are closed under the action of these algebras. We also know from(3.73) that |σ,n〉 are the highest weight vectors of WN ⊕ H, so we have a non-zeromap from the Verma module to Hσn, but this Verma module is generally irreducibleand has the same character tr qL0 , so we actually have an isomorphism.

Free fermions and representations of W-algebras

Let us now illustrate how can free fermions appear in the theory with WN -symmetryat integer central charges after inclusion of extra Heisenberg algebra. Constructionbelow is a straightforward generalization of the bosonization procedure from [ILTe].

It is well-known [FZ, FL] that conformal theory with WN -symmetry contains twodegenerate fields Vµ1(z) and VµN−1

(z), such that their W -charges are determined by

50


the highest weights of the fundamental (N) and antifundamental (N) representations,respectively. Their dimensions are

∆(µi) = 12µ2i =

N − 1

2N, i = 1, N − 1 (3.74)

and they have the following fusion rules with arbitrary primary field

[µ1]⊗ [σ] = ⊕Nα=1[σ + eα]

[µN−1]⊗ [σ] = ⊕Nα=1[σ − eα](3.75)

where ±eβ is the set of all weights of N and N. One can define now the vertexoperators

Ψα(z) =∑σ

Pσ+eαVµ1(z)Pσ, Ψα(z) =∑σ

Pσ−eαVµN−1(z)Pσ (3.76)

which, due to extra projector operators, act only from one Verma module to another,just extracting the corresponding term from the fusion rules (3.75). Using the generalstructure of the OPE of two initial degenerate fields

Vµ1(z)VµN−1(w) =

(1 · (z − w)

1−NN + # · (z − w)

1+NN T (w)

)+

+(z − w)1N

∑α∈roots(glN )

cαVα(w) + . . . ,(3.77)

one finds, that Ψα(z)Ψβ(w) = δαβ1 · (z − w)1−NN + reg., i.e. these fields look almost

like fermions, except for the wrong power in the OPE. To fix this let us add an extrascalar field φ(z), such that

φ(z)φ(w) = − 1

Nlog(z − w) + . . . (3.78)

and define the new, the true fermionic, vertex operators

ψα(z) = e−iφ(z)Ψα(z), ψα(z) = eiφ(z)Ψα(z), α = 1, . . . , N (3.79)

which have the canonical OPE (cf. with (3.52))

ψα(z)ψβ(w) =δαβz − w

+ reg.

ψα(z)ψβ(w) = reg. ψα(z)ψβ(w) = reg.

(3.80)

The rest is to understand, how to express the W-algebra generators in terms of thesefree fermions. One can easily write for the structure of the sum

(z − w)−1/N∑α

Ψα(z)Ψα(w) =1

z − w+ # · (z − w)(L−21)(w)+

+#(z − w)2 · (L−1L−21) + #(z − w)2 · (W−31)(w) + . . . =

=1

z − w+ # · (z − w)T (w) + # · (z − w)2∂T (w) + # · (z − w)2W (w) + ...

(3.81)

51


with some coefficients (and where we have used obvious notations for the descendants).We do not need their exact numeric values at the moment, just the very fact thatonly the unit operator 1 enters the r.h.s. of this OPE together with its descendants.Using additionally the OPE of the U(1) factors

(z − w)1/Neiφ(z)e−iφ(w) =

=: exp

(i(z − w)∂φ(w) +

1

2i(z − w)2∂2φ(w) +

1

6(z − w)3∂3φ(w)

):=

= 1 + (z − w)j(w) +1

2(z − w)2∂j(w) +

1

6(z − w)3∂2j(w)+

+1

2(z − w)2 : j(w)2 : +

1

2(z − w)3 : j(w)∂j(w) : +

1

6(z − w)3j(w)3 + . . .

(3.82)

one can get∑α

ψα(z)ψα(w) =1

z − w+ j(w) + (z − w)

(# · T (w) +

1

2j(w) +

1

2: j(w)2 :

)+

+(z − w)2

(# ·W (w) + # · j(w)T (w) +

1

6∂2j(w) +

1

2: j(w)∂j(w) : +

1

6: j(w)3 :

)+ ...

(3.83)This formula states, how the standard W-generators can be expressed via the fermionicbilinears by some triangular transformation, and its symmetric form is equivalent to(3.71), (3.72).

Vertex operators and Riemann-Hilbert problem

Vertex operators and monodromies

Let us now turn to general construction of the monodromy vertex operator 2

Vν(t) : Hσ → Hθ (3.84)

Actually one can define only the operator Vν(1) due to conformal Ward identity

Vν(t) = t−∆ν tL0Vν(1)t−L0 (3.85)

and the operator Vν(1) is defined by the following three properties:

• Vν(1) is a (quasi)-group element, i.e.

Vν(1)Hσ (Vν(1))−1 ⊆ Hθ, (Vν(1))−1HθVν(1) ⊆ Hσ

As we discussed already in sect. 3.2 this fact actually implies that all correlatorsof fermions in the presence of such an operator can be computed using the Wicktheorem.

2Notice, that we have here only the conservation of the “total charge”∑α σα +

∑α να =

∑α θα,

and apart of that their values are arbitrary.

52

3.4. Vertex operators and Riemann-Hilbert problem

• 〈θ|Vν(1)|σ〉 = 1, which is a kind of convenient normalization. Notice, however,that vertex operator is defined by the adjoint action only up to some diagonalfactor S = exp(β), β ∈ h ⊂ gl(N). In what follows we shall restore thesediagonal factors when necessary.

• All two-fermionic correlators give the solution for the 3-point Riemann-Hilbertproblem in the different regions

〈θ|Vν(1)ψσα (z)ψσβ (w)|σ〉 = Kαβ(z, w), |z| ≤ 1, |w| ≤ 1

〈θ|ψθα(z)ψθβ(w)Vν(1)|σ〉 = Kαβ(z, w), |z| ≥ 1, |w| ≥ 1

〈θ|ψθα(z)Vν(1)ψσβ (w)|σ〉 = Kαβ(z, w), |z| ≥ 1, |w| ≤ 1

−〈θ|ψθβ(w)Vν(1)ψσα (z)|σ〉 = Kαβ(z, w), |z| ≤ 1, |w| ≥ 1

(3.86)

In terms of some matrix kernels K(z, w) = Kν(z, w), where we have used α, βand α, β to denote matrix indices, corresponding to different bases, associatedwith the points z = 0 and z =∞ respectively.

By this moment the only claim is that this operator is uniquely defined by thproperties listed above, and this follows from the fact, that all matrix elements ofthe quasi-group Vν(1) element are given by certain determinants of the matrices withthe entries, constructed from K(z, w). Existence of this operator is therefore obvious,since one can compute all its matrix elements using the Wick theorem.

Now, we would like to specify the kernels K(z, w) first by their monodromy prop-erties. We associate the basis at z = 0 with the eigenvectors of M0 ∼ e2πiσ, while thebasis at z = ∞ with the eigenvectors of M∞ ∼ e2πiθ (only the conjugacy classes ofthese two matrices are fixed, and certainly in general [M0,M∞] 6= 0). We propose anexplicit form of the kernel

Kαβ(z, w) =[φ(z)φ(w)−1]αβ

z − w(3.87)

given in terms of the solution to the linear system

d

dzφ(z) = φ(z)

(A0

z+

A1

z − 1

)= φ(z)A(z) (3.88)

with A0 ∼ σ, A1 ∼ ν, A∞ ∼ θ and prescribed monodromies

γ0 : φαi(z) 7→∑β

(M0)αβφ(z)βi

γ∞ : φαi(z) 7→∑β

(M∞)αβφ(z)βi(3.89)

also implying monodromy around z = 1, i.e. γ1 : φαi(z) 7→∑

β(M1)αβφ(z)βi, with

M1 ∼ e2πiν and M0M1M∞ = 1. Solutions for a linear system (3.88) can be expressedthemselves in terms of a fermionic correlators, namely

φαγ(z) =z · 〈θ|Vν(1)ψα(z)| − 1γ,σ〉φ−1γβ (z) =z · 〈θ|Vν(1)ψβ(z)|1γ,σ〉

(3.90)

53


for some fixed normalization at z → 0. We are going to prove in next section, thatdefinitions (3.87) are indeed self-consistent and also consistent with (3.90), whichfollows from the generalized Hirota bilinear relations, satisfied by the monodromyvertex operators.

Actually, we have four different matrix kernels (3.87) with the indices αβ, αβ, αβ,αβ, corresponding to all possible combinations of different regions. When we changefrom one region to another one, then we have to change the basis of solutions, andthis transition can be given by some matrix Cα

α .

The expansion of these kernels, e.g. for 0 < z,w < 1

Kαβ(z, w) =δαβz − w

+∑p,q>0

〈θ|Vν(1)ψα−pψβ−q|σ〉zp−

12

+σαwq−12−σβ =

=δαβz − w

+∑p,q>0

Kαβpq z

p− 12

+σαwq−12−σβ

(3.91)

or at z, w > 1

Kαβ(z, w) =δαβz − w

+∑p,q>0

〈θ|ψαpψβq Vν(1)|σ〉z−p−12

+θαw−q−12−θβ =

=δαβz − w

+∑p,q>0

K αβpq z

−p− 12

+θαw−q−12−θβ

(3.92)

give the corresponding matrix elements for the fermionic modes. The correspondingmatrix elements (Kαβ

pα,qβand Kαβ

pα,qβ) are in fact defined up to the factors sαs

−1β which

comes from the ambiguity in normalization of the vertex operator. For any three mon-odromy matrices M0M1M∞ = 1 one can fix all their invariant functions (e.g. traces)and diagonalize M∞, but then one possible transformation survives: a simultaneousconjugation

Mi 7→ S−1MiS (3.93)

by diagonal S = diag(s1, . . . , sN). This gives vertex operators, actually different bysJ0 factor with corresponding multiplicative renormalization of their matrix elements.

For special vertex operators with ν = νNej these matrix elements can be expressedin terms of the products (3.50), for example

Kαβpα,qβ

= 〈θ|Vν(1)|pα, qβ;σ〉 = 〈θ|Vν(1)ψα−pαψβ−qβ |σ〉 =

=1

pα + qβ − σα + σβf1,α(σ,θ + ν, pα)f2,β(σ,θ + ν, qβ)

Kαβpα,qβ

= 〈pα, qβ;θ|Vν(1)|σ〉 = 〈θ|ψαpαψβqβVν(1)|σ〉 =

= − 1

pα + qβ − θα + θβf1,α(θ + ν,σ, pα)f2,β(θ + ν,σ, qβ)

(3.94)

We shall return to discussion of special case below in sect. 3.4.3.

54


The general formula for 2n-point fermionic correlator is given by the Wick formula

〈θ|N∏α=1

d′α∏i=1

ψθα(zα,i)ψθα(wα,i)Vν(1)

N∏α=1

dα∏i=1

ψσα (zα,i)ψσα (wα,i)|σ〉 =

= 〈θ|Vν(1)|σ〉 · det

(Kαβ(zα,i, wβ,j) Kαβ(zα,i, wβ,j)

Kαβ(zα,i, wβ,j) Kαβ(zα,i, wβ,j)

) (3.95)

On the punctured unit circle |z| = |w| = 1, z 6= 1, w 6= 1 one has

Kαβ(z, w) =∑α

CααKαβ(z, w)

Kαβ(z, w) =∑β

Kαβ(z, w)(C−1

)ββ

(3.96)

It follows then from (3.95), that there are two operator identities

ψθα(z)Vν(1) = Vν(1)∑α

Cαα ψ

σα (z)

ψθα(z)Vν(1) = Vν(1)∑α

ψσα (z)(C−1)αα(3.97)

Actually, these identities are enough to define the operator Vν(1). The simplestquantity to compute is

Vν(1)ψσα,rVν(1)−1 =

˛

|z|=1

dz

2πizr−

12

+σαVν(1)ψσα (z)Vν(1)−1

(3.98)

Using (3.97) one can rewrite this equivalently as

Vν(1)ψσα,rVν(1)−1 =∑β

Cβα

˛

|z|=1

dz

2πizr−

12

+σαψθβ(z) =

=∑β,s

Cβα

˛

|z|=1

dz

2πizr−s−1+σα−θβψθβ,s =

∑β,s

Cβα

2πˆ

0

dφ

2πe2πi(r−s+σα−θβ)φψθβ,s =

=∑β,s

Cβα

−i(e2πi(σα−θβ) − 1)

r − s+ σα − θβψθβ,s

(3.99)

In principle, this formula includes all possible information about Vν(t). Now it is easyto prove

Theorem 3.2. Vν(t) is a primary field of the conformal WN ⊕ H algebra with thehighest weights uk(ν).

Proof: First we notice that due to (3.97) and to the definitions (3.85), (3.65) onehas

Uθk (z)Vν(t) = Vν(t)Uσk (z) (3.100)

55


in the region |z| = t, z 6= t. This means that Uk(z) are actually single-valued operators(with trivial monodromies). Actually, we have already proved in Theorem 3.1 thatstates |σ〉 are highest weight vectors, so

〈θ| . . . Uk(z)|σ〉 =

(uk(σ)

zk+ less singular

)〈θ| . . . |σ〉 (3.101)

and, since (3.88) is symmetric under the permutation of the singular points, one canalso conclude, that for a different point

〈θ| . . . Uk(z)Vν(t) . . . |σ〉 =

(uk(ν)

(z − t)k+ less singular

)〈θ| . . . Vν(t) . . . |σ〉 (3.102)

so (Uk,n>0Vν)(t) = 0, and it means, that Vν(t) is just a primary field.

Generalized Hirota relations

Now consider any operator O with linear adjoint action on fermions

O−1ψα,rO =∑s,β

ROrα,sβψβ,s , O−1ψα,−rO =∑s,β

ψβ,−s(RO)−1

sβ,rα (3.103)

which is generally a relabeling of a GL(∞) transformation for a single fermion. Itleads to a standard statement of commutativity of two operators in H⊗H

O ⊗O∑r,α

ψα,−r ⊗ ψα,r =∑r,α

ψα,−r ⊗ ψα,rO ⊗O (3.104)

which is an operator form of the bilinear Hirota relation [MJD/KvdL, AZ].Let us now point out, that we have already introduced by (3.97) a particular

subclass of general transformations (3.103)

V −1ψα(z)V =∑α

(C−1)ααψα(z) , V −1ψα(z)V =∑α

Cαα ψα(z) (3.105)

where C and C−1 can be now interpreted as monodromy matrices: one can consider(3.105) as a linear relation between two analytic continuations of the fermionic fields

at |z| = 1 towards z →∞ and z → 0, preserving the OPE ψ(z)α(z)ψβ(z′) =δαβz−z′+. . ..

An immediate consequence of (3.105) is

Theorem 3.3. The Fourier modes of the bilinear operators

I(z) =∑α

ψα(z)⊗ ψα(z) =∑k∈Z

Ikzk+1

I†(z) =∑α

ψα(z)⊗ ψα(z) =∑k∈Z

I†kzk+1

(3.106)

commute with Vν(t)⊗ Vν(t) in the sense

Iθk · Vν(t)⊗ Vν(t) = Vν(t)⊗ Vν(t) · IσkI†θk · Vν(t)⊗ Vν(t) = Vν(t)⊗ Vν(t) · Iσk

(3.107)

56


Proof: First we notice that

Iθ(z) · Vν(t)⊗ Vν(t) = Vν(t)⊗ Vν(t) · Iσ(z) (3.108)

holds at |z| = t, z 6= t, due to (3.97)∑α

ψθα(z)⊗ ψθα(z) · Vν(t)⊗ Vν(t) = Vν(t)⊗ Vν(t)∑α,β,γ

(C−1)βαCγαψ

σβ

(z)⊗ ψσγ (z) =

= Vν(t)⊗ Vν(t)∑β

ψσβ

(z)⊗ ψσβ

(z)

(3.109)To continue this equality to z = t one has just to check that Iθ(z) · Vν(t) ⊗ Vν(t) isregular. Due to the symmetry of (3.88) this is the same as to check that Iσ(z)·|σ〉⊗|σ〉is regular. Since,

Iσ(z) · |σ〉 ⊗ |σ〉 =∑α

∑n<0

ψσα,n|σ〉zn+ 1

2+σα⊗∑m<0

ψσα,m|σ〉zm+ 1

2−σα

=

=∑α

ψσα,− 1

2|σ〉 ⊗ ψσ

α,− 12|σ〉+O(z)

(3.110)

this expression is regular, this completes the proof. Let us notice that we have also got the equalities

Iσk≥0 · |σ〉 ⊗ |σ〉 = 0, I†σk≥0 · |σ〉 ⊗ |σ〉 = 0 (3.111)

while, for example

I†−1|θ〉 ⊗ |θ〉 =∑α

ψα,−1/2 ⊗ ψα,−1/2|θ〉 ⊗ |θ〉 =∑α

|1α,θ〉 ⊗ | − 1α,θ〉

I−1|θ〉 ⊗ |θ〉 =∑α

ψα,−1/2 ⊗ ψα,−1/2|θ〉 ⊗ |θ〉 =∑α

| − 1α,θ〉 ⊗ |1α,θ〉(3.112)

but

〈θ| ⊗ 〈θ| · I†−1 = 〈θ| ⊗ 〈θ| · I−1 = 0 (3.113)

We shall see below, that existence of extra bilinear operator relations lead actually tothe infinite number of Hirota-like equations for the τ -function.

Let us also notice that operator tL0 belongs to the quasigroup, but it does notcommute with Ik:

tL0I(z)t−L0 = tI(tz) (3.114)

which means that tL0Ikt−L0 = t−k. So, in principle, vertex operator can contain somefactors tL0

i , but in such a combination with∏ti = 1.

Now we are ready to prove, that the correlation functions (3.86) (and in fact anycorrelation function 〈θ∞|Oψα(z)ψβ(w)|θ0〉 = 〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)ψα(z)ψβ(w)|θ0〉with two fermions) can be decomposed into two correlation functions with a single

57


fermion insertion. In addition to (3.112), (3.113) one has to compute commutator ofthis operator with ψ ⊗ ψ using the contour integral representation[

I−1, ψα(z)⊗ ψβ(w)]

=

˛z

+

˛

w

dx

x

∑γ

ψγ(x)⊗ ψγ(x) · ψα(z)⊗ ψβ(w) =

=

˛

z

dx

x

∑γ

δγαx− z

⊗ ψγ(x)ψβ(w) +

˛

w

dx

x

∑γ

ψγ(x)ψα(z)⊗ δγβx− w

=

=1

z· 1⊗ ψα(z)ψβ(w) +

1

w· ψβ(w)ψα(z)⊗ 1

(3.115)Inserting this operator identity inside the correlation functions, and using (3.112),(3.113) we get

0 = 〈θ∞| ⊗ 〈θ∞| · I−1 · O ⊗ O · ψα(z)⊗ ψβ(w) · |θ0〉 ⊗ |θ0〉 =

= 〈θ∞| ⊗ 〈θ∞| · O ⊗ O · ψα(z)⊗ ψβ(w)∑γ

| − 1γ,θ0〉 ⊗ |1γ,θ0〉+

+

(1

z− 1

w

)〈θ∞|O|θ0〉 · 〈θ∞|Oψα(z)ψβ(w)|θ0〉

(3.116)

The first term in the r.h.s. is equal to the bilinear combination of the correlationfunctions with a single fermion insertion, so one gets finally

〈θ∞|Oψα(z)ψβ(w)|θ0〉〈θ∞|O|θ0〉 =

=zw

z − w∑γ

〈θ∞|Oψα(z)| − 1γ,θ0〉〈θ∞|Oψβ(w)|1γ,θ0〉 (3.117)

which for O = Vν(1) gives the relation between (3.90) and (3.87). Substituting here

the OPE ψα(z)ψβ(w) =δαβz−w + reg. and taking residue at z → w one also proves that

matrices in (3.90) are indeed inverse to each other.

Riemann-Hilbert problem: hypergeometric example

A hypergeometric solution to the Riemann-Hilbert problem with three singular pointsat z = 0, 1,∞ can be given by the following formulas

φ(z) =

(zβF(α, β, ν|z) −z1+βC(α, β, ν)F(α, 1 + β, ν|z)

−z1−βC(α,−β, ν)F(α, 1− β, ν|z) z−βF(α,−β, ν|z)

),

φ−1(z) =

(z−βF(−α,−β,−ν|z) z1+βC(α, β, ν)F(α, 1 + β,−ν|z)

z1−βC(α,−β, ν)F(α, 1− β,−ν|z) zβF(α, β,−ν|z)

)where we have introduced F(α, β, ν|z) = 2F1

[−α+β−ν, α+β−ν2β

∣∣z] for a standard hyper-

geometric function and the constant C(α, β, ν) = (−α−β+ν)(α−β+ν)2β(2β+1)

.

These formulas give solution to the linear system (3.88) with the residues in thefollowing conjugacy classes:

A0 ∼ θ0 = σ = diag(β,−β), A∞ ∼ θ∞ = θ = diag(α,−α)

A1 ∼ θ1 = ν = diag(2ν, 0)(3.118)

58


According to (3.86), (3.87)

〈θ|Vνψα(z)ψβ(w)|σ〉 =[φ(z)φ(w)−1]αβ

z − w(3.119)

It means, for example, that in order to study the matrix elements with ψ1, ψ1 oneneeds to consider the function

K11(z, w) = z−βwβ[φ(z)φ(w)−1]11 = F(α,−β,−ν|z)F(α, β, ν|w)−

−

∏ε,ε′=±1

(εα + ε′β + ν)

4β2(4β2 − 1)zwF(α, 1− β,−ν|z)F(α, 1 + β, ν|w)

(3.120)

Already the simplest fact, that K11(z, z) = 1 becomes a non-trivial bilinear relationfor the hypergeometric function. However, our claim is much stronger: this functionis almost as nice as (3.40) since its expansion (3.91) is given by

K(z, w)11 =K11(z, w)

z − w=

1

z − w−

−∞∑

a,b=0

2β(α− β + ν)a+1(−α + β + ν)a+1(−α + β − ν)b+1(α + β − ν)b+1

(a+ b+ 1)(−α + β − ν)(α + β − ν)a!b!(2β)b+1(−2β)a+1

zbwa

(3.121)and it is indeed a generation function of the matrix elements we are interested in.One can substitute here a = q − 1

2, b = p− 1

2

〈(α,−α)|V(2ν,0)(1)ψ1,−pψ1,−q|(β,−β〉 =

=2β(α + β − ν)q+ 1

2(−α + β − ν)q+ 1

2(α− β + ν)p+ 1

2(−α− β + ν)p+ 1

2

(p+ q)(−α + β − ν)(α + β − ν)(p− 12)!(q − 1

2)!(2β)q+ 1

2(−2β)p+ 1

2

(3.122)

and compare this formula with (3.48)

f1,1(θ,θ′, p)f2,1(θ,θ′, q)

p+ q=

1

(p− 12)!

∏β

(θ′β − θ1)p+ 12√

θ′β − θ1

∏β 6=1

√θβ − θ1

(θβ − θ1)p+ 12

×

× 1

(q − 12)!

∏β

(θ1 − θ′β)q+ 12√

θ1 − θ′β

∏β 6=1

√θ1 − θβ

(θ1 − θβ)q+ 12

× 1

p+ q=

=(θ1 − θ2)(−θ1 + θ′1)p+ 1

2(−θ1 + θ′2)p+ 1

2(θ1 − θ′1)q+ 1

2(θ1 − θ′2)q+ 1

2

(p+ q)(−θ1 + θ′1)(θ1 − θ′2)(p− 12)!(q − 1

2)!(θ2 − θ1)p+ 1

2(θ2 − θ1)q+ 1

2

(3.123)

It is easy to see, that after the appropriate identification

θ1 = β, θ2 = −β, θ′1 = α + ν, θ′2 = −α + ν (3.124)

the r.h.s.’s in two last formulas coincide exactly.In addition to the hypergeometric case another explicit example can be provided

by the exact conformal blocks, considered in [GMtw]. We are planning to consider itin detail elsewhere.

59


Isomonodromic tau-functions and Fredholm deter-

minants

Isomonodromic tau-function

First we need to prove the simple

Lemma 3.4. Monodromies of ψβ(w) and ψα(z) in the matrix elements

〈Y ′,n′,θ|Vν(1)ψσα (z)ψσβ (w)|Y ,n,σ〉 (3.125)

do not depend on n,Y ,n′,Y ′.

Proof: All these matrix elements can be obtained from (3.95) by certain con-tour integration, producing fermionic modes from the fermionic fields. However, in(3.95) due to the Wick theorem factorization, all contributions have the factorizedform Kαγ(z, •) × . . ., where all other factors do not depend at all on z, so that allmonodromies comes from a single kernel K.

Now it is easy to prove

Theorem 3.5. Solution of the linear problem with n marked points is given by(z − w)Kαβ(z, w) with

Kαβ(z, w) =〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)ψθ0

α (z)ψθ0β (w)|θ0〉

〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)|θ0〉(3.126)

whereas its isomonodromic tau-function is defined by

τ(t1, . . . , tn−2) = 〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)|θ0〉 (3.127)

Proof: First, insert resolutions of unity between each two (radially-ordered) vertexoperators, e.g.

τ · Kαβ(z, w) =∑

Y 1,mi

〈θ∞|Vθn−2(tn−2)|Y n−3,mn−3,σn−3〉〈Y n−3,mn−3,σn−3|×

× . . .× 〈Y 2,m2,σ2|Vθ2(t2)|Y 1,m1,σ1〉〈Y 1,m1,σ1|Vθ1(t1)ψθ0α (z)ψθ0

β (w)|θ0〉(3.128)

for 0 < |z|, |w| < |t1| and similarly in the other regions. Due to Lemma 3.4 themonodromies of the fermionic fields do not depend on the intermediate states, but onlyon the vertex operators and the set of charges σ’s 3, therefore it is enough to reduce theproblem of computation of all monodromies to the collection of corresponding three-point problems with different vertex operators Vθj(tj) inserted. So we have proven that

3In addition to (n−3) time parameters (t1, . . . , tn modulo Mobius transformation, which alwaysallow to fix three of them to 0, 1,∞) and n sets of W-charges θj the isomonodromic tau-functiondepends upon the charges σk ∈ (R/Z)N−1, k = 1, . . . , n − 3 in the intermediate channels andtheir duals βk, which we had already discussed in the context of ambiguity in normalization ofthe vertex operators and their matrix elements.

60

3.5. Isomonodromic tau-functions and Fredholm determinants

(z − w)Kαβ(z, w) = [Φ(z)Φ−1(w)]αβ (to cancel extra singularity in (3.126)), actuallygives a solution to the multi-point Riemann-Hilbert problem.

In order to prove (3.127) consider∑α

ψα(z +t

2)ψα(z − t

2) =

N

t+ J(z) + tU2(z) + . . . (3.129)

so that

t TrK(z +t

2, z − t

2) = Tr Φ(z +

t

2)Φ(z − t

2)−1 =

= N + t〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)J(z)|θ0〉〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)|θ0〉

+ t2〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)U2(z)|θ0〉〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)|θ0〉

+ . . .

(3.130)where from (3.72) and the conformal Ward identities

〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)U2(z)|θ0〉〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)|θ0〉

=

=n∑i=1

(12θ2i

z − ti+

∂iz − ti

log〈θ∞|Vθn−2(tn−2) . . . Vθ1(t1)|θ0〉

) (3.131)

where we have extended this formula to include t1 = 0 and tn =∞.Now solving the linear system (3.88) with A(z) =

∑iAiz−ti we get

Φ(z +t

2)Φ(z − t

2)−1 = Φ(z)

(1 +

t

2A(z) +

t2

8(∂A(z) + A(z)2) + . . .

)×

×(

1 +t

2A(z) +

t2

8(−∂A(z) + A(z)2) + . . .

)Φ(z)−1 =

= Φ(z)

(1 + tA(z) +

t2

2A(z)2 + . . .

)Φ(z)−1

(3.132)

Therefore, due to the definition of the tau-function

Tr Φ(z +t

2)Φ(z − t

2)−1 =

t2

2

n∑i=1

(12θ2i

(z − ti)2+

∂iz − ti

log τ(t1, . . . , tn)

)+ . . .

(3.133)Comparing this formula with (3.131) completes the proof.

Fredholm determinant

Consider now the isomonodromic tau-function τ(t) = 〈θ∞|Vν1(1)Vνt(t)|θ0〉, corre-sponding to the problem on sphere with four marked points at z = 0, t, 1,∞. Insertingthe resolution of unity one can write

τ(t) = 〈θ∞|Vν1(1)Vνt(t)|θ0〉 =∑Y,m

〈θ∞|Vν1(1)|Y,m;σ〉〈Y,m;σ|Vνt(t)|θ0〉 =

=∑

pα,i,qα,i

〈θ∞|Vν1(1)|pα,i, qα,i;σ〉〈qα,i, pα,i;σ|Vνt(t)|θ0〉

(3.134)

61


Here we have used first just a particular case of the expansion (3.128), applying itto the simplest nontrivial isomonodromic tau-function. However, now it is useful tonotice, that summation over the basis in total spaceHσ =

⊕m∈ZN

Hσm can be performed

in Frobenius coordinates just forgetting restriction #pα = #qα for the states (3.56)in Hσm, hence there is no restriction in summation range in the r.h.s. of (3.134).

Now, one can still apply formulas (3.91), (3.92) for the matrix elements in (3.134).It gives

〈θ∞|Vν1(1)|pα,i, qα,i;σ〉 = detKxIyJ

〈pα,i, qα,i;σ|Vνt(t)|θ0〉 = det KxIyJ (t)

Kαβpα,qβ

(t) = tpα+qβ−σα+σβKαβpα,qβ

(3.135)

where we have used again the multi-indices ∪α(α, pα,i) = xI and ∪α(α, qα,i) =yJ. It means, that the tau-function (3.134) can be summed up into a single Fred-holm determinant

τ(t) =∑

pα,i,qα,i

〈θ∞|Vν1(1)|pα,i, qα,i;σ〉〈qα,i, pα,i;σ|Vνt(t)|θ0〉 =

=∑x,y

detKx,y · det Ky,x(t) =∞∑n=0

∑|x|=n|y|=n

detKx,y · det Ky,x(t) =

=∞∑n=0

Tr ∧n (KK(t)) = det(1 +KK(t)) = det (1 +Rt)

(3.136)

where basically only the Wick theorem has been used. One can also present the kernelof this operator Rt = KK(t) explicitly by the formula

R(x, z) =

˛

|y|=r

(φ(x)φ(y)−1 − xσy−σ)(S−1φ(y/t)φ(z/t)−1S − yσz−σ)

t−1(x− y)(y − z)dy (3.137)

(where S is the diagonal matrix introduced before), so that this integral operatoracts from the space of vector-valued functions f(z) = (f1(z), . . . , fN(z)) on the circle|z| = r, t < r < 1. These functions have the fractional Laurent expansion

fα(z) = zσα∑n∈Z

fα,nzn

(3.138)

otherwise their convolution with our kernel will be ill-defined.The representation in terms of the Fredholm determinant definitely requires fur-

ther careful investigation, and it could appear to be useful for practical computationswith isomonodromic tau-functions, which basically have no explicit representations.

Conclusion

We have considered in this chapter the free fermion formalism, which allows to studyrepresentations of the W-algebras at least at integer values of the central charges. The

62

3.6. Conclusion

vertex operators are defined by their two-fermion matrix elements, which are fixedby monodromies of auxiliary linear system, and can be obtained from solution of thecorresponding Riemann-Hilbert problem.

This chapter is just the first step of studying this relation (apart of the well-knownand effectively used for different applications Abelian case). A natural developmentof the above ideas is only outlined in sect. 3.5. We are going to return elsewhere tothe problem of rewriting the isomonodromic tau-functions in terms of the Fredholmdeterminants, which can be quite useful representations (though still not an explicitform) for these complicated objects. Another point, which has to be understoodbetter is the relation of class of the isomonodromic solutions to the Toda lattices,which have been defined above using the generalized Hirota bilinear relations, to theclass of solutions, obeying the Virasoro-W constraints.

63

4Fredholm determinant and Nekrasov sum

representations of isomonodromic taufunctions

Abstract

We derive Fredholm determinant representation for isomonodromic tau functions ofFuchsian systems with n regular singular points on the Riemann sphere and genericmonodromy in GL (N,C). The corresponding operator acts in the direct sum ofN (n− 3) copies of L2 (S1). Its kernel has a block integrable form and is expressed interms of fundamental solutions of n − 2 elementary 3-point Fuchsian systems whosemonodromy is determined by monodromy of the relevant n-point system via a de-composition of the punctured sphere into pairs of pants. For N = 2 these buildingblocks have hypergeometric representations, the kernel becomes completely explicitand has Cauchy type. In this case Fredholm determinant expansion yields multivari-ate series representation for the tau function of the Garnier system, obtained earliervia its identification with Fourier transform of Liouville conformal block (or a dualNekrasov-Okounkov partition function). Further specialization to n = 4 gives a seriesrepresentation of the general solution to Painleve VI equation.

Introduction

Motivation and some results

The theory of monodromy preserving deformations plays a prominent role in many ar-eas of modern nonlinear mathematical physics. The classical works [WMTB, JMMS,TW1] relate, for instance, various correlation and distribution functions of statisticalmechanics and random matrix theory models to special solutions of Painleve equa-tions. The relevant Painleve functions are usually written in terms of Fredholm orToeplitz determinants. Further study of these relations has culminated in the de-velopment by Tracy and Widom [TW2] of an algorithmic procedure of derivation ofsystems of PDEs satisfied by Fredholm determinants with integrable kernels [IIKS]restricted to a union of intervals; the isomonodromic origin of Tracy-Widom equa-tions has been elucidated in [Pal94] and further studied in [HI]. This raises a natural

65

4. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions

question:

?© Can the general solution of isomonodromy equations be expressed in terms of aFredholm determinant?

One of the goals of the present chapter is to provide a constructive answer to thisquestion in the Fuchsian setting. Let us consider a Fuchsian system with n regularsingular points a := a0, . . . , an−2, an−1 ≡ ∞ on P1 ≡ P1 (C):

∂zΦ = ΦA (z) , A (z) =n−2∑k=0

Akz − ak

, (4.1)

where A0, . . . , An−2 are N ×N matrices independent of z and Φ (z) is a fundamentalmatrix solution, multivalued on P1\a. The monodromy of Φ (z) realizes a represen-tation of the fundamental group π1 (P1\a) in GL (N,C). When the residue matricesA0, . . . , An−2 and An−1 := −

∑n−2k=0 Ak are non-resonant, the isomonodromy equations

are given by the Schlesinger system,∂aiAk =

[Ai, Ak]

ak − ai, i 6= k,

∂aiAi =∑k 6=i

[Ai, Ak]

ai − ak.

(4.2)

Integrating the flows associated to affine transformations, we may set without loss ofgenerality a0 = 0 and an−2 = 1, so that there remains n− 3 nontrivial time variablesa1, . . . , an−3. In the case N = 2, Schlesinger equations reduce to the Garnier systemGn−3, see for example [IKSY, Chapter 3] for the details. Setting further n = 4, we areleft with only one time t ≡ a1 and the latter system becomes equivalent to a nonlinear2nd order ODE — the Painleve VI equation.

The main object of our interest is the isomonodromic tau function of Jimbo-Miwa-Ueno [JMU]. It is defined as an exponentiated primitive of the 1-form

da ln τJMU :=1

2

n−2∑k=0

resz=ak TrA2 (z) dak. (4.3)

The definition is consistent since the 1-form on the right is closed on solutions of thedeformation equations (4.2). It generates the hamiltonians of the Schlesinger system.Dealing with the Garnier system, we will assume the standard gauge where TrA (z) =0 and denote the eigenvalues of Ak by ±θk with k = 0, . . . , n− 1. In the Painleve VIcase, it is convenient to modify this notation as (θ0, θ1, θ2, θ3) 7→ (θ0, θt, θ1, θ∞). Thelogarithmic derivative ζ (t) := t (t− 1) d

dtln τVI (t) then satisfies the σ-form of Painleve

VI,

(t(t−1)ζ ′′

)2

= −2 det

2θ20 tζ ′ − ζ ζ ′ + θ2

0 + θ2t + θ2

1 − θ2∞

tζ ′ − ζ 2θ2t (t− 1)ζ ′ − ζ

ζ ′ + θ20 + θ2

t + θ21 − θ2

∞ (t− 1)ζ ′ − ζ 2θ21

.

(4.4)

66

4.1. Introduction

Monodromy of the associated linear problems provides a complete set of conservedquantities for Painleve VI, the Garnier system and Schlesinger equations. By thegeneral solution of deformation equations we mean the solution corresponding togeneric monodromy data. The precise genericity conditions will be specified in themain body of the text.

In [Pal90], Palmer (developing earlier results of Malgrange [Mal]) interpreted theJimbo-Miwa-Ueno tau function (4.3) as a determinant of a singular Cauchy-Riemannoperator acting on functions with prescribed monodromy. The main idea of [Pal90]is to isolate the singular points a0, . . . , an−1 inside a circle C ⊂ P1 and represent theFuchsian system (4.1) by a boundary space of functions on C that can be analyticallycontinued inside with specified branching. The variation of positions of singularitiesgives rise to a trajectory of this space in an infinite Grassmannian. The tau functionis obtained by comparing two sections of an associated determinant bundle.

The construction suggested in the present chapter is essentially a refinement ofPalmer’s approach, translated into the Riemann-Hilbert framework. A single circle Cis replaced by the boundaries of n− 3 annuli which cut the n-punctured sphere P1\ainto trinions (pairs of pants), see e.g. Fig. 4.2a below. To each trinion is assigned aFuchsian system with 3 regular singular points whose monodromy is determined bymonodromy of the original system. We show that the isomonodromic tau function isproportional to a Fredholm determinant:

τJMU (a) = Υ (a) · det (1−K) , (4.5)

where the prefactor Υ (a) is a known elementary function. The integral operator Kacts on holomorphic vector functions on the union of annuli and involves projectionson certain boundary spaces.

The pay-off of a more complicated Grassmannian model is that the kernel of Kmay be written explicitly in terms of 3-point solutions1. In particular, for N = 2(i.e. for the Garnier system) the latter have hypergeometric expressions. The n = 4specialization of our result is as follows.

Theorem 4.1. Let the independent variable t of Painleve VI equation vary inside thereal interval ]0, 1[ and let C = z ∈ C : |z| = R, t < R < 1 be a counter-clockwiseoriented circle. Let σ, η be a pair of complex parameters satisfying the conditions

|<σ| ≤ 1

2, σ 6= 0,±1

2,

θ0 ± θt + σ /∈ Z, θ0 ± θt − σ /∈ Z, θ1 ± θ∞ + σ /∈ Z, θ1 ± θ∞ − σ /∈ Z.

General solution of the Painleve VI equation (4.4) admits the following Fredholm

1We would like to note that somewhat similar refined construction emerged in the analysis ofmassive Dirac equation with U (1) branching on the Euclidean plane [Pal93]. Every branch pointwas isolated there in a separate strip, which ultimately allowed to derive an explicit Fredholmdeterminant representation for the tau function of appropriate Dirac operator [SMJ]. In physicalterms, the determinant corresponds to a resummed form factor expansion of a correlation functionof U(1) twist fields in the massive Dirac theory. The paper [Pal93] was an important source ofinspiration for the present work, although it took us more than 10 years to realize that the stripsshould be replaced by pairs of pants in the chiral problem.

67


determinant representation:

τVI (t) = const · tσ2−θ20−θ2

t (1− t)−2θtθ1 det (1− U) , U =

(0 ad 0

), (4.6)

where the operators a, d ∈ End (C2 ⊗ L2 (C)) act on g =

(g+

g−

)with g± ∈ L2 (C) as

(ag) (z) =1

2πi

˛C

a (z, z′) g (z′) dz′ , (dg) (z) =1

2πi

˛C

d (z, z′) g (z′) dz′, (4.7)

and their kernels are explicitly given by

a (z, z′) =

(1− z′)2θ1

(K++ (z) K+− (z)K−+ (z) K−− (z)

)(K−− (z′) −K+− (z′)−K−+ (z′) K++ (z′)

)− 1

z − z′,

d (z, z′) =

1−(1− t

z′

)2θt

(K++ (z) K+− (z)K−+ (z) K−− (z)

)(K−− (z′) −K+− (z′)−K−+ (z′) K++ (z′)

)z − z′

,

(4.8)with

K±± (z) = 2F1

[θ1 + θ∞ ± σ, θ1 − θ∞ ± σ

±2σ; z

],

K±∓ (z) = ± θ2∞ − (θ1 ± σ)2

2σ (1± 2σ)z 2F1

[1 + θ1 + θ∞ ± σ, 1 + θ1 − θ∞ ± σ

2± 2σ; z

],

K±± (z) = 2F1

[θt + θ0 ∓ σ, θt − θ0 ∓ σ

∓2σ;t

z

],

K±∓ (z) = ∓ t∓2σe∓iηθ2

0 − (θt ∓ σ)2

2σ (1∓ 2σ)

t

z2F1

[1 + θt + θ0 ∓ σ, 1 + θt − θ0 ∓ σ

2∓ 2σ;t

z

].

(4.9)

Moreover, we demonstrate that for a special choice of monodromy in the PainleveVI case, U becomes equivalent to the hypergeometric kernel of [BO05] and therebyreproduces previously known family of Fredholm determinant solutions [BD]. Thehypergeometric kernel is known to produce other random matrix integrable kernelsin confluent limits.

Another part of our motivation comes from isomonodromy/CFT/gauge theorycorrespondence. It was conjectured in [GIL12] that the tau function associated tothe general Painleve VI solution coincides with a Fourier transform of 4-point c = 1Virasoro conformal block with respect to its intermediate momentum. Two indepen-dent derivations of this conjecture have been already proposed in [ILTe] and [BShch].The first approach [ILTe] also extends the initial statement to the Garnier system.Its main idea is to consider the operator-valued monodromy of conformal blocks withadditional level 2 degenerate insertions. At c = 1, Fourier transform of such confor-mal blocks reduces their “quantum” monodromy to ordinary 2 × 2 matrices. It cantherefore be used to construct the fundamental matrix solution of a Fuchsian system

68

4.1. Introduction

with prescribed SL (2,C) monodromy. The second approach [BShch] uses an embed-ding of two copies of the Virasoro algebra into super-Virasoro algebra extended byMajorana fermions to prove certain bilinear differential-difference relations for 4-pointconformal blocks, equivalent to Painleve VI equation. An interesting feature of thismethod is that bilinear relations admit a deformation to generic values of Virasorocentral charge.

Among other developments, let us mention the papers [GIL13, ILT14, Nag] whereasymptotic expansions of Painleve V, IV and III tau functions were identified withFourier transforms of irregular conformal blocks of different types. The study ofrelations between isomonodromy problems in higher rank and conformal blocks ofWN algebras has been initiated in [Gav, GMtw, GMfer].

The AGT conjecture [AGT] (proved in [AFLT]) identifies Virasoro conformalblocks with partition functions of N = 2 4D supersymmetric gauge theories. Thereexist combinatorial representations of the latter objects [Nek], expressing them assums over tuples of Young diagrams. This fact is of crucial importance for isomon-odromy theory, since it gives (contradicting to an established folklore) explicit seriesrepresentations for the Painleve VI and Garnier tau functions. Since the very firstpaper [GIL12] on the subject, there has been a puzzle to understand combinatorialtau function expansions directly within the isomonodromic framework. There havealso been attempts to sum up these series to determinant expressions; for example, in[Bal] truncated infinite series for c = 1 conformal blocks were shown to coincide withpartition functions of certain discrete matrix models.

In this work, we show that combinatorial series correspond to the principal minorexpansion of the Fredholm determinant (4.5), written in the Fourier basis of thespace of functions on annuli of the pants decomposition. Fourier modes which labelthe choice of rows for the principal minor are related to Frobenius coordinates ofYoung diagrams. It should be emphasized that this combinatorial structure is validalso for N > 2 where CFT/gauge theory counterparts of the tau functions have yetto be defined and understood.

We prove in particular the following result, originally conjectured in [GIL12] (thedetails of notation concerning Young diagrams are explained in the next subsection):

Theorem 4.2. General solution of the Painleve VI equation (4.4) can be written as

τVI(t) = const ·∑n∈Z

einη′B(~θ;σ + n; t

), (4.10)

where B(~θ, σ; t) is a double sum over Young diagrams,

B(~θ, σ; t

)= N θ1

θ∞,σN θtσ,θ0

tσ2−θ2

0−θ2t (1− t)2θtθ1

∑λ,µ∈Y

Bλ,µ(~θ, σ)t|λ|+|µ|,

Bλ,µ(~θ, σ)

=∏

(i,j)∈λ

((θt + σ + i− j)2 − θ2

0

) ((θ1 + σ + i− j)2 − θ2

∞)

h2λ(i, j)

(λ′j − i+ µi − j + 1 + 2σ

)2 ×

×∏

(i,j)∈µ

((θt − σ + i− j)2 − θ2

0

) ((θ1 − σ + i− j)2 − θ2

∞)

h2µ(i, j)

(µ′j − i+ λi − j + 1− 2σ

)2 ,

69


N θ2θ3,θ1

=

∏ε=±G (1 + θ3 + ε(θ1 + θ2))G (1− θ3 + ε(θ1 − θ2))

G(1− 2θ1)G(1− 2θ2)G(1 + 2θ3).

Here σ /∈ Z/2, η′ are two arbitrary complex parameters, and G (z) denotes the BarnesG-function.

The parameters σ play exactly the same role in the Fredholm determinant (4.6)and the series representation (4.10), whereas η and η′ are related by a simple trans-formation. An obvious quasiperiodicity of the second representation with respect tointeger shifts of σ is by no means manifest in the Fredholm determinant.

Notation

The monodromy matrices of Fuchsian systems and the jumps of associated Riemann-Hilbert problems appear on the left of solutions. These somewhat unusual conventionsare adopted to avoid even more confusing right action of integral and infinite matrixoperators. The indices corresponding to the matrix structure of rank N Riemann-Hilbert problem are referred to as color indices and are denoted by Greek letters, suchas α, β ∈ 1, . . . , N. Upper indices in square brackets, e.g. [k] in T [k], label differenttrinions in the pants decomposition of a punctured Riemann sphere. We denote byZ′ := Z + 1

2the half-integer lattice, and by Z′± = p ∈ Z′ | p ≷ 0 its positive and

negative parts. The elements of Z′,Z′± will be generally denoted by the letters p andq.

i

j

l3=3'

l2=5 | =l 17|

al( )=2

ll( )=1

hl( )=4

=(2,3)

Figure 4.1: Young diagram associated to the partition λ =6, 5, 4, 2.

The set of all partitions identified with Young diagrams is denoted by Y. Forλ ∈ Y, we write λ′ for the transposed diagram, λi and λ′j for the number of boxesin the ith row and jth column of λ, and |λ| for the total number of boxes in λ. Let = (i, j) be the box in the ith row and jth column of λ ∈ Y (see Fig. 4.1). Itsarm-length aλ () and leg-length lλ () denote the number of boxes on the right andbelow. This definition is extended to the case where the box lies outside λ by theformulae aλ () = λi − j and lλ () = λ′j − i. The hook length of the box ∈ λ isdefined as hλ () = aλ () + lλ () + 1.

Outline of the chapter

The chapter is organized as follows. Section 4.2 is devoted to the derivation of Fred-holm determinant representation of the Jimbo-Miwa-Ueno isomonodromic tau func-tion. It starts from a recast of the original rank N Fuchsian system with n regular

70

4.1. Introduction

singular points on P1 in terms of a Riemann-Hilbert problem. In Subsection 4.2.2we associate to it, via a decomposition of n-punctured Riemann sphere into pairsof pants, n − 2 auxiliary Riemann-Hilbert problems of Fuchsian type having only 3regular singular points. Section 4.2.3 introduces Plemelj operators acting on func-tions holomorphic on the annuli of the pants decomposition, and deals with theirbasic properties. The main result of the section is formulated in Theorem 4.11 ofSubsection 4.2.4, which relates the tau function of a Fuchsian system with prescribedgeneric monodromy to a Fredholm determinant whose blocks are expressed in termsof 3-point Plemelj operators. In Subsection 4.2.5, we consider in more detail the ex-ample of n = 4 points and show that the Fredholm determinant representation canbe efficiently used for asymptotic analysis of the tau function. In particular, Theo-rem 4.13 provides a generalization of the Jimbo asymptotic formula for Painleve VIvalid in any rank and up to any asymptotic order.

In Section 4.3 we explain how the principal minor expansion of the Fredholm de-terminant leads to a combinatorial structure of the series representations for isomon-odromic tau functions. Theorem 4.15 of Subsection 4.3.1 shows that 3-point Plemeljoperators written in the Fourier basis are given by sums of a finite number of infi-nite Cauchy type matrices twisted by diagonal factors. Combinatorial labeling of theminors by N -tuples of charged Maya diagrams and partitions is described in Subsec-tion 4.3.2.

Section 4.4 deals with rank N = 2. Hypergeometric representations of the appro-priate 3-point Plemelj operators are listed in Lemma 4.28 of Subsection 4.4.1. Theo-rem 4.30 provides an explicit combinatorial series representation for the tau functionof the Garnier system. In the final subsection, we explain how Fredholm determinantof the Borodin-Olshanski hypergeometric kernel arises as a special case of our con-struction. Appendix contains a proof of a combinatorial identity expressing Nekrasovfunctions in terms of Maya diagrams instead of partitions.

Perspectives

In an effort to keep the chapter of reasonable length, we decided to defer the studyof several straightforward generalizations of our approach to separate publications.These extensions are outlined below together with a few more directions for futureresearch:

1. In higher rank N > 2, it is an open problem to find integral/series representa-tions for general solutions of 3-point Fuchsian systems and to obtain an explicitdescription of the Riemann-Hilbert map. There is however an important excep-tion of rigid systems having two generic regular singularities and one singularityof spectral multiplicity (N − 1, 1); these can be solved in terms of generalized hy-pergeometric functions of type NFN−1. The spectral condition is exactly whatis needed to achieve factorization in Lemma 4.26. The results of Section 4.4can therefore be extended to Fuchsian systems with two generic singular pointsat 0 and∞, and n−2 special ones. The corresponding isomonodromy equations(dubbed GN,n−3 system in [Tsu]) are the closest higher rank relatives of PainleveVI and Garnier system. It is natural to expect their tau functions to be relatedon the 2D CFT and gauge theory side, respectively, to WN conformal blocks

71


with semi-degenerate fields [FLitv12, Bul] and Nekrasov partition functions of

4D linear quiver gauge theories with the gauge group U (N)⊗(n−3).

In the generic non-rigid case the 3-point solutions depend on (N − 1) (N − 2)accessory parameters and may be interpreted as matrix elements of a generalvertex operator for the WN algebra. They should also be related to the so-calledTN gauge theory without lagrangian description [BMPTY].

2. Fredholm determinants and series expansions considered in the present work areassociated to linear pants decompositions of P1 \n points , which means thatevery pair of pants has at least one external boundary component (see Fig. 4.2a).Plemelj operators assigned to each trinion act on spaces of functions on internalboundary circles only. To be able to deal with arbitrary decompositions, inaddition to 4 operators a[k], b[k], c[k], d[k] appearing in (4.20) one has to introduce5 more similar operators associated to other possible choices of ordered pairs ofboundary components.

a) b) c)

Figure 4.2: (a) Linear and (b) Sicilian pants decompositionof P1 \6 points ; (c) gluing 1-punctured torus from a pairof pants.

A (tri)fundamental example where this construction becomes important is knownin the gauge theory literature under the name of Sicilian quiver (Fig. 4.2b). Al-ready for N = 2 the monodromies along the triple of internal cicles of this pantsdecomposition cannot be simultaneously reduced to the form “1+rank 1 ma-trix” by factoring out a suitable scalar piece. The analog of expansion (4.87) inTheorem 4.30 will therefore be more intricate yet explicitly computable. Sincethe identification [ILTe] of the tau function of the Garnier system with a Fouriertransform of c = 1 Virasoro conformal block does not put any constraint on theemployed pants decomposition, Sicilian expansion of the Garnier tau functionmay be used to produce an analog of Nekrasov representation for the correspond-ing conformal blocks. It might be interesting to compare the results obtainedin this way against instanton counting [HKS].

Extension of the procedure to higher genus requires introducing additional sim-ple (diagonal in the Fourier basis) operators acting on some of the internalannuli. They give rise to a part of moduli of complex structure of the Riemannsurface and correspond to gluing a handle out of two boundary components.Fig. 4.2c shows how a 1-punctured torus may be obtained by gluing two bound-ary circles of a pair of pants. The gluing operator encodes the elliptic modulus,which plays a role of the time variable in the corresponding isomonodromic

72

4.1. Introduction

problem. Elliptic isomonodromic deformations have been studied e.g. in [K00],where the interested reader can find further references.

3. It is natural to wonder to what extent the approach proposed in the presentwork may be followed in the presence of irregular singularities, in particular,for Painleve I–V equations. The contours of appropriate isomonodromic RHPsbecome more complicated: in addition to circles of formal monodromy, theyinclude anti-Stokes rays, exponential jumps on which account for Stokes phe-nomenon [FIKN]. We will sketch here a partial answer in rank N = 2. Forthis it is useful to recall a geometric representation of the confluence diagramfor Painleve equations recently proposed by Chekhov, Mazzocco and Rubtsov[CM, CMR], see Fig. 4.3. To each of the equations (or rather associated linearproblems) is assigned a Riemann surface with a number of cusped boundarycomponents. They are obtained from Painleve VI 4-holed sphere using twosurgery operations: i) a “chewing-gum” move creating from two holes with kand l cusps one hole with k + l + 2 cusps and ii) a cusp removal reducing thenumber of cusps at one hole by 1. The cusps may be thought of as representingthe anti-Stokes rays of the Riemann-Hilbert contour.

VI

V

Vdeg

III(D )

III(D )

III(D )

IV

IIFN

IIJM

I

6

7

8

Figure 4.3: CMR confluence diagram for Painleve equa-tions.

An extension of our approach is straightforward for equations from the upperpart of the CMR diagram and, more generally, when the Poincare ranks ofall irregular singular points are either 1

2or 1. The associated surfaces may

be decomposed into irregular pants of three types corresponding to solvableRHPs: Gauss hypergeometric, Whittaker and Bessel systems (Fig. 4.4). Theyserve to construct local Riemann-Hilbert parametrices which in turn producethe relevant Plemelj operators.

The study of higher Poincare rank seems to require new ideas. Moreover, evenfor Painleve V and Painleve III Fredholm determinant expansions naturally giveseries representations of the corresponding tau functions of regular type, first

73


Whittaker BesselGauss

Figure 4.4: Some solvable RHPs in rank N = 2: Gauss hypergeo-metric (3 regular punctures), Whittaker (1 regular + 1 of Poincarerank 1) and Bessel (1 regular + 1 of rank 1

2).

proposed in [GIL13] and expressed in terms of irregular conformal blocks of[G, BMT, GT]. It is not clear to us how to extract from them irregular (long-distance) asymptotic expansions. Let us mention a recent work [Nag] whichrelates such expansions to irregular conformal blocks of a different type.

4. Given a matrix K ∈ CX×X indexed by elements of a discrete set X, it is almosta tautology to say that the principal minors detKY∈2X define a determinantalpoint process on X and a probability measure on 2X. Fredholm determinant rep-resentations and combinatorial expansions of tau functions thus generalize in anatural way various families of measures of random matrix or representation-theoretic origin, such as Z- and ZW -measures [BO05, BO01] (the former cor-respond to the scalar case N = 1 with n = 4 regular singular points, and thelatter are related to hypergeometric kernel considered in the last subsection).We believe that novel probabilistic models coming from isomonodromy deservefurther investigation.

5. Perhaps the most intriguing perspective is to extend our setup to q-isomonodromyproblems, in particular q-difference Painleve equations, presumably related tothe deformed Virasoro algebra [SKAO] and 5D gauge theories. Among the re-sults pointing in this direction, let us mention a study of the connection problemfor q-Painleve VI [Ma] based on asymptotic factorization of the associated linearproblem into two systems solved by the Heine basic hypergeometric series 2ϕ1,and critical expansions for solultions of q-P (A1) equation recently obtained in[JR].

Tau functions as Fredholm determinants

Riemann-Hilbert setup

The classical setting of the Riemann-Hilbert problem (RHP) involves two basic ingre-dients:

74

4.2. Tau functions as Fredholm determinants

• A contour Γ on a Riemann surface Σ of genus g con-sisting of a finite set of smooth oriented arcs that canintersect transversally. Orientation of the arcs definespositive and negative side Γ± of the contour in theusual way, see Fig. 4.5.

• A jump matrix J : Γ → GL (N,C) that satisfies suit-able smoothness requirements.

+ -

Figure 4.5: Ori-entation andlabeling of sidesof Γ

The RHP defined by the pair (Γ, J) consists in finding an analytic invertible matrixfunction Ψ : Σ\Γ→ GL (N,C) whose boundary values Ψ± on Γ± are related by Ψ+ =JΨ−. Uniqueness of the solution is ensured by adding an appropriate normalizationcondition.

In the present work we are mainly interested in the genus 0 case: Σ = P1. Let usfix a collection

a := (a0 = 0, a1, . . . , an−3, an−2 = 1, an−1 =∞)

of n distinct points on P1 satisfying the condition of radial ordering 0 < |a1| < . . . <|an−3| < 1. To reduce the amount of fuss below, it is convenient to assume thata1, . . . , an−3 ∈ R>0. The contour Γ will then be chosen as a collection

Γ =(⋃n−1

k=0γk

)∪(⋃n−2

k=0`k

)of counter-clockwise oriented circles γk of sufficiently small radii centered at ak, andthe segments `k ⊂ R joining the circles γk and γk+1, see Fig. 4.6.

a0

a1 a

2a

3

l0

l1

l2

l3

g0

g1

g2 g

3

g4

Figure 4.6: Contour Γ for n = 5

The jumps will be defined by the following data:

75


• An n-tuple of diagonal N×N matrices Θk = diag θk,1, . . . , θk,N ∈ CN×N (withk = 0, . . . , n−1) satisfying Fuchs consistency relation

∑n−1k=0 Tr Θk = 0 and hav-

ing non-resonant spectra. The latter condition means that θk,α − θk,β /∈ Z\ 0.

• A collection of 2n matrices Ck,± ∈ GL (N,C) subject to the constraints

M0→k := Ck,−e2πiΘkC−1

k,+ = Ck+1,−C−1k+1,+, k = 0, . . . , n− 3,

M0→n−2 := Cn−2,−e2πiΘn−2C−1

n−2,+ = Cn−1,−e−2πiΘn−1C−1

n−1,+,

M0→n−1 := 1 = Cn−1,−C−1n−1,+ = C0,−C

−10,+,

(4.11)

which are simultaneously viewed as the definition of M0→k ∈ GL (N,C). Onlyn of the initial matrices (for example, Ck,+) are therefore independent.

The jump matrix J that we are going to consider is then given by

J (z)∣∣∣`k

= M −10→k, k = 0, . . . , n− 2,

J (z)∣∣∣γk

= (ak − z)−Θk C−1k,±, =z ≷ 0, k = 0, . . . , n− 2,

J (z)∣∣∣γn−1

= (−z)Θn−1 C−1n−1,±, =z ≷ 0.

(4.12)

Throughout this chapter, complex powers will always be understood as zθ = eθ ln z,the logarithm being defined on the principal branch. The subscripts ± of Ck,± aresometimes omitted to lighten the notation.

A major incentive to study the above RHP comes from its direct connection tosystems of linear ODEs with rational coefficients. Indeed, define a new matrix Φ by

Φ (z) =

Ψ (z) , z outside γ0...n−1,

Ck (ak − z)Θk Ψ (z) , z inside γk, k = 0, . . . , n− 2,

Cn−1 (−z)−Θn−1 Ψ (z) , z inside γn−1.

(4.13)

It has only piecewise constant jumps JΦ (z)∣∣]ak,ak+1[

= M−10→k on the positive real axis.

The matrix A (z) := Φ−1∂zΦ is therefore meromorphic on P1 with poles only possibleat a0, . . . , an−1. It follows immediately that

∂zΦ = ΦA (z) , A (z) =n−2∑k=0

Akz − ak

, (4.14)

with Ak = Ψ (ak)−1 ΘkΨ (ak). Thus Φ (z) is a fundamental matrix solution for a class

of Fuchsian systems related by constant gauge transformations. It has prescribedmonodromy and singular behavior that are encoded in the connection matrices Ckand local monodromy exponents Θk. The freedom in the choice of the gauge reflectsthe dependence on the normalization of Ψ.

The monodromy representation ρ : π1 (P1\a) → GL (N,C) associated to Φ isuniquely determined by the jumps. It is generated by the matrices Mk = ρ (ξk)

76


a0

a1

a2

an-2

x0

x1

x2

xn-2

xn-1

Figure 4.7: Generators of π1 (P1\a)

assigned to counter-clockwise loops ξ0, . . . , ξn−1 represented in Fig. 4.7. They may beexpressed as

M0 = M0→0, Mk+1 = M0→k−1M0→k+1,

which means simply that M0→k = M0 . . .Mk−1Mk. It is a direct consequence of thedefinition (4.11) that the spectra of Mk coincide with those of e2πiΘk .

Assumption 4.3. The matrices M0→k with k = 1, . . . , n − 3 are assumed to bediagonalizable:

M0→k = Ske2πiSkS−1

k , Sk = diag σk,1, . . . , σk,N .

It can then be assumed without loss in generality that TrSk =∑k

j=0 Tr Θj and|< (σk,α − σk,β)| ≤ 1. We further impose a non-resonancy condition σk,α−σk,β 6= ±1.

In order to have uniform notation, we may also identify S0 ≡ Θ0, Sn−2 ≡ −Θn−1.Note that any sufficiently generic monodromy representation can be realized as de-scribed above.

Auxiliary 3-point RHPs

Consider a decomposition of the original n-punctured sphere into n− 2 pairs of pantsT [1], . . . , T [n−2] by n− 3 annuli A1, . . . ,An−3 represented in Fig. 4.8. The labeling isdesigned so that two boundary components of the annulus Ak that belong to trinionsT [k] and T [k+1] are denoted by C[k]

out and C[k+1]in . We are now going to associate to

the n-point RHP described above n − 2 simpler 3-point RHPs assigned to differenttrinions and defined by the pairs

(Γ[k], J [k]

)with k = 1, . . . , n− 2.

The curves C[k]in and C[k]

out are represented by circles of positive and negative ori-entation as shown in Fig. 4.9. For k = 2, . . . , n − 3, the contour Γ[k] of the RHPassigned to trinion T [k] consists of three circles C[k]

in , C[k]out, γk associated to boundary

components, and two segments of the real axis. For leftmost and rightmost trinionsT [1] and T [n−2], the role of C[1]

in and C[n−2]out is played respectively by the circles γ0 and

γn−1 around 0 and ∞.The jump matrix J [k] is constructed according to two basic rules:

77


A1T[1] T

[k]

Ak-1 Ak Ak+1 An-3T

[n-2]T[k+1]

g0

g1

gk g

k+1 gn-2

gn-1

C[k]

in C[k]

out C[k+1]

in C[k+1]

out C[n-2]

inC[1]

out

Figure 4.8: Labeling of trinions, annuli and boundary curves

a1

a2

a3

g0

g1

g2 g

3

g4 C

[2]

in C[1]

outC[2]

outC[3]

inak

gk

C[k]

inC

[k]

out

Figure 4.9: Contour Γ[k] (left) and Γ for n = 5 (right)

• The arcs that belong to original contour give rise to the same jumps:(J [k] − J

) ∣∣Γ[k]∩Γ

= 0.

• The jumps on the boundary circles C[k]out, C

[k+1]in mimic regular singularities char-

acterized by counter-clockwise monodromy matrices M0→k:

J [k]∣∣∣C[k]

out

= (−z)−Sk S−1k , J [k+1]

∣∣∣C[k+1]

in

= (−z)−Sk S−1k , k = 1, . . . , n− 3.

(4.15)

The solution Ψ[k] of the RHP defined by the pair(Γ[k], J [k]

)is thus related in a way

analogous to (4.13) to the fundamental matrix solution Φ[k] of a Fuchsian system with3 regular singular points at 0, ak and ∞ characterized by monodromies M0→k−1, Mk,M−1

0→k:

∂zΦ[k] = Φ[k]A[k] (z) , A[k] (z) =

A[k]0

z+

A[k]1

z − ak. (4.16)

We note in passing that the spectra of A[k]0 , A

[k]1 and A

[k]∞ := −A[k]

0 −A[k]1 coincide with

the spectra of Sk−1, Θk and −Sk. The non-resonancy constraint in Assumption 4.3

78


ensures the existence of solution Φ[k] with local behavior leading to the jumps (4.15)in Ψ[k].

It will be convenient to replace the n-point RHP described in the previous sub-

section by a slightly modified one. It is defined by a pair(

Γ, J)

such that (cf right

part of Fig. 4.9)

Γ =n−2⋃k=1

Γ[k], J∣∣∣Γ[k]

= J [k]. (4.17)

Constructing the solution Ψ of this RHP is equivalent to finding Ψ: it is plain that

Ψ (z) =

(−z)−Sk S−1k Ψ (z) , z ∈ Ak,

Ψ (z) , z ∈ P1\n−3⋃k=1

Ak.(4.18)

Our aim in the next subsections is to construct the isomonodromic tau function interms of 3-point solutions Φ[k]. This construction employs in a crucial way integralPlemelj operators acting on spaces of holomorphic functions on A :=

⋃n−3k=1 Ak.

Plemelj operators

Given a positively oriented circle C ⊂ C centered at the origin, let us denote by V (C)the space of functions holomorphic in an annulus containing C. Any f ∈ V (C) iscanonically decomposed as f = f+ + f−, where f+ and f− denote the analytic andprincipal part of f . Let us accordingly write V (C) = V+ (C) ⊕ V− (C) and denote byΠ± (C) the projectors on the corresponding subspaces. Their explicit form is

Π± (C) f (z) =1

2πi

˛C±,|z′|=|z|±0

f (z′) dz′

z′ − z,

where the subscript of C± indicates the orientation of C. Projectors Π± (C) are simpleinstances of Plemelj operators to be extensively used below.

Let us next associate to every trinion T [k] with k = 2, . . . , n − 3 the spaces ofvector-valued functions

H[k] =⊕

ε=in,out

(H[k]ε,+ ⊕H

[k]ε,−

), H[k]

ε,± = CN ⊗ V±(C[k]ε

).

With respect to the first decomposition, it is convenient to write the elements f [k] ∈H[k] as

f [k] =

(f

[k]in,−

f[k]out,+

)⊕

(f

[k]in,+

f[k]out,−

).

Here f[k]ε,± denote N -column vectors which represent the restrictions of analytic and

principal part of f [k] to boundary circle C[k]ε . Now define an operator P [k] : H[k] → H[k]

by

P [k]f [k] (z) =1

2πi

˛C[k]

in ∪C[k]out

Ψ[k]+ (z) Ψ

[k]+ (z′)

−1f [k] (z′) dz′

z − z′. (4.19)

79


The singular factors 1/ (z − z′) for z, z′ ∈ C[k]in,out are interpreted with the following

prescription: the contour of integration is deformed to appropriate annulus (e.g. Ak−1

for C[k]in and Ak for C[k]

out) as to avoid the pole at z′ = z. Matrix function Ψ[k] (z) is asolution of the 3-point RHP described in the previous subsection. Its normalizationis irrelevant as the corresponding factor cancels out in (4.19).

Lemma 4.4. We have(P [k]

)2= P [k] and kerP [k] = H[k]

in,+ ⊕H[k]out,−. Moreover, P [k]

can be explicitly written as

P [k] :

(f

[k]in,−

f[k]out,+

)⊕

(f

[k]in,+

f[k]out,−

)7→

(f

[k]in,−

f[k]out,+

)⊕

(a[k] b[k]

c[k] d[k]

)(f

[k]in,−

f[k]out,+

),

where the operators a[k], b[k], c[k], d[k] are defined by

(a[k]g

)(z) =

1

2πi

˛C[k]

in

[Ψ

[k]+ (z) Ψ

[k]+ (z′)

−1− 1

]g (z′) dz′

z − z′, z ∈ C[k]

in , (4.20a)

(b[k]g

)(z) =

1

2πi

˛C[k]

out

Ψ[k]+ (z) Ψ

[k]+ (z′)

−1g (z′) dz′

z − z′, z ∈ C[k]

in , (4.20b)

(c[k]g

)(z) =

1

2πi

˛C[k]

in

Ψ[k]+ (z) Ψ

[k]+ (z′)

−1g (z′) dz′

z − z′, z ∈ C[k]

out, (4.20c)

(d[k]g

)(z) =

1

2πi

˛C[k]

out

[Ψ

[k]+ (z) Ψ

[k]+ (z′)

−1− 1

]g (z′) dz′

z − z′, z ∈ C[k]

out. (4.20d)

Proof. Let us first prove that H[k]in,+,H

[k]out,− ⊂ kerP [k]. This statement follows from

the fact that Ψ[k]+ holomorphically extends inside C[k]

in and outside C[k]out, so that the

integration contours can be shrunk to 0 and ∞. To prove the projection property,decompose for example(P [k]f

[k]out,+

)out

(z) =

=1

2πi

˛C[k]

out

[Ψ

[k]+ (z) Ψ

[k]+ (z′)

−1− 1

]f

[k]out,+ (z′) dz′

z − z′+

1

2πi

˛C[k]

out,|z′|>|z|

f[k]out,+ (z′) dz′

z − z′.

The first integral admits holomorphic continuation in z outside C[k]out thanks to non-

singular integral kernel, and leads to (4.20d), whereas the second term is obviously

equal to f[k]out,+. The action of P [k] on f

[k]in,− is computed in a similar fashion.

The leftmost and rightmost trinions T [1] and T [n−2] play somewhat distinguishedrole. Let us assign to them boundary spaces

H[1] := H[1]out,+ ⊕H

[1]out,−, H[n−2] := H[n−2]

in,+ ⊕H[n−2]in,− ,

80


and the operators P [k] : H[k] → H[k] with k = 1, n− 2 defined by

P [1]f [1] (z) =1

2πi

˛C[1]

out

Ψ[1]+ (z) Ψ

[1]+ (z′)

−1f [1] (z′) dz′

z − z′,

P [n−2]f [n−2] (z) =1

2πi

˛C[n−2]

in

Ψ[n−2]+ (z) Ψ

[n−2]+ (z′)

−1f [n−2] (z′) dz′

z − z′.

Analogously to the above, one can show that

P [1] : f[1]out,+ ⊕ f

[1]out,− 7→ f

[1]out,+ ⊕ d[1]f

[1]out,+,

P [n−2] : f[n−2]in,− ⊕ f

[n−2]in,+ 7→ f

[n−2]in,− ⊕ a[n−2]f

[n−2]in,− ,

where the operators d[1], a[n−2] are given by the same formulae (4.20a), (4.20d). Note

in particular that P [1] and P [n−2] are projections along their kernels H[1]out,− and H[n−2]

in,+ .Let us next introduce the total space

H :=n−2⊕k=1

H[k].

It admits a splitting that will play an important role below. Namely,

H = H+ ⊕H−,

H± := H[1]out,± ⊕

(H[2]

in,∓ ⊕H[2]out,±

)⊕ . . .⊕

(H[n−3]

in,∓ ⊕H[n−3]out,±

)⊕H[n−2]

in,∓ .(4.21)

Combine the 3-point projections P [k] into an operator P⊕ : H → H given by thedirect sum

P⊕ = P [1] ⊕ . . .⊕ P [n−2].

Clearly, we have

Lemma 4.5. P2⊕ = P⊕ and kerP⊕ = H−.

Another important operator PΣ : H → H is defined using the solution Ψ (z) (de-fined by (4.17)) of the n-point RHP in a way similar to construction of the projection(4.19):

PΣf (z) =1

2πi

˛CΣ

Ψ+ (z) Ψ+ (z′)−1f (z′) dz′

z − z′, CΣ :=

n−3⋃k=1

C[k]out ∪ C

[k+1]in . (4.22)

We use the same prescription for the contours: whenever it is necessary to interpretthe singular factor 1/ (z − z′), the contour of integration goes clockwise around thepole.

Let HA be the space of boundary values on CΣ of functions holomorphic on A =⋃n−3k=1 Ak.

Lemma 4.6. P2Σ = PΣ and HA ⊆ kerPΣ.

81


Proof. Given f ∈ HA, the integration contours C[k]out and C[k+1]

in in (4.22) can bemerged thanks to the absence of singularities inside Ak, which proves the secondstatement. To show the projection property, it suffices to notice that

P2Σf

[k] (z) =1

(2πi)2

‹CΣ

Ψ+ (z) Ψ+ (z′′)−1f [k] (z′′) dz′dz′′

(z − z′) (z′ − z′′).

Because of the ordering of contours prescribed above, the only obstacle to mergingC[k]

out and C[k]in in the integral with respect to z′ is the pole at z′ = z. The result follows

by residue computation.

Lemma 4.7. PΣP⊕ = P⊕ and P⊕PΣ=PΣ.

Proof. Similar to the proof of Lemma 4.6. Use that Ψ−1Ψ[k] has no jumps on Γ[k]

to compute by residues the intermediate integrals in PΣP⊕ and P⊕PΣ.

The above suggests to introduce the notation

HT := imP⊕ = imPΣ. (4.23)

The space HT ⊂ H can be thought of as the subspace of functions on the union ofboundary circles C[k]

in , C[k]out that can be continued inside

⋃n−2k=1 T [k] with monodromy

and singular behavior of the n-point fundamental matrix solution Φ (z). The onlyexception is the regular singularity at ∞ where the growth is slower.

The structure of elements of HT is described by Lemma 4.4. Varying the positionsof singular points, one obtains a trajectory of HT in the infinite-dimensional Grass-mannian Gr (H) defined with respect to the splitting H = H+ ⊕H−. Note that eachof the subspaces H± may be identified with N (n− 3) copies of the space L2 (S1) offunctions on a circle; the factor n− 3 corresponds to the number of annuli and N isthe rank of the appropriate RHP.

We can also write

H = HT ⊕H−. (4.24)

The operator P⊕ introduced above gives the projection on HT along H−. Similarly,the operator PΣ is a projection on HT along kerPΣ ⊇ HA. We would like to expressit in terms of 3-point projectors. To this end let us regard f

[k]in,−, f

[k]out,+ as coordinates

on HT . Suppose that f ∈ H can be decomposed as f = g + h with g ∈ HT andh ∈ HA. The latter condition means that

h[k]out,± = h

[k+1]in,± , k = 1, . . . , n− 3,

which can be equivalently written as a system of equations for components of g:

g[k]in,− − c[k−1]g

[k−1]in,− − d[k−1]g

[k−1]out,+ = f

[k]in,− − f

[k−1]out,−,

g[k]out,+ − a[k+1]g

[k+1]in,− − b[k+1]g

[k+1]out,+ = f

[k]out,+ − f

[k+1]in,+ ,

(4.25)

where g[1]in,− = 0, g

[n−2]out,+ = 0. The first and second equations are valid in sufficiently

82


narrow annuli containing C[k]in and C[k]

out, respectively. Define

gk =

(g

[k]out,+

g[k+1]in,−

), fk =

(f

[k]out,+ − f

[k+1]in,+

f[k+1]in,− − f

[k]out,−

),

Uk =

(0 a[k+1]

d[k] 0

), k = 1, . . . , n− 3,

Vk =

(b[k+1] 0

0 0

), Wk =

(0 00 c[k+1]

), k = 1, . . . , n− 4,

K =

U1 V1 0 . 0W1 U2 V2 . 00 W2 U3 . .. . . . Vn−4

0 0 . Wn−4 Un−3

, ~g =

g1

g2...

gn−3

, ~f =

f1

f2...

fn−3

.

(4.26)The system (4.25) can then be rewritten in a block-tridiagonal form

(1−K)~g = ~f. (4.27)

The decomposition H = HT ⊕HA thus uniquely exists provided that 1−K is invert-ible.

Let us prove a converse result and interpret K in a more invariant way. Considerthe operators P⊕,+ : H+ → HT and PΣ,+ : H+ → HT defined as restrictions of P⊕and PΣ to H+. The first of them is invertible, with the inverse given by the projectionon H+ along H−. Hence one can consider the composition L ∈ End (H+) defined by

L := P⊕,+−1PΣ,+. (4.28)

We are now going to make an important assumption which is expected to hold gener-ically (more precisely, outside the Malgrange divisor). It will soon become clear thatit is satisfied at least in a sufficiently small finite polydisk D ⊂ Cn−3 in the variablesa1, . . . , an−3, centered at the origin.

Assumption 4.8. PΣ,+ is invertible.

Proposition 4.9. For g ∈ H+, let gk and ~g be defined by (4.26). In these coordinates,L−1 = 1−K.

Proof. Rewrite the equation L−1f ′ = f as P⊕,+f ′ = PΣ,+f . Setting f = P⊕,+f ′+h,the latter equation becomes equivalent to PΣh = 0. The solution thus reduces toconstructing h ∈ HA such that (h+ P⊕,+f ′)− = 0, where the projection is taken withrespect to the splitting H = H+ ⊕H−. This can be achieved by setting

h[k]out,+ =h

[k+1]in,+ = − (P⊕,+f ′)[k+1]

in,+ ,

h[k]out,− =h

[k+1]in,− = − (P⊕,+f ′)[k]

out,− .

It then follows that f = f ′ + h+ = (1−K) f ′.

83


Tau function

Definition 4.10. Let L ∈ End (H+) be the operator defined by (4.28). We define thetau function associated to the Riemann-Hilbert problem for Ψ as

τ (a) := det(L−1

). (4.29)

In order to demonstrate the relation of (4.29) to conventional definition [JMU] ofthe isomonodromic tau function and its extension [ILP], let us compute the logarith-mic derivatives of τ with respect to isomonodromic times a1, . . . , an−3. At this pointit is convenient to introduce the notation

∆k =1

2Tr Θ2

k, ∆k =1

2TrS2

k. (4.30)

Recall that ∆0 ≡ ∆0 and ∆n−2 ≡ ∆n−1.

Theorem 4.11. We have

τ (a) = Υ (a)−1τJMU (a) , (4.31)

where τJMU (a) is defined up to a constant independent of a by

da ln τJMU =∑

0≤k<l≤n−2

TrAkAl d ln (ak − al) , (4.32)

and the prefactor Υ (a) is given by

Υ (a) =n−3∏k=1

a∆k−∆k−1−∆k

k . (4.33)

Proof. We will proceed in several steps.

Step 1. Choose a collection of points a0 close to a in the sense that the same annulican be used to define the tau function τ (a0). The collection a0 will be considered fixedwhereas a varies. Let us compute the logarithmic derivatives of the ratio τ (a) /τ (a0).First of all we can write

τ (a)

τ (a0)= det

(P⊕,+

(a0)−1PΣ,+

(a0)PΣ,+ (a)−1P⊕,+ (a)

)(4.34)

Note that since PΣ,+ (a) : H+ → HT (a) can be viewed as a projection of elements ofH along HA, the composition

Pa0→a := PΣ,+ (a)PΣ,+

(a0)−1

: HT(a0)→ HT (a)

is also a projection along HA. It therefore coincides with the restriction PΣ

∣∣HT (a0)

.

One similarly shows that

Fa0→a := P⊕,+ (a)P⊕,+(a0)−1

= P⊕∣∣HT (a0)

.

84


The exterior logarithmic derivative of (4.34) can now be written as

da lnτ (a)

τ (a0)= − TrHT (a0)

da (Fa→a0Pa0→a) · Pa→a0Fa0→a

=

= − TrHT (a0)

Fa→a0 · daPa0→a · Pa→a0Fa0→a

= (4.35)

= − TrH

P⊕(a0)· daPΣ (a) · PΣ

(a0)P⊕ (a)

.

The possibility to extend operator domains as to have the second equality is a conse-quence of (4.23). Furthermore, using once again the projection properties, one showsthat

PΣ (a)(1− PΣ

(a0))

= 0, P⊕ (a)(1− P⊕

(a0))

= 0.

which reduces the equation (4.35) to

da ln τ (a) = −TrH

P⊕daPΣ

= −

n−2∑k=1

TrH[k]

P [k]⊕ daPΣ

. (4.36)

Step 2. Let us now proceed to calculation of the right side of (4.36). Computationsof the same type have already been used in the proofs of Lemmata 4.6 and 4.7. Theidea is that Ψ[k] and Ψ have the same jumps on the contour Γ[k] which reduces theintegrals in (4.19), (4.22) to residue computation. In particular, for f [k] ∈ H[k] withk = 2, . . . , n− 3 we have

P [k]⊕ daPΣf

[k] (z) =1

(2πi)2

‹C[k]

in ∪C[k]out

Ψ[k]+ (z) Ψ

[k]+ (z′)

−1da

(Ψ+ (z′) Ψ+ (z′′)

−1)f [k] (z′′) dz′dz′′

(z − z′) (z′ − z′′).

(4.37)

The integrals are computed with the prescription that z is located inside the contourof z′, itself located inside the contour of z′′, and then passing to boundary values. But

since the function (z′ − z′′)−1 da

(Ψ+ (z′) Ψ+ (z′′)

−1)

has no singularity at z′′ = z′, the

contours of z′ and z′′ can be moved through each other. This identifies the trace ofthe integral operator on the right of (4.37) with

Tr(P [k]⊕ daPΣ

)=

= − 1

(2πi)2

‹C[k]

in ∪C[k]out

Tr

Ψ[k]+ (z) Ψ

[k]+ (z′)

−1da

(Ψ+ (z′) Ψ+ (z)

−1)

dz dz′

(z − z′)2 =

= − 1

(2πi)2

‹C[k]

in ∪C[k]out

Tr

Ψ[k]+ (z′)

−1daΨ+ (z′) · Ψ+ (z)

−1Ψ

[k]+ (z)

dz dz′

(z − z′)2

− 1

(2πi)2

‹C[k]

in ∪C[k]out

Trda

(Ψ+ (z)

−1)·Ψ[k]

+ (z) Ψ[k]+ (z′)

−1Ψ+ (z′)

dz dz′

(z − z′)2 ,

where z is considered to be inside the contour of z′. The first term vanishes since thecontours C[k]

in and C[k]out in the integral with respect to z can be merged. In the second

85


term the integral with respect to z′ is determined by the residue at z′ = z, whichyields

Tr(P [k]⊕ daPΣ

)=

1

2πi

˛C[k]

in ∪C[k]out

Trda

(Ψ+ (z)

−1)·Ψ[k]

+ (z) · ∂z(

Ψ[k]+ (z)

−1Ψ+ (z)

)dz.

Recall that Ψ+, Ψ[k]+ are related to fundamental matrix solutions Φ, Φ[k] of n-point

and 3-point Fuchsian systems by

Ψ+ (z)∣∣∣C[k]

in

= S−1k−1 (−z)−Sk−1 Φ (z) , Ψ+ (z)

∣∣∣C[k]

out

= S−1k (−z)−Sk Φ (z) ,

Ψ[k]+ (z)

∣∣∣C[k]

in

= S−1k−1 (−z)−Sk−1 Φ[k] (z) , Ψ

[k]+ (z)

∣∣∣C[k]

out

= S−1k (−z)−Sk Φ[k] (z) .

This leads to

Tr(P [k]⊕ daPΣ

)=

1

2πi

˛C[k]

in ∪C[k]out

Trda(Φ−1

)· Φ[k] · ∂z

(Φ[k]−1

Φ)

dz =

= resz=ak TrdaΦ · Φ−1

(∂zΦ · Φ−1 − ∂zΦ[k] · Φ[k]−1

). (4.38)

The contributions of the subspacesH[1] andH[n−2] to the trace (4.36) can be computed

in a similar fashion. The only difference is that instead of merging C[k]in with C[k]

out one

should now shrink the contour C[1]out to 0 and C[n−2]

in to ∞. The result is given by thesame formula (4.38).

Step 3. To complete the proof, it now remains to compute the residues in (4.38).

Note that near the regular singularity z = ak the fundamental matrices Φ, Φ[k] arecharacterized by the behavior

Φ (z → ak) =Ck (ak − z)Θk

(1 +

∞∑l=1

gk,l (z − ak)l)Gk, (4.39a)

Φ[k] (z → ak) =Ck (ak − z)Θk

(1 +

∞∑l=1

g[k]1,l (z − ak)l

)G

[k]1 . (4.39b)

The coinciding leftmost factors ensure the same local monodromy properties. Therightmost coefficients appear in the n-point and 3-point RHPs as Gk = Ψ (ak), G

[k]1 =

Ψ[k] (ak). It becomes straightforward to verify that as z → ak, one has

∂zΦ · Φ−1 − ∂zΦ[k] · Φ[k]−1=Ck (ak − z)Θk

[gk,1 − g[k]

1,1 +O (z − ak)]

(ak − z)−Θk C−1k ,

daΦ · Φ−1 =Ck (ak − z)Θk

[−Θkdakz − ak

+O (1)

](ak − z)−Θk C−1

k .

In combination with (4.36), (4.38), this in turn implies that

da ln τ (a) =n−3∑k=1

Tr Θk

(gk,1 − g[k]

1,1

)dak. (4.40)

86


Substituting local expansion (4.39a) into the Fuchsian system (4.14), we may re-cursively determine the coefficients gk,l. In particular, the first coefficient gk,1 satisfies

gk,1 + [Θk, gk,1] = G−1k

(n−2∑

l=0,l 6=k

Alak − al

)Gk, (4.41)

so thatn−3∑k=1

Tr (Θkgk,1) dak =n−3∑k=1

n−2∑l=0,l 6=k

TrAkAlak − al

dak = da ln τJMU. (4.42)

The 3-point analog of the relation (4.41) is

g[k]1,1 +

[Θk, g

[k]1,1

]= G

[k]1

A[k]0

akG

[k]1

−1,

which gives

Tr(

Θkg[k]1,1

)=

TrA[k]0 A

[k]1

ak=

Tr(A

[k]∞

2− A[k]

0

2− A[k]

1

2)

2ak=

∆k − ∆k−1 −∆k

ak. (4.43)

Combining (4.40) with (4.42) and (4.43) finally yields the statement of the theorem.

Corollary 4.12. Jimbo-Miwa-Ueno isomonodromic tau function τJMU (a) admits ablock Fredholm determinant representation

τJMU (a) = Υ (a) · det (1−K) , (4.44)

where the operator K is defined by (4.26). Its N×N subblocks (4.20) are expressed interms of solutions Ψ[k] of RHPs associated to 3-point Fuchsian systems with prescribedmonodromy.

Example: 4-point tau function

In order to illustrate the developments of the previous subsection, let us considerthe simplest nontrivial case of Fuchsian systems with n = 4 regular singular points.Three of them have already been fixed at a0 = 0, a2 = 1, a3 = ∞. There remains asingle time variable a1 ≡ t. To be able to apply previous results, it is assumed that0 < t < 1.

The monodromy data are given by 4 diagonal matrices Θ0,t,1,∞ of local monodromyexponents and connection matrices C0, Ct,±, C1,±, C∞ satisfying the relations

M0 ≡ C0e2πiΘ0C−1

0 = Ct,−C−1t,+, e2πiS = Ct,−e

2πiΘtC−1t,+ = C1,−C

−11,+

Observe that, in the hope to make the notation more intuitive, it has been slightlychanged as compared to the general case. The indices 0, 1, 2, 3 are replaced by0, t, 1,∞. Also, for n = 4 there is only one nontrivial matrix M0→k (namely, withk = 1). Therefore it becomes convenient to work from the very beginning in a

87


t0 1

( )t-z Ct,-

-1-Qt

( )t-z +Ct,

-1-Qt (1 )-z C1,+

-1-Q1

(1 )-z C1,-

-1-Q1

( )-z C0

-1( ) C-z

-Q0-1

8

Q 8

( )-z-s

( )-z-s

M0

-1M 8e

-2 ip se

-2 ip s

Figure 4.10: Contour Γ and jump matrices J for the 4-punctured sphere

distinguished basis where M0→1 is given by a diagonal matrix e2πiS with TrS =Tr (Θ0 + Θt) = −Tr (Θ1 + Θ∞). In terms of the previous notation, this correspondsto setting S1 = S and S1 = 1. The eigenvalues of S will be denoted by σ1, . . . , σN .Recall (cf Assumption 4.3) that S is chosen so that these eigenvalues satisfy

|< (σα − σβ)| ≤ 1, σα − σβ 6= ±1. (4.45)

The 4-punctured sphere is decomposed into two pairs of pants T [L], T [R] by oneannulus A as shown in Fig. 4.10. The space H is a sum

H = H+ ⊕H−, H± = H[L]out,± ⊕H

[R]in,∓. (4.46)

Both subspacesH± may thus be identified with the spaceHC := CN⊗L2 (C) of vector-valued square integrable functions on a circle C centered at the origin and belongingto the annulus A. It will be very convenient for us to represent the elements of HCby their Laurent series inside A,

f (z) =∑p∈Z′

fpz−12

+p, f p ∈ CN . (4.47)

In particular, the first and second component of H+ in (4.46) consist of functionswith vanishing negative and positive Fourier coefficients, respectively, i.e. they maybe identified with Π+HC and Π−HC. At this point the use of half-integer indices p ∈ Z′for Fourier modes may seem redundant, but its convenience will quickly become clear.

When n = 4, the representation (4.44) reduces to

τJMU (t) = t12

Tr(S2−Θ20−Θ2

t) det (1− U) , U =

(0 ad 0

)∈ End (HC) , (4.48)

88


1

(1 )-z C1,+

-1-Q1

(1 )-z C1,-

-1-Q1

( ) C-z-1

8

Q 8 M 8

e-2 ip s

(1 )-z Ct,+

-1-Qt

( )-z C0

-1-Q0( )-z

-s

M0

-1

e-2 ip s

( )-z-s

010

(1 )-z Ct,-

-1-Qt

Figure 4.11: Contours and jump matrices for Ψ[L] (left) and Ψ[R] (right)

where the operators a ≡ a[R] ≡ a[2] : Π−HC → Π+HC and d ≡ d[L] ≡ d[1] : Π+HC →Π−HC are given by

(ag) (z) =1

2πi

˛C

a (z, z′) g (z′) dz′ , a (z, z′) =Ψ[R] (z) Ψ[R] (z′)

−1 − 1

z − z′,

(4.49a)

(dg) (z) =1

2πi

˛C

d (z, z′) g (z′) dz′ , d (z, z′) =1−Ψ[L] (z) Ψ[L] (z′)

−1

z − z′.

(4.49b)

The contour C is oriented counterclockwise, which is the origin of sign difference in theexpression for d as compared to (4.20d). In the Fourier basis (4.47), the operators aand d are given by semi-infinite matrices whose N×N blocks a p

−q , d−pq are detereminedby

a (z, z′) =∑p,q∈Z′+

a p−q z

− 12

+pz′−12

+q, d (z, z′) =∑p,q∈Z′+

d−pq z− 1

2−pz′−

12−q. (4.50)

It should be emphasized that the indices of a p−q and d−pq belong to different ranges,

since in both cases p, q are positive half-integers.The matrix functions Ψ[L] (z), Ψ[R] (z) appearing in the integral kernels of a and

d solve the 3-point RHPs associated to Fuchsian systems with regular singularities at0, t,∞ and 0, 1,∞, respectively. In order to understand the dependence of the 4-pointtau function on the time variable t, let us rescale the fundamental solution of the firstsystem by setting

Φ[L] (z) = Φ[L](zt

). (4.51)

The rescaled matrix Φ[L] (z) solves a Fuchsian system characterized by the same mon-odromy as Φ[L] (z) but the corresponding singular points are located at 0, 1,∞. Denoteby Ψ[L] (z) the solution of the RHP associated to Φ[L] (z). To avoid possible confusionof the reader, we explicitly indicate the contours and jump matrices for RHPs for Ψ[L]

89


and Ψ[R] in Fig. 4.11; note the independence of jumps on t. In particular, inside thedisk around ∞ we have Φ[L] (z) = (−z)S Ψ[L] (z). Since the annulus A belongs to thedisk around∞ in the RHP for Ψ[L], the formula (4.51) yields the following expressionfor Ψ[L] inside A:

Ψ[L] (z)∣∣∣A

= (−z)−S Φ[L] (z) = t−SΨ[L](zt

)= t−S

(1 +

∞∑k=1

g[L]k tkz−k

)G[L]∞ , (4.52a)

where the N ×N matrix coefficients g[L]k are independent of t. Analogous expression

for Ψ[R] (z) inside A does not contain t at all:

Ψ[R] (z)∣∣∣A

=

(1 +

∞∑k=1

g[R]k zk

)G

[R]0 . (4.52b)

The formulae (4.52) allow to extract from the determinant representation (4.48) theasymptotics of 4-point Jimbo-Miwa-Ueno tau function τJMU (t) as t→ 0 to any desiredorder. We are now going to explain the details of this procedure.

Rewrite the integral kernel d (z, z′) as

d (z, z′) = t−S1− Ψ[L]

(zt

)Ψ[L]

(z′

t

)−1

z − z′tS.

The block matrix elements of d in the Fourier basis are therefore given by

d−pq = t−Sd−pqtS · tp+q, p, q ∈ Z′+, (4.53)

where N × N matrix coefficients d−pq are independent of t. They can be extractedfrom the Fourier series

1− Ψ[L] (z) Ψ[L] (z′)−1

z − z′=∑p,q∈Z′+

d−pqz− 1

2−pz′−

12−q, (4.54)

and are therefore expressed in terms of the coefficients of local expansion of the 3-point solution Φ[L] (z) around z = ∞ by straightforward algebra. For instance, thefirst few coefficients are given by

d− 1

212

= g[L]1 ,

d− 1

232

= g[L]2 − g

[L]1

2, d

− 3212

= g[L]2 ,

d− 1

252

= g[L]3 − g

[L]2 g

[L]1 − g

[L]1 g

[L]2 + g

[L]1

3, d

− 3232

= g[L]3 − g

[L]2 g

[L]1 , d

− 5212

= g[L]3 ,

. . . . . . . . . . . . . . . . . .

Different lines above contain the coefficients of fixed degree p+q ∈ Z>0 which appearsin the power of t in (4.53). Very similar formulas are also valid for matrix elementsof a:

a12

− 12

= g[R]1 , a

12

− 32

= g[R]2 − g

[R]1

2, a

32

− 12

= g[R]2 , . . .

90

4.3. Fourier basis and combinatorics

The crucial point for the asymptotic analysis of τ (t) is that for small t the operatord becomes effectively finite rank. Indeed, fix a positive integer Q. To obtain a uniformapproximation of d (z, z′) up to order O

(tQ), it suffices to take into account its Fourier

coefficients d−pq with p+q ≤ Q; recall that the eigenvalues of S are chosen as to satisfy(4.45). Since here p, q ∈ Z′+, the total number of relevant coefficients is finite and equalto Q (Q− 1) /2. It follows that the only terms in the Fourier expansion of a (z, z′)that contribute to the determinant (4.48) to order O

(tQ)

correspond to monomials

zp−12 z′q−

12 with p+ q ≤ Q. This is summarized in

Theorem 4.13. Let Q ∈ Z>0. The 4-point tau function τJMU (t) has the followingasymptotics as t→ 0:

τJMU (t) ' t12

Tr(S2−Θ20−Θ2

t)[det (1− UQ) +O

(tQ)], UQ =

(0 aQ

dQ 0

). (4.55)

Here UQ denotes a 2NQ×2NQ finite matrix whose NQ×NQ-dimensional blocks aQand dQ are themselves block lower and block upper triangular matrices of the form

aQ =

aQ− 1

2

− 12

0 · · · 0

... aQ− 3

2

− 32

· ...

a32

− 12

· . . . 0

a12

− 12

a12

− 32

· · · a12

12−Q

, dQ = t−S

d− 1

2

Q− 12

tQ · · · d− 1

232

t2 d− 1

212

t

0. . . · d

− 3212

t2

... · d32−Q

32

tQ...

0 · · · 0 d12−Q12

tQ

tS,

where a p−q , d−pq are determined by (4.49a), (4.50), (4.54), and the conjugation by tS in

the expression for dQ is understood to act on each N ×N block of the interior matrix.Moreover, strengthening the condition (4.45) to strict inequality |< (σα − σβ)| < 1improves the error estimate in (4.55) to o

(tQ).

Remark 4.14. The above theorem gives the asymptotics of τJMU (t) to arbitrary finiteorder Q in terms of solutions Φ[R] (z), Φ[L] (z) of two 3-point Fuchsian systems withprescribed monodromy around regular singular points 0, 1, ∞. For Q = 1 and underassumption |< (σα − σβ)| < 1, its statement may be rewritten as

τJMU (t) ' t12

Tr(S2−Θ20−Θ2

t)[det(1− g[R]

1 t1−Sg[L]1 tS

)+ o (t)

]. (4.56)

A result equivalent to this last formula has been recently obtained in [ILP, Proposition3.9] by a rather involved asymptotic analysis based on the conventional Riemann-Hilbert approach. For N = 2, the leading term in the expansion of the determinantappearing in (4.56) gives Jimbo asymptotic formula [Jimbo] for Painleve VI.

Fourier basis and combinatorics

Structure of matrix elements

Let us return to the general case of n regular singular points on P1. We have alreadyseen in the previous subsection certain advantages of writing the operators which

91


appear in the Fredholm determinant representation (4.44) of the tau function in theFourier basis. This motivates us to introduce the following notation for the integralkernels of the 3-point projection operators a[k], b[k], c[k], d[k] from (4.20):

a[k] (z, z′) :=Ψ

[k]+ (z) Ψ

[k]+ (z′)

−1− 1

z − z′=∑p,q∈Z′+

a[k] p−q z

− 12

+pz′−12

+q, z, z′ ∈ C[k]in ,

(4.57a)

b[k] (z, z′) := −Ψ[k]+ (z) Ψ

[k]+ (z′)

−1

z − z′=∑p,q∈Z′+

b[k]pq z− 1

2+pz′

− 12−q, z ∈ C[k]

in , z′ ∈ C[k]

out,

(4.57b)

c[k] (z, z′) :=Ψ

[k]+ (z) Ψ

[k]+ (z′)

−1

z − z′=∑p,q∈Z′+

c[k]−p−q z

− 12−pz′−

12

+q, z ∈ C[k]out, z

′ ∈ C[k]in ,

(4.57c)

d[k] (z, z′) :=1−Ψ

[k]+ (z) Ψ

[k]+ (z′)

−1

z − z′=∑p,q∈Z′+

d[k]−pqz− 1

2−pz′−

12−q, z, z′ ∈ C[k]

out.

(4.57d)

Just as before in (4.49b), the overall minus signs in the expressions for b[k] (z, z′) and

d[k] (z, z′) are introduced to absorb the negative orientation of C[k]out.

Our task in this subsection is to understand the dependence of matrix elementsa[k] p−q , b[k]p

q , c[k]−p−q , d[k]−p

q on their indices p, q ∈ Z′+. To this end recall that (cf (4.15))

Ψ[k]+ (z) =

(−z)−Sk−1 S−1

k−1Φ[k] (z) , z ∈ C[k]in ,

(−z)−Sk S−1k Φ[k] (z) , z ∈ C[k]

out.(4.58)

where Φ[k] (z) denotes the fundamental solution of the 3-point Fuchsian system (4.16).

Theorem 4.15. Denote by r[k] the rank of the matrix A[k]1 which appears in the Fuch-

sian system (4.16). Let u[k]r , v

[k]r ∈ CN with r = 1, . . . , r[k] be the column and row

vectors giving the decomposition

akA[k]1 = −

r[k]∑r=1

u[k]r ⊗ v[k]

r . (4.59)

Let(ψ

[k]r

)p,(ψ

[k]r

)p,(ϕ

[k]r

)−p,(ϕ

[k]r

)−p ∈ CN be the coefficients of the Fourier expansions

92


Ψ[k]+ (z)u

[k]r

z − ak=∑p∈Z′+

(ψ[k]r

)pz−

12

+p,

v[k]r Ψ

[k]+ (z)

−1

z − ak=∑p∈Z′+

(ψ[k]r

)pz−

12

+p,

z ∈ C[k]in , (4.60a)

Ψ[k]+ (z)u

[k]r

z − ak=∑p∈Z′+

(ϕ[k]r

)−pz−

12−p,

v[k]r Ψ

[k]+ (z)

−1

z − ak=∑p∈Z′+

(ϕ[k]r

)−pz

− 12−p,

z ∈ C[k]out. (4.60b)

Then the operators a[k], b[k], c[k], d[k] can be represented as sums of a finite number ofinfinite-dimensional Cauchy matrices with respect to the indices p, q, explicitly givenby

a[k] p;α−q;β =

r[k]∑r=1

(ψ

[k]r

)p;α(ψ

[k]r

)q;β

p+ q + σk−1,α − σk−1,β

, (4.61a)

b[k]p;αq;β =

r[k]∑r=1

(ψ

[k]r

)p;α(ϕ

[k]r

)−q;β

q − p− σk−1,α + σk,β, (4.61b)

c[k]−p;α−q;β =

r[k]∑r=1

(ϕ

[k]r

)−p;α(ψ

[k]r

)q;β

q − p+ σk,α − σk−1,β

, (4.61c)

d[k]−p;αq;β =

r[k]∑r=1

(ϕ

[k]r

)−p;α(ϕ

[k]r

)−q;β

p+ q − σk,α + σk,β, (4.61d)

where the color indices α, β = 1, . . . , N correspond to internal structure of the blocksa[k] p−q , b[k]p

q, c[k]−p−q, d[k]−p

q.

Proof. The Fuchsian system (4.16) can be used to differentiate the integral kernels(4.57) with respect to z and z′. Consider, for instance, the operator

L0 = z∂z + z′∂z′ + 1.

It is easy to check that2 L01

z−z′ = 0. Combining this with (4.57a), (4.58) and (4.16),one obtains e.g. that

L0a[k](z, z′

)=

(z∂z + z′∂z′) Ψ[k]+ (z) Ψ

[k]+ (z′)

−1

z − z′=[a[k](z, z′

),Sk−1

]−

Ψ[k]+ (z)

z − akakA

[k]1

Ψ[k]+ (z′)

−1

z′ − ak,

where z, z′ ∈ C[k]in . The crucial point here is that the dependence of the second term

on z and z′ is completely factorized. Indeed, it follows from the last identity, the

2The reader with acquintance with two-dimensional conformal field theory will recognize in thisequation the dilatation Ward identity for the 2-point correlator of Dirac fermions.

93


form of L0 and the notation (4.60a) that N ×N matrix a[k] p−q from (4.57a) satisfies the

equation (p+ q + adSk−1

)a[k] p−q =

r[k]∑r=1

(ψ[k]r

)p ⊗ (ψ[k]r

)q.

The formula (4.61a) is nothing but a rewrite of this identity. The proof of Cauchy typerepresentations (4.61b)–(4.61d) for the other three operators is completely analogous.

Combinatorics of determinant expansion

This subsection develops a systematic approach to the computation of multivariateseries expansion of the Fredholm determinant τ (a) = det (1−K). Recall that, ac-cording to Theorem 4.11, the isomonodromic tau function τJMU (a) coincides withτ (a) up to an elementary explicit prefactor.

Let A ∈ CX×X be a matrix indexed by a discrete and possibly infinite set X. Ourbasic tool for expanding τ (a) is the von Koch’s formula:

det (1 + A) =∑Y∈2X

detAY, (4.62)

where detAY denotes the |Y| × |Y| principal minor obtained by restriction of A to asubset Y ⊆ X. Of course, the series in (4.62) terminates when X is finite.

In our case, the role of the matrix A is played by the operator K written in theFourier basis. The elements of X are multi-indices which encode the following data:

• the positions of the blocks a[k], b[k], c[k], d[k] in K defined by (4.26);

• a half-integer Fourier index of the appropriate block;

• a color index taking its values in the set 1, . . . , N.

It is useful to combine Fourier and color indices into one multi-index ı = (p, α) ∈ N :=Z′ × 1, . . . , N. Unordered sets ı1, . . . , ım ∈ 2N of such multi-indices are denotedby capital Roman letters I or J . Given a matrix M ∈ CN×N, we denote by MJ

I its|I| × |J | restriction to rows I and columns J .

Principal submatrices ofK may be labeled by pairs(~I, ~J)

, where ~I = (I1, . . . , In−3),

~J = (J1, . . . , Jn−3) and I1...n−3, J1...n−3 ∈ 2N. Namely, define

K~I, ~J :=

0(a[2]

)I1

J1

(b[2]

)I1

I20 0 0 · · 0 0(

d[1])J1

I10 0 0 0 0 · · 0 0

0 0 0(a[3]

)I2

J2

(b[3]

)I2

I30 · · 0 0

0(c[2]

)J2

J1

(d[2]

)J2

I20 0 0 · · 0 0

0 0 0 0 0(a[4]

)I3

J3· · · ·

0 0 0(c[3]

)J3

J2

(d[3]

)J3

I30 · · · ·

· · · · · · · ·(b[n−3]

)In−2

In−30

· · · · · · · · 0 0

0 0 0 0 · · 0 0 0(a[n−2]

)In−3

Jn−3

0 0 0 0 · · 0(c[n−3]

)Jn−3

Jn−4

(d[n−3]

)Jn−3

In−30

94


For reasons that will become apparent below, the pairs(~I, ~J)

will be referred to as

configurations. It is useful to keep in mind that the lower index in Ik, Jk correspondsto the annulus Ak, and the blocks of K are acting between spaces of holomorphicfunctions on the appropriate annuli.

Definition 4.16. A configuration(~I, ~J)∈(2N)×2(n−3)

is called

• balanced if |Ik| = |Jk| for k = 1, . . . , n− 3;

• proper if all elements of Ik (and Jk) have positive (resp. negative) Fourierindices for k = 1, . . . , n− 3.

The sets of all balanced and proper balanced configurations will be denoted by Confand Conf+, respectively.

Definition 4.17. For(~I, ~J)∈ Conf, define

ZIk−1,Jk−1

Ik,Jk

(T [k]

):= (−1)|Ik| det

(a[k])Ik−1

Jk−1

(b[k])Ik−1

Ik(c[k])JkJk−1

(d[k])JkIk

, k = 1, . . . , n− 2.

(4.63a)In order to have uniform notation, here we set I0 = J0 = In−2 = Jn−2 ≡ ∅, so that

Z ∅, ∅I1,J1

(T [1]

)= (−1)|I1| det

(d[1])J1

I1, Z

In−3,Jn−3

∅, ∅(T [n−2]

)= det

(a[n−2]

)In−3

Jn−3.

(4.63b)

Proposition 4.18. The principal minor D~I, ~J := detK~I, ~J vanishes unless(~I, ~J)∈

Conf+, in which case it factorizes into a product of n − 2 finite (|Ik−1|+ |Ik|) ×(|Ik−1|+ |Ik|) determinants as

D~I, ~J =n−2∏k=1

ZIk−1,Jk−1

Ik,Jk

(T [k]

). (4.64)

Proof. For k = 1, . . . , n − 3, exchange the (2k − 1)-th and 2k-th block row of thematrix K~I, ~J . As such permutation can only change the sign of the determinant, theproposition for balanced configurations follows immediately from the block structureof the resulting matrix. The sign change is taken into account by the factor (−1)|Ik|

in (4.63a).The only non-zero Fourier coefficients of a[k], b[k], c[k], d[k] are given by (4.57).

Therefore, if a configuration(~I, ~J)∈ Conf is not proper, then at least one of the

factors on the right of (4.64) vanishes due to the presence of zero rows or columns inthe relevant matrices.

Corollary 4.19. Fredholm determinant τ (a) is given by

τ (a) =∑

(~I, ~J)∈Conf+

n−2∏k=1

ZIk−1,Jk−1

Ik,Jk

(T [k]

). (4.65)

95


Proof. Another useful consequence of the block structure of the operator K is thatTrK2m+1 = 0 for m ∈ Z≥0. This implies that det (1−K) = det (1 +K). It now suf-fices to combine this symmetry with von Koch’s formula (4.62) and Proposition 4.18.

Let us now give a combinatorial description of the set Conf+ of proper balancedconfigurations in terms of Maya diagrams and charged partitions.

Definition 4.20. A Maya diagram is a map m : Z′ → −1, 1 subject to the conditionthat m (p) = ±1 for all but finitely many p ∈ Z′±. The set of all Maya diagrams willbe denoted by M.

A convenient graphical representation of m ∈M is obtained by replacing −1’s and1’s by white and black circles located at the sites of half-integer lattice, see bottom partof Fig. 4.12 for an example. The white circles in Z′+ and black circles in Z′− are referredto as particles and holes in the Dirac sea, which itself corresponds to the diagram m0

defined by m0

(Z′±)

= ±1. An arbitrary diagram is completely determined by asequence p (m) = (p1, . . . , pr) of strictly decreasing positive half-integers p1 > . . . > prgiving the positions of particles, and a sequence h (m) = (−q1, . . . ,−qs) of strictlyincreasing negative half-integers −q1 < . . . < −qs corresponding to the positions ofholes. The integer Q (m) := |p (m)| − |h (m)| is called the charge of m.

Given a configuration(~I, ~J)∈ Conf+, consider a pair of its multi-indices (Ik, Jk)

associated to the annulus Ak. Recall that the Fourier indices of elements of Ik (andJk) are positive (resp. negative). They can therefore be interpreted as positions ofparticles and holes of N different colors. This yields a bijection between the set ofpairs (Ik, Jk) verifying the balance condition |Ik| = |Jk| and the set

MN0 =

(m(1), . . . ,m(N)

)∈MN

∣∣∣∑N

α=1Q(m(α)

)= 0

of N -tuples of Maya diagrams with vanishing total charge. We thereby obtain aone-to-one correspondence

Conf+∼= MN

0 × . . .×MN0︸︷︷︸

n−3 factors

.

Definition 4.21. A charged partition is a pair Y = (Y,Q) ∈ Y × Z. The integer Qis called the charge of Y .

There is a well-known bijection between Maya diagrams and charged partitions,whose construction is illustrated in Fig. 4.12. Given a Maya diagram m ∈M, we startfar on the north-west axis and draw a segment directed to the south-east above eachblack circle and a segment directed north-east above each white circle. The resultingpolygonal line defines the outer boundary of the Young diagram Y corresponding tom. The charge Q = Q (m) of Y is the signed distance between Y and the north-eastaxis. In the case Q (m) = 0, the sequences p (m) and −h (m) give the Frobeniuscoordinates of Y .

96


... ...

NE

NW

Q

12

32

12

-52

32

52

- -72

92

72

-92

-

Figure 4.12: The correspondence betweenMaya diagrams and charged partitions;here the charge Q (m) = 2 and the positions of particles andholes are given by p (m) =

(132, 7

2, 3

2, 1

2

)and h (m) =

(−5

2,−1

2

).

Let us write N -tuples(Y (1), . . . , Y (N)

)of charged partitions as

(~Y , ~Q

), with

~Y =(Y (1), . . . , Y (N)

)∈ YN and ~Q =

(Q(1), . . . , Q(N)

)∈ ZN . The set of such N -

tuples with zero total charge can be identified with MN0∼= YN × QN , where QN

denotes the AN−1 root lattice:

QN :=~Q ∈ ZN

∣∣∣∑N

α=1Q(α) = 0

.

This suggests to introduce an alternative notation for elementary finite determinantfactors in (4.65). For |Ik−1| = |Jk−1| and |Ik| = |Jk|, we define

Z~Yk−1, ~Qk−1

~Yk, ~Qk

(T [k]

):= Z

Ik−1,Jk−1

Ik,Jk

(T [k]

), (4.66)

where(~Yk−1, ~Qk−1

),(~Yk, ~Qk

)∈ YN × QN are associated to N -tuples of Maya di-

agrams describing subconfigurations (Ik−1, Jk−1), (Ik, Jk). In what follows, the twonotations are used interchangeably.

The structure of the expansion of τ (a) may now be summarized as follows.

Theorem 4.22. Fredholm determinant τ (a) giving the isomonodromic tau functionτJMU (a) can be written as a combinatorial series

τ (a) =∑

~Q1,... ~Qn−3∈QN

∑~Y1,...~Yn−3∈YN

n−2∏k=1

Z~Yk−1, ~Qk−1

~Yk, ~Qk

(T [k]

), (4.67)

where Z~Yk−1, ~Qk−1

~Yk, ~Qk

(T [k]

)are expressed by (4.66), (4.63) in terms of matrix elements of

3-point Plemelj operators in the Fourier basis.

Example 4.23. Let us outline simplifications to the above scheme in the case N = 2,

n = 4 corresponding to the Painleve VI equation. Here a configuration(~I, ~J)∈ Conf+

is given by a single pair (I, J) of multi-indices whose structure may be described asfollows: I (and J) encode the positions of particles (resp. holes) of two colors +,−,

97


and the total number of particles in I coincides with the total number of holes inJ . Relative positions of particles and holes of each color are described by two Youngdiagrams Y+, Y− ∈ Y. The vectors (Q+, Q−) ∈ Q2 of the charge lattice are labeled bya single integer n = Q+ = −Q− ∈ Z. In the notation of Subsection 4.2.5, the series(4.65) can be rewritten as

τ (t) =∑n∈Z

∑p+,p−∈2

Z′+ ; h+,h−∈2Z′−

|p+|−|h+|=|h−|−|p−|=n

(−1)|p+|+|p−| det ap+,p−h+,h−

det d h+,h−p+,p− =

=∑n∈Z

∑Y+,Y−∈Y

ZY+,Y−,n

(T [L]

)ZY+,Y−,n

(T [R]

),

(4.68)

where ZY+,Y−,n

(T [L]

)= (−1)|p+|+|p−| det d h+,h−

p+,p− and ZY+,Y−,n(T [R]

)= det ap+,p−

h+,h−. In

these equations, the particle/hole positions (p+, h+) and (p−, h−) for the 1st and 2ndcolor are identified with a pair of Maya diagrams, subsequently interpreted as chargedpartitions (Y+, n) and (Y−,−n).

Remark 4.24. Describing the elements of Conf+ in terms of N -tuples of Young di-agrams and vectors of the AN−1 root lattice is inspired by their appearance in thefour-dimensional N = 2 supersymmetric linear quiver gauge theories. Combinatorialstructure of the dual partition functions of such theories [Nek, NO] coincides with thatof (4.67). These partition functions can in fact be obtained from our construction orits higher genus/irregular extensions by imposing additional spectral constraints onmonodromy. It will shortly become clear that the multiple sum over QN is responsiblefor a Fourier transform structure of the isomonodromic tau functions. This structurewas discovered in [GIL12, ILT13] for Painleve VI, understood for N = 2 and arbitrarynumber of punctures within the framework of Liouville conformal field theory [ILTe],and conjectured to appear in higher rank in [Gav]. It might be interesting to men-tion the appearance of a possibly related structure in the study of topological stringpartition functions [GHM, BGT].

Rank two case

For N = 2, the elementary 3-point RHPs can be solved in terms of Gauss hyperge-ometric functions so that Fredholm determinant representation (4.44) becomes com-pletely explicit. Being rewritten in Fourier components, the blocks of K may bereduced to single infinite Cauchy matrices acting in `2 (Z). We are going to use this

observation to calculate the building blocks Z~Yk−1, ~Qk−1

~Yk, ~Qk

(T [k]

)of principal minors of K

in terms of monodromy data, and derive thereby a multivariate series representationfor the isomonodromic tau function of the Garnier system.

98

4.4. Rank two case

Gauss and Cauchy in rank 2

The form of the Fuchsian system (4.14) is preserved by the following non-constantscalar gauge transformation of the fundamental solution and coefficient matrices:

Φ (z) 7→ Φ (z)n−2∏l=0

(z − al)κl ,

Al 7→ Al + κl1, l = 0, . . . , n− 2.

Under this transformation, the monodromy matrices Ml are multiplied by e−2πiκl , andthe associated Jimbo-Miwa-Ueno tau function transforms as

τJMU (a) 7→ τJMU (a)∏

0≤k<l≤n−2

(al − ak)−Nκkκl+κk Tr Θl+κl Tr Θk .

The freedom in the choice of κ0, . . . , κn−2 allows to make the following assumption.

Assumption 4.25. One of the eigenvalues of each of the matrices Θ0, . . . ,Θn−2 isequal to 0.

This involves no loss in generality and means in particular that the ranks r[k] of thecoefficient matrices A

[k]1 in the auxiliary 3-point Fuchsian systems (4.16) are at most

N − 1.For r[k] = 1, the factor Z

Ik−1,Jk−1

Ik,Jk

(T [k]

)in (4.67) can be computed in explicit form.

In this case the sums such as (4.59) or (4.61) contain only one term, and the index r

can therefore be omitted. The matrix A[k]1 ∈ CN×N may be written as

akA[k]1 = −u[k] ⊗ v[k].

The crucial observation is that the blocks (4.61) are now given by single Cauchymatrices conjugated by diagonal factors (instead of being a sum of such matrices). In

order to put this to a good use, let us introduce two complex sequences(x

[k]ı

)ı∈Ik−1tJk

,(y

[k]

)∈Jk−1tIk

of the same finite length |Ik−1| + |Ik|. Their elements are defined by

shifted particle/hole positions:

x[k]ı :=

p+ σk−1,α, ı ≡ (p, α) ∈ Ik−1,

−p+ σk,α, ı ≡ (−p, α) ∈ Jk,(4.69a)

y[k] :=

−q + σk−1,β, ≡ (−q, β) ∈ Jk−1,

q + σk,β, ≡ (q, β) ∈ Ik.(4.69b)

Lemma 4.26. If r[k] = 1, then ZIk−1,Jk−1

Ik,Jk

(T [k]

)can be written as

ZIk−1,Jk−1

Ik,Jk

(T [k]

)= ±

∏(p,α)∈Ik−1

(ψ[k])p;α ∏

(−p,α)∈Jk−1

(ψ[k])p;α

∏(−p,α)∈Jk

(ϕ[k])−p;α∏

(p,α)∈Ik

(ϕ[k])−p;α×

×

∏ı,∈Ik−1tJk;ı<

(x[k]ı − x[k]

) ∏ı,∈Jk−1tIk;ı<

(y[k] − y[k]

ı

)∏

ı∈Ik−1tJk

∏∈Jk−1tIk

(x[k]ı − y[k]

) .

(4.70)

99


Proof. The diagonal factors in (4.61) produce the first line of (4.70). It remains tocompute the determinant

det

[

1

p+ σk−1,α + q − σk−1,β

] (p,α)∈Ik−1

(−q,β)∈Jk−1

[1

p+ σk−1,α − q − σk,β

](p,α)∈Ik−1

(q,β)∈Ik[1

−p+ σk,α + q − σk−1,β

](−p,α)∈Jk

(−q,β)∈Jk−1

[1

−p+ σk,α − q − σk,β

](−p,α)∈Jk

(q,β)∈Ik

,

(4.71)

which already includes the sign (−1)|Ik| in (4.63a). The ± sign in (4.70) depends onthe ordering of rows and columns of the determinant (4.63a). This ambiguity doesnot play any role as the relevant sign appears twice in the full product (4.64).

On the other hand, the notation introduced above allows to rewrite (4.71) as a(|Ik−1|+ |Ik|)× (|Ik−1|+ |Ik|) Cauchy determinant

det

(1

x[k]ı − y[k]

)ı∈Ik−1tJk

∈Jk−1tIk,

and the factorized expression (4.70) easily follows.

We now restrict ourselves to the case N = 2, where the condition r[1] = . . . =r[n−2] = 1 does not lead to restrictions on monodromy. Let us start by preparing asuitable notation.

• The color indices will take values in the set +,− and will be denoted by ε, ε′.

• Recall that the spectrum of A[k]1 coincides with that of Θk. According to As-

sumption 4.25, the diagonal matrix Θk has a zero eigenvalue for k = 0, . . . , n− 2.Its second eigenvalue will be denoted by −2θk. Obviously, there is a relation

2θkak = v[k] · u[k], k = 1, . . . , n− 2,

where v ·u = v+u+ + v−u− is the standard bilinear form on C2. The eigenvaluesof the remaining local monodromy exponent Θn−1 may be parameterized as

θn−1,ε =n−2∑k=0

θk + εθn−1, ε = ±.

• Also, the spectra of A[k]0 and A

[k]∞ = −A[k]

0 −A[k]1 coincide with the spectra of Sk−1

and −Sk. Since furthermore TrSk =∑k

j=0 Tr Θj, we may write the eigenvaluesof Sk as

σk,ε = −k∑j=0

θj + εσk, ε = ±, k = 0, . . . , n− 2, (4.72)

where σ0 ≡ θ0 and σn−2 ≡ −θn−1.

100

4.4. Rank two case

The non-resonancy of monodromy exponents and Assumption 4.3 imply that

2θk /∈ Z\ 0 , k = 0, . . . , n− 1,

|<σk| ≤1

2, σk 6= ±

1

2, k = 1, . . . , n− 3.

To simplify the exposition, we add to this extra genericity conditions.

Assumption 4.27. For k = 1, . . . , n− 2, we have

σk−1 + σk ± θk /∈ Z, σk−1 − σk ± θk /∈ Z.

It is also assumed that σk 6= 0 for k = 0, . . . , n− 2.

Let us introduce the space

MΘ =

[M0, . . . ,Mn−1] ∈ (GL (N,C))n/ ∼

∣∣M0 . . .Mn−1 = 1, Mk ∈ [e2πiΘk ] for k = 0, . . . , n− 1

of conjugacy classes of monodromy representations of the fundamental group withfixed local exponents. The parameters σ1, . . . , σn−3 are associated to annuliA1, . . . ,An−3

and provide n−3 local coordinates onMΘ (that is, exactly one half of dimMΘ = 2n− 6).The remaining n− 3 coordinates will be defined below.

Our task is now to find the 3-point solution Ψ[k] explicitly. The freedom in thechoice of its normalization allows to pick any representative in the conjugacy class[A

[k]0 , A

[k]1

]for the construction of the 3-point Fuchsian system (4.16). An important

feature of the N = 2 case is that this conjugacy class is completely fixed by localmonodromy exponents Sk−1, Θk and −Sk. We can set in particular

A[k]0 = diag σk−1,+, σk−1,− , akA

[k]1 = −u[k] ⊗ v[k],

with σk−1,± parameterized as in (4.72) and

u[k]± =

(σk−1 ± θk)2 − σ2k

2σk−1

ak, v[k]± = ±1.

As in Subsection 4.2.5, one may first construct the solution Φ[k] of the rescaledsystem

∂zΦ[k] = Φ[k]

(A

[k]0

z+

A[k]1

z − 1

), (4.73)

having the same monodromy around 0, 1, ∞ as the solution Φ[k] of the originalsystem (4.16) has around 0, ak and ∞. To write it explicitly in terms of the Gauss

hypergeometric function 2F1

[a, bc ; z

], we introduce a convenient notation,

χ

[θ2

θ1 θ3; z

]:= 2F1

[θ1 + θ2 + θ3, θ1 + θ2 − θ3

2θ1; z

],

φ

[θ2

θ1 θ3; z

]:=

θ23 − (θ1 + θ2)2

2θ1 (1 + 2θ1)z 2F1

[1 + θ1 + θ2 + θ3, 1 + θ1 + θ2 − θ3

2 + 2θ1; z

].

(4.74)

101


The solution of (4.73) can then be written as

Φ[k] (z) = Sk−1 (−z)Sk−1 Ψ[k]in (z) , (4.75)

where Sk−1 is a constant connection matrix encoding the monodromy (cf (4.15)), and

Ψ[k]in is given by (

Ψ[k]in

)±±

(z) =χ

[θk

±σk−1 σk; z

],(

Ψ[k]in

)±∓

(z) =φ

[θk

±σk−1 σk; z

].

(4.76)

It follows that Φ[k] (z) = Φ[k](zak

)and

Ψ[k]+ (z) = a

−Sk−1

k Ψ[k]in

(z

ak

), z ∈ C[k]

in . (4.77a)

Let us also note that det Φ[k] (z) = const · (−z)TrA[k]0 (1− z)TrA

[k]1 implies that det Ψ

[k]in (z) =

(1− z)−2θk , which in turn yields a simple representation for the inverse matrix

Ψ[k]+ (z)

−1=

(1− z

ak

)2θk

(

Ψ[k]in

)−−

(zak

)−(

Ψ[k]in

)+−

(zak

)−(

Ψ[k]in

)−+

(zak

) (Ψ

[k]in

)++

(zak

) a

Sk−1

k , z ∈ C[k]in .

(4.77b)The equations (4.75)–(4.76) are adapted for the description of local behavior of

Ψ[k] (z) inside the disk around 0 bounded by the circle C[k]in , cf left part of Fig. 4.9. To

calculate Ψ[k]+ (z) inside the disk around∞ bounded by C[k]

out, let us first rewrite (4.75)using the well-known 2F1 transformation formulas. One can show that

Φ[k] (z) = Sk−1C[k]∞ (−z)Sk Ψ

[k]out (z)G[k]

∞ , (4.78)

where (Ψ

[k]out

)±±

(z) =χ

[θk

∓σk σk−1; z−1

],(

Ψ[k]out

)±∓

(z) =φ

[θk

∓σk σk−1; z−1

],

(4.79)

and

G[k]∞ =

1

2σk

(−θk + σk−1 + σk θk + σk−1 − σk−θk + σk−1 − σk θk + σk−1 + σk

), (4.80)

C [k]∞ =

Γ (2σk−1) Γ (1 + 2σk)

Γ (1 + σk−1 + σk − θk) Γ (σk−1 + σk + θk)−

Γ (2σk−1) Γ (1− 2σk)

Γ (1 + σk−1 − σk − θk) Γ (σk−1 − σk + θk)

−Γ (−2σk−1) Γ (1 + 2σk)

Γ (1− σk−1 + σk − θk) Γ (θk − σk−1 + σk)

Γ (−2σk−1) Γ (1− 2σk)

Γ (1− σk−1 − σk − θk) Γ (θk − σk−1 − σk)

.

(4.81)

As a consequence,

Ψ[k]+ (z) = D[k]

∞a−Skk Ψ

[k]out

(z

ak

)G[k]∞ , z ∈ C[k]

out, (4.82a)

102

4.4. Rank two case

whereD[k]∞ = diag

d

[k]∞,+, d

[k]∞,−

is a diagonal matrix expressed in terms of monodromy

as

D[k]∞ = S−1

k Sk−1C[k]∞ .

Analogously to (4.77b), it may be shown that for z ∈ C[k]out

Ψ[k]+ (z)

−1=(

1− akz

)2θkG[k]∞−1

(

Ψ[k]out

)−−

(zak

)−(

Ψ[k]out

)+−

(zak

)−(

Ψ[k]out

)−+

(zak

) (Ψ

[k]out

)++

(zak

) aSkk D[k]

∞−1.

(4.82b)

We now have at our disposal all quantities that are necessary to compute theexplicit form of the integral kernels of a[k], b[k], c[k], d[k] in the Fredholm determinantrepresentation (4.44) of the Jimbo-Miwa-Ueno tau function, as well as of diagonalfactors ψ[k], ϕ[k], ψ[k], ϕ[k] in the building blocks (4.70) of its combinatorial expansion(4.67).

Lemma 4.28. For N = 2, the integral kernels (4.57) can be expressed as

a[k] (z, z′) = a−Sk−1

k

(1− z′

ak

)2θk(K++ (z) K+− (z)K−+ (z) K−− (z)

)(K−− (z′) −K+− (z′)−K−+ (z′) K++ (z′)

)− 1

z − z′aSk−1

k ,

(4.83a)

b[k] (z, z′) = − a−Sk−1

k

(1− ak

z′

)2θk ( K++ (z) K+− (z)K−+ (z) K−− (z)

)G

[k]∞−1(

K−− (z′) −K+− (z′)−K−+ (z′) K++ (z′)

)z − z′

aSk

k D[k]∞−1,

(4.83b)

c[k] (z, z′) = D[k]∞ a−Sk

k

(1− z′

ak

)2θk(K++ (z) K+− (z)K−+ (z) K−− (z)

)G

[k]∞

(K−− (z′) −K+− (z′)−K−+ (z′) K++ (z′)

)z − z′

aSk−1

k ,

(4.83c)

d[k] (z, z′) = D[k]∞ a−Sk

k

1−(1− ak

z′

)2θk ( K++ (z) K+− (z)K−+ (z) K−− (z)

)(K−− (z′) −K+− (z′)−K−+ (z′) K++ (z′)

)z − z′

aSk

k D[k]∞−1,

(4.83d)

where we introduced a shorthand notation K (z) = Ψ[k]in

(zak

), K (z) = Ψ

[k]out

(zak

); the

matrices Ψ[k]in,out (z) and G

[k]∞ are defined by (4.76), (4.78) and (4.80).

Proof. Straightforward substitution.

Lemma 4.29. Under genericity assumptions on parameters formulated above, the

103


Fourier coefficients which appear in (4.60) are given by(ψ[k])p;ε

=

∏ε′=± (θk + εσk−1 + ε′σk)p+ 1

2(p− 1

2

)! (2εσk−1)p+ 1

2

a∑k−1j=0 θk−εσk−1−(p− 1

2)k (−ε) , (4.84a)

(ψ[k])p;ε

=

∏ε′=± (1− θk − εσk−1 + ε′σk)p− 1

2(p− 1

2

)! (1− 2εσk−1)p− 1

2

a−∑k−1j=0 θk+εσk−1−(p+ 1

2)k (−ε) , (4.84b)

(ϕ[k])−p;ε

=

∏ε′=± (θk + ε′σk−1 − εσk)p+ 1

2(p− 1

2

)! (−2εσk)p+ 1

2

a∑kj=0 θk−εσk+(p+ 1

2)k d[k]

∞,εε, (4.84c)

(ϕ[k])−p;ε =

∏ε′=± (1− θk + ε′σk−1 + εσk)p− 1

2(p− 1

2

)! (1 + 2εσk)p− 1

2

a−∑kj=0 θk+εσk+(p− 1

2)k d[k]

∞,ε−1ε, (4.84d)

where ε = ± and (c)l :=Γ (c+ l)

Γ (c)denotes the Pochhammer symbol.

Proof. From the first equation in (4.60a), the representation (4.77a) for Ψ[k]+ (z) on

C[k]in , and hypergeometric contiguity relations such as

2F1

[a, bc

; z

]+(z − 1) 2F1

[a+ 1, b+ 1

c+ 1; z

]=

(c− a) (c− b)c (c+ 1)

z 2F1

[a+ 1, b+ 1

c+ 2; z

],

it follows that

∑p∈Z′+

(ψ[k]

)pz−

12 +p = −a−Sk−1

k

(θk + σk−1)2 − σ2

k

2σk−12F1

[1 + θk + σk−1 + σk, 1 + θk + σk−1 − σk

1 + 2σk−1;z

ak

](θk − σk−1)2 − σ2

k

2σk−12F1

[1 + θk − σk−1 + σk, 1 + θk − σk−1 − σk

1− 2σk−1;z

ak

] .

This in turn implies the equation (4.84a). The proof of three other identities issimilar.

The Cauchy determinant in (4.70) remains invariant upon simultaneous translation

of all x[k]ı and y

[k] by the same amount. Let us use this to replace the notation (4.69)

in the case N = 2 by

x[k]ı :=

p+ εσk−1, ı ≡ (p, ε) ∈ Ik−1,

−p− θk + εσk, ı ≡ (−p, ε) ∈ Jk,(4.85a)

y[k] :=

−q + εσk−1, ≡ (−q, ε) ∈ Jk−1,

q − θk + εσk, ≡ (q, ε) ∈ Ik.(4.85b)

Define a notation for the charges

mk := |( · ,+) ∈ Ik|−|( · ,+) ∈ Jk| = |( · ,−) ∈ Jk|−|( · ,−) ∈ Ik| , k = 1, . . . , n−3,

and combine them into a vector m := (m1, . . . ,mn−3) ∈ Zn−3. We will also writeσ := (σ1, . . . , σn−3) ∈ Cn−3 and further define

η := (η1, . . . , ηn−3) , eiηk :=d

[k]∞,−

d[k]∞,+

. (4.86)

The parameters η provide the remaining n− 3 local coordinates on the spaceMΘ ofmonodromy data. The main result of this section may now be formulated as follows.

104

4.4. Rank two case

Theorem 4.30. The isomonodromic tau function of the Garnier system admits thefollowing multivariate combinatorial expansion:

τGarnier (a) = const · a−θ20

1

n−3∏k=1

a−θ2

kk

∏1≤k<l≤n−2

(1− ak

al

)−2θkθl

×

×∑

m∈Zn−3

eim·η∑

~Y1,...,~Yn−3∈Y2

n−3∏k=1

(akak+1

)(σk+mk)2+|~Yk| n−2∏k=1

Z~Yk−1,mk−1

~Yk,mk

(T [k]

),

(4.87)

where ~Yk stands for the pair of charged Young diagrams associated to (Ik, Jk), the total

number of boxes in ~Yk is denoted by∣∣∣~Yk∣∣∣, and

Z~Yk−1,mk−1

~Yk,mk

(T [k]

)=

=∏

(p,ε)∈Ik−1

∏ε′=± (θk + εσk−1 + ε′σk)p+ 1

2(p− 1

2

)! (2εσk−1)p+ 1

2

∏(−p,ε)∈Jk−1

∏ε′=± (1− θk − εσk−1 + ε′σk)p− 1

2(p− 1

2

)! (1− 2εσk−1)p− 1

2

×

×∏

(−p,ε)∈Jk

∏ε′=± (θk + ε′σk−1 − εσk)p+ 1

2(p− 1

2

)! (−2εσk)p+ 1

2

∏(p,ε)∈Ik

∏ε′=± (1− θk + ε′σk−1 + εσk)p− 1

2(p− 1

2

)! (1 + 2εσk)p− 1

2

×

×

∏ı,∈Ik−1tJk;ı<

(x[k]ı − x[k]

) ∏ı,∈Jk−1tIk;ı<

(y[k] − y[k]

ı

)∏

ı∈Ik−1tJk

∏∈Jk−1tIk

(x[k]ı − y[k]

) .

(4.88)

Proof. Consider the product in the first line of (4.70). The balance conditions |Ik| =|Jk| imply that the factors such as e

∑k−1j=0 θk in (4.84) cancel out from Z

Ik−1,Jk−1

Ik,Jk

(T [k]

).

The factors of the form ±ε also compensate each other in the product of elemen-

tary determinants in (4.65). The factors d[k]∞,ε±1

in (4.84c) and (4.84d) produce theexponential eim·η in (4.87).

The total power in which the coordinate ak appears in (4.70) is equal to

2mkσk − 2mk−1σk−1 −∑

(p,ε)∈Ik−1

p−∑

(−p,ε)∈Jk−1

p+∑

(−p,ε)∈Jk

p+∑

(p,ε)∈Ik

p =

=(

2mkσk +m2k +

∣∣∣~Yk∣∣∣)− (2mk−1σk−1 +m2k−1 +

∣∣∣~Yk−1

∣∣∣) .The last equality is demonstrated graphically in Fig. 4.13. The prefactor in the firstline of (4.87) comes from two sources: i) the shifts of (initially traceless) Garniermonodromy exponents Θk by −θk1 making one of their eigenvalues equal to 0 and ii)the prefactor Υ (a) from Theorem 4.11.

In the Appendix, we show that the formula (4.88) can be rewritten in terms ofNekrasov functions. In the Painleve VI case (n = 4), this transforms Theorem 4.30into Theorem 4.2 of the Introduction.

105


Q>0 Q<0

Figure 4.13: A charged Maya diagram m and the associatedpartition Y (m) for positive and negative charges Q (m). Giventhe positions p (m) = (p1, . . . , pr) and q (m) = (−q1, . . . ,−qs) ofparticles and holes, the red and green areas represent the sums∑pk and

∑qk. We clearly have

∑pk +

∑qk = Q(m)2

2+ |Y (m) |

in both cases.

Hypergeometric kernel

Recall that the matrices Θ0, . . . ,Θn−1 are by convention diagonal with eigenvaluesdistinct modulo non-zero integers. However, all of the results of Section 4.2 remainvalid if the diagonal parts corresponding to the degenerate eigenvalues are replacedby appropriate Jordan blocks.

In this subsection we will consider in more detail a specific example of this typeby revisiting the 4-point tau function. We will thus follow the notational conventionsof Subsection 4.2.5. Fix n = 4, N = 2 and assume furthermore that the monodromyrepresentation ρ[L] : π1 (P1\ 0, t,∞) → GL (2,C) associated to the internal trinionT [L] is reducible, whereas its counterpart ρ[R] : π1 (P1\ 0, 1,∞)→ GL (2,C) for theexternal trinion T [R] remains generic. For instance, one may set

Θ0 = S =

(0 00 −2σ

), Θt =

(0 01 0

),

so that the monodromy matrices M0, Mt can be assumed to have the lower triangularform

M0 =

(1 0

−2πiκe−2πiσ e−4πiσ

), Mt =

(1 0

2πiκe2πiσ 1

), M0Mt = e2πiS.

(4.89)The solution Ψ[L] (z) of the appropriate internal 3-point RHP may be constructed

from the fundamental solution of a Fuchsian system

∂zΦ[L] = Φ[L]

0 0%t

z (z − t)−2σ

z

, (4.90)

with a suitably chosen value of the parameter %. Taking into account the diagonalmonodromy around ∞, such a solution Φ[L] (z) on C\R≥0 can be written as

Φ[L] (z) =

1 0

%t (−z)−2σ−1

1 + 2σl2σ(tz

)(−z)−2σ

= C0

1 0

% (−z)−2σ

2σl−1−2σ

(zt

)(−z)−2σ

,

(4.91)

106

4.4. Rank two case

where la (z) := 2F1

[1 + a, 12 + a ; z

], and the modified connection matrix C0 is lower-

triangular:

C0 =

1 0

− π%t−2σ

sin 2πσ1

.

The monodromy matrix around 0 is clearly equal to M0 = C0e2πiΘ0C−1

0 . This allowsto relate the monodromy parameter κ to the coefficient ρ of the Fuchsian system(4.90) as

κ = %t−2σ. (4.92)

The 3-point RHP solution Ψ[L] (z) inside the annulus A is thus explicitly given by

Ψ[L] (z)∣∣∣A

=

(1 0

0 (−z)2σ

)Φ[L] (z) =

1 0

− %t

(2σ + 1) zl2σ(tz

)1

. (4.93)

This formula leads to substantial simplifications in the Fredholm determinant repre-sentation (4.48) of the tau function τJMU (t). It follows from from the structure of(4.93) and (4.49b) that

d−+ (z, z′) =%

1 + 2σ

tzl2σ(tz

)− t

z′l2σ(tz′

)z − z′

(4.94)

is the only non-zero element of the 2 × 2 matrix integral kernel d (z, z′) (note thatthe lower indices here are color and should not be confused with half-integer Fouriermodes). This in turn implies that the only entry of a (z, z′) contributing to thedeterminant is

a+− (z, z′) =1

det Ψ[R] (z′)

Ψ[R]+− (z) Ψ

[R]++ (z′)−Ψ

[R]++ (z) Ψ

[R]+− (z′)

z − z′. (4.95)

Therefore, (4.48) reduces to

τJMU (t) = det (1− a+−d−+) . (4.96)

The action of the operators a+−, d−+ involves integration along a circle C ⊂ A.The kernel a+− (z, z′) extends to a function holomorphic in both arguments insideC. Therefore in the computation of contributions of different exterior powers to thedeterminant one may try to shrink all integration contours to the branch cut B :=[0, t] ⊂ R. The latter comes from two branch points 0, t of d−+ (z, z′) defined by(4.94).

Lemma 4.31. Let |<σ| < 12. For m ∈ Z≥0, denote Xm = Tr (a+−d−+)m. We have

Xm = TrKmF ,

where KF denotes an integral operator on L2 (B) with the kernel

KF (z, z′) = −κ (zz′)σ

a+− (z, z′) . (4.97)

107


Proof. Let us denote by Bup and Bdown the upper and lower edge of the branch cutB. After shrinking of the integration contours in the multiple integral Ik to B, theoperators a+−, d−+ should be interpreted as acting on W = L2 (Bup) ⊕ L2 (Bdown)instead of L2 (C). Here L2 (Bup,down) arise as appropriate completions of spaces ofboundary values of functions holomorphic inside DC\B, where DC denotes the diskbounded by C. The space W can be decomposed as W = W+ ⊕ W−, where theelements of W+ are continuous across the branch cut, whereas the elements of W−have opposite signs on its two sides:

W± = f ∈ W : f (z + i0) = ±f (z − i0) , z ∈ B .

We will denote by pr± the projections on W± along W∓.Since a+− (z, z′) is holomorphic in z, z′ inside C, it follows that im a+− ⊆ W+ ⊆

ker a+−. Therefore Xk remains unchanged if a+− is replaced by pr+ a+− pr−. Thisis in turn equivalent to replacing d−+ by pr− d−+ pr+. Given f = g⊕ g ∈ W+ withg ∈ L2 (B), the action of d−+ on f is given by

(d−+f) (z) =1

2πi

ˆ t

0

[d−+ (z, z′ − i0)− d−+ (z, z′ + i0)] g (z′) dz′ =

=%t

2πi (1 + 2σ)

ˆ t

0

l2σ(

tz′+i0

)− l2σ

(t

z′−i0

)z′ (z − z′)

g (z′) dz′.

An important consequence of the lower triangular monodromy is that the jump ofl2σ(tz′

)on B yields an elementary function, cf (4.91):

l2σ

(t

z′ + i0

)− l2σ

(t

z′ − i0

)= −2πi (2σ + 1)

(z′

t

)2σ+1

.

Substituting this jump back into the previous formula and using (4.92), one obtains

(d−+f) (z) = κ

ˆ t

0

z′2σg (z′) dz′

z′ − z, z ∈ DC\B.

Next we have to compute the projection pr− of this expression onto W−. Writepr− d−+f = h⊕ (−h), with h ∈ L2 (B). Then

h (z) =1

2[(d−+f) (z + i0)− (d−+f) (z − i0)] = πiκz2σg (z) , z ∈ B.

Finally, write a+− pr− d−+f as g⊕ g ∈ W+. It follows from the previous expressionfor h (z) that

g (z) = −κˆ t

0

a+− (z, z′) z′2σg (z′) dz′, z ∈ B.

The minus sign in front of the integral is related to orientation of the contour C inthe definition of a. We have thereby computed the action of a+− pr− d−+ on W+.Raising this operator to an arbitrary power k ∈ Z≥0 and symmetrizing the factorsz′2σ under the trace immediately yields the statement of the lemma.

108

4.4. Rank two case

Theorem 4.32. Given complex parameters θ1, θ∞, σ satisfying previous genericityassumptions, let

ϕ (x) :=xσ (1− x)θ1 2F1

[σ + θ1 + θ∞, σ + θ1 − θ∞

2σ;x

], (4.98a)

ψ (x) :=x1+σ (1− x)θ1 2F1

[1 + σ + θ1 + θ∞, 1 + σ + θ1 − θ∞

2 + 2σ;x

]. (4.98b)

Define the continuous 2F1 kernel by

KF (x, y) :=ψ (x)ϕ (y)− ϕ (x)ψ (y)

x− y, (4.99)

and consider Fredholm determinant

D (t) := det(1− λKF

∣∣(0,t)

), λ ∈ C. (4.100)

Then D (t) is a tau function of the Painleve VI equation with parameters~θ = (θ0 = σ, θt = 0, θ1, θ∞). The conjugacy class of monodromy representation for theassociated 4-point Fuchsian system is generated by the matrices (4.89) and

M1 =e−2πiθ1

i sin 2πσ

(cos 2πθ∞ − e−2πiσ cos 2πθ1 s−1e−2πiσ [cos 2πθ∞ − cos 2π (θ1 − σ)]

se2πiσ [cos 2π (θ1 + σ)− cos 2πθ∞] e2πiσ cos 2πθ1 − cos 2πθ∞

),

(4.101a)

M∞ =e−2πiθ∞

i sin 2πσ

(cos 2πθ1 − e−2πiσ cos 2πθ∞ s−1 [cos 2π (θ1 − σ)− cos 2πθ∞]s [cos 2πθ∞ − cos 2π (θ1 + σ)] e2πiσ cos 2πθ∞ − cos 2πθ1

)= M−1

1 e−2πiS.

(4.101b)

where

λ = κ(θ1 + σ)2 − θ2

∞2σ (2σ + 1)

, (4.102)

s = −Γ (1− 2σ) Γ (θ1 + σ + θ∞) Γ (θ1 + σ − θ∞)

Γ (1 + 2σ) Γ (θ1 − σ + θ∞) Γ (θ1 − σ − θ∞). (4.103)

Proof. To prove that D (t) is a Painleve VI tau function with λ and κ related by(4.102), it suffices to combine the determinant representation (4.96) with Lemma 4.31,and substitute into the formula (4.95) for a+− (z, z′) explicit hypergeometric expres-sions (4.76).

The formula (4.101b) follows from M∞ = C∞e2πiΘ∞C−1

∞ , where C∞ is obtainedfrom the connection matrix (4.81) by replacements (θk, σk−1, σk)→ (θ1, σ,−θ∞). Theexpression (4.101a) for M1 is then most easily deduced from the diagonal form of theproduct M1M∞ = e−2πiS.

Remark 4.33. The 2F1 kernel is related to the so-called ZW -measures [BO05] arising inthe representation theory of the infinite-dimensional unitary group U (∞). It producesvarious other classical integrable kernels (such as sine and Whittaker) as limiting cases.The first part of Theorem 4.32, namely the Painleve VI equation for D (t), was provedby Borodin and Deift in [BD]. Monodromy data for the associated Fuchsian system

109


have been identified in [Lis]. To facilitate the comparison, let us note that indroducinginstead of λ and κ a new parameter σ defined by

λ =sin π (σ − θ1) sinπ (σ + θ1)

π2

∏ε,ε′=± Γ (1 + σ + εθ1 + ε′θ∞)

Γ (1 + 2σ) Γ (2 + 2σ),

we have in particular that TrM∞M0 = 2e−2πi(σ+θ∞) cos 2πσ and TrMtM1 = 2e2πi(σ+θ∞) cos 2πσ.The relation between parameters z, z′, w, w′ of [BD] and ours is

(z, z′, w, w′)[BD] = (σ + θ1, σ − θ1, σ − σ + θ∞, σ − σ − θ∞) .

Appendix

Relation to Nekrasov functions

Here we demonstrate that the formula (4.88) can be rewritten in terms of Nekrasovfunctions. This rewrite is conceptually important for identification of isomonodromictau functions with dual partition functions of quiver gauge theories [NO]. It is alsouseful from a computational point of view: naively, the formula (4.88) may producepoles in the tau function expansion coefficients when θk ± σk ±′ σk−1 ∈ Z. Ourcalculation shows that these poles actually cancel.

The statement we are going to prove3 is the relation

Z~Y ′,m′

~Y ,m(T ) = (−1)lsgn(~Y ′,m′)+lsgn(~Y ,m)Z

~Y ′, ~Q′

~Y , ~Q(T ) , (4.104)

where

Z~Y ′, ~Q′

~Y , ~Q(T ) =

∏Nα,β C (σ′α − σβ|Q′α, Qβ)∏N

α<β C(σ′α − σ′β

∣∣Q′α, Q′β)C(σα − σβ|Qα, Qβ

)×× eiδ~η

′· ~Q′+iδ~η· ~Q∏Nα

∣∣Z bif

(0∣∣Yα, Yα)∣∣ 1

2∣∣Z bif

(0∣∣Y ′α, Y ′α)∣∣ 1

2

×

×∏N

α,β Z bif (σ′α +Q′α − σβ −Qβ|Y ′α, Yβ)∏Nα<β Z bif

(σ′α +Q′α − σ′β −Q′β

∣∣Y ′α, Y ′β)Z bif (σα +Qα − σβ −Qβ|Yα, Yβ).

(4.105)

The notation used in these formulas means the following:

• ~Q = (m,−m), ~Q′ = (m′,−m′), though the right side of (4.105) is defined evenwithout this specialization.

• Y ′ and Y are identified, respectively, with Yk−1 and Yk in (4.88). Similar con-ventions will be used for all other quantities. We denote, however, σ′± = ±σk−1

and σ± = −θk ± σk; T stands for T [k].

3In the present chapter we do it only for N = 2 but the generalization is relatively straightforward.

110

4.5. Relation to Nekrasov functions

• lsgn(~Y ,m

)∈ Z/2Z means the “logarithmic sign”,

lsgn(~Y ,m

):= |q+| · |p+|+

∑i

(q+,i +

1

2

)+∑i

(p−,i +

1

2

). (4.106)

Here, for example, |p+| denotes the number of coordinates p+,i of particles in theMaya diagram corresponding to the charged partition (Y+,m). The logarithmic

signs cancel in the product∏n−2

k=1 Z~Yk−1,mk−1

~Yk,mkwhich appears in the representation

(4.87) for the Garnier tau function.

• δ~η and δ~η′ are some explicit functions which are computed below. They justshift Fourier transformation parameters and their relevant combinations areexplicitly given by

eiδη′+−iδη′− =

1

2σk−1

(θk + σk−1)2 − σ2k

(θk − σk−1)2 − σ2k

,

eiδη+−iδη− =−1

2σk

(θk + σk)2 − σ2

k−1

(θk − σk)2 − σ2k−1

.

(4.107)

• Z bif (ν|Y ′, Y ) is the Nekrasov bifundamental contribution

Z bif (ν|Y ′, Y ) :=∏∈Y ′

(ν + 1 + aY ′ () + lY ()

) ∏∈Y

(ν − 1− aY ()− lY ′ ()

).

(4.108)

In particular, we have |Z bif (0|Y, Y )|12 =

∏∈Y hY ().

• The three-point function C (ν|Q′, Q) is defined by

C (ν|Q′, Q) ≡ C (ν|Q′ −Q) =G (1 + ν +Q′ −Q)

G (1 + ν) Γ (1 + ν)Q′−Q , (4.109)

where G (x) is the Barnes G-function. The only property of this function essen-tial for our purposes is the recurrence relation G (x+ 1) = Γ (x)G (x).

• Using the formula (4.88), we assume a concrete ordering: p′+, p′−, q+, q−, p1 >

p2 > . . ., and in (4.105) we suppose that + < −.

An important feature of the product (4.105) is that the combinatorial part in the 2ndline depends only on combinations such as σα +Qα, σ′α +Q′α. This is most crucial forthe Fourier transform structure of the full aswer for the tau function τGarnier (a).

Let us now present the plan of the proof, which will be divided into several self-contained parts. Most computations will be done up to an overall sign, and sometimeswe will omit to indicate this. In the end we will consider the limit θk → +∞,σk, σk−1 θk, σk, σk−1 → +∞ to recover the correct sign.

1. First we will rewrite the formula (4.88) as

Z~Y ′,m′

~Y ,m(T ) = ±eiδ1~η′· ~Q′+iδ1~η· ~Q ˆ

Z~Y ′, ~Q′

~Y , ~Q(T ) ,

111


whereˆZ~Y ′, ~Q′

~Y , ~Q(T ) is expressed in terms of yet another function Z bif

(ν∣∣Q′, Y ′;Q, Y ),

ˆZ~Y ′, ~Q′

~Y , ~Q(T ) =

N∏α

∣∣Zbif

(0∣∣Qα, Yα, Qα, Yα

)∣∣− 12∣∣Zbif

(0∣∣Q′α, Y ′α, Q′α, Y ′α)∣∣− 1

2 ×

×∏N

α,β Zbif

(σ′α − σβ

∣∣Q′α, Y ′α;Qβ, Yβ)∏N

α<β Z bif

(σ′α − σ′β

∣∣Q′α, Y ′α;Q′β, Y′β

)Z bif

(σα − σβ

∣∣Qα, Yα;Qβ, Yβ) ,

(4.110)

which is defined as

Z bif

(ν∣∣Q′, Y ′;Q, Y ) =

∏i

(−ν)q′i+ 12

∏i

(ν + 1)qi− 12

∏i

(−ν)pi+ 12

∏i

(ν + 1)p′i− 12×

×∏

i,j (ν − q′i − pj)∏

i,j (ν + p′i + qj)∏i,j (ν − q′i + qj)

∏i,j (ν + p′i − pj)

.

(4.111)

2. At the second step, we prove that Z bif

(ν∣∣0, Y ′; 0, Y

)≡ Z bif

(ν∣∣Y ′, Y ) = ±Z bif

(ν∣∣Y ′, Y ).

3. Next it will be shown that

Z bif

(ν∣∣Q′, Y ′;Q, Y ) = C

(ν∣∣Q′, Q)Z bif

(ν +Q′ −Q

∣∣Y ′, Y ) . (4.112)

4. Finally, we check the overall sign and compute extra contribution to ~η to absorbit.

A realization of this plan is presented below.

Step 1

It is useful to decompose the product (4.88) into two different parts: a “diagonal”one, containing the products of functions of one particle/hole coordinate, and a “non-diagonal” part containing the products of pairwise sums/differences. Careful compar-ison of the formulas (4.88) and (4.110) shows that their non-diagonal parts actuallycoincide. Further analysis of (4.110) shows that its diagonal part is given by∏

(p′,ε)∈I′ψp′,ε

∏(−q′,ε)∈J ′

ψq′,ε∏

(−q,ε)∈J

ϕq,ε∏

(p,ε)∈I

ϕp,ε.

with

ψp′,ε =(1 + εσk−1 + θk − σk)p′− 1

2(1 + εσk−1 + θk + σk)p′− 1

2[ε = + : (1 + 2σk−1)p′− 1

2; ε = − : (−2σk−1)p′+ 1

2

] (p′ − 1

2

)!,

ψq′,ε =(−εσk−1 − θk + σk)q′+ 1

2(−εσk−1 − θk − σk)q′+ 1

2[ε = + : (−2σk−1)q′+ 1

2; ε = − : (1 + 2σk−1)q′− 1

2

] (q′ − 1

2

)!,

ϕq,ε =(σk−1 + θk − εσk + 1)q− 1

2(−σk−1 + θk − εσk + 1)q− 1

2[ε = + : (−2σk)q+ 1

2; ε = − : (1 + 2σk)q− 1

2

] (q − 1

2

)!,

ϕp,ε =(−σk−1 − θk + εσk)p+ 1

2(σk−1 − θk + εσk)p+ 1

2[ε = + : (1 + 2σk)p− 1

2; ε = − : (−2σk)p+ 1

2

] (p− 1

2

)!.

(4.113)

112


The notation [ε = + : X; ε = − : Y ] means that we should substitute this constructionby X when ε = + and by Y when ε = −. Comparing these expressions with (4.88),

we may compute the ratios of diagonal factors which appear in Z~Y ′,m′

~Y ,m

/ ˆZ~Y ′, ~Q′

~Y , ~Q:

δψp′,ε =(εσk−1 + θk − σk) (εσk−1 + θk + σk)

[ε = + : 2σk−1; ε = − : 1],

δψq′,ε =(−εσk−1 − θk + σk)

−1 (−εσk−1 − θk − σk)−1

[ε = + : (−2σk−1)−1 ; ε = − : 1],

δϕq,ε =(σk−1 + θk − εσk) (−σk−1 + θk − εσk)

[ε = + : 1; ε = − : 2σk],

δϕp,ε =(−σk−1 − θk + εσk)

−1 (σk−1 − θk + εσk)−1

[ε = + : 1; ε = − : (−2σk)−1]

.

(4.114)

Since |p±| − |q±| = Q±, these formulas allow to determine the corrections δ1η±:

eiδ1η′+ =

(θk + σk−1)2 − σ2k

2σk−1

, e−iδ1η+ = (θk − σk)2 − σ2k−1,

eiδ1η′− = (θk − σk−1)2 − σ2

k, e−iδ1η− =(θk + σk)

2 − σ2k−1

2σk.

(4.115)

One could notice that some minus signs should also be taken into account, so that

Z~Y ′,m′

~Y ,m(T ) = (−1)|q

′+|+|p−| eiδ1~η

′· ~Q′+iδ1~η· ~Q ˆZ~Y ′, ~Q′

~Y , ~Q(T ) .

This is however not essential, as these signs will be recovered at the last step. Amore important thing to note is that in the reference limit described by θk → +∞,σk, σk−1 θ, σk, σk−1 → +∞ one has

sgn(eiδ1η±

)= sgn

(eiδ1η

′±)

= 1.

Step 2

Let us now formulate and prove combinatorial

Theorem 4.34. Z bif

(ν∣∣0, Y ′; 0, Y

)≡ Z bif

(ν∣∣Y ′, Y ) = ±Z bif

(ν∣∣Y ′, Y ).

This statement follows from the following two lemmas.

Lemma 4.35. Equality Z bif = ±Z bif holds for the diagrams Y ′, Y ∈ Y iff it holds forY ′, Y with added one column of admissible height L.

Proof. Let us denote the new value of Z bif by Z∗bif4, then

Z∗bif =(1 + ν)L

∏i

(L+ p′i + 1

2+ ν)∏

i

(L− q′i + 1

2+ ν) (1− ν)L

∏i

(L+ pi + 1

2− ν)∏

i

(L− qi + 1

2− ν) Z bif . (4.116)

The extra factor comes only from the product over 2L new boxes. To explain how itsexpression is obtained, we will use the conventions of Fig. 4.14.

4Everywhere in this appendix X∗ denotes the value of a quantity X after appropriate transfor-mation.

113


Figure 4.14: A Young diagram Y ′∗ obtained from Y ′ =6, 4, 4, 2, 2 by addition of a column of length L = 7.

To compute the contribution from the red boxes it is enough just to multiply thecorresponding shifted hook lengths, which yields

∏i

(L+ p′i + 1

2+ ν). To compute the

contribution from the green boxes one has to first write down the product of numbersfrom ν+L to ν+1 (i.e. the Pochhammer symbol (1 + ν)L in the numerator), keepingin mind that each step down by one box decreases the leg-length of the box by atleast one. Then one has to take into account that some jumps in this sequence aregreater than one: this happens exactly when we meet some rows of the transposeddiagram. We mark with the green crosses the boxes whose contributions should becancelled from the initial product: they produce the denominator.

Next let us check what happens with Z bif . We have

Z∗bif (ν|Y ′, Y ) =∏i

(−ν)q′∗i + 12

∏i

ν−1 (ν)q∗i + 12

∏i

(−ν)p∗i+ 12

∏i

ν−1 (ν)p′∗i + 12×

×∏

i,j

(ν − q′∗i − p∗j

)∏i,j

(ν + p′∗i + q∗j

)∏i,j

(q′∗i − q∗j − ν

)∏i,j

(p′∗i − p∗j + ν

) ,where

q∗i =

(L− 1/2) , (q1 − 1) , . . . , (qd−1 − 1) , ˜(qd − 1),

p∗i =

(p1 + 1) , . . . , (pd + 1) , 1/2,

q′∗i =

(L− 1/2) , (q′1 − 1) , . . . ,(q′d′−1 − 1

),

˜(q′d′ − 1)

,

p′∗i =

(p′1 + 1) , . . . , (p′d′ + 1) ,˜1/2,

and d, d′ denote the number of boxes on the main diagonals of Y, Y ′. The abovenotation means that one has either to simultaneously include or not to include thecoordinates tilded in the same way. These numbers are included in the case when bothof them are positive (it implies that qd 6= 1

2or q′d′ 6= 1

2). Fig. 4.15 below illustrates the

difference between these two cases.

We may now consider one by one four possible options, namely: i) qd 6= 12, q′d′ 6= 1

2;

ii) qd = q′d′ = 12; iii) qd 6= 1

2, q′d′ = 1

2; iv) qd = 1

2, q′d′ 6= 1

2. For instance, for qd 6= 1

2,

114


Figure 4.15: Possible mutual configurations of main di-agonals of Y , Y ∗; qd = 1

2(left) and qd 6= 1

2(right).

q′d′ 6= 12

after massive cancellations one obtains

Z∗bifZ bif

=

d′∏i=1

1

−ν + q′i − 12

(−ν)L

d∏i=1

1

ν + qi − 12

ν−1 (ν)L

d∏i=1

(−ν + pi +

1

2

)(−ν)1

d′∏i=1

(ν + p′i +

1

2

)×

×∏i

(ν − L− 1

2 − pi)∏

i

(ν − q′i + 1

2

)∏i

(ν − 1

2 + qi)∏

i

(ν + p′i + L+ 1

2

)∏i

(L+ 1

2 − qi − ν)∏

i

(q′i − L− 1

2 − ν)∏

i

(p′i + 1

2 + ν)∏

i

(− 1

2 − pi + ν) (ν − L) (ν + L)

ν2=

= (1− ν)L (1 + ν)L

∏i(ν − L−

12 − pi)

∏i(ν + p′i + L+ 1

2 )∏i(L+ 1

2 − qi − ν)∏i(q′i − L− 1

2 − ν)=Z∗bifZ bif

,

where the first line of the first equality corresponds to the ratio of diagonal parts andthe second to non-diagonal ones. The proof in the other three cases is analogous.

Corollary 4.36. Z bif = Z bif for arbitrary Y, Y ′ ∈ Y iff Z bif = ±Z∗bif for diagramswith qi =

12, . . . , L− 1

2

(that is, for the diagrams containing a large square on the

left).

Lemma 4.37. The equality Z bif = Z bif holds for given diagrams Y, Y ′ ∈ Y with alarge square iff it holds for the diagrams with a large square and one deleted box.

Proof. Suppose that we have added one box to the ith row of Y ′. The only boxes

j =2

Figure 4.16: A pair of Young diagrams (red andgreen) with a large square (black) and added box(blue square).

whose contribution to Z bif depends on the added box lie on its left in the diagram Y ′

and above it in the diagram Y , see Fig. 4.16. The contribution from the boxes on theleft (green circles) was initially given by

Z leftbif =

(ν)p′i+L+ 12∏

j≥j (p′i − pj + ν) · (ν)j−i+1

,

115


where j = min [j|pj + j ≤ p′i + i+ 1 ∪ L] (notice that we can move j in the rangewhere pj+j = p′i+i+1). The contribution from the boxes on the top (red circles) wasZtop

bif =∏

j<j (−ν + pj − p′i − 1). After addition of one box (blue square) it transforms

into Z∗topbif =

∏j<j (−ν + pj − p′i), whereas the previous part becomes

Z∗leftbif =

(ν)p′i+L+ 32∏

j≥j (p′i − pj + 1 + ν) · (ν)j−i+1

.

The ratio of the transformed and initial functions is then given by

Z∗bifZ bif

=

(p′i + L+ 1

2 + ν)∏

j<j (p′i − pj + ν)∏j≥j (p′i − pj + ν)∏

j≥j (p′i − pj + 1 + ν)∏j<j (−ν + pj − p′i − 1)

=

(p′i + L+ 1

2

)∏j (p′i − pj + ν)∏

j (p′i − pj + 1 + ν).

On the other hand, the ratio Z∗bif/Z bif is easier to compute since the addition ofone box to the ith row of Y ′ simply shifts one coordinate, p′i 7→ p′i + 1. From (4.111)and the large square condition qi =

12, . . . , L− 1

2

it follows that

Z∗bifZ bif

=

(p′i +

1

2+ ν

)×(p′i + 1

2+ L+ ν

)∏j (p′i − pj + ν)(

p′i + 12

+ ν)∏

j (p′i − pj + 1 + ν)=Z∗bifZ bif

,

which finishes the proof.

Using two inductive procedures described above, any pair of diagrams Y, Y ′ ∈ Ycan be reduced to equal squares, in which case the statement of Theorem 4.34 can bechecked directly.

Step 3

Let us move to the third part of our plan and prove

Theorem 4.38. Z bif

(ν∣∣Q′, Y ′;Q, Y ) = C

(ν∣∣Q′ −Q)Z bif

(ν +Q′ −Q

∣∣Y ′, Y ).Proof. It is useful to start by computing Z bif for the “vacuum state”

pα = pQα :=1

2,3

2, . . . , Q(α) − 1

2

, qα = ∅ for Q(α) > 0,

pα = ∅, qα = qQα :=1

2,3

2, . . . ,−Q(α) − 1

2

for Q(α) < 0.

One obtains

Z bif

(ν∣∣pQ′ , ∅; pQ, ∅

)= (−1)Q(Q+1)/2

Q′∏i=1

ν−1 (ν)i

Q∏i=1

(−ν)i

Q′∏i=1

Q∏j=1

(ν + i− j)−1 =

= (−1)Q(Q+1)/2Q′∏i=1

Γ (ν + i)

Γ (ν + 1)

Q∏i=1

Γ (i− ν)

Γ (−ν)

Q∏j=1

Γ (ν − j + 1)

Γ (ν − j +Q′ + 1)=

=G (1 + ν +Q′)

G (1 + ν)

G (1− ν +Q)

G (1− ν)

(−1)Q(Q+1)/2

Γ (ν + 1)Q′Γ (−ν)Q

G (ν + 1)

G (ν + 1−Q)

G (ν +Q′ + 1−Q)

G (ν +Q′ + 1)=

= (−1)Q(Q+1)/2 G (1− ν +Q)

G (1 + ν −Q)

G (1 + ν +Q′ −Q)

G (1− ν) Γ (1 + ν)Q′Γ (−ν)Q

.

116


Using the recurrence relation

G (1− ν +Q)

G (1 + ν −Q)= (−1)Q(Q−1)/2 G (1− ν)

G (1 + ν)

( π

sin πν

)Q,

and the reflection formula Γ (−ν) Γ (1 + ν) = − πsinπν

, the last expression can be rewrit-ten as

C (ν|Q′ −Q) := Z bif

(ν∣∣pQ′ , ∅; pQ, ∅

)=

G (1 + ν +Q′ −Q)

G (1 + ν) Γ (1 + ν)Q′−Q .

Next let us rewrite the expression for Z bif (ν|Y ′, Q′;Y,Q) for charged Young di-agrams in terms of uncharged ones. To do this, we will try to understand how thisexpression changes under the following transformation, shifting in particular all par-ticle/hole coordinates associated to Y ′:

p′i 7→ p′i + 1, q′i 7→ q′i − 1, ν 7→ ν − 1.

It should also be specified that if we had q′ = 12, then this value should be dropped

from the new set of hole coordinates; if not, we should add a new particle at p′ = 12.

Looking at Fig. 4.12, one may understand that this transformation is exactly the shiftQ′ 7→ Q′ + 1 preserving the form of the Young diagram.

Now compute what happens with Z bif (ν|Y ′, Q′;Y,Q). One should distinguish twocases:

1. If there is no hole at q′ = 12

in (Y ′, Q′), then it follows from (4.111) that

Z bif (ν − 1|Q′ + 1, Y ′;Q, Y )

Z bif (ν|Q′, Y ′;Q, Y )=

=

∏i

(ν − 1

2+ qi

)∏i

(ν − 1

2− pi

) ∏i

ν

ν + qi − 12

∏i

−ν + pi + 12

ν× ν |p′|ν−|q′| = νQ

′−Q.

2. Similarly, if there is a hole at q′ = 12

to be removed, then

Z bif(ν − 1|Q′ + 1, Y ′;Q, Y )

Z bif (ν|Q′, Y ′;Q, Y )=

= ν−1

∏i

(ν − 1

2+ qi

)∏j

(ν − 1

2− pj

) × ν |p′|ν−|q′|+1∏i

ν − pi − 12

ν

∏i

ν

ν − 12

+ qi= νQ

′−Q.

The computation of the shift of Q is absolutely analogous thanks to the symmetryproperties of Z bif .

Introducing

Z?bif (ν|Q′, Y ′;Q, Y ) =

Z bif (ν|Q′, Y ′;Q, Y )

C (ν|Q′ −Q),

it is now straightforward to check that

Z?bif (ν − 1|Q′ + 1, Y ′;Q, Y )

Z?bif (ν|Q′, Y ′;Q, Y )

=Z?

bif (ν + 1|Q′, Y ′;Q+ 1, Y )

Z?bif (ν|Q′, Y ′;Q, Y )

= 1,

117


and therefore Z?bif (ν|Q′, Y ′;Q, Y ) = Z?

bif (ν +Q′ −Q|0, Y ′; 0, Y ). Finally, combiningthis recurrence relation with C (ν|0) = 1, one obtains the identity

Z bif (ν|Q′, Y ′;Q, Y )

C (ν|Q′ −Q)= Z bif (ν +Q′ −Q|Y ′, Y ) ,

which is equivalent to the statement of the theorem.

Step 4

At this point, we have already shown that

Z~Y ′,m′

~Y ,m(T ) = ±ei(δ1η′+−δ1η′−)m′+i(δ1η+−δ1η−)mZ

~Y ′, ~Q′

~Y , ~Q(T ) .

It remains to check the signs in the reference limit described above. Note that

sgn(Z)

= 1, since sgn (C (ν|Q′, Q)) = 1 and sgn (Z bif (ν|Y ′, Y )) = 1 as ν → ∞.

Everywhere in this subsection the calculations are done modulo 2.First let us compute the sign of the non-diagonal part of Z. To do this, one has

to fix the ordering as

xI : p′+ + σk−1, p′− − σk−1, −q+ − θk + σk, −q− − θk − σk,

yI : − q′+ + σk−1, −q′− − σk−1, p+ − θk + σk, p− − θk − σk.

The variables in each of these groups are ordered as p1, p2, . . . where p1 > p2 > . . .This gives

lsgn(Z|non−diag

)= |p′−| · |q′+|+ |q+| ·

(|q′+|+ |q′−|+ |p+|

)+ |q−| ·

(|q′+|+ |q′−|+ |p+|+ |p−|

)+

+|q+| (|q+| − 1)

2+|q−| (|q−| − 1)

2+|p+| (|p+| − 1)

2+|p−| (|p−| − 1)

2+

+|q′+| · (|q′−|+ |p+|+ |p−|) + |q′−| · (|p+|+ |p−|) + |p+| · |p−|.

Using the charge balance conditions

|p+| − |q+| = |q−| − |p−| = m,

|p′+| − |q′+| = |q′−| − |p′−| = m′,

the above expression can be simplified to

lsgn(Z|non−diag

)= m+m′ +m|p+|+m′|p′+|+ |p+|+ |p−|.

Next compute the sign of the diagonal part,

lsgn(Z|diag

)=∑

(p′− + q′+ + q+ + p−) +|p′−| − |q′+|+ |q+| − |p−|

2.

Combining these two expressions, after some simplification we get

lsgn (Z) = |p+|·|q+|+|p′+|·|q′+|+∑(

q+ +1

2

)+∑(

q′+ +1

2

)+∑(

p− +1

2

)+∑(

p′− +1

2

)+m.

118


This expression can be represented as

lsgn (Z) =: lsgn (p, q) + lsgn (p′, q′) +m.

To get the desired formula, one has to absorb m by adding extra shift eiδ2η+ = −1.Combining this shift with the previous formulas (4.115), we deduce the full shift(4.107) of the Fourier transformation parameters. The final formula for the relativesign is

lsgn(Z/Z

)= lsgn (p, q) + lsgn (p′, q′) ,

which completes our calculation.

119

5Exact conformal blocks for the W-algebras,

twist fields and isomonodromicdeformations

Abstract

We consider the conformal blocks in the theories with extended conformal W-symmetryfor the integer Virasoro central charges. We show that these blocks for the generalizedtwist fields on sphere can be computed exactly in terms of the free field theory on thecovering Riemann surface, even for a non-abelian monodromy group. The generalizedtwist fields are identified with particular primary fields of the W-algebra, and we pro-pose a straightforward way to compute their W-charges. We demonstrate how theseexact conformal blocks can be effectively computed using the technique arisen fromthe gauge theory/CFT correspondence. We discuss also their direct relation withthe isomonodromic tau-function for the quasipermutation monodromy data, whichcan be an encouraging step on the way of definition of generic conformal blocks forW-algebra using the isomonodromy/CFT correspondence.

Introduction

An interest to conformal field theories (CFT) with extended nonlinear W-symmetrygenerated by the higher spin holomorphic currents has long history, starting from theoriginal work [ZamW]. These theories resemble many features of ordinary CFT (withonly Virasoro symmetry), like free field representation and degenerate fields [FZ, FL],but it already turns to be impossible to construct in generic situation their conformalblocks [BW] (or the blocks for the algebra of higher spin W-currents) which are themain ingredients in the bootstrap definition of the physical correlation functions.

This interest has been seriously supported in the context of rather nontrivial corre-spondence between two-dimensional CFT and four-dimensional supersymmetric gaugetheory [LMN, NO, AGT], where the conformal blocks have to be compared with theNekrasov instanton partition functions [Nek, NP] producing in the quasiclassical limitthe Seiberg-Witten prepotentials [SW]. This correspondence meets serious difficultiesbeyond the level of the SU(2) gauge quivers on gauge theory side, i.e. for the higherrank gauge groups, which should correspond to the not yet defined generic blocks of

121

5. Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations

the W-conformal theories. It is already clear, however, that the technique developedin two-dimensional CFT can be applied to four-dimensional gauge theories, and viceversa. Following [KriW, Mtau, GMqui] we are going to demonstrate how it can saveefforts for the computation of the exact conformal blocks for the twist fields in theorieswith W-symmetry.

Even in the Virasoro case generic conformal block is a very nontrivial special func-tion [BPZ], but there exists two important particular cases where the answer is knownalmost in explicit form – the correlation functions containing degenerate fields (whichare related to the integrals of hypergeometric type) and the exact Zamolodchikovblocks for a nontrivial (though c = 1) theory [ZamAT86, ZamAT87, ApiZam] 1. Thefirst class can be generalized to the case of W-algebras, where similar hypergeometricformulas arise in the case of so-called completely degenerate fields [FLitv07]. Thealgebraic definition still exists when degeneracy is not complete, and in this casethe most effective way of computation comes from use of the gauge theory Nekrasovfunctions.

Below we are going to study the W-analogs of the Zamolodchikov conformal blocks,which do not belong to the class of algebraic ones. They can be nevertheless computedexactly, partially using the methods of gauge theories and corresponding integrablesystems. We are going to demonstrate also their direct relations with exactly knownisomonodromic τ -functions [SMJ], which confirms therefore their role as an impor-tant example of a generic W-block which can be possible defined (for integer centralcharges) in terms of corresponding isomonodromic problem [Gav].

The exact conformal blocks of the W-algebras are closely related to the correlationfunctions of the twist fields, studied long ago in the context of perturbative stringtheory (see e.g. [Knizhnik, BR, DFMS]). However, unlike [ZamAT87], the correlatorsof the twist fields in these papers were not really expressed through the conformalblocks, and therefore their relation to the W-algebras remained out of interest, so weare going to fill partially this gap.

The chapter is organized as follows. In sect. 6.4 we define the correlators of cur-rents on sphere in presence of the twist fields, and show how they can be computedin terms of free conformal field theory on the cover. In sect. 5.3 we identify the twistfields with the primary fields of the W-algebra and propose a way to extract the valuesof their quantum numbers from the previously computed correlation functions of thecurrents. We also show there that these W-charges have obvious meaning in terms ofthe eigenvalues of the quasipermutation monodromy matrices. In sect. 5.4 we definethe result for the exact conformal block in terms of integrable systems. In particular,we show that the main classical contribution to the result satisfies the well-knownSeiberg-Witten (SW) period equation [SW, KriW], moreover, in this case they can beimmediately solved, which gives the most effective way to express the answer throughthe period matrix and the prime form on the covering surface. Next, in sect. 5.5 wediscuss the connection of the W-algebra conformal blocks with the τ -function of theisomonodromic problem, and show that the W-blocks we have constructed correspondin this context to the τ -function for the case of quasipermutation monodromy data.In sect. 5.6 we construct some explicit examples, and some extra technical information

1Strictly speaking the CFT-Painleve correspondence [GIL12] gives rise to a collection of newexact conformal blocks, coming from the algebraic solutions of Painleve VI.

122

5.2. Twist fields and branched covers

(the recursion procedure we have used for construction of correlators of the higherW-currents, the discussion of their OPE with the stress-tensor, and the computationof the asymptotics of the period matrix on the cover and its relation with the struc-ture constants in the expansion of the isomonodromic τ -function) is located in theAppendix.

Twist fields and branched covers

Definition

We start now with the construction of the conformal blocks of W (slN) = WN alge-bra at integer Virasoro central charges c = N − 1 following the lines of [ZamAT87,Knizhnik, BR]. It is well-known [FL] that WN algebra has free-field representation interms of N − 1 bosonic fields with the currents Ja(z) = i∂φa(z) satisfying operatorproduct expansion (OPE)

Ja(z)J b(z′) =Kab

(z − z′)2+ reg. (5.1)

where Kab is the scalar product in the Cartan subalgebra h ⊂ g = slN . For the

current J(z) =N−1∑a=1

haJa(z) = i∂φ(z), where ha is the basis in h, it is useful to

introduce explicit components

Ji(z) = (ei, J(z)), i = 1, . . . , N (5.2)

with ei being the weights of the first fundamental or vector representation, so that

Ji(z)Jj(z′) =

(ei, ej)

(z − z′)2+ reg. =

δij − 1N

(z − z′)2+ reg. (5.3)

All high-spin currents of the WN -algebra at c = N − 1 are elementary symmetricpolynomials of Ji(z) (

∑i

Ji(z) = 0), e.g. the first three are

T (z) = −W2(z) =1

2: (J(z), J(z)) :=

1

2

∑i

: Ji(z)2 :

W (z) = W3(z) =∑i<j<k

: Ji(z)Jj(z)Jk(z) :=1

3

∑i

: Ji(z)3 :

W4(z) =∑

i<j<k<l

: Ji(z)Jj(z)Jk(z)Jl(z) :=1

8:

(∑i

J2i (z)

)2

: −1

4

∑i

: J4i (z) :

(5.4)

and the primary fields for the current algebra are exponentials of φ(z) ∈ h

Vθ(z) = ei(θ,φ(z)) (5.5)

with the corresponding eigenvalues wk(θ) of the zero modes of the Wk(z)-generatorsgiven by symmetric functions of (ei,θ).

123


Now we are going to introduce new fields Os(z), which are still primary for allhigh-spin currents Wk(z), but not for the currents Ji(z). They can be realized asmonodromy fields

γq : J(z)Os(q) 7→ s(J(z))Os(q) (5.6)

for some contours γq encircling the point q on the base curve, where s ∈ WslN = SN isan element of the corresponding Weyl group. The particular cases of this constructionwere known for the Abelian monodromy group of the cover [BR, ZamAT87], but eventhere in the cases with N > 2 they were not identified with WN primary fields.

Now we are going to construct the particular conformal block (on P1 with globalcoordinate z), where all monodromy fields can be grouped as Os(q2i+1)Os−1(q2i+2) atq2i+1 → q2i+2, so that one can take an OPE

Os(z)Os−1(z′) =∑θ

Cs,θ(z − z′)∆(θ)−2∆(s) (Vθ(z′) + descendants) (5.7)

and fix the quantum numbers in the intermediate channels, where there are onlythe fields with definite h = u(1)N−1 charges 1

2πi

¸zdζJ(ζ)Vθ(z) = θ · Vθ(z). In or-

der to do this consider G0(q) = G0(q1, ..., q2L), together with 1-form Gi1(z|q)dz =Gi1(z|q1, ..., q2L)dz and bidifferential Gij2 (z, z′|q)dzdz′ = Gij2 (z, z′|q1, ..., q2L)dzdz′, where

G0(q1, ..., q2L) = 〈Os1(q1)Os−11

(q2)...OsL(q2L−1)Os−1L

(q2L)〉Gi1(z|q1, ..., q2L) = 〈Ji(z)Os1(q1)Os−1

1(q2)...OsL(q2L−1)Os−1

L(q2L)〉

Gij2 (z, z′|q1, ..., q2L) = 〈Ji(z)Jj(z′)Os1(q1)Os−1

1(q2)...OsL(q2L−1)Os−1

L(q2L)〉

(5.8)

which become single-valued on the cover π : C → P1 with the branch points qα andcorresponding monodromies sα. The indices i, j are just labels of the sheets of thiscover, so the multi-valued differentials (5.8) on P1 are now expressed in terms of thesingle-valued G1(ξ|q1, ..., q2L)dξ and G2(ξ, ξ′|q1, ..., q2L)dξdξ′ on the covering surface C:

Gi1(z|q1, ..., q2L)dz = G1(zi|q1, ..., q2L)dzi

Gij2 (z, z′|q1, ..., q2L)dzdz′ = G2(zi, z′j|q1, ..., q2L)dzidz′j(5.9)

where zi = π−1(z)i is the coordinate at i’th preimage of the point z, not the power(note that number i is not defined globally due to the presence of monodromies). Weshould also point out that only local deformations of the positions of the branch pointsqα are allowed, since the global ones – due to nontrivial monodromies – can changethe global structure of the cover π : C → P1. This leads in particular to the fact thatin the case of non-abelian monodromy group the positions of the branch points qαcannot play the role of the global coordinates on the corresponding Hurwitz space.2

The picture of the 3-sheeted cover with the most simple branch cuts looks likeat fig.5.1, where we have shown explicitly three (dependent) cycles in H1(C) corre-sponding to the cuts between the positions of the fields, labeled by mutually inversepermutations. To understand our notations better we present also at fig.5.2 the pic-ture of the vicinity of the branch-point (of the 6-sheeted cover) of the cyclic types ∼ [3, 1, 2] with several independent permutation cycles.

2Although, sometimes the Hurwitz space of our interest occurs to be rational, and in this caseone can choose some global coordinates – but not the positions of the branch points. An explicitexample is considered below in sect.5.6.

124


Figure 5.1: Covering Riemann surface C with simplest cuts between the positions ofcolliding twist-fields. Sum of the shown cycles of A-type vanishes in H1(C).

Figure 5.2: Vicinity of a ramification point of a general type.

125


Correlators with the current

Consider a permutation of the cyclic type s ∼ [l1, ..., lk], which corresponds to theramification at z = q (for simplicity we put q = 0) with k preimages qi, π(qi) = qwith multiplicities li. The coordinates in the vicinity of these points can be chosen asξi = z1/li . One can write down a general expression for the expansion of current J(z)on the cover

J(z) =k∑i=1

li−1∑vi=1

∑n∈Z

a(i)n−vi/li · hi,viz1+n−vi/li

+k−1∑j=1

∑n∈Z

b(j)n ·Hj

zn+1(5.10)

where hi,vi and Hj form the orthogonal basis in h out of the eigenvectors of thepermutation s, and in coordinates (related to the weights ei) they have the form

h1,v1 = (1, e2πi·v1/l1 , ..., e2πi(l1−1)·v1/l1 ; 0, ..., 0; ...; 0, ..., 0)

h2,v2 = (0, ...0; 1, e2πi·v2/l2 , ..., e2πi(l2−1)·v2/l2 ; 0, ..., 0; ...; 0, ..., 0)

Hj = (y(1)j , ..., y

(1)j ; y

(2)j , ..., y

(2)j ; ....; y

(k)j , ..., y

(k)j )

∑i

liy(i)j = 0

(5.11)

with hi,vi , corresponding to non-zero eigenvalues of the permutation cycles si, whileHj – to the trivial permutations.

The expansion modes satisfy usual Heisenberg commutation relations [a(i)u , a

(j)v ] =

uδu+vδij, [b(i)u , b

(j)v ] = uδu+vδij, up to possible inessential numerical factors which can

be extracted from the singularity of the OPE J(z)J(z′). The condition that fieldOs(q) is primary for the W-currents means in terms of the corresponding state that

a(i)ui|s〉 = b(j)

n |s〉 = 0, ui > 0, n > 0, ∀ i, j (5.12)

and this state is also an eigenvector of the zero modes b(j)0 ∀ j. The corresponding

eigenvalues are extra quantum numbers – the charges, which have to be included intothe definition of the state |s〉 → |s, r〉 (and Os(q) → Os,r(q)) and fixed by expansionof the h-valued 1-form dzJ(z)|s〉 at z → 0, i.e.

dz

zJ(z)|s, r〉 =

dz

z

N∑i=1

riei|s, r〉+ reg. (5.13)

where r1 = . . . = rl1 , rl1+1 = . . . = rl1+l2 , etc: the U(1) charges are obviously thesame for each point of the cover, they also satisfy the slN condition

N∑i=1

riα = 0, ∀ α (5.14)

for each branch point q ∈ qα. It means that G1(z)dz on the cover C has only poleswith prescribed by (5.13) singularities, so one can write

G1(ξ|q)dξ

G0(q)=

2L∑α=1

dΩrα +

g∑I=1

aIdωI = dS (5.15)

126


and we shall call this 1-form as the Seiberg-Witten (SW) differential, since its periodsover the cycles in H1(C) play important role in what follows. Here dωI, I = 1, . . . , gare the canonically normalized first kind Abelian holomorphic differentials

1

2πi

˛AI

dωJ = δIJ

(in slightly unconventional normalization of [Dub] as compare to [Fay, Mumford]),while

dΩrα =N∑i=1

riαdΩqiα,p0

is the third kind meromorphic Abelian differential with the simple poles at all preim-

ages of qα (with the expansion dΩrα =piα

liαriαdξiαξiα

+ reg. in corresponding local coor-

dinates) and vanishing A-periods. We denote by qiα = π−1(qα), i = 1, . . . , N thepreimages on C of the point qα, with such conventions the point of multiplicity liα hasto be counted liα times ( Res piαdΩrα = liαr

iα).

The A-periods of the differential (5.15)

aI =1

2πi

˛AI

dS =1

2πi

˛AI

dξG1(ξ|q)G0(q)

, I = 1, . . . , g (5.16)

are determined by fixed charges in the intermediate channels due to (5.7). The numberof these constraints is ensured by the Riemann-Hurwitz formula χ(C) = N · χ(P1)−#BP for the cover π : C → P1, or

g =L∑α=1

kα∑j=1

(lαj − 1

)−N + 1 =

L∑α=1

(N − kα)−N + 1 (5.17)

where kα stands for the number of cycles in the permutation sα. One can easily seethis in the “weak-coupling” regime, when we can apply (5.7) in the limit q2α−1 → q2α,so that

G0(q1, ..., q2L)|θ = 〈Os1(q1)Os−11

(q2)∣∣∣θ1

. . . OsL(q2L−1)Os−1L

(q2L)∣∣∣θL〉 ∼

∼q2α−1→q2α

〈L∏α=1

Vθα(q2α)〉+ . . .(5.18)

and the charge conservation law∑L

α=1 θα = 0 gives exactly N − 1 constraints to

the parameters θα, whose total number is∑L

α=1 (N − kα), since for each pair ofcolliding ends of the cut (i.e. α = 1, . . . , L) there are kα linear relations for the Nintegrals over the contours, encircling two colliding ramification points, see fig.5.1(this procedure also gives a way to choose convenient basis in H1(C) as shown on thispicture). For the dual B-periods of (5.15) one gets

aDI =

˛

BI

dS = TIJaJ + UI , I = 1, . . . , g (5.19)

127


where the last term can be transformed using the Riemann bilinear relations (RBR)as

UJ =∑α

˛

BJ

dΩrα =∑α,m

rmα AJ(qmα ), J = 1, . . . , g (5.20)

where AJ(p) =´ pp0dωJ is the Abel map of a point p ∈ C, and UJ do not depend on

the reference point p0 ∈ C due to (5.14).

Stress-tensor and projective connection

Similarly the 2-differential from (5.8) is fixed by its analytic properties and one canwrite

G2(p′, p|q)

G0(q)dξp′dξp = dS(p′)dS(p) +K(p′, p)− 1

NK0(p′, p) (5.21)

where

K(p′, p) = dξp′dξp logE(p′, p) =dξp′dξp

(ξp′ − ξp)2+ reg.,

˛AI

K(p′, p) = 0 (5.22)

is the canonical meromorphic bidifferential on C (the double logarithmic derivative ofthe prime form, see [Fay]), normalized on vanishing A-periods in each of two variables,while

K0 =dπ(ξ)dπ(ξ′)

(π(ξ)− π(ξ′))2(5.23)

is just the pull-back π∗ of the bidifferential dzdz′

(z−z′)2 from P1. Formula (5.21) is fixedby the following properties: in each of two variables it has almost the same structureas G1(ξ)dξ, but with extra singularity on diagonal p′ = p, which comes from (5.3), italso satisfies an obvious condition

∑i

Gij2 (z, z′) =∑j

Gij2 (z, z′) = 0

Now one can define [Fay] the projective connection tx(p) by subtracting the sin-gular part of (5.22)

tx(p)dx2 =

1

2

(K(p′, p)− dx(p′)dx(p)

(x(p′)− x(p))2

)∣∣∣∣p′=p

(5.24)

It depends on the choice of the local coordinate x(p), and it is easy to check that

tx(p)dx2 − tξ(p)dξ2 =

1

12ξ, xdx2 (5.25)

where ξ, x = (Sξ)(x) = ξxxxξx− 3

2

(ξxxξx

)2

is the Schwarzian derivative.

It is almost obvious that expression (5.24) is directly related with the averageof the Sugawara stress-tensor T (z) (5.4) of conformal field theory (with extendedW-symmetry), since normal ordering of free bosonic currents exactly results in sub-traction of its singular part. One gets in this way from (5.21) that

〈: 12Ji(z)Ji(z) : Os1(q1)Os−1

1(q2) . . .OsL(q2L−1)Os−1

L(q2L)〉

〈Os1(q1)Os−11


(q2L)〉=

= tz(zi) +

1

2

(dS(zi)

dz

)2(5.26)

128

5.3. W-charges for the twist fields

where z = z(p) is the global coordinate on P1, and we have used that after subtraction(5.24) one can substitute K 7→ 2tz(p)dz

2 and K0 7→ 0, leading to

〈T (z)〉O =〈T (z)Os1(q1)Os−1

1(q2) . . .OsL(q2L−1)Os−1

L(q2L)〉

〈Os1(q1)Os−11

(q2) . . .OsL(q2L−1)Os−1L

(q2L)〉=

=∑π(p)=z

(tz(p) +

1

2

(dS(p)

dz

)2) (5.27)

where sum in the r.h.s. computes the pushforward π∗, appeared here as a result ofsummation in (5.4).

W-charges for the twist fields

Conformal dimensions for quasi-permutation operators

Using the OPE with the stress-tensor T (z)

T (z)Os,r(q) =∆(s, r)Os,r(q)

(z − q)2+∂qOs,r(q)z − q

+ reg. (5.28)

one can extract from the singularities of (5.27) the dimensions of the twist fields.Following [BR] we first notice from (5.24) that near the branch point (e.g. at q = 0)the local coordinate is ξi = z1/li , so that

tz(p) = tξ(p)

(dξ

dz

)2

+1

12ξ, z = tξ(p)z

2/li−2 +l2 − 1

24l21

z2(5.29)

The first term in the r.h.s. cannot contain 1z2 -singularity, since tξ(p) is regular in local

coordinate on the cover C. The second source of the second-order pole in (5.27) comesfrom the poles of the Seiberg-Witten differential (5.15), which look as

dS ≈ rilidξiξi

+ reg. = ridz

z+ reg. (5.30)

Taking them into account together with (5.29) one comes finally to the formula

∆(s, r) =k∑i=1

l2i − 1

24li+

k∑i=1

1

2lir

2i (5.31)

which gives the full conformal dimension for the twist fields with r-charges.Since we are going to use this formula intensively below, let us illustrate first, how

it works in the first two nontrivial cases:

• N = 2: there are only two possible cyclic types:

– s ∼ [1, 1], then l1 = l2 = 1, r1 = −r2 = r, so ∆(s, r) = r2 is only given bythe r-charges;

129


– s ∼ [2], then the only l1 = 2, the single r-charge must vanish, so one justgets here the original Zamolodchikov’s twist field with ∆(s, r) = 1

16.

• N = 3: here one has three possible cyclic types:

– s ∼ [1, 1, 1], then l1 = l2 = l3 = 1, r1+r2+r3 = 0, ∆(s, r) = 12

(r21 + r2

2 + r23)

– s ∼ [2, 1], then l1 = 2, l3 = 1, r1 = r2 = r, r3 = −2r, ∆(s, r) = 116

+ 3r2

– s ∼ [3], then l1 = 3, the single r-charge again should vanish, so that thedimension is ∆(s, r) = 1

9.

Quasipermutation matrices

The hypothesis of the isomonodromy-CFT correspondence [Gav] relates the con-structed above twist fields to the quasipermutation monodromies (we return to thisissue in more details later). This correspondence relates the WN charges of the twistfields to the symmetric functions of eigenvalues of the logarithms of the quasipermu-tation monodromy matrices

Mα ∼ e2πiθα , α = 1, . . . , 2L , (5.32)

being the elements of the semidirect product SN n (C×)N

(here we consider only thematrices with detMα = 1). An example of the quasipermutation matrix of cyclictype s ∼ [3, 2] is

M =

0 a1e

2πir1 0 0 00 0 a2e

2πir1 0 0a3e

2πir1 0 0 0 00 0 0 0 b1e

2πir2

0 0 0 b2e2πir2 0

(5.33)

where a1a2a3 = 1, b1b2 = −1, 3r1 + 2r2 = 0 to get detM = 1. A generic quasipermu-tation is decomposed into several blocks of the sizes li, each of these blocks is givenby

e2πiri × eiπliε(li)sli , i = 1, . . . , k

where sli is the cyclic permutation of length li, ε(l) = 0 for l-odd and ε(l) = 1 forl-even. It is easy to check that eigenvalues of such matrices are

λi,vi = e2πiθi,vi = e2πi(ri+

vili

), i = 1, . . . , k

vi =1− li

2,1− li

2+ 1 . . . ,

li − 1

2− 1,

li − 1

2

(5.34)

According to relation (5.32) the conformal dimension of the corresponding field is

∆(M) =1

2

∑θ2i,vi

=1

2

∑(ri +

vili

)2

=k∑i=1

l2i − 1

24li+

k∑i=1

1

2lir

2i (5.35)

130


where we have used that∑vi = 0 for any fixed i = 1, . . . , k, and

l(l2 − 1)

12=

(l−1)/2∑−(l−1)/2

v2 l = 2m+ 1 (v ∈ Z)

(l−1)/2∑−(l−1)/2

v2 l = 2m (v ∈ Z + 12)

(5.36)

for both even or odd l ∈ li. The calculation (5.35) for the quasipermutation matricesreproduces exactly the CFT formula (5.31), confirming the correspondence.

W3 current

One can also perform a similar relatively simple check for the first higher W3-current.An obvious generalization of (5.35) gives

w3(M) =∑a<b<c

(ra +vala

)(rb +vblb

)(rc +vclc

) =1

3

∑a

(ra +vala

)3 =

=1

3

∑a

r3a +

∑a

rav2a

l2a=

k∑i=1

1

3lir

3i +

k∑i=1

ril2i − 1

12li

(5.37)

To extract such formulas from conformal field theory one has to analyze the multi-current correlation functions in presence of twist operators and action of the corre-sponding modes of the Wk(z) currents. For W = W3(z), following (5.8) one can firstdefine

Gijk3 (z, z′, z′′|q)dzdz′dz′′ = Gijk3 (z, z′, z′′|q1, ..., q2L)dzdz′dz′′ =

= 〈Ji(z)Jj(z′)Jk(z

′′)Os1(q1)Os−11


(q2L)〉dzdz′dz′′(5.38)

and write, similarly to (5.21)

G3(p′′, p′, p|q)

G0(q)dξp′′dξp′dξp = dS(p′′)dS(p′)dS(p)+

+dS(p′′)

(K(p′, p)− 1

NK0(p′, p)

)+ dS(p′)

(K(p′′, p)− 1

NK0(p′′, p)

)+

+dS(p)

(K(p′′, p′)− 1

NK0(p′′, p′)

) (5.39)

where the r.h.s. has appropriate singularities at all diagonals and correct A-periods ineach of three variables. Extracting singularities and using (5.4), (5.24) one can write

〈W (z)〉O =∑π(p)=z

(1

3

(dS(p)

dz(p)

)3

+ 2tz(p)dS(p)

dz(p)

)(5.40)

It is easy to see that due to (5.29), (5.30) this formula gives the same result as (5.37).

131


Formula (5.34) also shows, how the charges of the twist fields can be seen withinthe context of W-algebras. It is important, for example, that for the complete cyclepermutation one would get its WN charges w2(θ), w3(θ), . . . , wN(θ), where

θ =ρ

N=

1

N

(N − 1

2,N − 1

2− 1, . . . ,

1−N2

+ 1,1−N

2

)(5.41)

i.e. the vector of charges is proportional to the Weyl vector of g = slN . Such fieldsare non-degenerate from the point of view of the WN algebra, since for degeneratefields the charge vector always satisfy the condition (θ, α) ∈ Z for some root α.It means that here we are beyond the algebraically defined conformal blocks, andfurther investigation of descendants W−1O etc can shed light on the structure ofgeneric conformal blocks for the W-algebras. We are going to return to this issueelsewhere.

Higher W-currents

For the higher W-currents (Wk(z) with k > 3) the situation becomes far more com-plicated. We discuss here briefly only the case of W4(z), which already gives a hint onwhat happens in generic situation. An analog of (5.35), (5.37) gives for the quasiper-mutation matrices

w4(M) =∑

a<b<c<d

(ra +vala

)(rb +vblb

)(rc +vclc

)(rd +vdld

) =1

2∆(M)2 − 1

4A (5.42)

with ∆(M) given by (5.35) and

A =N∑a=1

(ra +

vala

)4

=k∑i=1

lir4i + 6

k∑i=1

r2i

l2i − 1

12li+

k∑i=1

(l2i − 1)(3l2i − 7)

240l3i(5.43)

To get this from CFT one needs just the most singular part of the correlation function

〈W4(z)〉O(dz)4 =z→q

w4

(dz

z − q

)4

+ . . . (5.44)

which is a particular case of the current correlators

Ri1,...in(z1, . . . zn) = 〈: Ji1(z1), . . . Jin(zn) :〉Odz1 . . . dzn (5.45)

and the technique of calculation of such expressions is developed in Appendix 5.8.From the definition of the W4(z) current (5.4) it is clear, that one should take only

the most singular parts of the correlation functions of four currents

Riiii(z, z, z, z) =ttiittiittiittiittiittii+ +6 · 3 · =

= dS(zi)4 + 6dS(zi)2Kii(z, z) + 3Kii(z, z)2 (5.46)

and

132


Riijj(z, z, z, z) =ttiittjjttiittjjttiittjj+ + +

ttiittjjttiittjjttiittjj+ + +4 · 2 · =

= dS(zi)2dS(zj)2 + Kii(z, z)dS(zj)2 + Kjj(z, z)dS(zi)2+

+4Kij(z, z)dS(zi)dS(zj) + Kii(z, z)Kjj(z, z) + 2Kij(z, z)2(5.47)

taken at the coinciding values of all arguments. It means, that one has to substitute

dS(zi) = ridz

z+ . . . (5.48)

(we again put here q = 0 for simplicity) and do the same for the propagator Kij(z1, z2) =K(zi1, z

j2)− δijK0(z1, z2) (see Appendix 5.8 for details), i.e. to substitute into (5.46),

(5.47)

Kii(z, z) =dz1/ldz1/l

(z1/l − z1/l)2− dzdz

(z − z)2

∣∣∣∣z→z

+ . . . =l2 − 1

12l2(dz)2

z2+ . . .

Kij(z, z) =ζ idz1/lζjdz1/l

(ζ iz1/l − ζjz1/l)2+ . . . =

1

l2ζ i−j

(1− ζ i−j)2

(dz)2

z2+ . . .

(5.49)

where ζ = exp(

2πil

). In order to compute −1

4

∑i

Riiii(z, z, z, z) + 18

∑i,j

Riijj(z, z, z, z)

it is useful to move the term Kij(z, z)2 from the second expression to the first one,which gives ∑

i

dS(zi)4 + 6∑i

dS(zi)2Kii(z, z) + 3∑i

Kii(z, z)2−

−∑ij

Kij(z, z)2 →(5.48),(5.49)

A

(dz

z

)4 (5.50)

while the rest from (5.47) gives rise to(∑i

dS(zi)2 +∑i

Kii(z, z)

)2

+ 4∑ij

Kij(z, z)dS(zi)dS(zj) →(5.48),(5.49)

4∆2N

(dz

z

)4

(5.51)after using (5.48), (5.49) and several nice formulas like

1

l

l−1∑j=1

ζj

(1− ζj)2=

1

l

l−1∑j=1

e2πij/l

(1− e2πij/l)2= −

(l−1)/2∑v=(1−l)/2

v2

l2

1

l3

l−1∑j=1

ζ2j

(1− ζj)4=

1

l3

l−1∑j=1

e4πij/l

(1− e2πij/l)4=

2

l

(l−1)/2∑v=(1−l)/2

v2

l2

2

−(l−1)/2∑v=(1−l)/2

v4

l4

(5.52)

133


Here the sum over the roots of unity can be performed using the contour integral

l−1∑j=1

ζjm

(1− ζj)2m=

1

2πi

˛

z 6=1

d logzl − 1

z − 1· zm

(1− z)2m=

= Res z=1d logz − 1

zl − 1· zm

(1− z)2m

(5.53)

and the result indeed allows to identify the coefficients at maximal singularities in(5.50), (5.51) with the expressions (5.43). It means that the conformal charge (5.44)of the twist field indeed coincide with the corresponding symmetric function (5.42) ofthe eigenvalues of the permutation matrix, but it comes here already from a nontrivialcomputation.

It is known from long ago that already a definition of the higher W-currents is anontrivial issue (see e.g. [FZ, FL, Bil, MarMor, FLitv12]). Here it was important toconsider the particular (normally ordered) symmetric function of the currents (5.4),since, for example, another natural choice

∑i

: J4i (z) : is even not contained in the

algebra generated by T (z), W3(z) and W4(z). However, the so defined W4(z)-currentis not a primary field of conformal algebra, we discuss this issue in Appendix 5.9.

Conformal blocks and τ-functions

Consider now the next singular term from the OPE (5.28), which immediately allowsto extract from (5.27) the accessory parameters

∂

∂qαlog G0(q1, ..., q2L) =

∑π(qiα)=qα

Res qiαtzdz +1

2

∑π(qiα)=qα

Res qiα(dS)2

dz (5.54)

Computing residues in the r.h.s. one gets the set of differential equations (α =1, . . . , 2L), which define the correlation function of the twist fields G0(q1, ..., q2L) itself.A non-trivial statement [KriW, GMqui, KK04, KK06] is that these equations arecompatible, moreover (5.54) defines actually two different functions τSW (q) and τB(q),where

∂

∂qαlog τSW (q1, ..., q2L) =

1

2

∑π(qiα)=qα

Res qiα(dS)2

dz (5.55)

and∂

∂qαlog τB(q1, ..., q2L) =

∑π(qiα)=qα

Res qiαtzdz (5.56)

so that G0(q) = τSW (q) · τB(q), and the claim of [Mtau, KK04, KK06] is that boththem are well-defined separately.

Seiberg-Witten integrable system

Let us concentrate attention on τSW = τSW (a,q) or the Seiberg-Witten prepotentialF = log τSW , which is the main contribution to conformal block, and the only one,

134

5.4. Conformal blocks and τ -functions

which depends on the charges in the intermediate channel. According to [Mtau,GMqui] F(a,q), up to some possible only a-dependent term, satisfies also another setof equations

∂

∂aIlog τSW = aDI , I = 1, . . . , g (5.57)

where the dual periods aDI are defined in (5.19). The total system of equations (5.55),(5.57) is also integrable [KriW, Mtau, GMqui] due to the Riemann bilinear relations.Moreover, in our case this system of equations can be easily solved due to

Theorem 5.1. Function

log τSW = 12

∑I,J

aITIJaJ +∑I

aIUI + 12Q(r) (5.58)

solves the system (5.55), iff Q(r) solves the system ∂Q(r)∂qα

=∑

π(qiα)=qα

Res qiα(dΩ)2

dzfor

α = 1, . . . , 2L, dΩ =∑α

dΩrα and other ingredients in the r.h.s. are given by (5.16),

(5.20) and the period matrix of C.

One can check this statement explicitly, using the definitions (5.15) and (5.20)∑π(qiα)=qα

Res qiαdωIdωJdz

= −∑

π(qiα)=qα

Res qiα∂ωI∂qα

dωJ = −˛∂C

∂ωI∂qα

dωJ =

=∂

∂qα

˛BI

dωJ =∂TIJ∂qα

,

(5.59)

where we have first applied the formula ∂ωI∂qα

= −dωIdz

+hol. and then the RBR. Similarly,for the second term:∑

π(qiα)=qα

Res qiαdωIdΩrα

dz= −

∑π(qiα)=qα

Res qiα∂Ωrα

∂qαdωI = −

˛∂C

∂Ωrα

∂qαdωI =

=∂

∂qα

˛BI

dΩrα =∂UI∂qα

,

(5.60)

while the last term Q(r), vanishing after taking the a-derivatives, should be computedseparately, and the proof will be completed in next section.

Quadratic form of r-charges

In the limit aI = 0 equation (5.55) gives us the formula

∂

∂qαQ(r) =

∑qiα∈π−1(qα)

Res qiαdΩ2

dz (5.61)

wheredΩ =

∑α

dΩrα =∑α,i

riαdΩqiα,p0 (5.62)

135


Figure 5.3: Integration path for Q(r)~ε

Theorem 5.2. Regularized expression for Q(r)

Q(r)~ε =∑α,i

riα

ˆ (qiα)εα

p0

dΩ (5.63)

satisfies (5.61) in the limit ε→ 0

Proof: It is useful to introduce the differential with shifted poles

dΩ~ε =∑α,i

riαdΩ(qiα)εα ,p0 (5.64)

Note that due to conditions (5.14) nothing depends on the reference points p0, p0.The regularized points (qαi )εα are defined in such a way that

z ((qαi )εα) = z (qαi )− εα = qα − εα (5.65)

and this is the only place where the coordinate z on P1 enters the definition ofQ(r). Allother parts of τSW do not depend explicitly on the choice of the coordinate z becausethey are given by the periods of some meromorphic differentials on the covering curve.Expression (5.63) can now be rewritten equivalently

Q(r)~ε = − 1

2πi

˛C

Ω~ε dΩ (5.66)

where contour C (see fig.5.3) encircles the branch-cuts of Ω~ε, while the poles of dΩare left outside. Taking the derivatives one gets

∂

∂qαQ(r)~ε =

1

2πi

˛C

[∂Ω

∂qαdΩ~ε −

∂Ω~ε

∂qαdΩ

](5.67)

where each of the terms in r.h.s. contains only the poles at the points qiα and (qiα)εαcorrespondingly. One can therefore shrink the contour of integration in the firstterm onto the points qiα (up to the integration over the boundary of cut Riemannsurface, which vanishes due to the Riemann bilinear relations for the differentialswith vanishing A-periods), and in the second – to the points (qiα)εα , hence

∂

∂qαQ(r)~ε = −

∑i

Res qiα∂Ω

∂qαdΩ~ε −

∑i

Res (qiα)εα

∂Ω~ε

∂qαdΩ (5.68)

136


Near the point piα one can choose the local coordinate ξ such that z = qα + ξl, so thatexpansion of Abelian integrals can be written as

Ω = ri log(z − qα) + c0(q) + c1(q)(z − qα)1/l + c2(q)(z − qα)2/l + . . .

Ω~ε = c0(q) + c1(q)(z − qα)1/l + c2(q)(z − qα)2/l + . . .(5.69)

giving rise to∂Ω

∂qα= −dΩ

dz+∂c0(q)

∂qα+O

((z − qα)1/l

)∂Ω~ε

∂qα= −dΩ

dz+∂c0(q)

∂qα+O

((z − qα − εα)1/l

) (5.70)

Since the differential dΩ~ε is regular near z = qα, one can ignore the regular part whencomputing the residues:

∂

∂qαQ(r)~ε =

∑i

Res qiαdΩ

dzdΩ~ε +

∑i

Res (qiα)εα

dΩ~ε

dzdΩ =

=∑i

1

2πi

˛qiα,(q

iα)εα

dΩdΩ~ε

dz

(5.71)

The r.h.s. of this formula has a limit when εα → 0, so extracting the singular partfrom Q(r)~ε (easily found from the explicit formula below)

Q(r) = Q(r)~ε − 2∑

∆α log εα (5.72)

one gets from (5.71) exactly the formula (5.61). This also completes (together with(5.59), (5.60)) the proof of (5.58).

Using the integration formula for the third kind Abelian differentials [Fay]

ˆ b

a

dΩc,d = logE(c, b)E(d, a)

E(c, a)E(d, b)

one gets from (5.63) an explicit expression

Q(r)~ε =∑α,i,β,j

riαrjβ log

E((qiα)εα , qjβ)E(p0, p0)

E((qiα)εα , p0)E(p0, qjβ)

=∑α,i,β,j

riαrjβ logE((qiα)εα , q

jβ) =

=∑qiα 6=q

jβ

riαrjβ logE(qiα, q

jβ) +

∑α,i

(riα)2liα logE((qiα)εα , qiα)

(5.73)

The first term in the r.h.s. is regular, while for the second one can use

E((qiα)εα , qiα) =

(z − qα + εα)1/liα − (z − qα)1/liα√d(z − qα + εα)1/liαd(z − qα)1/liα

≈ ε1/liαα

d[(z − qα)1/liα

]∣∣∣∣∣z→qα

(5.74)

Therefore

Q(r) =∑qiα 6=q

jβ

riαrjβ logE(qiα, q

jβ)−

∑α,i

(riα)2liα log d[(z − qα)1/l]

∣∣∣∣∣z→qα

(5.75)

137


Substituting expression of the prime form

E(p, p′) =Θ∗(A(p)− A(p′))

h∗(p)h∗(p′)(5.76)

in terms of some odd theta-function Θ∗, the already defined above Abel map A(p),and holomorphic differential

h2∗(p) =

∑I

∂Θ∗(0)

∂AIdωI(p) (5.77)

one can write more explicitly

Q(r) =∑qiα 6=q

jβ

riαrjβ log Θ∗(A(qiα)− A(qjβ))−

∑qiα

(riα)2liα logd(z(q)− qα)1/liα

h2∗(q)

∣∣∣∣∣∣q=qiα

(5.78)If cover C has zero genus g(C) = 0 itself, the prime form is just E(ξ, ξ′) = ξ−ξ′√

dξ√dξ′

in

terms of the globally defined coordinate ξ, and formula (5.78) acquires the form

Q(r) =∑piα 6=p

jβ

riαrjβ log(ξiα − ξ

jβ)−

∑ξiα

(riα)2liα logd(z(ξ)− qα)1/liα

dξ

∣∣∣∣∣∣ξ=ξiα

(5.79)

Below we are going to apply this formula to explicit calculation of a particular examplefor a genus zero cover, but with a non-abelian monodromy group. The result of thecomputation clearly shows that τ -function (5.79) cannot be expressed already in suchcase as a function of positions of the ramification points z = qα on P1, which meansthat the corresponding formula for Q(r) from [K04] can be applied only in the caseof Abelian monodromy group.

Bergman τ-function

The Bergman τ -function, was studied extensively for the different cases [Knizhnik,BR, ZamAT87] from early days of string theory, mostly using the technique of freeconformal theory. Modern results and formalism for this object can be found in [KK04,KK06]. Already from its definition (5.56) τB can be identified with the variation w.r.t.moduli of the complex structure of the one-loop effective action in the free field theoryon the cover.

We are not going to present here an explicit formula for the general Bergmanτ -function, it can be found in [KK06, formula 1.7]. We would like only to point out,that for our purposes of studying the conformal blocks this is the less interestingpart, since it does not depends on quantum numbers of the intermediate channels(it means in particular, that it can be computed just in free field theory). Below insect. 5.6 we present the result of its direct computation in the simplest case with non-abelian monodromy group. The result shows that it arises just as some quasiclassicalrenormalization of the term (5.79) in the classical part.

138


However, as for the SW tau-function, the definition (5.56) is easily seen to beconsistent. Taking one more derivative one gets from this formula

∂2 log τB(q)

∂qα∂qβ=

∂

∂qβ

∑π(p)=qα

Resp

1

dz(p)limp′→p

(K(p′, p)− dz(p′)dz(p)

(z(p′)− z(p))2

)=

=∑

π(p)=qα

Resp

1

dz(p)limp′→p

∂K(p′, p)

∂qβ=

∑π(p)=qα

Resp

1

dz(p)×

× limp′→p

∑π(p′′)=qβ

ResP

K(p′, p′′)K(p, p′′)

dz(p′′)=

∑π(p)=qαπ(p′′)=qβ

Resp,p′′

K(p, p′′)2

dz(p)d(p′′),

(5.80)

where we have used the Rauch variational formula [Fay92, formula 3.21] for the canon-ical meromorphic bidifferential, computed in the points p and p′ with fixed projections

∂K(p′, p)

∂qβ=

∑π(P )=qβ

ResP

K(p′, P )K(p, P )

dz(P ) (5.81)

so that the expression in r.h.s. of (5.80) is symmetric w.r.t. α↔ β.This is certainly a well-known fact, but we would like just to point out here,

that the Rauch formula (5.81), which ensures integrability of (5.56) can be easilyderived itself from the Wick theorem, using the technique, developed in sect. 6.4 andAppendix 5.8. Indeed,

∂K(z′i, zj)

∂qβ=

∂

∂qβ

Gij2 (z′, z|q)

G0(q)dz′idzj =

=

(∂∂qβGij2 (z′, z|q)

G0(q)− G

ij2 (z′, z|q)

G0(q)

∂∂qβG0(q)

G0(q)

)dz′idzj

(5.82)

as follows from (5.21) for the conformal block with two currents inserted Gij2 (z′, z|q) =Gij2 (z′, z|q)0 = 〈Ji(z′)Jj(z)O(q)〉0 when projected to the vanishing a-periods (5.16) orthe charges in the intermediate channels (note, that the Bergman tau-function doesnot depend on these charges). Proceeding with (5.82) and using ∂

∂qβ= Lβ−1 one gets

therefore

∂K(z′i, zj)

∂qβ=

(〈Ji(z′)Jj(z)Lβ−1O(q)〉0

〈O(q)〉0− 〈Ji(z

′)Jj(z)O(q)〉0〈O(q)〉0

〈Lβ−1O(q)〉0〈O(q)〉0

)dz′idzj

(5.83)where we have used the obvious notations

〈O(q)〉0 = 〈2L∏α=1

Oα(qα)〉0 = 〈Os1(q1)Os−11


(q2L)〉0

〈Lβ−1O(q)〉0 = 〈(L−1Oβ(qβ))∏α 6=β

Oα(qα)〉0 = 12

˛qβ

∑k

dζ〈: J2k (ζ) : O(q)〉0

〈Ji(z′)Jj(z)Lβ−1O(q)〉0 = 12

˛qβ

∑k

dζ〈Ji(z′)Jj(z) : J2k (ζ) : O(q)〉0

(5.84)

139


where the integration¸qβdζ is performed on the base P1. Applying now in the r.h.s.

the Wick theorem (see Appendix 5.8 for details), one gets

12〈Ji(z′)Jj(z) : J2

k (ζ) : O(q)〉0〈O(q)〉0 = 12〈Ji(z′)Jj(z)O(q)〉0〈: J2

k (ζ) : O(q)〉0+

+〈Ji(z′)Jk(ζ)O(q)〉0〈Jj(z)Jk(ζ)O(q)〉0(5.85)

which means for (5.83), that

∂K(z′i, zj)

∂qβ=

˛qβ

∑k

dζ〈Ji(z′)Jk(ζ)O(q)〉0

〈O(q)〉0〈Jj(z)Jk(ζ)O(q)〉0

〈O(q)〉0dz′idzj =

=

˛qβ

∑k

K(z′i, ζk)K(zj, ζk)

dζ=

∑π(P )=qβ

ResP

K(z′i, P )K(zj, P )

dz(P )

(5.86)

where we have used that¸qβ

∑k

=∑

π(P )=qβ

ResP

. Hence, the same methods, which give

rise to explicit formula for the main part τSW (a,q) of the exact conformal block,ensure also the consistency of definition of the quasiclassical correction τB(q).

Isomonodromic τ-function

The full exact conformal block equals therefore

G0(q|a) = τB(q) exp

(12

∑IJ

aITIJ(q)aJ +∑I

aIUI(q, r) + 12Q(r)

)(5.87)

According to [GIL12, Gav] the τ -functions of the isomonodromy problem [SMJ] onsphere with four marked points 0, q, 1,∞ can be decomposed into a linear combinationof the corresponding conformal blocks 3. This expansion looks as

τIM(q) =∑

w∈Q(slN )

e(b,w)C(0q)w (θ0,θq,a, µ0q, ν0q)C

(1∞)w (θ1,θ∞,a, µ1∞, ν1∞)×

×q12

(σ0t+w,σ0t+w)− 12

(θ0,θ0)− 12

(θt,θt)Bw(θi,a, µ0q, ν0q, µ1∞, ν1∞; q)

(5.88)

and can be tested, both numerically and exactly for some degenerate values of theW-charges θ of the fields [Gav, GavIL]. In (5.88) the normalization of conformal

block Bw(•; q) is chosen to be Bw(•; q) = 1 + O(q) and C(•)w (•) as usually denote the

corresponding 3-point structure constants (all these quantities in the case of W (slN) =WN blocks with N > 2 depend on extra parameters µ, ν, being the coordinates onthe moduli space of flat connections on 3-punctured sphere, and for their genericvalues the conformal blocks Bw(•; q) are not defined algebraically, see [Gav] for moredetails).

3This relation has been predicted in [Knizhnik], see also [Nov] for a slightly different observationof the same kind.

140

5.6. Examples

We now conjecture that such decomposition exists also for conformal blocks con-sidered above. Moreover, then a natural guess is, that the structure constants havesuch a form that

C(0q)w (θ0,θq,a, µ0q, ν0q)C

(1∞)w (θ1,θ∞,a, µ1∞, ν1∞)q

12

(a+w,a+w)− 12

(θ0,θ0)− 12

(θq ,θq)··Bw(θi,a, µ0q, ν0q, µ1∞, ν1∞; q) = G0(θi,a+w; q)

(5.89)i.e. they are absorbed into our definition of the W-block of the twist fields, and thiscan be extended from four to arbitrary number of even 2L points on sphere. Thisconjecture can be easily checked in the N = 2 case, where the structure constantsfor the values, corresponding to the Picard solution [GIL12, ILTe], coincide exactlywith given by degenerate period matrices in (5.87), when applied to the case of theZamolodchikov conformal blocks [GMqui] (see sect. 5.6 and Appendix 5.10).

It means that in order to get isomonodromic τ -functions from the exact conformalblocks (5.87) one has just to sum up the series (for the arbitrary number of points onehas to replace the root lattice of Q(slN) = ZN−1 by the lattice Zg, where g = g(C) isthe genus of the cover)

τIM(q|a, b) =∑n∈ZgG0(q|a+ n)e(n,b) = τB(q) exp

(12Q(r)

)×

×∑n∈Zg

exp

(1

2(a + n, T (a + n)) + (U ,a+ n) + (b,n)

)=

= τB(q) exp(

12Q(r)

)Θ

[ab

](U)

(5.90)

which is easily expressed through the theta-function. One gets in this way exactly theKorotkin isomonodromic τ -function, where the only difference of this expression withproposed in [K04, formula 6.10] is in the term Q(r), which is not expressed globallythrough the coordinates of the branch points in the case of non-abelian monodromygroup. This fact supports both our conjectures: about the form of the structureconstants, and about the general correspondence between the isomonodromic defor-mations and conformal field theory.

Formula (5.90) has also clear meaning in the context of gauge theory/topologicalstring correspondence. It has been noticed yet in [NO], that the CFT free fermionrepresentation exists only for the dual partition function, which is obtained from thegauge-theory matrix element (conformal block) by a Fourier transform 4. We plan toreturn to this issue separately in the context of the free fermion representation for theexact W-conformal blocks.

Examples

There are several well-known examples of the conformal blocks corresponding toAbelian monodromy groups. All of them basically come from the Zamolodchikov

4The fact, that only the Fourier-Legendre transformed quantity can be identified with partitionfunction in string theory has been established recently in quite general context from their transfor-mation properties in [CWM].

141


exact conformal block [ZamAT87, formula 3.29] for the Ashkin-Teller model, definedon the families of hyperelliptic curves

y2 =2L∏α=1

(z − qα) (5.91)

with projection π : (y, z) 7→ z. Parameters r are absent here, so the result is just

G0(q) = τB(q) exp

(12

∑IJ

aITIJ(q)aJ

), where for the hyperelliptic period matrices

one gets from (5.59) the well-known Rauch formulas (see e.g. [GMqui] and referencestherein).

When the hyperelliptic curve degenerates (see Appendix 5.10), this formula gives

G0(q) ≈ 4−∑a2I−(

∑aI)2

g∏I=1

(q2I − q2I−1)a2I−

18

g∏I>J

(q2I − q2J)2aIaJR−(∑aI)2

≈

≈ 4−∑a2I−(

∑aI)2

·g∏I=1

εa2I−

18

I R−(∑aI)2

·g∏

I>J

(q2I − q2J)2aIaJ

(5.92)

Here in the r.h.s. the second factor comes from the OPE (5.7), i.e. O(q2I−εI)O(q2I) ∼εa2I−

18

I VaI (q2I) + . . ., while the third one is just the correlator 〈∏VaI (q2I)〉. Hence, the

first most important factor corresponds to the non-trivial product of the structureconstants in (5.89), which acquires here a very simple form. The main point of thisobservation is that normalization of (5.87) automatically contains not only q# factors,but also the structure constants, and we have already exploited such conjecture forgeneral situation in sect. 5.5, since the argument with degenerate tau-function can beeasily extended.

These observations have an obvious generalization for the ZN -curves

yN =2L∏α=1

(z − qα)kα (5.93)

with the same projection π : (y, z) 7→ z. The main contribution to the answer

τSW = exp

(12

∑IJ

aITIJ(q)aJ

)comes just from a general reasoning as in sect. 6.4

and to make it more explicit one can use the Rauch formulas for ZN -curves, whichexpress everything in terms of the coordinates q on the projection, since there is nosumming over preimages in formulas like (5.59).

Let us now turn to an elementary new example with non-abelian monodromygroup. Notice, first, that a simple genus g(C) = 0 curve

y3 = x2(1− x) (5.94)

gives rise to the curve with non-abelian monodromy group if one takes a different(from Z3-option πx : (y, x) 7→ x) projection πy : (y, x) 7→ y. For the curve C, whichis just a sphere or P1 itself, one gets here two essentially different (and unrelated!)setups, corresponding to differently chosen functions x or y.

142

5.6. Examples

In the first case our construction leads, for example, to the formulas

〈T (x)〉O =〈T (x)Os(0)Os−1(1)〉〈Os(0)Os−1(1)〉

=1

4ξ;x =

1

9x2(x− 1)2(5.95)

where x = 11+ξ3 in terms of the global coordinate ξ on C, and this formula fixes the

insertions at x = 0, 1 to be the twist operators for s = (123), with ∆(s) = l2−124l

=l=3

19.

However, for a similar correlator on y-sphere

〈T (y)〉O =〈T (y)

∏A=0,1,2,3 O(yA)〉

〈∏

A=0,1,2,3 O(yA)〉=

1 + 54y3

(27y3 − 4)2y2=

=∑

A=0,1,2,3

(1

16(y − yA)2+

uAy − yA

)y0 = u0 = 0, 3yk = −8uk = 22/3e2πi(k−1)/3, k = 1, 2, 3

(5.96)

one has to insert the twist operators for s = (12)(3) of dimension ∆(s) = l2−124l

=l=2

116

.

The r.h.s. here follows from summation of

1

12ξ; y =

ξ(ξ3 + 4)(1 + ξ3)4

2(2ξ3 − 1)4=

ξ5(3y + ξ)

2y(2ξ − 3y)4(5.97)

where

y =ξ

1 + ξ3, ξ ∈ C = P1 (5.98)

To get (5.96) one has to sum (5.97) over π(ξ) = y, or three solutions of the equationR(ξ) = ξ3 − 1

yξ + 1 = 0, i.e.

〈T (y)〉C =1

12

∑β

ξ(β); y =∑β

resξ=ξ(β)

(ξ5(3y + ξ)

2y(2ξ − 3y)4d logR(ξ)

)=

= − 1

2y

(resξ=3y/2 + resξ=∞

)(ξ5(3y + ξ)R′(ξ)

(2ξ − 3y)4R(ξ)dξ

)=

1 + 54y3

(27y3 − 4)2y2

(5.99)

in contrast to the sum over three sheets of the cover π(ξ) = x, which gives only afactor 〈T (x)〉O = 3 · ξ;x/12.

To analyze the simplest nontrivial τ -function for non-abelian monodromy group,let us consider the deformation of the formula from (5.98) for z = 1/y = ξ2 + 1/ξ, i.e.the cover π : C = P1

ξ → P1z given by 1-parametric family

z =(2ξ − t2 + 1)2(ξ − 4)

(t− 3)2(t2 − 2t− 3)ξ(5.100)

The parametrization is adjusted in the way that the branching points dz = 0 are at

ξ = 12(t2 − 1), z = 0

ξ = 1 + t, z = 1

ξ = 1− t, z = q(t) =(t+ 3)3(t− 1)

(t− 3)3(t+ 1)

(5.101)

143


together with ξ =∞, z =∞.

One also has non-branching points above the branched ones ξ = 4, z = 0; ξ =(t− 1)2, z = 1; ξ = (t+ 1)2, z = q(t). Now we rewrite these points in our notation

ξ10 = 4, ξ2

0 = 12(t2 − 1), ξ3

0 = 12(t2 − 1)

ξ1q = (t+ 1)2, ξ2

q = 1− t, ξ3q = 1− t

ξ11 = (t− 1)2, ξ2

1 = 1 + t, ξ31 = 1 + t

ξ1∞ = 0, ξ2

∞ =∞, ξ3∞ =∞

(5.102)

Using an explicit formula (5.79) and the definition (5.56) of τB one can write downthe result for the τ -function

τ(t) = τB(t) exp

(1

2Qr(t)

)=

= (t− 3)δ3−13 (t− 1)δ1−

18 tδ0+ 1

24 (t+ 1)δ−1(t+ 3)δ−3+ 124

(5.103)

where δν = δν(r) are given by some particular quadratic forms

δ3 = 9r2q − 9r2

∞

δ1 = r20 − 4r0r1 + 4r2

1 + 8r0rq − 4r1rq + r2q − 4r0r∞ + 8r1r∞ − 4rqr∞ + 4r2

∞

δ0 = −9r21 − 9r2

q

δ−1 = 4r20 + 8r0r1 + 4r2

1 − 4r0rq + 7r2q − 4r0r∞ − 4r1r∞ + 8rqr∞ + r2

∞

δ−3 = −9r20 − 9r2

q

(5.104)

while their “semiclassical” shifts come from the Bergman τ -function. Notice thatisomonodromic function (5.103) looks very similar to the tau-functions of algebraicsolutions of the Painleve VI equation [GIL12, examples 5-7], but depends on essen-tially more parameters.

An interesting, but yet unclear observation is that in this example τB(t) itself canbe represented as

τB(t) = exp

(1

2Q(r)

)(5.105)

for several particular choices of parameters r, e.g.

(r0, rq, r1, r∞) = (

√7

12√

3,−√

7

12√

3,

i

4√

3,

√7

12√

3)

(r0, rq, r1, r∞) = (i

12√

3,

i

12√

3,

i

12√

3,

√5

12)

(5.106)

whereas all other (altogether eight) solutions are obtained after the action of theGalois group generated by

√3 7→ −

√3,√

5 7→ −√

5 and i 7→ −i. Notice that thisstatement is nevertheless nontrivial because we express five variables δi in terms ofonly four variables ri.

144

5.7. Conclusions

Conclusions

We have presented above an explicit construction of the conformal blocks of the twistfields in the conformal theory with integer central charges and extended W-symmetry.We have computed the W-charges of these twist fields and show that their Verma mod-ules are non-degenerate from the point of view of W-algebra representation theory.The obtained exact formulas for the corresponding conformal blocks were derivedintensively using the correspondence between two-dimensional conformal and four-dimensional supersymmetric gauge theory. We also checked that so constructed exactconformal blocks, when considered in the context of isomonodromy/CFT correspon-dence, give rise to the isomonodromic τ -functions of the quasipermutation type.

We believe that it is only the beginning of the story and, finally, would like topresent a list (certainly not complete) of unresolved yet problems. For the conformalfield theory side these obviously include:

• What is the algebraic structure of the W-algebra representations correspondingto the twist-field vertex operators, and in particular – what are the form-factorsor matrix elements of these operators?

• Already for the twist fields representations the analysis of this chapter shouldbe supplemented by study of the W-analogs of the higher-twist representations[ApiZam] and of the W-representations at “dual values” of the central charges(an example of such block for the Virasoro case can be found in [ZamAT86]).

• Finally, perhaps the most intriguing question is – what is the constructive gen-eralization of these vertex operators to non-exactly-solvable case?

However, the main intriguing part still corresponds to the side of supersymmetricgauge theory, where the resolution of these problems can help to understand theirproperties in the “unavoidable” regime of strong coupling, where even the Lagrangianformulation is not known. We are going to return to these questions elsewhere.

Appendix

Diagram technique

In order to compute the correlators of the currents (5.45) the first useful observationis that one can embed slN ⊂ glN and introduce an extra current h(z), commutingwith Ji(z), such that

h(z)h(z′) =1/N

(z − z′)2+ reg., h(z)O(q) = reg. (5.107)

Introduce the glN currentsJi(z) = Ji(z) + h(z) (5.108)

which satisfy the OPE

Ji(z)Jk(z′) =

δjk(z − z′)2

+ reg. (5.109)

145


and their normally-ordered averages are the same as for Ji(z) since

〈: Ji1(z1) . . . Jim(zm)h(zm+1) . . . h(zn) :〉O =

= 〈: Ji1(z1) . . . Jim(zm) :〉O · 〈: h(zm+1) . . . h(zn) :〉 = 0(5.110)

Hence, one can simply to replace Ji(z) → Ji(z) in (5.45), so below we just computethe averages for the glN currents.

The normal ordering for two currents at colliding points is given by

: Ji(z)Jj(z′) : dz dz′ = Ji(z)Jj(z

′)dz dz′ − δijdz dz′

(z − z′)2=

= Ji(z)Jj(z′)dz dz′ − δijK0(z, z′)

(5.111)

i.e. it is defined by subtracting the canonical meromorphic bidifferential on the basecurve, since it corresponds to the vacuum expectation value of the Gaussian fields.Normal ordering for the correlators of many currents is defined, as usual, by the Wicktheorem.

Similarly to (5.8) consider now

〈Ji1(z1) : Ji2(z2) . . . Jin(zn) :〉O dz1 . . . dzn =

= dS(zi11 )〈: Ji2(z2) . . . Jin(zn) :〉O dz2 . . . dzn+

+n∑j=2

K(zi11 , zijj )〈: Ji2(z2) . . . Jij(zj) . . . Jin(zn) :〉O dz2 . . . dzj . . . dzn

(5.112)

where by zik = π−1(zk)i we have denoted the preimages on the cover. This formula

is again obtained just from the analytic structure of this expression as 1-form in thefirst variable. The next formula comes from the application of the Wick theorem and(5.111)

〈Ji1(z1) : Ji2(z2) . . . Jin(zn) :〉Odz1 . . . dzn = 〈: Ji1(z1) . . . Jin(zn) :〉Odz1 . . . dzn+

+n∑j=2

δijK0(z1, zj)〈: Ji2(z2) . . . Jij(zj) . . . Jin(zn) :〉Odz2 . . . dzj . . . dzn

(5.113)Subtracting them, one gets the recurrence relation

〈: Ji1(z1) . . . Jin(zn) :〉Odz1 . . . dzn = dS(zi11 )〈: Ji2(z2) . . . Jin(zn) :〉Odz2 . . . dzn+

+n∑j=2

Ki1ij(z1, zj)〈: Ji2(z2) . . . Jij(zj) . . . Jin(zn) :〉Odz2 . . . dzj . . . dzn

(5.114)where we have introduced the “propagator”

Kij(z1, z2) = K(zi1, zj2)− δijK0(z1, z2) (5.115)

Graphically for the result this recurrence produces one can write

146

5.9. W4(z) and the primary field

Ri(z1) = ti = dS(zi1)

Rij(z1, z2) =ttij +ttij = dS(zi1)dS(zj2) + Kij(z

i1, z

j2)

Rijk(z1, z2, z3) =ttij tkttij tkttij tkttij tk+ + +

""

HH =

= dS(zi1)dS(zj2)dS(zk3 ) + Kij(z1, z2)dS(zk3 ) + Kjk(z2, z3)dS(zi1) + Kik(z1, z3)dS(zj2)

These expressions have very simple meaning: the full correlation function is ex-pressed through the only possible connected parts Rc, which are Rc

i(z1) = dS(zi1),Rcij(z1, z2) = Kij(z1, z2), while Rc

ijk(z1, z2, z3) = 0 and all higher connected partsvanish. The so constructed four point functions Rijkl(z1, z2, z3, z4) at coinciding ar-guments (and at least pairwise coinciding labels of the sheets of the cover) were usedin sect. 5.3.4 for computation of the higher W-charges.

W4(z) and the primary field

Here we study the OPE of W4(z) with T (z) and show an explicit correction whichmakes this field primary.

W4(z) =∑ijkl

Cijkl : Ji(z)Jj(z)Jk(z)Jl(z) : (5.116)

where Cijkl is completely symmetric tensor, Cijkl = 124

when i 6= j 6= k 6= l andCijkl = 0 otherwise.

T (z)W4(z′) =6

(z − z′)4

∑ijkl

(δij −

1

N

)Cijkl : Jk(z)Jl(z) : +

+4W4(z′)

(z − z′)2+∂W4(z′)

z − z′+ reg.

(5.117)

The first sum equals

6∑ij

(δij −

1

N

)Cijkl = −(N − 2)(N − 3)

4N(1− δij) (5.118)

Using now the fact that∑i

Ji(z) = 0 we get

T (z)W4(z′) =(N − 2)(N − 3)

8N

T (z′)

(z − z′)4+

4W4(z′)

(z − z′)2+∂W4(z′)

z − z′+ reg. (5.119)

147


Figure 5.4: Degenerate hyperelliptic curve with chosen basis in H1.

There is also another well-known field Λ(z) = (TT )(z)− 310∂2T (z), where (TT )(z) =¸

zdww−zT (w)T (z), with the OPE

T (z)Λ(z′) =

(c+

22

5

)T (z′)

(z − z′)4+

4Λ(z′)

(z − z′)2+∂Λ(z′)

z − z′+ reg. (5.120)

One can therefore cancel an anomalous term in (5.119) just introducing

W4(z) = W4(z)− (N − 2)(N − 3)

8(N + 175

)Λ(z) (5.121)

which is already a primary conformal field. Its charge therefore is given by the formula

w4 = w4 −(N − 2)(N − 3)

8(5N + 17)∆(5∆ + 1) (5.122)

Degenerate period matrix

Here we compute the period matrix of the genus g hyperelliptic curve (see fig. 5.4)

y2 = (z −R)

g∏I=1

(z − q2I)(z − q2I + εI) = (z −R)

g∏I=1

(z − q2I)(z − q2I−1) (5.123)

in the degenerate limit εI → 0, R→∞ up to the terms of order O(εI) and O(

1R

)(this

equivalence will be denoted by “≈”). The normalized first kind Abelian differentialswith such accuracy are

dωI =√q2I −R

∏K 6=I

(z − q2K)dz

y,

1

2πi

˛

AJ

dωI ≈ δIJ (5.124)

since z−qI√(z−qI)(z−qI+εI)

≈ 1 when z goes far from qI . First we compute the off-diagonal

matrix element TIJ for J > I

TIJ =

˛

BJ

dωI ≈ −2

R

qJ

√q2I −Rz −R

dz

z − q2I

≈ −2 log 4 + 2 logq2J − q2I

R(5.125)

148

5.10. Degenerate period matrix

and then a little bit more complicated diagonal element

TII =

˛

BI

dωI ≈ −2

R

q2I

√q2I −Rz −R

dz√(z − q2I)(z − q2I + εI)

≈

≈ 2

R

q2I

(1−

√q2I −Rz −R

)dz

z − q2I

− 2

R

q2I

dz√(z − q2I)(z − q2I + εI)

=

= −2 log 4− 2 log 4 + 2 logεIR

(5.126)

where we have used the fact, that for our purposes in the expressions f(z)√(z−q2I)(z−q2I+εI)

one can drop εI if f(q2I) = 0.Now using (5.87) we can compute in this limit

τSW = exp

(1

2

∑I<J

aITIJaJ

)≈

≈ ·4−∑a2I−(

∑aI)2

g∏I=1

(q2I − q2I−1)a2I

g∏I>J

(q2I − q2J)2aIaJR−(∑aI)2

(5.127)

The result for τB(q) in this simple hyperelliptic example can be taken from [ZamAT87]

τB(q) =

2g+1∏i<j

(qi − qj)−18 ×

detIJ

1

2πi

˛

AI

zJ−1dz

y

− 12

(5.128)

where the determinant can be easily computed using (5.123)

detIJ

1

2πi

˛

AI

zJ−1dz

y≈ det

IJ

qJ−12I√

R∏J 6=I

(q2I − q2J)= R−

g2

∏I>J

(qI − qJ)−1

(5.129)

Altogether this gives the formula (5.92) for the degenerate form of the hyperellipticZamolodchikov exact conformal block.

149

6Twist-field representations of W-algebras,

exact conformal blocks and characteridentities

Abstract

We study twist-field representations of the W-algebras and generalize the constructionof the corresponding vertex operators to D- and B-series. We demonstrate how thecomputation of characters of such representations leads to the nontrivial identitiesinvolving lattice theta-functions. We propose a construction of their exact conformalblocks, which for D-series express them in terms of geometric data of correspondingPrym variety.

Introduction

Representation theory for the W-algebras [ZamW, FZ, FL] is still the subject withmany open questions. These questions often arise in the context of a two-dimensionalconformal theory (CFT) with extended symmetry, and due to a nontrivial recentlyfound correspondence [LMN, NO, AGT, Nek15] may be important for multidimen-sional supersymmetric gauge theories.

The main object of this study is a conformal block, which for generic W-algebrabeyond the Virasoro case is not fixed by its algebraic properties. Actually there areat least two different meanings of the term “conformal block” in the literature:

• Space of all functionals on the product of Verma modules at given points of theRiemann surface that solve Ward identities (see e.g. [FBZv] where the languageof coinvariants was used). Such spaces form a bundle over moduli space ofcurves with fixed points. We prefer to call this a ”space of conformal blocks”,and reserve the word ”conformal block” for another meanings following [BPZ].

• Concrete section of this bundle. Usually the corresponding functionals are com-puted on the highest weights in each of the Verma modules. For the Virasoro al-gebra the conformal blocks are specified by definite intermediate dimensions, or,equivalently, by the asymptotic behavior of the conformal block on the boundaryof the moduli space;

151

6. Twist-field representations of W-algebras, exact conformal blocks and character identities

The first object (which we call the “space of conformal blocks”) is defined for anyvertex algebra, but the problem is to specify a concrete section of this bundle, it isno longer enough to do this by fixing quantum numbers in the intermediate channels.Even for three points on sphere, the vector space of conformal blocks becomes infinitedimensional for WN algebras with N > 2.

However, for certain particular cases this conformal block can be constructed ex-plicitly applying some extra machinery. In what follows we first restrict ourselves tothe case of integer and sometimes half-integer Virasoro central charges, when rep-resentations W-algebras are more directly related to the representations of the cor-responding Kac-Moody (KM) algebras (of level k = 1), and the corresponding fieldtheories can be directly described by free fields [FK].

Even in such situation the most general case cannot be formulated explicitly,one of the recent methods [GMfer] reduces the problem here to a Riemann-Hilbertproblem, arising in the context of the isomonodromy/CFT correspondence [GIL12,ILTe, Gav]. Below, following [GMtw], we are going to restrict ourselves to the case ofso-called twist fields [ZamAT87, ZamAT86, ApiZam], when the representations of theW-algebras become related to the twisted representations of the corresponding KMalgebras 1 [KacBook, BK].

The chapter is organized as follows. We start from the formulation of the repre-sentations of KM and W-algebras in terms of free bosons and fermions, remind firstthe GL(N) case and extend it to the D- and B- series, using real fermions. We definethen the twist representations, and show that they are parameterized by the conjugacyclasses in the correspondent Cartan’s normalizer NG(h). We classify the conjugacyclasses g ∈ NG(h) for G = GL(N) and G = O(n) (for n = 2N and n = 2N + 1) anddefine the twist fields Og in terms of the boundary conditions in corresponding freetheory.

Bosonization rules allow to compute easily the characters χg(q) of the correspond-ing representations. For the twist fields of “GL(N) type” this goes back to the oldresults of Al. Zamolodchikov and V. Knizhnik, and we develop here similar techniquein the case of real fermions and another class of twist fields, arising in D- and B-series. The character formulas include summations over the root lattices, reflectingthe fact that we deal here with the class of lattice vertex algebras. Dependently of theconjugacy class g ∈ NG(h) of a twist field the lattice can be reduced to its projectionto the Weyl-invariant part, in this case the “smaller” lattice theta functions show up,or we find even a kind of “exchange” between those for D- and B- series.

If two different classes g1,2 ∈ NG(h) are nevertheless conjugated g1 ∼ g2 in G (butnot in NG(h)) this gives a nontrivial identity χg1(q) = χg2(q) between two characters,involving lattice theta-functions. Such character identities go back to 1970’s (see[Mac], [Kac78]) and even to Gauss, but our derivation gives probably the new ones,involving in particular the theta functions for D- and B-root lattices.

We propose construction of the exact conformal blocks of the twist fields for W-algebras of D-series, generalizing approach of [ZamAT87, ZamAT86, ApiZam, GMtw],and obtain an explicit formula, expressing multipoint blocks in terms of the algebro-

1To prevent the reader’s confusion we should notice that “twist field representation” is differentfrom “twisted representation”: the latter one implies that the algebra itself is changed (twisted),wheres the first one only reflects the way – how this representation was constructed.

152

6.3. W-algebras and KM algebras at level one

geometric objects on the branched cover with extra involution.

W-algebras and KM algebras at level one

Boson-fermion construction for GL(N)

We start from standard complex fermions

ψ∗α(z) =∑p∈ 1

2+Z

ψ∗α,p

zp+12

, ψα(z) =∑p∈ 1

2+Z

ψα,p

zp+12

(6.1)

with the operator product expansions (OPE’s)

ψ∗α(z)ψβ(w) = −ψβ(w)ψ∗α(z) =δαβz − w

+ reg.

ψα(z)ψβ(w) = ψ∗α(z)ψ∗β(w) = reg.(6.2)

equivalent to the following anticommutation relations

ψ∗α,p, ψβ,q = δαβδp+q,0, ψα,p, ψβ,q = ψ∗α,p, ψ∗β,q = 0, p, q ∈ 12

+ Z (6.3)

One can introduce the Kac-Moody gl(N)1 algebra by the currents

Jαβ(z) =: ψ∗α(z)ψβ(z) : (6.4)

where the free fermion normal ordering moves all ψr and ψ∗r with r > 0 to theright. These currents have standard OPE’s:

Jαβ(z)Jγδ(w) =δβγδαδ

(z − w)2+δβγJαδ(w)− δαδJβγ(w)

z − w+ reg. (6.5)

and when expanded into the (integer!) powers of z

Jαβ(z) =∑n∈Z

Jαβ,nzn+1 (6.6)

we get the standard Lie-algebra commutation relations

[Jαβ,n, Jγδ,m] = nδn+m,0δβγδαδ + δβγJαδ,m+n − δαδJβγ,m+n, n,m ∈ Z (6.7)

This set contains zero modes Jαβ,0, generating the subalgebra gl(N) ⊂ gl(N)1. TheW (gl(N)) = WN ⊕H algebra can be defined in a standard way - as a commutant of

gl(N) in the (completion of the) universal enveloping U(gl(N)1).This basis of the generators of W (gl(N)) = WN ⊕ H algebra can be chosen in

several different ways. In what follows the most convenient for our purposes is to usefermionic bilinears

N∑α=1

ψ∗α(z + 1

2t)ψα(z − 1

2t)

=N

t+∞∑k=1

tk−1

(k − 1)!Uk(z) (6.8)

153


or, using the Hirota derivative Dnz f(z) · g(z) = (∂z1 − ∂z2)n f(z1)g(z2)|z1=z2=z,

Uk(z) = Dk−1z

N∑α=1

: ψ∗α(z) · ψα(z) : (6.9)

while another useful basis is the bosonic representation

Wk(z) =∑

α1<...<αk

: Jα1α1(z) . . . Jαkαk(z) :≡∑

α1<...<αk

: Jα1(z) . . . Jαk(z) : (6.10)

The formula (6.10) is equivalent to quantum Miura transform from [ZamW, FZ, FL].To explain that the formula (6.9) is actually equivalent to (6.10) one can use descrip-tion of W (gl(N)) as centralizer of screening operators which coincide with gl(N) inthis case. It is already proven, that (6.10) is centralizer of screening operators [FF],so it remains to show that (6.9) is centralizer as well, what can be done in severalsteps:

1. Consider all normally-ordered fermionic monomials :∏

i ∂kiψαi(z)

∏i ∂

liψ∗βi(z) :,which transform as tensors under the action of GL(N). By First fundamentaltheorem of invariant theory [Weyl] the only invariants in such representation aregiven by all possible contractions, so they can be written as :

∏i(∑α

∂kiψα(z)∂liψ∗α(z)) :.

2. Any such expression can be obtained by taking regular products of the “elemen-tary elements” :

∑α

∂kψα(z)∂lψ∗α(z) :, since

:∏i

(∑α

∂kiψα(z)∂liψ∗α(z)

)::∑β

∂kψβ(z)∂lψ∗β(z) :=

=:∏i

(∑α

∂kiψα(z)∂liψ∗α(z)

)∑β

∂kψβ(z)∂lψ∗β(z) : + lower terms

(6.11)

Therefore one can perform this procedure iteratively and express everything asregular products of bilinears.

3. Any element :∑α

∂kψα(z)∂lψ∗α(z) : can be expressed as a linear combination of

∂l′+1Uk′(z) for different l′ and k′ with l′ + k′ = l + k.

Hence, the generators Uk(z) are expressible in terms of Wk(z) (and vice versa)by some non-linear triangular transformations, but we do not need here these explicitformulas 2.

Formally there is an infinite number of generators in (6.8) and (6.9), since all ofthem are expressed in terms of N generators (6.10)they are not independent: we have

UN+n(z) = Pn(∂kUl≤N) (6.12)

2The fact, that nonlinear W-algebra generators can be expressed through just bilinear fermionicexpressions is well-known, and was already exploited in [LMN, NO] (see also [GMfer] and referencestherein).

154

6.3. W-algebras and KM algebras at level one

for some polynomials Pn, and this is the origin of the non-linearity of the W-algebra [FKRW]. The relation between fermions and bosons are given by well-known[FK, KVdL] bosonization formulas

ψ∗α(z) = exp

(−∑n<0

Jα,nnzn

)exp

(−∑n>0

Jα,nnzn

)eQαzJα,0εα(J0) =

= eiϕα,−(z)eiϕα,+(z)eQαzJα,0εα(J0)

ψα(z) = exp

(∑n<0

Jα,nnzn

)exp

(∑n>0

Jα,nnzn

)e−Qαz−Jα,0εα(J0) =

= e−iϕα,−(z)e−iϕα,+(z)e−Qαz−Jα,0εα(J0)

(6.13)

where εα(J0) =∏α−1

β=1(−1)J0,β and diagonal Jαα,n ≡ Jα,n form the Heisenberg algebra

[Jα,n, Jβ,m] = nδαβδm+n,0, [J0,α, Qβ] = δαβ (6.14)

One can also express all other generators in terms of (positive and negative parts of)the bosons

iϕ+,α(z) = −∑n>0

Jα,nnzn

, iϕ−,α(z) = −∑n<0

Jα,nnzn (6.15)

namely

Jαβ(z) = eiϕ−,α−iϕ−,βeiϕ+,α−iϕ+,βeQα−QβzJα,0−Jβ,0(−1)

β−1∑γ=α−1

Jγ,0+θ(β−α)

, α 6= β

Jαα(z) = Jα(z) = i∂ϕ+,α(z) + i∂ϕ−,α(z)

(6.16)

Real fermions for D- and B- series

Now we can almost repeat the same construction for the orthogonal series, BN andDN , which correspond to the W-algebras W (so(2N+1)) and W (so(2N)), respectively.The corresponding Kac-Moody algebras at level one can be realized in terms of thereal fermions (see e.g. [AWM]) with the OPE’s

Ψi(z)Ψj(w) =δij

z − w+ reg., i, j = 1, . . . , n (6.17)

(here dependently on the case we put either n = 2N or n = 2N+1), which correspondsto anti-commutation relations

Ψi,p,Ψj,q = δijδp+q,0, p, q ∈ 12

+ Z (6.18)

One can say that these OPE’s and commutation relations are define by the metrics on

n-dimensional space given by δij, or symbolically by ds2 =n∑i=1

dΨ2i . The Kac-Moody

currents are again expressed by bilinear combinations

J(1)ik (z) =: Ψi(z)Ψj(z) : (6.19)

155


and satisfy usual commutation relations together with J(1)ij (z) = −J (1)

ji (z). It is alsoconvenient to pass to the complexified fermions (α = 1, . . . , N)

ψ∗α(z) =1√2

(Ψ2α−1(z) + iΨ2α(z)) , ψα(z) =1√2

(Ψ2α−1(z)− iΨ2α(z)) (6.20)

which due to (6.17) have the standard OPE’s given by (6.2). Let us point out thatBN -series (i, j = 1, . . . , 2N +1) differs from DN -series (i, j = 1, . . . , 2N) by remainingsingle real fermion Ψ2N+1(z) = Ψ(z).

Using the complexified fermions the generators (6.19) can be re-written as

Jαβ =: ψ∗α(z)ψβ(z) :=1

2(J

(1)2α−1,2β−1 + J

(1)2α,2β) +

i

2(J

(1)2α,2β−1 + J

(1)2β,2α−1) (6.21)

together with

Jαβ = ψ∗α(z)ψ∗β(z), Jαβ = ψα(z)ψβ(z)

Jα,Ψ = ψ∗α(z)Ψ(z), Jα,Ψ = ψα(z)Ψ(z)(6.22)

so that we see explicitly gl(N)1 ⊂ so(n)1. Note also, that elements Jαα(z) = Jα(z)again form the Heisenberg algebra, and its zero modes Jα,0 correspond to the Cartansubalgebra of so(n).

As before, we define the W-algebra W (so(n)) as commutant of so(n) ⊂ so(n)1.In contrast to the simple-laced cases we find this commutant for B-series not in com-

pletion of the U( so(2N + 1)1), but in the entire fermionic algebra. An inclusion ofalgebras gl(N) ⊂ so(2N), acting on the same space, leads to inverse inclusion

W (so(2N)) ⊂ W (gl(N)) (6.23)

Similarly to (6.9) one can present the generators of the W (so(n))-algebra explicitly,using the real fermions

Uk(z) =1

2Dk−1z

n∑j=1

: Ψj(z) ·Ψj(z) :, V (z) =n∏j=1

Ψj(z) (6.24)

The last current is bosonic in DN case and fermionic for BN . These expressions areobtained analogously to (6.9) with the help of invariant theory, the only importantdifference is that for SO(n) case there is also completely antisymmetric invarianttensor. We can rewrite these expressions using complex fermions (for the DN caseone should just put here Ψ(z) = 0 in the expressions for U -currents and Ψ(z) = 1 inthe expressions for the V -current)

Uk(z) = 12Dk−1z

N∑α=1

(ψ∗α(z) · ψα(z) + ψα(z) · ψ∗α(z)) + 12Dk−1z Ψ(z) ·Ψ(z)

V (z) =N∏α=1

: ψ∗α(z)ψα(z) : Ψ(z)

(6.25)

156

6.4. Twist-field representations from twisted fermions

It is easy to see that odd generators vanish U2k+1 = 0, while even generators in DN

case coincide with those in W (gl(N)) algebra

U2k(z) = D2k−1z

N∑α=1

: ψ∗α(z) · ψα(z) : +12D2k−1z Ψ(z) ·Ψ(z), k = 1, 2, . . . (6.26)

So finally we have the following sets of independent generators:

U1(z), U2(z), . . . , UN(z) for W (gl(N))

U2(z), U4(z), . . . , U2N−2(z), V (z) for W (so(2N))

U2(z), U4(z), . . . , U2N(z), V (z) for W (so(2N + 1))

(6.27)

Twist-field representations from twisted fermions

Fermions and W-algebras

For any current algebra, generated by currents AI(z), the commutation relationsfollow from their local OPE’s

AI(z)AJ(w) =z→w

∑K

(AIAJ)K(w)

(z − w)K(6.28)

However, to define the commutation relations in addition to local OPE’s one shouldalso know the boundary conditions for the currents: in radial quantization – theanalytic behaviour of AI(z) around zero. Any vertex operator Vg(0), e.g. sitting atthe origin 3, can create nontrivial monodromy for our currents:

AI(e2πiz)Vg(0) =

∑j

gIJAJ(z)Vg(0) (6.29)

for some linear automorphism of the current algebra.Perhaps the simplest example of such nontrivial monodromy is the diagonal trans-

formation of the fermionic fields

ψ∗α(e2πiz) = e2πiθαψ∗α(z), ψα(e2πiz) = e−2πiθαψα(z), α = 1, . . . , N (6.30)

which just shifts the mode expansion index

ψ∗α(z) =∑p∈Z+ 1

2

ψ∗α,p

zp+12−θα

, ψα(z) =∑p∈Z+ 1

2

ψ∗α,p

zp+12

+θα (6.31)

Instead of the OPE (6.2) one gets therefore

ψ∗α(z)ψβ(w)→ zθαw−θβψ∗α(z)ψβ(w) =zθαw−θβ

z − w+ zθαw−θβ : ψ∗α(z)ψβ(w) : =

=z→w

1

z − w+θαδαβw

+ : ψ∗α(w)ψβ(w) : +reg.

(6.32)

3For nontrivial boundary conditions we assume presence of such field by default, when obvious –not indicating it explicitly.

157


which means that for the shifted fermions (6.31) one should use different normalordering:

(ψ∗α(z)ψβ(z)) =θαδαβz

+ : ψ∗α(z)ψβ(z) : (6.33)

This implies that for the diagonal components gl(N)1 algebra one has extra shiftJα(z)→ Jα(z) + σα

z, while for the non-diagonal currents we obtain

Jαβ(z) =∑n∈Z

Jαβ,nzn+1+θα−θβ (6.34)

so that the commutation relations for this “twisted” Kac-Moody algebra become

[Jαβ,n, Jγδ,m] = (n− θα + θβ)δn+m,0δβγδαδ + δβγJαδ,m+n − δαδJβγ,m+n (6.35)

We see that these commutation relations differ from the conventional ones (6.7) onlyby the extra shift which can be hidden into the Cartan generators Jαα,0. However,

in the twisted case gl(N)1 does not contain zero modes, and we cannot think aboutthe W-algebra as about commutant of some gl(N). But nevertheless we define thecurrents

Uk(z) = Dk−1z

N∑α=1

(ψ∗α(z) · ψα(z)) (6.36)

One can still use two basic facts:

• since Uk(e2πiz) = Uk(z), they are expanded in integer powers of z as before;

• they satisfy the same algebraic relations for all values of monodromies σα,because the OPE’s of ψα, ψ

∗α (and so the OPE’s of Uk) do not depend on these

monodromy parameters.

Twist fields and Cartan’s normalizers

Consider now more general situation, when

ψ∗α(e2πiz) =N∑β=1

gαβψ∗β(z), ψα(e2πiz) =

N∑β=1

g−1βαψβ(z) (6.37)

i.e. compare to (6.30) the monodromy is no longer diagonal 4. It is clear that then

the action on gl(N)1 isJαβ(z) 7→ gαα′g

−1β′βJα′β′(z) (6.38)

The most general transformation we consider in the O(n) case mixes ψ and ψ∗:

ψα(e2πiz) =N∑

β=−N

gαβψβ(z), α = −N, . . . , N (6.39)

4This element should preserve the structure of the OPEs, so it should preserve symmetric form onfermions, and lies therefore in O(2N). Notice that it automatically implies that all even generatorsof the W-algebra U2k(w) are also preserved. To preserve odd generators U2k+1(z) one should havealso g ∈ Sp(2N), but O(2N) ∩ Sp(2N) = GL(N), so g ∈ GL(N).

158


where it is convenient to introduce conventions ψ∗−α = ψα, α > 0, and ψ0 can beabsent. Matrix g here should preserve the anticommutation relations.

Definition 1. We call the vertex operator Vg = Og a twist field when g lies in thenormalizer of Cartan h ⊂ g, i.e. g ∈ NG(h) iff

ghg−1 = h (6.40)

Such elements also preserve the Heisenberg subalgebra h = 〈J1(z), . . . , Jrank g(z)〉 ⊂g1

ghg−1 = h (6.41)

We are going now to discuss the structure of the Cartan normalizers NGL(N)(h) andNO(n)(h), which classify the twist fields for the WN = W (gl(N)) and W (so(n)) (foreven n = 2N and odd n = 2N + 1) correspondingly.

Structure of the Cartan normalizer for gl(N). Let us choose the Cartan subal-gebra in a standard way h ⊃ diag(x1, . . . , xN), so conjugation (6.40) can only permutethe eigenvalues. Therefore we conclude that

g = s · (λ1, . . . , λN) ∈ SN n(C×)N

= NGL(N)(h) (6.42)

or just g is a quasipermutation.Let us now find the conjugacy classes in this group. Any element of NGL(N)(h)

has the form g = (c1 . . . ck, (λ1, . . . , λN)), where ci are the cyclic permutations – theironly parameters are lengths lj = l(cj). It is enough to consider just a single cycle ofthe length l = l(c)

g = (c, (λ1, . . . , λl)) (6.43)

since any g can be decomposed into a product of such elements. Conjugation of thiselement by diagonal matrix is given by

(1, (µ1, . . . , µl)) · (c, (λ1, . . . , λl)) · (1, (µ1, . . . , µl))−1 =

= (c, (λ1µ1

µ2

, λ2µ2

µ3

, . . . , λlµlµ1

))(6.44)

Therefore one can always adjust µi to get rid of all λi except for one, e.g. to put

λi 7→ λ =∏l

i=1 λ1/li = e2πir, these “averaged over a cycle” parameters have been called

as r-charges in [GMtw]. Hence, all elements of g ∈ NGL(N)(h) can be conjugated tothe products over the cycles

[g] ∼K∏j=1

[lj, λj] =K∏j=1

[lj, e2πirj ] (6.45)

Structure of NO(n)(h). Using complexification of fermions (6.20) we rewrite the

quadratic form ds2 =n∑i=1

dΨ2i as ds2 =

N∑α=1

dψ∗αdψα + dΨ2 =N∑α=1

dψ−αdψα + dΨ2 (the

last term is present only for theBN -series). In this basis the so(n) algebra (the algebra,preserving this form) becomes just the algebra of matrices, which are antisymmetric

159


under the reflection w.r.t. the anti-diagonal. In particular, the Cartan elements aregiven by

h 3 diag(x1, . . . , xN , 0,−xN , . . . ,−x1) (6.46)

for BN -series (and for the DN -series 0 in the middle just should be removed). Theaction of an element from NO(n)(h) should preserve the chosen quadratic form, and,when acting on the diagonal matrix (6.46), it can only permute some eigenvalues, alsodoing it simultaneously in the both blocks, or interchange xα with −xα (the same asto change the sign of xα). It is defined in this way up to a subgroup of diagonalmatrices themselves. In other words

NO(2N)(h) = SN n (Z/2Z)N n (C×)N

NO(2N+1)(h) = NO(2N)(h)× Z/2Z(6.47)

where the last factor Z/2Z comes from changing sign of the extra fermion Ψ. This

triple (s, ~σ,~λ) ∈ NO(n)(h), with s ∈ SN , σα ∈ Z/2Z and λα ∈ C×, is embedded intoO(n) as follows

SN : (α 7→ s(α), 1, 1) = ψα 7→ ψs(α); ψ∗α 7→ ψ∗s(α)

(Z/2Z)N : (1, ~σ, 1) = ψα 7→ ψσαα(C×)N : (1, 1, ~λ) = ψα 7→ λαψα; ψ∗α 7→ λ−1

α ψ∗α(6.48)

and in these formulas ψ−α = ψ∗α and ψ∗−α = ψα is again implied. The structure of theactions in the semidirect product has the obvious from:

~σ : λα 7→ λσαα , s : (σα, λα) 7→ (σs(α), λs(α)) (6.49)

Notice that normalizer of Cartan in SO(n)

NSO(n)(h) = SO(n) ∩ NO(n)(h) (6.50)

is distinguished by condition that∏N

α=1 σα = 1, and the Weyl group is given as thefactor of this normalizer by the Cartan torus

W(so(n)) = NSO(n)(h)/H (6.51)

Consider now the conjugacy classes in NO(n)(h). First, conjugating an arbitrary ele-

ment (s, ~σ,~λ) by permutations, we again reduce the problem to the case when s = cis just a single cycle. Then one can further conjugate this element by (Z/2Z)N :

(1,~ε, 1) · (c, ~σ, •) · (1,~ε, 1)−1 7→ (c, (σ1 · ε1ε2, σ2ε2ε3, . . . , σN · ε1εN), •) (6.52)

and solving equations for εα remove all σα = −1) except for, maybe, one. Hence:

• For σ = (1, . . . , 1) the situation is the same as in gl(N) case: we can transform~λ to (λ, . . . , λ). These conjugacy classes are therefore (denoted by [l, λ]+)

(c, 1, ~λ) ∼ [l(c),∏

λ1/l(c)i ]+ (6.53)

160


• For, say, σ = (−1, 1, . . . , 1) let us conjugate this element by (1, 1, ~µ):

(1, 1, ~µ)(c, (−1, 1, . . . 1), ~λ)(1, 1, ~µ)−1 = (c, (−1, 1, . . . , 1), ~λ′)

~λ′ = (λ1µ1µ−12 , λ2µ2µ

−13 , . . . , λl−1µl−1µ

−1l , λlµlµ1)

(6.54)

In contrast to the previous case, here one can put all λ′i = 1, since one canput first µ2

1 =∏

i λ−1i , and then solve N − 1 equations µi+1 = λiµi not being

restricted by any boundary conditions. It means that

(c, (−1, 1, . . . , 1), ~λ) ∼ [l(c)]− (6.55)

Therefore we can formulate:

Lemma 6.1. One gets for the conjugacy classes

NO(2N)(h) : g ∼K∏j=1

[lj, λj]+ ·K′∏j=1

[lj]−

NO(2N+1)(h) : g ∼ [ε] ·K∏j=1

[lj, λj]+ ·K′∏j=1

[lj]−

(6.56)

and we are now ready to describe the twist fields in detail.Corollary : Formulas (6.45) and (6.56) give also classification of the conjugacy classesin the Weyl groups W(gl(N)) = SN and W(so(n)). It is enough just to drop the λ-dependence and to consider only the even number of minus-cycles (the latter conditioncorresponds to the fact that (extension of) the Weyl group lies inside the connectedcomponent of identity in O(n), whereas another component corresponds to exteriorautomorphism of the Dynkin diagram). In [BK] the Weyl group has been extended toW = Wn (Z/2Z)N−1 ⊂ NG(h), which corresponds to breaking the Cartan torus H ⊂ Gdown to (Z/2Z)N−1.

Twist fields and bosonization for gl(N)

Take an element (6.45), whose action on fermions (in the fundamental and antifun-damental representations), say for a single cycle, is

g : (ψ∗α(z), ψα(z)) 7→ (e2πirψ∗α+1(z), e−2πirψα+1(z)), mod l (6.57)

while the corresponding (adjoint) action on the Cartan is just

gAdj : Jα(z) 7→ Jα+1(z), mod l (6.58)

Such formulas have simple geometric interpretation [Knizhnik]: there is the branchedcover in the vicinity of the point z = 0 given by ξl = z, so that all these (fermionicand bosonic) fields are actually defined on different sheets ξ(α) = z1/le2πiα/l of thecover:

ψ∗α(z)√dz = ψ∗(ξ(α))

√dξ(α), ψα(z)

√dz = ψ(ξ(α))

√dξ(α)

Jα(z)dz = J(ξ(α))dz = J(ξ(α))dξ(α)(6.59)

161


Using these formulas one can write down expansions for the fields on the cover, whoseOPE’s would be locally given by

ψ∗(ξ)ψ(ξ′) =1

ξ − ξ′+ reg., J(ξ)J(ξ′) =

1

(ξ − ξ′)2+ reg. (6.60)

Now one write for the mode expansion

ψ(z) =

√dξ

dzψ(ξ) =

z12l− 1

2

√l

∑p∈Z+ 1

2

ψp

ξp+12

+σ=

1√l

∑p∈Z+ 1

2

ψp

z12

+ 1l(p+σ)

ψ∗(z) =

√dξ

dzψ∗(ξ) =

z12l− 1

2

√l

∑p∈Z+ 1

2

ψ∗p

ξp+12−σ

=1√l

∑p∈Z+ 1

2

ψ∗p

z12

+ 1l(p−σ)

(6.61)

Due to (6.57) one should have ψ∗(e2πilz) = e2πilrψ∗(z) and ψ(e2πilz) = e−2πilrψ(z),therefore one can take, for example, σ = lr + 1−l

2, so that:

ψ(z) =1√l

∑p∈Z+ 1

2

ψp

z1l( 1

2+p)+r

ψ∗(z) =1√l

∑p∈Z+ 1

2

ψ∗p

z1l( 1

2+p)−r

ψp, ψ∗p′

= δp+p′,0

(6.62)

or the mode expansion is shifted by the r-charges, corresponding to given cycles.The same procedure gives for the twisted bosons

J(z) =1

lz

1l−1J(ξ) =

1

lz

1l−1∑n∈Z

Jn/lξn+1

=1

lz

1l−1∑n∈Z

Jn/l

z1l(n+1)

=1

l

∑n∈Z

Jn/l

z1ln+1 (6.63)

with the commutation relations between the modes of these bosons being[Jn/l, Jm/l

]= nδn+m,0 n,m ∈ Z (6.64)

These twisted bosons provide one of the convenient languages for the twist field rep-resentations. The other one is provided by bosonization of the constituent fermionswith the fixed fractional parts of the power expansions in (6.62)

ψ(z) =1√l

∑a∈Z/lZ

ψ(a)(z), ψ(a)(e2πiz) = e−2πir−2πia

l ψ(a)(z)

ψ∗(z) =1√l

∑a∈Z/lZ

ψ∗(a)(z), ψ∗(a)(e2πiz) = e2πir+2πia

l ψ∗(a)(z)(6.65)

The corresponding bosons (see (6.266) in Appendix)

I(a)(z) =(ψ∗(a)(z)ψ(a)(z)

)=∑n∈Z

I(a)n

zn+1+

1

z

(r +

a

l

)(6.66)

always have integer mode expansion.

162


Twist fields and bosonization for so(n)

Let us mention first, that there is a difference between the groups NO(n)(h) andNSO(n)(h), since the action of the first one can also map V (z) 7→ −V (z), so thatone of the generators of the W-algebra V (e2πiz) = −V (z) becomes a Ramond field,and we allow this extra minus sign below 5.

In addition to the conjugacy classes [l, λ]+, similar to those of gl(N), we now alsohave to study [l]−’s. First one has to identify the action of NO(n)(h) on the fermions,where just by definition:

σα = −1 : (1, ~σ, 1) : ψα 7→ ψ∗α (6.67)

This means that the element of our interest is the complete cycle

[l]− : ψ1 7→ ψ2 7→ . . . 7→ ψl 7→ ψ∗1 7→ . . . 7→ ψ∗l 7→ ψ1 (6.68)

Therefore 2N complex fermions can be realized as a pushforward of a single realfermion η(ξ), living on a 2l-sheeted branched cover

ψα(z)√dz = η(ξ(α))

√dξ

ψ∗α(z)√dz = η(ξ(l+α))

√dξ

(6.69)

Here the branched cover z = ξ2l can be realized as a sequence of two covers π2 : ξ 7→ζ = ξ2 and πl : ζ 7→ ζ l = z, and it leads to more tricky global construction of theexact conformal blocks, see sect. 6.7 below.

An important fact is that there is an element σ ∈ (NO(n)(h)/H) in the center ofthis group

σ = (1, (−1,−1, . . . ,−1)) (6.70)

which generates the global automorphism of the cover of order two, which is contin-ued to the global automorphism of algebraic curve during the consideration of exactconformal blocks in sect. 6.7. It acts locally by ξ 7→ −ξ. Using this element one canwrite the OPE of η(ξ) in the form:

η(ξ)η(σ(ξ′)) =1

ξ − ξ′+ reg. (6.71)

Now the analytic structure of this field can be obtained

ψ(z) =

√dz

dξη(ξ) =

z14l− 1

2

√2l

∑p∈Z+ 1

2

ηp+ 12

z12l

(p+ 12

+σ)=

1√2l

∑p∈Z+ 1

2

ηp+ 12

z12l

(p+σ)+ 12

ψ∗(z) = ψ(e2πilz)

(6.72)

5Note that in case of n = 2N the action of NSO(2N)(h) on h is given by Weyl group action, butadditional element from NO(2n)(h) gives external (diagram) automorphism. Corresponding twisted

representations could be viewed as a representation of twisted affine Lie algebra D(2)N .

163


In order to ensure right monodromies (6.68) for ψ, ψ∗ one should get powers 12lZ in

the r.h.s., which means, that σ ∼ l − 12∼ 1

2, and η(ξ) turns to be a Ramond fermion

with the extra ramification

η(ξ) =∑n∈Z

ηn

ξn+ 12

, ψ(z) =1√2l

∑n∈Z

ηn

zn2l

+ 12

, ψ∗(z) =(−)l√

2l

∑n∈Z

(−)nηn

zn2l

+ 12

(6.73)

Let us now construct (a twisted!) boson from this fermion by

J(z) = (ψ∗(z)ψ(z)) =(ψ(e2iπlz)ψ(z)

)(6.74)

This boson behaves like follows under the action of twist field:

J1 7→ J2 7→ . . . 7→ Jl 7→ −J1 7→ . . . 7→ −Jl (6.75)

To realize this situation we may take the Ramond boson on the cover in variable ζ:

J(z) =dζ

dz

∑r∈Z+ 1

2

Jr/lζr+1

=z

1l−1

l

∑r∈Z+ 1

2

Jr/l

z1l(r+1)

=1

l

∑r∈Z+ 1

2

Jr/l

zrl+1 (6.76)

where commutation relations of modes are given by[Jr/l, Jr′/l

]= rδr+r′,0 r, r′ ∈ Z + 1

2(6.77)

Inverse bosonization formula for this real fermion looks like

√zψ(z) =

∑n∈Z

ηn

zn2l

=σ1√

2eiφ−(z)eiφ+(z)

(6.78)

with the Pauli matrix σ1 =

(0 11 0

), and it is discussed in detail in Appendix (6.10.1).

Characters for the twisted modules

Now we turn directly to the computation of characters, using bosonization rules. Inorder to do this one has to apply the following heuristic “master formula” for thetrace

χg(q) = trHgqL0 “ = ”

χZM(q)∏k

∞∏n=1

(1− qθAdj,k(g)+n)(6.79)

over the space Hg which is the minimal space closed under the action of both W-algebra and twisted Kac-Moody algebra. For simply-laced cases, gl(N) and so(2N),Hg is the module of corresponding Kac-Moody algebra, whereas in the so(2N + 1)case it should be entire fermionic Fock module due to presence of the fermionic W-current. Explicit descriptions of Hg are the following: for gl(N) it is the subspacewith fixed total fermionic charge, for so(2N) it is the subspace with fixed parity oftotal fermionic charge, and for so(2N + 1) it is entire space.

164

6.5. Characters for the twisted modules

Denominator of (6.79) collects the contributions from the Fock descendants oftwisted bosons (parameters θAdj,k(g) are the eigenvalues of adjoint action of g on theCartan subalgebra), and the numerator – contribution of the zero modes. This formulais heuristic, moreover in some important cases we also get contribution from extrafermion, sometimes it is more informative to consider super-characters etc. Below weprove the following

Theorem 6.2. The characters of twisted representations are given by the formulas(6.85), (6.88), (6.95), (6.97).

gl(N) twist fields

To be definite, let us fix an element g =K∏j=1

[lj, e2πirj ] from (6.45) which, according

to (6.57) performs the permutation of fermions with simultaneous multiplication bye±2πirj . In this setting N fermions can be bosonized in terms of K twisted bosons(see detail in Appendix 6.10.3), and here we just present the final formulas

ψ∗α(z) =z

1−l2l

√leiφ

(j)− (e2πiαz)eiφ

(j)+ (e2πiαz)eQ

(j)

(e2πiαz)1lJ

(j)0 (−1)

∑k<j

J(k)0

ψα(z) =z

1−l2l

√le−iφ

(j)− (e2πiαz)e−iφ

(j)+ (e2πiαz)e−Q

(j)

(e2πiαz)−1lJ

(j)0 (−1)

∑k<j

J(k)0

(6.80)

for α ∈ Z/ljZ, labeling the fields within [lj]-cycle. For the conformal dimensionone gets therefore (see (6.262), and computation by alternative methods in (6.139),(6.191))

L0 =K∑j=1

l2j − 1

24lj+

K∑j=1

1

li(J

(j)0 )2 + . . . (6.81)

and since we are computing character of the space, obtained by the action of gl(N)1,we have to take into account all vacua arising after the action of the shift operatorseQ

(i)−Q(j), i.e. labeled by AK−1 root lattice. Hence, the character (6.79) for this case

is given by

χg(q) = q

K∑j=1

l2j−1

24lj

∑n1+...+nK=0

q

K∑i=1

12li

(rili+ni)2

K∏j=1

∞∏k=1

(1− qk/lj)(6.82)

In this formula the numerator collects contribution from the highest vectors χZM (they

differ by the value of zero modes J(i)0 ) of the Heisenberg algebras with generators J

(i)n/li

,whereas the denominator contains the contributions from the descendants.

165


so(2N) twist fields, K ′ = 0

Consider now the twist fields (6.56) for g ∈ NO(2N)(h), and take first K ′ = 0, so ourtwist has no minus-cycles

g =K∏j=1

[lj, e2πirj ]+ (6.83)

The only difference from the previous situation with the gl(N) case is that now onealso has extra currents Jαβ = ψ∗α(z)ψ∗β(z) and Jαβ = ψα(z)ψβ(z). It means that due

to bosonization (6.266), (6.80) possible charge’s shifts now include e±(Q(i)+Q(j)), so thefull lattice of the zero-mode charges (one zero mode for each cycle [li, e

2πiri ]+) containsall points with

K∑i=1

ni ∈ 2Z, ni ∈ ZK (6.84)

or is just the root lattice QDK . After corresponding modification of numerator andthe same contribution of the twisted Heisenberg algebra to denominator, the formulafor the character in this case acquires the form

χg(q) = q

K∑j=1

l2j−1

24lj

∑~n∈QDK

q

K∑j=1

12lj

(nj+ljrj)2

K∏j=1

∞∏n=1

(1− qn/lj)

(6.85)

so(2N) twist fields, K ′ > 0

Take

g =K∏j=1

[lj, e2πirj ]+

K′∏j=1

[l′j]− (6.86)

Now we have extra cycles of type [l′i]−, so we have extra η-fermions that have to bebosonized in a different way (6.241):

ηi(z) =z−

12

2√leiφ−(z

12li )eiφ+(z

12li )(−1)

∑kJ

(k)0γi (6.87)

where γi, γj = 2δij are gamma-matrices (or generators of the Clifford algebraClK′(C)) in the smallest possible representation, which make different fermions anti-commuting. Due to presence of K ′ cycles of type [l′i]−, the zero-mode χZM(q) gener-

ating operators include now γjeQ(i)

, which perform integer shifts of i-th bosonic zeromode together with inessential action on fermionic vacua – now we do not have toimply that the number of shifts by eQ

(i)should be even. Hence, instead of DK-lattice

from (6.85) the numerator includes now summation over the root lattice QBK , i.e.

χg(q) = q∆0g

2[K′+12

]−1∑

~n∈QBK

q

K∑i=1

12li

(ni+liri)2

K∏i=1

∞∏k=1

(1− qk/li)K′∏i=1

∞∏k=0

(1− q(k+ 12

)/l′i)

(6.88)

166


where factor 2[K′+12

]−1 corresponds to the dimension of the smallest representation ofso(K ′), generated by γiγj. Another simple factor q∆0

g contains the minimal conformaldimension (without contribution of the “r-charges”)

∆0g =

K∑i=1

l2i − 1

24li+

K′∑i=1

2l′2i + 1

48l′i(6.89)

which will be computed below in (6.142), (6.191). Numerator of (6.88) contains Kcontributions from twisted bosons corresponding to plus-cycles, and K ′ contributionsfrom twisted Ramond bosons corresponding to minus-cycles.

so(2N + 1) twist fields

The W-algebra W (so(2N+1)) contains fermionic operator V (z) = Ψ1(z) . . .Ψ2N+1(z),which cannot be expressed in terms of generators of so(2N+1)1 since they are all evenin fermions. It means that to construct a module of the W -algebra one should useentire fermionic algebra. Taking into account the fermionic nature of this W-algebraone can consider Z/2Z graded modules and define two different characters

χ+(q) = tr qL0 , χ−(q) = tr (−1)F qL0 (6.90)

where F is the fermionic number:

(−1)FUk(z) = Uk(z)(−1)F , (−1)FV (z) = −V (z)(−1)F (6.91)

One of the characters vanishes χ−(q) = 0 if at least one fermionic zero mode exists,since each state gets partner with opposite fermionic parity. Such fermionic zeromodes are always present for the Ramond fermions and η-fermions, so the only casewith non-trivial χ−(q) corresponds to:

g = [1]K∏i=1

[li, e2πiri ]+ (6.92)

In this case our computation works as follows: take bosonization for the [l]+-cycles interms of K twisted bosons (6.266), (6.80), then the fermionic operators produce the

zero-mode shifts e±Q(i)

with the fermionic number F = F b + F f = F b = 1, and theHeisenberg generators J

(i)n/li

with the fermionic number F = F b = 0. Moreover, we

also have an extra “true” fermion Ψ(z) with F = F f = 1. Therefore the total tracecan be computed, separating bosons and fermions, as

χ−(q) = tr qL0(−1)F = tr qLb0(−1)F

b · tr qLf0 (−1)F

f(6.93)

where the traces over bosonic and fermionic spaces are given by

tr qLb0(−1)F

b

=

∑n1,...,nK∈Z

q

K∑i=1

12li

(ni+liri)2

(−1)

K∑i=1

ni

K∏i=1

∞∏n=1

(1− qn/li)

tr qLf0 (−1)F

f

=∞∏n=0

(1− qn+ 12 )

(6.94)

167


Hence, the final answer for this character is given by

χ−g (q) = q∆0g

∑~n∈QDK

q

K∑i=1

12li

(ni+liri)2

−∑

~n∈QD′K

q

K∑i=1

12li

(ni+liri)2

∞∏k=0

(1− qk+ 12 )

K∏i=1

∞∏k=1

(1− qk/li)

(6.95)

where D- and D′-lattices are defined in (6.205).

Let us now turn to the computation of χ+(q). Choose an element from NO(2N+1)(h)

g = [(−1)a+1]K∏i=1

[li, e2πiri ]+

K′∏i=1

[l′i]− (6.96)

where a = 0, 1. The bosonized fermions eiϕ(i)(z) contain elements of eQ

(i). generating

the BK root lattice, which together with contribution of the fermionic and Heisenbergmodes finally give

χ+g (q) = q∆0

g

2[K′

2]∑

~n∈QBK

q

K∑i=1

12li

(ni+liri)2 ∞∏k=0

(1 + qk+a2 )

K∏i=1

∞∏k=1

(1− qk/li)K′∏i=1

∞∏k=0

(1− q(k+ 12

)/l′i)

(6.97)

where

∆0g =

δa,016

+K∑i=1

l2i − 1

24li+

K′∑i=1

2l′2i + 1

48l′i(6.98)

Here the only new part, compare to the DN -case, is extra factor corresponding to (Ror NS)

χf (q) = qδa,016

∞∏k=0

(1 + qa2

+k) (6.99)

fermionic contribution.

Character identities

In sect. 6.4 we have classified the twist fields by conjugacy classes in NG(h). However itis possible, that two different elements g1, g2 ∈ NG(h) in the normalizer of Cartan arenevertheless conjugated in the group G. Such twisted representations are isomorphic,and it gives an obvious

Theorem 6.3. If g1 ∼ g2 in G for different g1, g2 ∈ NG(h), then χg1(q) = χg2(q).

This leads sometimes to non-trivial identities and product formulas for the latticetheta-functions, and below we examine such examples.

168


gl(N) case. Here any element is conjugated to a product of cycles of length one:

[l, e2πir

]∼

l−1∏j=0

[1, e2πivj ] , (6.100)

where vj = r + 1−l+2j2l

. One gets therefore an identity

∑k1+...+kN=0

q12

N∑i=1

(vi+ki)2

η(q)N=

∑n1+...+nK=0

q

K∑i=1

12li

(ni+liri)2

K∏i=1

η(q1/li)

(6.101)

where all conformal dimensions for vanishing r-charges are conveniently absorbed bythe Dedekind eta-functions η(q) = q1/24

∏∞n=1(1− qn).

This equality of characters can be checked by direct computation, see (6.221) inAppendix 6.9 for S = 0. For a single cycle K = 1 this gives a product formula forthe lattice AN−1-theta function (6.220), which for N = 2

q116∏

k≥0 (1− qk+1/2)=

∑n∈Z q

(n+1/4)2∏n>0(1− qn)

(6.102)

was known yet to Gauss and has been originally used by Al. Zamolodchikov in thecontext of twist-field representations of the Virasoro algebra.

so(2N) case. For the conjugacy classes of the first type we have again (6.100), or

[l, e2πir

]+∼

l−1∏j=0

[1, e2πivj ]+ (6.103)

which leads to very similar identities to the gl(N)-case. For example, one can easilyrederive the product formula [Mac] for the D-lattice theta function

∑~n∈QDN

q12

(~n+~v)2

= ΘDN (~v|q) =η(q)N+1η(q1/(N−1))

η(q1/2)η(q1/2(N−1)) (6.104)

for ~v = ~ρh, where the structure of product in the r.h.s. again comes from the character-

istic polynomial of the Coxeter element of the Weyl group W(DN). Here h = 2(N − 1)is the Coxeter number, and ~ρ = (N − 1, N − 2, . . . , 1, 0) is the Weyl vector, corre-

sponding to the twist field with dimension ∆ = ∆0 = N(2N−1)48(N−1)

, and the easiest way to

derive (6.104) is to use (6.223) from Appendix 6.9.For another type of the conjugacy classes [l]−, the situation is more tricky. The

corresponding η-fermion

η(z) = z−12

∑k∈Z

ηk

zk2l

(6.105)

169


can be separated into the parts with fixed monodromies around zero:

η(a) = z−12

∑k∈Z

ηa+2l·k

za2l

+k, (6.106)

so that the only non-trivial OPE is between η(a) and η(2l−a). In particular, η(0) andη(l) are self-conjugated Ramond (R) and Neveu-Schwarz (NS) fermions, which canbe combined into new η fermion, whereas all other components can be considered ascharged twisted fermions ψ, ψ∗:

ψ(a)(z) = η(a)(z), ψ∗(z) = η(2l−a)(z), a = 1, . . . , l − 1

η(z) = η(0)(z) + η(l)(z)(6.107)

Therefore one gets equivalence

[l]− ∼ [1]− ·l−1∏j=1

[1, e2πivj ] , (6.108)

where vj = j2l

.

Moreover, if we take the product of two cycles [1]−, then we can combine a pair ofR-fermions and a pair of NS-fermions into two complex fermions with charges 0 and12, therefore

[1]− [1]− ∼ [1, 1]+[1,−1]+ (6.109)

This means literally that pair of η-fermions is equivalent to two charged bosons: onewith charge v = 0 and another one with charge v = 1

2. Equivalence between these

two representations leads to the simple identity (6.247), (6.248):

2q18

∞∏n=1

(1− qn+ 12 )2

=

∑k,n∈Z

q12n2+ 1

2(k+ 1

2)2

∞∏n=1

(1− qn)2(6.110)

Using this identity we can remove a pair of [1]− cycles from (6.88) shifting K ′ 7→ K ′−2,and add two more directions to the lattice of charges BK 7→ BK+2 with correspondingr-charges 0 and 1

2.

so(2N) case, K ′ = 0. We have the consequence of identity (6.221) for the caseS = 2Z: ∑

~k∈QDN

q12

N∑i=1

(vi+ki)2

=K∏i=1

η(q)li

η(q1li )·∑

~n∈QDK

q

K∑i=1

12li

(ni+liri)2

(6.111)

so(2N) case, K ′ > 0; so(2N + 1), K ′ > 0. In these cases everything can beexpressed in factorized form using (6.223) and checked explicitly, so these cases arenot very interesting.

170


so(2N + 1) case, NS fermion. Here in addition to all identities that we had inthe so(2N) case, we have two more identities that appear because of the fact that wecan combine NS (or R) fermion with a pair of NS,R fermions to get one complexfermion with twist 0 (or twist 1

2) and one R-fermion (or NS-fermion). Thus

[1] · [1]− ∼ [−1] · [1, 1]+

[−1] · [1]− ∼ [1] · [1,−1]+(6.112)

Thanks to these identities in the cases K ′ 6= 0 we can transform any character withNS fermion to a character with R fermion, and vice versa.

Twist representations and modules of W-algebras

By definition, all our twisted representations are representations of the W -algebra.As was explained in previous section it is sufficient to consider the case g ∈ H (otherelements of NG(h) are conjugated to H). In this case subspaces of Hg with all fixedfermion charges become representations of W-algebra6. The r-charges of the corre-sponding representations are given by shifts of the vector ~r = log g

2πiby root lattice

of g.

The explicit formulas are given below, but we want first to comment the irre-ducibility of representations. The Verma modules of W -algebras are irreducible if

(α, r) 6∈ Z, (6.113)

see [FKW], [Arakawa] (in particular Theorem 6.6.1) or [FL] (eq (4.4)). For genericr this condition is satisfied and all modules arising in the decomposition (subspacesof Hg with all fixed fermion charges) are Verma modules due to coincidence of thecharacters.

If g comes from the element of NG(h) with nontrivial cyclic structure then r is notnecessarily generic. For gl(N) case, as follows (6.100), the r-charges corresponding toa single cycle do satisfy (6.113), and for different cycles this condition also holds pro-vided r are generic (no relations between r from different cycles). The same argumentworks for so(N) with “plus-cycles”, but if we have at least two “minus-cycles” thecorresponding r-charges can violate condition (6.113), and not only Verma modulesarise in the decomposition over irreducible representations.

In any case we have an identity of characters

χg(q) = χ0(q)χg(q) (6.114)

where χ0(q) is the character of Verma module, and χg(q) is the character of the spaceof highest vectors. Hence, there is a non-trivial statement, that all coefficients of thepower expansion of the ratios χg(q)/χ0(q) are positive integers, which can be provenusing identities, derived in previous section.

The list of characters of the Verma modules, appeared above is:

6This is a common well-known procedure, see e.g. [MMMO] and references therein.

171


• gl(N), so(2N) (NS sector). Algebra is generated by N bosonic currents, eachof them producing 1∏

n>0(1−qn), so the character is

χ0(q) =1∏∞

n=1(1− qn)N(6.115)

• so(2N) (R sector). One of these currents, V (z), becomes Ramond, with half-integer modes:

χ0(q) =1∏∞

n=1(1− qn)N−1∏∞

n=0(1− q 12

+n)(6.116)

• so(2N + 1) (NS sector). One current, V (z) becomes Neveu-Schwarz fermion, sotaking into account its parity we get

χ±0 (q) =

∏∞n=0(1± q 1

2+n)∏∞

n=1(1− qn)N(6.117)

• so(2N + 1) (R sector). In the case of Ramond fermion V (z) character χ−0 (q)vanishes because fermionic zero mode produces equal numbers of states withopposite fermionic parities:

χ+0 (q) = 2

∏∞n=1(1 + qn)∏∞n=1(1− qn)N

χ−0 (q) = 0

(6.118)

gl(N) case. Any element is conjugated to a product of cycles of length 1, so

χg(q) = q∆0g

∑~n∈AN−1

q12

(v+~n)2

(6.119)

so(2N) case, K ′ = 0. Any element is conjugated to∏N

j=1[1, e2πvj ]+, so

χg(q) = q∆0g

∑~n∈DN

q12

(v+~n)2

(6.120)

so(2N) case, K ′ > 0, NS-sector. Again, any element is conjugated to∏

[1, e2πvj ]+,so

χg(q) = 2K′2−1q∆0

g

∑~n∈BN

q12

(v+~n)2

(6.121)

so(2N) case, R-sector. Here any element is conjugated to [1]−∏N−1

j=1 [1, e2πvj ]+, so

χg(q) = 2[K′

2]q∆0

g

∑~n∈BN−1

q12

(v+~n)2

(6.122)

because contribution from the cycle [1]− to the denominator cancels contribution fromthe Ramond boson V (z).

172

6.6. Characters from lattice algebras constructions

so(2N + 1) case, K ′ = 0, NS fermion. Here one has two non-trivial characters

χ+g (q) = q∆0

g

∑~n∈BN

q12

(v+~n)2

χ−g (q) = q∆0g

∑~n∈DN

q12

(v+~n)2 −∑~n∈D′N

q12

(v+~n)2

(6.123)

so(2N + 1) case, K ′ > 0 This case gives nothing interesting as compared to DN

situation.

χ+g (q) = 2[K

′2

]q∆0g

∑~n∈BN

q12

(v+~n)2

χ−g (q) = 0

(6.124)

Characters from lattice algebras constructions

Twisted representation of g1

Now we reformulate the results of previous sections using the notion of twisted repre-sentations of vertex algebras. Recall the corresponding setting (following, for example,[BK]). Let V be a vertex algebra (equivalently vacuum representation of the vertexalgebra), and σ be a automorphism of V of finite order l. Then V = ⊕Vk, whereVk = v ∈ V |σv = exp(2πik/l)v. The σ-twisted module is a vector-space M en-dowed with a linear map from V to the space of currents

v 7→ Av(z) =∑m∈ 1

lZ

am(v)z−m−1, v ∈ V, am(v) ∈ End(M).(6.125)

Such correspondence should be σ-equivariant, namely

Aσv(z) = Av(e2πiz) (6.126)

giving the boundary conditions for the currents, and agree with the vacuum vectorand relations in V . In particular, it follows from the σ-equivariancy (6.126), that ifv ∈ Vk then Av(z) ∈ z−k/lC[[z, z−1]].

Consider now a Lie group G (either GL(N) or SO(2N), N ≥ 2), with g = Lie(G)being the corresponding Lie algebra. Denote by V(g) the irreducible vacuum repre-sentation of g of the level one. This space has a structure of the vertex algebra i.e.for any v ∈ V(g) one can assign the current Av(z), this space of currents is generatedby the currents Jαβ(z) from sect. 6.3.

The vertex algebra V(g) is a lattice vertex algebra. Let Qg denote the root latticeof g, and introduce rank of g bosonic fields with the OPE ϕi(z)ϕj(w) = −δij log(z −w) + reg, and the stress-energy tensor T (z) = −1

2

∑j : ∂ϕj(z)2 :, then the currents of

V(g) can be presented in the bosonized form

:∏i,m

∂ai,mϕi exp(i∑

αiϕi(z)) :, (6.127)

173


where α = (α1, . . . , αn) ∈ Qg and ai,m are any positive integers, while the stress-energytensor corresponding to standard conformal vector 1

2

∑J2j,−1|0〉 = τ ∈ V(g) (here Jj,n

are modes of the field i∂ϕj(z)). The group G acts on V (g), and in order to use latticealgebra description we consider only the subgroup NG(h) ⊂ G which preserves theCartan subalgebra.

In [BK] the representations of the lattice vertex algebra, twisted by automor-phisms, arise from isometries of the lattice Qg. Here we restrict ourself to the isome-tries provided by action of the Weyl group W (this case was actually considered in [KP]without language of twisted representations). Let s ∈ W be an element of the Weylgroup, by g we denote its lifting to G, in other words g ∈ NG(h) such that adjointaction g on h coincides with s. We consider representation twisted by such g. Settingof [BK] and [KP] works for special g, for example such g should have finite order,but we will expand this to the generic g ∈ NG(h). Clearly, the conformal vector τ isinvariant under the adjoint action of NG(h).

The g-twisted representations of V (g) in [BK] are defined as a direct sum of twisted

representations of h. By e2πiθAdj,k we denote eigenvalues of s, or of the adjoint actiongadj on h, we set −1 < θAdj,k ≤ 0, by Jk ∈ h - the corresponding eigenvectors, anddefine the currents

Jk(z) =∑n∈Z

Jk,θAdj,k+nz−θAdj,k−n−1

(6.128)

A g-twisted representations of the Heisenberg algebra h is a Fock module Fµ with thehighest weight vector vµ

Jk,θAdj,k+nvµ = 0, n > 0, Jk,0vµ = µ(Jk)vµ, if θAdj,k = 0. (6.129)

generated by creation operators Jk,θAdj,k+n, n ≤ 0. Here µ ∈ h∗0, where h0 is gadj-invariant subspace of h.

It has been proven in [BK] that twisted representations of V (g) have the structure

M(s, µ0) = ⊕µ∈µ0+πsQgFµ ⊗ Cd(s) (6.130)

for certain finite set of µ0 ∈ h∗0. Here πs denotes projection from h∗ to h∗0, corre-sponding to the element s ∈ W for the chosen adjoint action gadj. For any root α thecorresponding current Jα(z) acts from Fµ to Fµ+πsα and equals to the linear combi-nation of vertex operators. Number d(s) denotes the defect of the element s ∈ W , itssquare is defined by

d(s)2 = |(Qg ∩ h⊥0 )/(1− s)Pg|. (6.131)

Here Pg denotes weight lattice of g, h⊥0 denotes the space of linear functions vanishingon h0, | · | stands for number of elements in the group. It can be proven that for anys the numbers d(s) is integer. In our case (GL(N) and SO(n) groups) this numberalways equals to some power of 2.

Formula (6.130) allows to calculate the character of module M i.e. the trace ofqL0 . First, notice that the character of the Fock module Fµ equals

χµ(q) =q∆µ∏

i

∏∞n=1(1− qθAdj,i+n)

(6.132)

174


where ∆µ is an eigenvalue of L0 on the vector vµ. The value of ∆µ consists of twocontributions. The first comes from the terms with θAdj = 0, and, as follows from(6.129), is equal to 1

2(µ, µ). The second contribution comes from the normal ordering.

The vectors Jk ∈ h, corresponding to θAdj,k 6= 0 can be always arranged into orthogonalpairs (J1, J1′), (J2, J2′), . . . with complementary eigenvalues θAdj,k + θAdj,k′ = −1 7.After normal ordering of the corresponding currents one gets

Jk(z)Jk′(w) =∑n,m∈Z

Jk,n+θ

zn+θ+1

Jk′,m−θwm−θ+1

=∑

n∈Z,m≥0

Jk,n+θ

zn+θ+1


+

+∑

n∈Z,m<0


Jk,n+θ

zn+θ+1+∑n>0

(n+ θ)wn+θ−1

zn+θ+1

(6.133)

where θ = θAdj,k. The last term in the r.h.s., which appears due to [Jk,n+θ, Jk′,m−θ] =(n+ θ)δn+m,0 also gives a nontrivial contribution to the action of L0 on highest vectorvµ, since ∑

n>0

(n+ θ)wn+θ−1

zn+θ+1=

(1 + θ)wθz−θ + (−θ)w1+θz−1−θ

(z − w)2=

=z→w

1

(z − w)2− θ(1 + θ)

2w2+ reg

(6.134)

Altogether one gets

∆µ =1

2(µ, µ)−

∑k

θAdj,k(1 + θAdj,k)

4(6.135)

and therefore, finally for the character of (6.130)

TrqL0

∣∣∣M(s,µ0)

= q−14

∑k θAdj,k(1+θAdj,k)

d(s)∑

µ∈µ0+πwQq

12

(µ,µ)∏Ni=1

∏∞n=1(1− qθAdj,i+n)

(6.136)

Recall that in the the initial weight µ0 in the setting of [BK] should belong to thefinite set in h∗0 (or h∗0/πWQ). But we will generalize such representations and take anyµ0 ∈ h∗0. This can be viewed as a twisting by more general elements g ∈ NG(h), whichcan have infinite order. Actually the corresponding elements are representatives ofthe conjugacy classes of NG(h) used in sect. 6.4.

Calculation of characters

GL(N) case The root lattice Qgl(N) = QAN−1is generated by vectors ei − ej,

where e1, . . . , eN denote the vectors of orthonormal basis in RN . Assume that s ∈ W

is product of disjoint cycles of lengths l1, . . . , lK , then without loss of generality theaction of such elements can be defined as (e1 7→ e2 7→ . . . 7→ el1 7→ e1), (el1+1 7→el1+2 7→ . . . 7→ el1+l2 7→ el1+1), . . ..

In this case h∗0 (the s-invariant part of h∗) is generated by the vectors

f1 = e1 + . . .+ el1 , f2 = el1+1 + . . .+ el1+l2 , . . . (6.137)

7There is also “degenerate” case Jk = Jk′ for θAdj,k = θAdj,k′ = − 12 .

175


while πsQgl(N) is generated by vectors 1lifi − 1

ljfj, so one can present any element

of πsQgl(N) as∑

1ljnjfj with

∑nj = 0 and identify with that from Qgl(K). Let

µ0 =∑

j rjfj. Then the formula (6.136) takes here the form

Tr(qL0)∣∣M(s,µ0)

= q∆0s

∑Qgl(K)

q∑j

12lj

(nj+ljrj)2

∏Kj=1

∏∞n=1(1− qn/lj)

, (6.138)

where, since for any length l cycle θAdj,k = −k/l,

∆0s =

K∑j=1

lj∑i=1

i(lj − i)4l2j

=K∑j=1

l2j − 1

24lj(6.139)

This formula coincides with (6.82), and the reason is that the corresponding elementfrom NGL(N)(h) is exactly (6.45), g =

∏Kj=1[lj, e

2πirj ]. Indeed, let α = ea − eb, wherea belongs to the cycle j and b belongs to the cycle j′ then the current Jα(z) shifts L0

grading by rj − rj′ + [rational number with denominator lj, lj′ ].

SO(2N) case The root lattice Qso(2N) = QDN is generated by the vectors ei −ej, ei+ej, where again e1, . . . , eN denote the basis in RN . As we already discussed insect. 6.4, there are two types of the Weyl group elements, the first type just permutesei, while the second type permutes ei together with the sign changes.

The first case almost repeats the previous paragraph, without loss of generalitywe assume that the Weyl group element acts as (e1 7→ e2 7→ . . . 7→ el1 7→ e1), (el1+1 7→el1+2 7→ . . . 7→ el1+l2 7→ el1+1), . . ., where l1, . . . , lK are again the lengths of the cycles.The s-invariant part of h∗0 is generated by the same “averaged” vectors (6.137), whileπsQDN is generated by the vectors 1

lifi− 1

ljfj,

1lifi+

1ljfj. In other words πsQDN consist

of vectors∑ nj

ljfj, where (n1, . . . , nk) ∈ Qso(2K). Let µ0 =

∑j rjfj, then the character

formula (6.136) for this case acquires the form


= q∆0s

∑Qso(2K)

q∑j

12lj

(nj+ljrj)2

∏Kj=1

∏∞n=1(1− qn/lj)

(6.140)

and coincides with (6.85). Here ∆0s is defined in (6.139). The corresponding element

from NSO(2N)(h) has the form∏K

j=1[lj, e2πirj ]+ in the notations of sect. 6.4 (see (6.56)).

For the second type (the corresponding element from NSO(2N)(h) has the form∏Kj=1[lj, e

2πirj ]+ ·∏K′

j′=1[lj′ ]−) one can present the Weyl group element as product ofK disjoint cycles of lengths l1, . . . , lK which just permutes ei and K ′ cycles of lengthsl1′ , . . . , lK′ which permutes ei with signs, see (6.56). Now, without loss of generality,we assume that s acts as (e1 7→ e2 7→ . . . 7→ el1 7→ e1), (el1+1 7→ el1+2 7→ . . . 7→el1+l2 7→ el1+1), . . ., (eL+1 7→ e2 7→ . . . 7→ eL+l1′

7→ −e1), (eL+l1′+1 7→ eL+l1′+2 7→. . . 7→ eL+l1′+l2′

7→ −eL+l1′+1), . . ., where L = l1 + . . . + lK . The s-invariant part ofh∗0 is generated by the same vectors (6.137), while πsQDN is generated by the vectors1lifi. One can say that πsQDN consists of the vectors

∑ njljfj, where (n1, . . . , nk) ∈

176


Qso(2K+1) = QBN , so that for the character formula one gets


= q∆0s

2K′/2−1

∑Qso(2K+1)

q∑j

12lj

(nj+ljrj)2

∏Kj=1

∏∞n=1(1− qn/lj)

∏K′

j=1

∏∞n=1(1− q(2n−1)/2l′j)

, (6.141)

where, since in addition to [l]+-cycles with θAdj,k = −k/l one now has [l′]−-cycles withθ′Adj,k = −(k − 1

2)/l′,

∆0s =

K∑j=1

lj∑i=1

i(lj − i)4l2j

+K′∑j=1

l′j∑i=1

(2i− 1)(2l′j − 2i+ 1)

16l′2j=

=K∑j=1

l2j − 1

24lj+

K′∑j=1

2l′2j + 1

48l′j

(6.142)

This formula coincides with (6.88). The number 2K′/2−1 equals to d(σ), this is the

first case where this number is nontrivial. Note, that we consider here only internalautomorphisms, i.e. K ′ is even.

Recall also (see sect. 6.5.5) that if g, g′ ∈ NG(h) are conjugate in G then corre-sponding characters Tr(qL0)

∣∣M(s,µ0)

and Tr(qL0)∣∣M(s′,µ′0)

are equal.

Characters from principal specialization of the Weyl-Kac for-mula

Fix element g ∈ G of finite order l. The g-twisted representations of V (g) are repre-sentations if the affine Lie algebra twisted by g. Recall that these twisted affine Liealgebras L(g, g) are defined in [KacBook, Sec 8] as g invariant part of g[t, t−1] ⊕ Ckwhere g acts as

g(tj ⊗ J) = ε−jtj ⊗(gJg−1

), where ε = exp(2πi/l), g(k) = k. (6.143)

By definition g is an internal automorphism, therefore the algebra L(g, g) is isomorphic

to g (see Theorem [KacBook, 8.5]), though natural homogeneous grading on L(g, g)differs from the homogeneous grading on g.

Therefore the g-twisted representations of V (g) as a vector spaces are integrablerepresentations of g 8. Their characters can be computed using the Weyl-Kac char-acter formula. This formula has simplest form in the principal specialization, i.e.computed on the element qρ

∨ ∈ G. Here ρ∨ ∈ h ⊕ Ck such that αi(ρ∨) = 1, for all

affine simple roots αi (including α0) . Then the character of integrable highest weightmodule with the highest weight Λ equals (see [KacBook, eq. (10.9.4)])

Tr(qρ∨/h)

∣∣LΛ

= qΛ(ρ∨)/h∏

α∨∈∆∨+

(1− q(Λ+ρ,α∨)/h

1− q(ρ,α∨)/h

)mult(α∨)

, (6.144)

8Note that we get only level 1 integrable representation of g since V (g) was defined above as alattice vertex algebra i.e. vacuum representation of the level k = 1

177


where ∆∨+ is the set of all positive (affine) coroots. Here h is the Coxeter number, itwill be convenient to use qρ

∨/h instead of qρ∨. The weight ρ is defined by (ρ, α∨i ) = 1

for all simple coroots αi (including affine root α0).The grading above in this section was the L0 grading and it was obtained using

the twist by the element g ∈ NGh. Now we take certain g such that g-twisted L0

grading coincides with principal grading in (6.144). We take g in Cartan subgroup Hand as was explained above choice g corresponds to the choice of µ0 in (6.136).

In the principal grading used in (6.144) degEαi = 1h

for all simple roots Eαi(including affine root α0). Therefore µ0 ∈ Pg + 1

hρ, where Pg is the weight lattice for

g, ρ is defined by the formula (ρ, αi) = 1 for all simple roots9.Below we write explicit formulas for characters of twisted representation corre-

sponding to such g (and such µ). In the simply laced case, computing the charactersusing two formulas (6.136) and (6.144) one gets an identity, which is actually theMacdonald identity [Mac].

In notation for root system we follow [Mac] and [KacBook]. Below we considerroots as vectors in the linear space, generated by e1, . . . , en, δ,Λ0, and coroots – in thespace generated by e∨1 , . . . , e

∨n , K, d. The pairing between these dual spaces given by

(ei, e∨j ) = δij, (Λ0, K) = (δ, d) = 1 while all other vanish.

GL(N) case. Root system is A(1)N−1 (affine AN−1) dual root system is also A

(1)N−1.

Simple roots: α0 = δ − e1 + eN , αi = ei − ei+1, 1 ≤ i ≤ N − 1

Simple coroots: α∨0 = K + e∨N − e∨1 , α∨i = e∨i − e∨i+1, 1 ≤ i ≤ N − 1

Real coroots: mK + e∨i − e∨j , m ∈ Z, i 6= j

Imaginary coroots: mK of multiplicity N, m ∈ Z.

Level k = 1 weights: Λ0, Λj = Λ0 +

j∑i=1

ei, 1 ≤ j ≤ N − 1

h = N, ρ = 12

N∑i=1

(N − 2i+ 1)ei +NΛ0, ρ = 12

N∑i=1

(N − 2i+ 1)ei.

(6.145)

Note the multiplicity of imaginary roots in N instead on N − 1 since we considerG = GL(N) instead of SL(N), and the corresponding affine algebra differs by oneadditional Heisenberg algebra.

The computation of the denominator in (6.144), using (6.145) gives∏α∨∈∆∨+

(1− q(ρ,α∨)/h)mult(α∨) =∞∏k=1

(1− qk/N)N (6.146)

while for the numerator (the same for all level k = 1 weights) one gets∏α∨∈∆∨+

(1− q(Λ+ρ,α∨)/h)mult(α∨) =∞∏k=1

(1− qk/N)N−1(6.147)

9Note the difference between ρ and ρ): first was defined by pairing with simple coroots (includingaffine one) and the second is defined by scalar products with (non affine) roots. In the simply lacedcase conditions in terms of roots and coroots are equivalent and we have ρ = ρ+ hΛ0

178


so that the character (6.144) in principal specialization

q−Λ(ρ∨)/hTr(qρ∨/h)

∣∣LΛ

=1∏∞

k=1(1− qk/N)(6.148)

One can compare the last expression with the formula (6.136) using the choice of µ0,as explained above. We get an identity∑

α∈Qsl(N)

q12

(α+ 1Nρ,α+ 1

Nρ)

∏∞k=1(1− qk)N

=q

12N2 (ρ,ρ)∏∞

k=1(1− qk/N).

(6.149)

which is a particular case of formula (6.101) from sect. 6.5.5, and again reproducesthe product formula for the lattice AN−1-theta function (6.220).

Recall that the r.h.s. of (6.149) also has an interpretation of a character of thetwisted Heisenberg algebra. This twist of the Heisenberg algebra emerges in therepresentation twisted by g with gAdj acting as the Coxeter element of the Weyl group,hence r.h.s. of (6.149) equals to the r.h.s. of (6.138) for a single cycle K = 1, l = N .This g is conjugate to used above in computing of l.h.s., therefore the characters of thetwisted modules should be the same. The construction of level one representations interms of principal Heisenberg subalgebra is well-known, see [LW, KKLW]. Anotherinterpretation of the l.h.s in (6.149) is the sum of characters of the W -algebra namelyW algebra of gl(N), (see sect. 6.5.6).

SO(2N) case. Root system D(1)N (affine DN), the dual root system is also D

(1)N .

Simple roots: α0 = δ−e1−e2, αi = ei−ei+1, 1 ≤ i < N, αN = eN−1+eN

Simple coroots: α∨0 = K − e∨1−e∨2 , α∨i = e∨i −e∨i+1, 1 ≤ i < N, αN = e∨N−1+e∨NReal coroots: mK+e∨i −e∨j , mK + e∨i + e∨j , mK−e∨i −e∨j ,m ∈ Z, i 6= j

Imaginary coroots: mK of multiplicity N, m ∈ Z

k = 1 weights: Λ0, Λ1=e1+Λ0, ΛN−1=12

N∑i=1

ei+Λ0, ΛN=12

N∑i=1

ei−eN+Λ0

h = 2N − 2, ρ =N∑i=1

(N − i)ei + (2N − 2)Λ0, ρ =N∑i=1

(N − i)ei.

(6.150)Now we again just compute the denominator∏

α∨∈∆∨+

(1− q(Λ+ρ,α∨)/h)mult(α∨) =∞∏k=1

(1− qk/(2N−2))N (6.151)

and the numerator (the same for all k = 1 weights)∏α∨∈∆∨+

(1− q(ρ,α∨)2N−2 )mult(α∨) =

=N−1∏j=1

∞∏k=1

(1− qk−2j−12N−2 )N+1(1− qk−

2j2N−2 )N ·

∞∏k=1

(1− qk−12 )

(6.152)

179


in (6.144), giving for the character

q−Λ(ρ∨)/hTr(qρ∨/h)

∣∣LΛ

=1∏N−1

j=1

∏∞k=1(1− qk−

2j−12N−2 ) ·

∏∞k=1(1− qk− 1

2 ). (6.153)

As in previous case, comparing this with the formula (6.130), one gets an identity∑α∈QDN

q12

(α+ 1hρ,α+ 1

hρ)∏∞

k=1(1− qk)N=

q1

2h2 (ρ,ρ)∏N−1j=1

∏∞k=1(1− qk−

2j−12N−2 ) ·

∏∞k=1(1− qk− 1

2 ). (6.154)

where the r.h.s. side can be interpreted as a character of the representation Heisenbergalgebra twisted by g such that gAdj is Coxeter element. Again, this is the same asconstruction of level k = 1 representation in terms of principal Heisenberg subalgebrafrom [LW, KKLW]. The l.h.s formula (6.154) can be also interpreted as the sum ofcharacters of the W (so(2N))-algebra, (see sect. 6.5.6).

By now in this section we have considered only the simply laced case – the onlyone, when the algebra V (g) is lattice algebra or, in other words, when the level k = 1representations can be constructed as a sum of representations of the Heisenbergalgebra. However, the formula (6.144) is valid for any affine Kac-Moody algebra.Below we consider the case G = SO(2N + 1), where the level k = 1 representationscan be constructed using free fermions.

SO(2N + 1), N > 2 case. Root system is B(1)N (affine BN), the dual root system is

B(1,∨)N = A

(2)2N−1 (affine twisted A2N−1)

Simple roots: α0 = δ−e1−e2, αi = ei−ei+1, 1 ≤ i ≤ N−1, αN = eN .

Simple coroots: α∨0 = K − e∨1−e∨2 , α∨i = e∨i −e∨i+1, 1 ≤ i ≤ N−1, αN = 2e∨N .

Real coroots: 2mK±2ei, mK±ei∓ej, mK±ei±ej, 1 ≤ i < j ≤ N, m ∈ Z.Imaginary coroots: (2m− 1)K of multiplicity N − 1, m ∈ Z

2mK of multiplicity N,m ∈ Z \ 0.

k = 1 weights: Λ0, Λ1 = Λ0 + e1, ΛN = Λ0 + 12

∑N

i=1ei

h = 2N, ρ =∑N

j=1(N − j + 1

2)ej + (2N − 1)Λ0, ρ =

∑N

j=1(N − j + 1)ej.

(6.155)Compute again the denominator∏

α∨∈∆∨+

(1− q

(ρ,α∨)2N

)mult(α∨)

=∞∏k=1

(1− qk

2N )N ·∞∏k=1

(1− q2k−12N ) (6.156)

and the numerator in the formula (6.144). Now the numerator for Λ = Λ0 and Λ = Λ1

is the same ∏α∨∈∆∨+

(1− q

(ρ+Λ0,α∨)

2N

)=

∏α∨∈∆∨+

(1− q

(ρ+Λ1,α∨)

2N

)=

=∞∏k=1

(1− qk

2N )N ·∞∏k=1

(1 + qk)

(6.157)

180


but for Λ = ΛN it is different

∏α∨∈∆∨+

(1− q

(ρ+ΛN,α∨)

2N

)=∞∏k=1

(1− qk

2N )N ·∞∏k=1

(1 + qk−12 ), (6.158)

Here we used the identities (6.235) and∏∞

k=1(1 − q2k−1)(1 − qk−1/2)−1 =∏∞

k=1(1 +qk−1/2). It is convenient to consider the direct sums of two representations LΛ0 ⊕ LΛ1

and LΛN⊕LΛN since these sums have construction in terms of fermions. Using (6.144)one gets

q−Λ0(ρ∨)/hTr(qρ∨/h)

∣∣LΛ0

+ q−Λ1(ρ∨)/hTr(qρ∨/h)

∣∣LΛ1

= 2∞∏k=1

(1 + qk)

(1− q 2k−12N )

,

q−ΛN (ρ∨)/hTr(qρ∨/h)

∣∣LΛN

= 2∞∏k=1

(1 + qk−12 )

(1− q 2k−12N )

.

(6.159)

The r.h.s. of these equations suggest the existence of the construction of these rep-resentation in terms of N -component twisted (principal) Heisenberg algebra and ad-ditional fermion (in NS and R sectors correspondingly), exactly this construction hasbeen considered in sect. 6.4.4.

On the other hand these characters can be rewritten in terms of the simplestB-lattice theta-functions just using the Jacobi triple product identity

2∞∏k=1

(1 + qk)

(1− q 2k−12N )

=∞∏k=1

2N∏i=0

(1 + qk−i

2N ) =

=∑

n1,...,nN∈Z

q12

∑Nj=1(n2

j+jNnj)

∞∏k=1

(1 + qk−12 )

(1− qk)N=

= q−(N+1)(2N+1)

48N

∑α∈QBN

q12

(α+ 12N

ρ,α+ 12N

ρ)

∞∏k=1

(1 + qk−12 )

(1− qk)N,

(6.160)

and

2∞∏k=1

(1 + qk−12 )

(1− q 2k−12N )

= 2∞∏k=1

2N−1∏i=0

(1 + qk−i

2N )∞∏k=1

(1 + qk−12 ) =

=∑

n1,...,nN∈Z

q12

∑Nj=1(n2

j+(j−1)N

nj) ·∞∏k=1

(1 + qk−1)

(1− qk)N=

= q−(N−1)(2N−1)

48N

∑α∈QBN+ΛN−Λ0

q12

(α+ 12N

ρ,α+ 12N

ρ)

∞∏k=1

(1 + qk−1)

(1− qk)N.

(6.161)

where ΛN −Λ0 is the highest weight of the spinor representation of SO(2N + 1). Ther.h.s. of these formulas are the characters of sums of nontwisted representations of N -component Heisenberg algebra with additional infinite-dimensional Clifford algebra(or real fermion). Another point of view that the r.h.s. are the characters of sums ofrepresentations of W (BN)-algebra [Luk].

181


Finally, let us point out, that for the root system B(1)2 = C

(1)2 (affine B2), the dual

roots system is C(1),∨2 = D

(2)3 (affine twisted D3).

Simple roots: α0 = δ − 2e1, α1 = e1 − e2, α2 = 2e2.

Simple coroots: α∨0 = K − e∨1 , α∨1 = e∨1 − e∨2 , α∨2 = e∨2 .

Real coroots: mK±e∨1 , mK±e∨2 , 2mK±e∨1±e∨2 , 2mK±e∨1∓e∨2 , m ∈ Z.Imaginary coroots: (2m− 1)K of multiplicity 1, m ∈ Z

2mK of multiplicity 2, m ∈ Z \ 0.k = 1 weights: Λ0, Λ1 = ε1 + Λ0, Λ2 = Λ0 + ε1 + ε2

h = 4, ρ = 2e1 + e2 + 3Λ0, ρ =3

2e1 +

1

2e2.

(6.162)

the computation leads to result, coinciding with formulas (6.156), (6.157), (6.158) forN = 2. Though the root system here has a bit different combinatorial structure, thefermionic construction is the same, using 5 real fermions.

Exact conformal blocks of W (so(2N)) twist fields

Global construction

It has been shown in [GMtw], that conformal block of the generic W (gl(N)) twistfields is given by explicit formula, analogous to the famous Zamolodchikov’s conformalblocks of the Virasoro twist fields with dimensions ∆ = 1

16[ZamAT87, ZamAT86,

ApiZam]. To generalize the construction of [GMtw] to all twist fields Og|g ∈ NG(h)considered in this chapter, one needs to glue local data in the vicinity of all twist fieldto some global structure. We consider below such construction for G = O(2N), sinceit can be entirely performed in terms of twisted bosons.

First, let us remind the local data in the vicinity of Og(0) already discussed insect. 6.4:

• 2l-fold cover z = ξ2l with holomorphic involution σ : ξ 7→ −ξ without stablepoints except for the twist field position.

• Fermionic field η(ξ) with exotic OPE η(ξ)η(σ(ξ′)) ∼ 1ξ−ξ′ . On the sheets, con-

nected to each other by [l, e2πir]+, one can identify η(ξ) with ordinary complexfermion ψ(ξ) = η(ξ), η(σ(ξ)) = ψ∗(ξ), in this case σ permutes ψ ↔ ψ∗.

• Bosonic field J(z) = (η(σ(z))η(z)), which is antisymmetric J(σ(z)) = −J(z)under the action of involution σ, and has first-order poles coming from zero-mode charges in the branch-points corresponding to cycles of type [l, e2πir]+.

To compute spherical 2M -point conformal block 10

G0(q1, . . . , q2M) = 〈Oh1·g1(q1)Og−11

(q2) . . .OhM ·gM (q2M−1)Og−1M

(q2M)〉 (6.163)

10In principle, we may choose any monodromies, though in this way we will get complicated twistedrepresentations in the intermediate channels, but as in [GMtw] we restrict ourselves to simpler, butstill quite general case of pairwise inverse (up to diagonal factors hi monodromies.

182

6.7. Exact conformal blocks of W (so(2N)) twist fields

we forget about fermion and consider only the twisted boson with current J(z). Nowlet us list the field-theoretic properties which fix this conformal block uniquely.

Considering the action of 1-form J(z)dz onto the highest weight vector |0〉g of themodule of twist-field Og of order l, due to Jk/l>0|0〉g = 0 one gets, that the mostsingular term

J(z)dz ∼z→0

rdz

z+ . . . (6.164)

in the vicinity of the twist field can be simple pole – in presence of r-charge or a zeromode.

Notice, that for two fields with opposite (up to diagonal factor h = diag (e2πia1 , . . . , e2πian))monodromies

Oh·g(z)Og−1(z′) ∼z→z′

(z − z′)∆h−2∆gVh(z′) + descendants (6.165)

where Vh(z′) is a field with fixed charges ~a ∈ h. Hence

1

2πi

˛

Cjz,z′

J(ξ)dξOh·g(z)Og−1(z′) = ajOg(z)Og−1(z′)(6.166)

where contour Cjz,z′ is the j-th preimage of the contour encircling two points z, z′ onthe base. We identify such contours with the A-cycles on the cover, and correspondinga’s with A-periods of 1-form J(z)dz.

The standard OPE of two currents

J(z)J(z′)dzdz′ =z→z′

dzdz′

(z − z′)2+ 4T (z′) + . . . (6.167)

gives the stress-energy tensor

T (z) =∑

π2N (ξ)=z

T (ξ)

T (z)Og(0) =∆g

z2Og(0) +

1

z∂Og(0) + . . .

(6.168)

and non-standard coefficient (4 instead of 2) arises due to involution σ. Summarizingthese facts we get:

• 2N -sheet branched cover π2N : Σ → P1 with the branch points q1, . . . , q2Mand ramification structure defined by the elements g1, g

−11 . . . , gM , g

−1M . In

particular, Σ is a disjoint union of two curves when all gi do not contain [l]−cycles.

• Involution of this cover σ : Σ → Σ with the stable points coinciding with [li]−cycles

Σ Σ CP1π2

π2N

σπN (6.169)

Projections and involution are shown on the diagram: π2N = πN π2, π2σ = π2.

183


• Odd meromorphic differential dS(σ(ξ)) = −dS(ξ) with the poles in preimagesof qi and residues given by corresponding r-charges.

• Symmetric bidifferential dΩ(ξ, ξ′), satisfying dΩ(σ(ξ), ξ′) = −dΩ(ξ, ξ′), with twopoles:

dΩ2(ξ, ξ′) ∼ξ→ξ′

dξdξ′

(ξ − ξ′)2, dΩ2(ξ, ξ′) ∼

ξ→σ(ξ′)− dξdξ′

(ξ − σ(ξ′))2(6.170)

and vanishing A-periods.

Using this data one can write for two auxiliary correlators

G1(ξ|q1, . . . , q2M) = dξ〈J(ξ)Oh1·g1(q1)Og−11

(q2) . . .OhM ·gM (q2M−1)Og−1M

(q2M)〉G2(ξ, ξ′) = dξdξ′〈J(ξ)J(ξ′)Oh1·g1(q1)Og−1

1(q2) . . .OhM ·gM (q2M−1)Og−1

M(q2M)〉

(6.171)

their explicit expressions

G1(ξ)G−10 = dS(ξ), G2(ξ, ξ′)G−1

0 = dS(ξ)dS(ξ′) + dΩ2(ξ, ξ′) (6.172)

fixed uniquely by their analytic behaviour. Now let us study in detail the structureof the curve Σ in order to construct all these objects.

Curve with holomorphic involution

Involution σ defines the two-fold cover π2 : Σ → Σ with the total number of branchpoints being 2K ′ = 2

∑Mi=1 K

′i, or exactly the total number of [l]− cycles in all elements

gi, g−1i . The Riemann-Hurwitz formula χ(Σ) = 2 · χ(Σ)−#BP then gives for the

genus

g(Σ) = 2g(Σ) +K ′ − 1 (6.173)

Then a natural way to specify the A-cycles on Σ is the following [Fay]: first to take

A(1)1 , . . . , A

(1)g , A

(2)1 , . . . , A

(2)g on each copy of Σ, where g = g(Σ); and second, all other

A-cycles that correspond to the branch cuts of the cover, connecting the branch pointsof π2: A

(0)1 , . . . , A

(0)K′−1. The action of involution on these cycles is obviously given by

σ(A(1)i ) = A

(2)i , σ(A

(2)i ) = A

(1)i , i = 1, . . . , g

σ(A(0)j ) = −A(0)

j , j = 1, . . . , K ′ − 1(6.174)

thus we have the decomposition of the real-valued first homology group into the evenand odd parts

H1(Σ,R) = H1(Σ,R)+ ⊕H1(Σ,R)−

dimH1(Σ,R)+ = g(Σ) = g

dimH1(Σ,R)− = g +K ′ − 1 = g−

(6.175)

Compute now g = g(Σ), using the Riemann-Hurwitz formula for the cover of P1. LetK =

∑Mi=1 Ki be the total number of [l, e2πir]+-type cycles in all elements gi, as well

184


as K ′ serves for the type [l′]−. Then χ(Σ) = N · χ(P1) − #BP gives (cf. with theformula (2.17) of [GMtw])

g = 1−N +K∑i=1

(li − 1) +K′∑i=1

(l′i − 1) (6.176)

so that

g− = g +K ′ − 1 =K∑i=1

(li − 1) +K′∑i=1

l′i −N (6.177)

and

g = 1− 2N + 2K∑i=1

(li − 1) + 2K′∑i=1

(l′i − 12) (6.178)

For our purposes the most essential is the odd part H1(Σ,R)− of the homology. Onecan see these g− A-cycles explicitly as follows: two mutually inverse permutations of

type [l]+ produce l pairs of A-cycles A(1,2)i with constraints

∑iA

(1,2)i = 0. These cycles

are permuted by σ (6.174), so they actually form l−1 independent odd combinations,giving contribution to the r.h.s. of (6.177). For two mutually inverse elements ofthe type [l′]− one gets instead 2l′ A-cycles with constraint

∑iAi = 0, and with

the action of involution σ : Ai 7→ Ai+l′ , giving l′ independent odd combinationsAi −Ai+l′, arising in the r.h.s. of (6.177), while the extra term −N corresponds tocharge conservation in the infinity.

Hence, we got g− odd A-cycles, whose projections to P1 encircle pairs of thecolliding twist fields Oh·g(q2i−1)Og−1(q2i) for i = 1, . . . ,M , so that the integrals of

1

2πi

˛AI

dS = aI , I = 1, . . . , g− (6.179)

give the W-charges in the intermediate channels of conformal block (6.163). ThereforedS can be expanded

dS =

g−∑I=1

aIdωI +2M∑i=1

dSri (6.180)

over the odd holomorphic differentials, and meromorphic differentials of the 3-rd kindcorresponding to the nonvanishing r-charges.

Now, for the bidifferential dΩ2(ξ, ξ′) one can write

dΩ2(ξ, ξ′) = K(ξ, ξ′)−K(σ(ξ), ξ′) = 2K(ξ, ξ′)− K(ξ, ξ′) (6.181)

where K(ξ, ξ′) is the canonical meromorphic bidifferential on Σ (the double logarith-mic derivative of the prime form, see [Fay]), normalized on vanishing A-periods ineach of two variables, and

K(ξ, ξ′) = K(ξ, ξ′) +K(σ(ξ), ξ′) (6.182)

is actually a pullback of the canonical meromorphic bidifferential on Σ. Indeed, con-sider

δK(ξ, ξ′) = K(ξ, ξ′)−K(σ(ξ), σ(ξ′)) (6.183)

185


which is already holomorphic at ξ = ξ′, and¸AiδK(ξ, ξ′) = 0, since due to (6.174)

normalization conditions do not change under involution. Thus δK(ξ, ξ′) = 0 and thecanonical bidifferential is σ-invariant

K(σ(ξ), σ(ξ′)) = K(ξ, ξ′) (6.184)

Moreover, since

K(ξ, ξ′) = K(σ(ξ), ξ′) = K(ξ, σ(ξ′)) (6.185)

expression (6.182) actually defines the canonical bidifferential on Σ.

Computation of conformal block

Now we use the technique from [ZamAT87, ZamAT86, ApiZam, GMtw] to computethe conformal block (6.163). For the vacuum expectation value of the stress-energytensor (6.168) one gets from (6.172), (6.181)

〈T (z)Oh1·g1(q1)Og−11

(q2) . . .OhM ·gM (q2M−1)Og−1M

(q2M)〉G−10 =

=∑

π2N (ξ)=z

tz(ξ)−∑

πN (ζ)=z

tz(ζ) +1

4

(dS

dz

)2(6.186)

where tz and tz are the regularized parts of the bidifferentials K and K on diagonalin coordinate z:

tz(ξ)dξ2 =

1

2

(limξ→ξ′

K(ξ′, ξ)− dπ2N(ξ)dπ2N(ξ′)

(π2N(ξ′)− π2N(ξ))2

)tz(ζ)dζ2 =

1

2

(limζ→ζ′

K(ζ ′, ζ)− dπN(ζ)dπN(ζ ′)

(πN(ζ ′)− πN(ζ))2

) (6.187)

Expanding (6.186) at z → qi one gets

tz(ζ) =z→qi

1

12ζ; z+ reg. =

1

(z − qi)2

l2 − 1

24l2+ reg.

tz(ξ) =z→qi

1

12ξ; z+ reg. =

1

(z − qi)2

4l′2 − 1

96l′2+ reg.

(6.188)

186


in local co-ordinates ξ2l′ = ζ l = z− qi, which gives for the conformal dimensions 11 ofthe fields Og (with generic o(2N) twist field of the type (6.86))

∆g =K∑j=1

l2j − 1

24lj+

K′∑j=1

2l′2j + 1

48l′j+

K∑i=1

1

2lir

2i = ∆0

g +K∑i=1

1

2lir

2i , (6.191)

where the last term in the r.h.s. comes from the expansion dS ≈ ridzz−qi + . . .. Without

contributions of r-charges this formula is equivalent to (6.142), (6.191).

From the first order poles we obtain

∂qi log G0(q1, . . . , q2M) =∑

π2N (ξ)=qi

Res tz(ξ)dξ −∑

πN (ζ)=qi

Res tz(ζ)dζ+

+1

4

∑π2N (ξ)=qi

Res(dS)2

dz, i = 1, . . . , 2M

(6.192)

This system of equations for conformal block is obviously solved, so that we canformulate:

Theorem 6.4. Conformal blocks (6.163) for generic W (o(2N)) twist fields are givenby

G0(a, r, q) = τB(Σ|q)τ−1B (Σ|q)τSW (a, r, q) (6.193)

where


π2N (ξ)=qi

Res tz(ξ)dξ


πN (ζ)=qi

Res tz(ζ)dζ

i = 1, . . . , 2M

(6.194)

and

∂qi log τSW (a, r, q) =1

4

∑π2N (ξ)=qi

Res(dS)2

dz, i = 1, . . . , 2M

∂

∂aIlog τSW =

˛BI

dS, AI BJ = δIJ , I, J = 1, . . . , g−

(6.195)

11The counting here works as

tz − tz → 2

K∑j=1

ljl2j − 1

24l2j−

K∑j=1

ljl2j − 1

24l2j=

K∑j=1

l2j − 1

24lj(6.189)

for the [l]+-cycles, and

tz − tz →K′∑j=1

2l′j4l′2j − 1

96l′2j−

K′∑j=1

l′jl′2j − 1

24l′2j=

K′∑j=1

2l′2j + 1

48l′j(6.190)

for the [l′]−-cycles.

187


Equations (6.194) define so-called Bergmann tau-functions [KK04] for the curvesΣ and Σ respectively, while the so-called Seiberg-Witten tau-function (6.195) can beread literally from [GMtw]

log τSW (a, r, q) =1

4

g−∑I,J=1

aITIJaJ +1

2

g−∑I=1

aIUI(r) +1

4Q(r) (6.196)

where TIJ is the g−× g− “odd block” of the period matrix of Σ, or the period matrixof corresponding Prym variety [Fay], the “odd” vector

UJ(r) =

˛

BJ

dΩr =∑i,α

rαi AJ(qαi ), J = 1, . . . , g− (6.197)

where qαi are preimages of qi and rαi – corresponding r-charges, and

Q(r) =∑qαi 6=q

βj

rαi rβj log θ∗(A(qαi )− A(qβj ))−

∑qαi

lαi (rαi )2 logd(z(q)− qi)1/lαi

h2∗(q)

∣∣∣∣q=qαi

(6.198)where θ∗ is some odd Riemann theta-function for the curve Σ, A is the Abel map,and

h2∗(z) =

g∑I=1

∂θ∗(0)

∂ZIdωI(z) (6.199)

Relation between W (so(2N)) and W (gl(N)) blocks

It is interesting to compare the formulas from previous section with the formulasfrom [GMtw] for the exact W (gl(N)) conformal blocks. Since, as we already dis-cussed W (so(2N)) ⊂ W (gl(N)), any vertex operator of the W (gl(N)) algebra is avertex operator of its subalgebra W (so(2N)), and it is clear from our construction,that twist fields Og for the elements g ∼

∏[l, e2πir]+, are also the twist fields for

W (gl(N)). Moreover, the corresponding Verma modules, generated by W (so(2N))and by W (gl(N)), actually coincide 12, and it means that corresponding conformalblocks of such fields in these two theories should coincide as well.

Indeed, in such a case Σ = ΣtΣ, and therefore K(ξ, ξ′) = 0 if ξ′, ξ are on differentcomponents, and K(ξ, ξ′) = K(ξ, ξ′) if they are on the same component, hence

tz(z) = 2tz(z) (6.200)

For holomorphic and meromorphic differentials, one has in this case in natural basis

aI =

˛

A(1)I

dS =

˛

A(2)I

dS, I = 1, . . . , g

rαi = Res qαi dS = Res σ(qαi )dS

(6.201)

12These two modules coincide due to dimensional argument: they are both irreducible and havethe same characters. Irreducibility follows from the fact that null-vector condition can be written as(α, log g

2πi

)∈ Z for a simple root α and generic r’s, see also comments in sect. 6.5.6.

188

6.8. Conclusion

for the preimages qαi on Σ, and the period matrix of Σ consists of two nonzero g× gblocks:

T (11) = T (22) = T (6.202)

Under such conditions formula (6.4) turns into

G0(a, r, q) = τB(Σ|q)τSW (a, r, q) (6.203)

where

log τSW (a, r, q) = 12

g∑I,J=1

aI TIJaJ +

g∑I=1

aIUI(r) + 12Q(r) (6.204)

with corresponding obvious modifications of formulas (6.197) and (6.198), which givesexactly the W (gl(N)) conformal block in terms of the data on smaller curve Σ.

Conclusion

We have considered in this chapter the twist fields for the W-algebras with integer Vi-rasoro central charges, which are labeled by conjugacy classes in the Cartan normaliz-ers NG(h) of corresponding Lie groups. In addition to the most common WN -algebras,corresponding to A-series (or W (gl(N)) = WN ⊕ H, coming from G = GL(N)), wehave extended this construction for the G = O(n) case, which includes in addition toD-series the non simply-laced B-case with the half-integer Virasoro central charge.

In terms of two-dimensional conformal field theory our construction is based onthe free-field representation, where generalization to the D-series and B-series exploitsthe theory of real fermions, which in the odd B-case cannot be fully bosonized, sothat in addition to modules of the twisted Heisenberg algebra one has to take intoaccount those of infinite-dimensional Clifford algebra. This construction producesrepresentations of the W-algebras (that are at the same time twisted representationsof corresponding Kac-Moody algebras), which can be decomposed further into Vermamodules. To find this decomposition we have computed the characters of twistedrepresentations, using two alternative methods.

The first one comes from bosonization of the W-algebra or corresponding Kac-Moody algebra at level one, dependently on particular element from NG(h) it identifiesthe representation space with a collection of the Fock modules for untwisted or twistedbosons. The essential new phenomenon, which appears in the case of orthogonalgroups is presence of different [l]− cycles in g ∈ NG(h) and necessity to use in suchcases “exotic” bosonization for the Ramond-type fermions with non-local OPE on thecover.

Alternative method for computation of the characters uses pure algebraic con-struction of the twisted Kac-Moody algebras and the Weyl-Kac formula in principalgradation.

There are examples of elements g1, g2 that are not conjugated in NG(h), but con-jugated in G. Since two different constructions with elements g1 and g2 give differentformulations of the same representation, computation of corresponding charactersχg1(q) and χg2(q) leads to some simple but nontrivial identities for the correspond-ing lattice theta-functions, χg1(q) = χg2(q), which have been also proven by directmethods.

189


We have also derived an exact formula for the general conformal block of the twistfields in D-case, which directly generalizes corresponding construction for commonWN -algebra. The result, as is usual for Zamolodchikov’s exact conformal block, isexpressed in terms of geometry of covering curve (here with extra involution), andcan be factorized into the classical “Seiberg-Witten” part, totally determined by theperiod matrix of the corresponding Prym variety, and the quasiclassical correction,expressed now in terms of two different canonical bi-differentials. In order to expandthis method for the B-case one has to learn more about the theory of “exotic fermions”on Riemann surfaces, probably along the lines of [FSZ, DVV], and we postpone thisfor a separate publication.

Another set of open problems is obviously related with generalization to otherseries and twisted fields related with external automorphisms. Here only the E-casesseem to be straightforward, since standard bosonization can be immediately appliedin the simply-laced case, and there should be not many problems with the fermionconstruction. However, it is not easy to predict what happens in the situation whenKac-Moody algebras at level k = 1 have fractional central charges, and the directapplication of the methods developed in this chapter is probably impossible. It isstill not very clear, what is the role of these exact conformal blocks in the context ofmulti-dimensional supersymmetric gauge theories, since generally there is no Nekrasovcombinatorial representation in most of the cases. We hope to return to these issuesin the future.

Finally, there is an interesting question of possible generalization of our approachto the twisted representations with k 6= 1, which has been already considered in[FSS]. Some overlap with our formulas with sect. 8 of this chapter suggests that suchgeneralization could exist. We hope to return to this problem elsewhere.

Appendix

Identities for lattice Θ-functions

Here we present few rigorously proved identities, used to verify representation-theoreticconsiderations at the level of computations of characters.

First identity for AN−1 and DN Θ-functions

One can describe the lattices AN−1, DN and D′N in a similar way:

AN−1 =k1, . . . , kN

∣∣∣∣∣N∑i=1

ki = 0

DN =k1, . . . , kN

∣∣∣∣∣N∑i=1

ki ∈ 2Z

D′N =k1, . . . , kN

∣∣∣∣∣N∑i=1

ki ∈ 2Z + 1

(6.205)

190

6.9. Identities for lattice Θ-functions

The last lattice is actually just DN lattice, but shifted by vector (1, 0, . . . , 0). So allthese definitions can be rewritten as

LS = k1, . . . , kN

∣∣∣∣∣N∑i=1

ki ∈ S (6.206)

where S ⊆ Z: in our cases it should be chosen to be 0, 2Z, and 2Z+1, respectively.Notice also that for S = Z we get the simplest BN lattice.

By definition

ΘLS(~v; q) =∑

k1+...kN∈S

q12

(~v+~k)2

(6.207)

For our purposes we need this function computed for the vector 13

~v = (r1 +l1 − 1

2l1, r1 +

l1 − 3

2l1, . . . , r1 +

1− l12l1

)⊕

⊕(r2 +l2 − 1

2l2, r2 +

l2 − 3

2l2, . . . , r2 +

1− l22l2

)⊕ . . .⊕

⊕(rK +lK − 1

2lK, rK +

lK − 3

2lK, . . . , rK +

1− lK2lK

)

(6.208)

where l1 + . . .+ lK = N . Let us parameterize vector ~k as follows:

~k = (n1, . . . , n1)⊕ . . .⊕ (nK , . . . , nK) + ω(l1)a1⊕ . . .⊕ ω(lK)

aK+

+(a1

l1, . . . ,

a1

l1)⊕ . . .⊕ (

aKlK, . . . ,

aKlK

) + ~m1 ⊕ . . .⊕ ~mK

(6.209)

where ~mi ∈ Ali−1, and

ω(l)a = (

l − al

, . . . ,l − al

,−al, . . . ,−a

l) (6.210)

so that the first number is repeated a times, whereas the second one l − a times.Hence, vectors ~k ∈ LS are parameterized by vectors ~mi ∈ Ali−1 and integer numbersni ∈ Z; ai ∈ Z/liZ, restricted by

K∑i=1

(nili + ai) ∈ S (6.211)

The algorithm of decomposition (6.209) works as follows: first we sum up all compo-

nents of ~k inside each cycle – each number divided by li gives ni, whereas remaindergives ai. Subtracting (ni, . . . , ni) + ω

(li)ai , we are left with the vectors ~mi with van-

ishing sums of components.Now it is easy to see that

Θ(~v + ω(l1)a1⊕ ω(l2)

a2⊕ . . .⊕ ω(lK)

aK; q) = Θ(~v; q) (6.212)

13Notation ~v ⊕ ~u means (v1, . . . , vk)⊕ (u1, . . . um) = (v1, . . . , vk, u1, . . . , um).

191


which follows from the fact that Θ(~v; q) = Θ(σ(~v); q), where σ is a permutation. Forexample, take σa to be a-th power of the cyclic permutation, then:

σa

(1− l

2l, . . . ,

l − 1

2l

)=

(l + 1− 2a

2l,l + 3− 2a

2l, . . . ,

l − 1

2l,1− l

2l, . . . ,

l − 1− 2a

2l

)=

=

(1− l

2l, . . . ,

l − 1

2l

)+ ω(l)

a

(6.213)

and therefore any vector ~v + ω(l1)a1 ⊕ ω

(l1)a2 ⊕ . . . ⊕ ω

(lK)aK can be obtained by several

permutation of components of ~v, so the corresponding Θ-functions are equal. Thus

ΘLS(~v; q) =∑

K∑i=1

(nili+ai)∈S

~mi∈QAli−1

q12

(~v+~m1⊕...⊕~mK+(n1+a1l1,...,n1+

a1l1

)⊕...⊕(nK+aKlK

,...,nK+aKlK

))2

(6.214)turns into the sum over several orthogonal sublattices

ΘLS(~v; q) =∑

~mi∈Ali−1

q12

(ρ(l1)⊕...⊕ρ(lK )+~m1⊕...⊕~mK)2 ·∑

K∑i=1

(nili+ai)∈S

q12

K∑i=1

li(ni+aili

+ri)2

=

=K∏i=1

ΘAli−1(ρ(li); q) ·

∑n′1+...+n′K∈S

q

K∑i=1

12li

(n′i+rili)2

(6.215)where

ρ(l) = (l − 1

2l,l − 3

2l, . . . ,

1− l2l

) (6.216)

One can identify the last factor in the r.h.s. with the contribution of zero modes,related to the r-charges [GMtw].

Product formula for AN−1 Θ-functions

Apply (6.215) to the simplest case of ΘBN (ρ(N); q) with S = Z

ΘBN (ρ(N); q) = ΘAN−1(ρ(N); q) ·

∑n∈Z

qn2

2N (6.217)

Using definition (6.216) and Jacobi triple product formula we get

ΘBN (ρ(N); q) = qN2−124N

N−1∏a=0

∑k∈Z

qk2

2+N−1−2a

2Nk =

= qN2−124N

∞∏k=1

(1 + q1N

(k− 12

))2

∞∏n=1

(1− qn)N(6.218)

as well as ∑n∈Z

qn2

2N =∞∏k=1

(1 + q1N

(k− 12

))2

∞∏n=1

(1− qnN ) (6.219)

192

6.10. Exotic bosonizations

Substituting into (6.217) one obtains

ΘAN−1(ρ(N); q) = q

N2−124N

∞∏k=1

(1− qk)N

∞∏k=1

(1− q kN )=η(q)N

η(q1N )

(6.220)

or the product formula [Mac] for ΘAN−1(ρ(N); q), where the r.h.s. is expressed in terms

of the Dedekind functions. Substituting this into (6.215) we get it in its final form

ΘLS(~v; q) =∑

k1+...+kN∈S

q12

N∑i=1

(vi+ki)2

=K∏i=1

η(q)li

η(q1li )·

∑n1+...+nK∈S

q

K∑i=1

12li

(ni+liri)2

(6.221)

An identity for DN and BN Θ-functions

Here we show how ΘDN (~v∗; q) can be simplified if ~v∗ contains at least one component12. One has then

ΘDN (~v∗; q) = ΘDN ((12, v2, . . . , vn); q) =

∑k1+...+kn∈2Z

q12

(~v∗+~k)2

=

= ΘDN ((−12, v2, . . . , vn); q) = ΘDn(~v∗ − (1, 0, . . . , 0); q)

(6.222)

Since for the lattices DN t DN − (1, 0, . . . , 0) = BN , it follows from (6.222) that

ΘDN (~v∗; q) = 12ΘBN (~v∗; q) (6.223)

Exotic bosonizations

Here we present some details of the bosonization procedures, used in the main text.

NS ×RConsider, first, construction [Ber, BBT] relating pair (of NS and R!) fermions to atwisted boson 14

φ(t) = i∑r∈Z+ 1

2

Jrrtr

= i√

2∑n∈Z

a2n+1

(2n+ 1)ξ2n+1= φ(ξ) (6.224)

with differently normalized oscillator modes [aM , aN ] = MδM+N,0 (M,N ∈ 2Z + 1).Compute the correlator

−〈φ(ξ)φ(ζ)〉 = 2∑〈a2n+1a2m+1〉ξ−2n−1ζ−2m−1 = −2

∞∑n=0

(ζ/ξ)2n+1

2n+ 1=

= 2 log

(1− ζ

ξ

)− log

(1− ζ2

ξ2

)= − log

ξ + ζ

ξ − ζ= −[φ+(ξ), φ−(ζ)]

(6.225)

14It is more convenient to use in this section coordinate ξ =√t, so analytic continuation in t

around 0 maps ξ to −ξ.

193


assuming |ξ| > |ζ|. Now introduce

η(ξ) =1√2

: eiφ(ξ) :=1√2eiφ−(ξ)eiφ+(ξ) (6.226)

so that for |ξ| > |ζ|

η(ξ)η(ζ) =1

2e−(φ−(ξ)+φ−(ζ))ei(φ+(ξ)+φ+(ζ))e−[φ+(ξ),φ−(ζ)] =

=1

2: ei(φ(ξ)+φ(ζ)) :

ξ − ζξ + ζ

(6.227)

while for |ξ| < |ζ|η(ζ)η(ξ) =

1

2: ei(φ(ξ)+φ(ζ)) :

ζ − ξξ + ζ

(6.228)

It means that OPE of the η-fields has fermionic nature:

η(ξ)η(−ζ) =1

2

ξ + ζ

ξ − ζ: e(φ(ξ)−φ(ζ)) :∼ 1

2

ξ + ζ

ξ − ζ+ reg. ∼ ζ

ξ − ζ+ reg. (6.229)

and in the anticommutator of components η(ξ) =∑k∈Z

ηkξk

ηk, (−1)lηl =

˛ζ l−1dζ

˛ζ

ζ

ξ − ζξk−1dξ = δk+l,0 (6.230)

one gets unusual sign factor.It is interesting to point our that the Ramond zero mode η2

0 = 12

has bosonicrepresentation

√2η0 =

˛dξ

ξeiφ−(ξ)eiφ+(ξ) =

= 1− 2a−1a1 + a2−1a

21 −

2

9(a−3 + a3

−1)(a3 + a31) + . . .

(6.231)

For example, the action of this operator on low-level vectors gives√

2η0 · |0〉 = |0〉,√

2η0 · a−1|0〉 = −a−1|0〉,√

2η0 · a2−1|0〉 = a2

−1|0〉√

2η0 · a−3|0〉 =1

3a−3|0〉 −

2

3a3−1|0〉,

√2η0 · a3

−1|0〉 = −4

3a−3|0〉 −

1

3a3−1|0〉

(6.232)

Here in the second line one gets the matrix 13

(1 −2−4 −1

)with the eigenvalues ±1.

We also haveη0ηk = −ηkη0, k 6= 0 (6.233)

so one can identify√

2η0 = (−1)F−F0 , where F is fermionic parity. Generally, algebra,generated by ηk, has two representations with the vacua |0〉±, such that η0|0〉± =±|0〉±. One can also take direct sum of such representations: bosonization formula inthis representation looks as

η(ξ) =σ1√

2eiφ−(ξ)eiφ+(ξ)

(6.234)

194


Existence of this bosonization at the level of characters gives us obvious identity

∞∏k=0

1

1− q2k+1=∞∏k=1

(1 + qk) (6.235)

Notice that above consideration actually concerns R and NS fermions because onecan construct two combinations

1√2

(η(z)− η(−z)) =∑p∈Z+ 1

2

η2p

zp= iψNS(z)

1√2

(η(z) + η(−z)) =∑n∈Z

η2n

zn= ψR(z)

(6.236)

then

J(z) =1

z

(ψ∗(√z)ψ(√z))

= iψNS(z)ψR(z) =∑p∈Z+ 1

2

Jpzp+1

Jp = i∑n+q=p

η2qη2n

(6.237)

here t−12 ψNS(

√t) and t−

12 ψR(

√t) are usual Ramond and Neveu-Schwarz fermions.

Here we consider fermion corresponding to the branch point of type [l]−. Thismeans that we should have

η(z)η(σ(w)) ∼ 1

z − w, (6.238)

and such monodromy that η(e4πilz) = ±η(z). Let us use the construction form (6.10.1)

η(z) =z−

12

√2lη(z

12l ) (6.239)

Therefore

η(z)η(σ(w)) ∼ z−12w−

12

2l

w12l

z12l − w 1

2l

∼ 1

z − w(6.240)

So final construction states that one should have

η(z) = σ1z−

12

2√leiφ−(z

12l )eiφ+(z

12l ) (6.241)

R×RLet us take two Ramond fermions ψ(1), ψ(2) and introduce

ψ(z) =1√2

(ψ(1)(z) + iψ(2)(z)

)=∑n∈Z

ψn

zn+ 12

ψ∗(z) =1√2

(ψ(1)(z)− iψ(2)(z)

)=∑n∈Z

ψ∗n

zn+ 12

(6.242)

195


Since there are two zero modes ψ∗0 and ψ0, one expects to have four vacua |0〉, ψ0|0〉,ψ∗0|0〉, ψ∗0ψ0|0〉.

We can mimic expansion (6.242) using fractional powers

ψ(z) =∑p∈Z+ 1

2

ψNS,p

zp+12

+σ, ψ∗(z) =

∑p∈Z+ 1

2

ψ∗NS,p

zp+12−σ (6.243)

with σ = 12, i.e. ψn = ψNS,n− 1

2and ψ∗n = ψ∗

NS,n+ 12

. It means that after standard

bosonization

ψ(z) = e−iφ−(z)e−iφ+(z)e−Qz−J0 , ψ∗(z) = eiφ−(z)eiφ+(z)eQzJ0

J0|0〉 = σ|0〉 = 12|0〉

(6.244)

one gets ψ∗0|0〉 = 0, and only one half of the vacuum states survive. To identify this

representation with something well-known, consider the eigenvectors√

2ψ(1)0 |0〉± =

±|0〉± of√

2ψ(1)0 = ψ0 + ψ∗0:

|0〉+ =1√2

(|0〉+ ψ0|0〉), |0〉− =i√2

(|0〉 − ψ0|0〉) (6.245)

Acting by√

2ψ(2)0 = i(ψ∗0 − ψ0) one gets

√2ψ

(2)0 |0〉+ = |0〉−,

√2ψ

(2)0 |0〉− = |0〉+ (6.246)

The character of such module is given by

2∞∏k=1

(1 + qk)2 = q−18

∑n∈Z

q12

(n+ 12

)2

∞∏k=1

(1− qk)(6.247)

where in the l.h.s. we have two Ramond fermions with two vacuum states, whereasthe r.h.s. corresponds to sum over bosonic modules with half-integer vacuum J0

charges. This formula is a simple consequence of the Jacobi triple product identity.Analogously we have similar formula for the bosonization of NS ×NS fermions

∞∏k=0

(1 + q12

+k)2 =

∑n∈Z

q12n2

∏∞k=1(1− qk)

(6.248)

It is the consequence of Jacobi triple product identity as well.

l twisted charged fermions

For the twisted boson

iφ(z) = −∑n6=0

Jn/lnzn/l

+1

lJ0 log z +Q (6.249)

196


with [Jn/l, Jm/l

]= nδn+m,0 [J0, Q] = 1 (6.250)

one has for |z| > |w|[φ+(z)− i

lJ0 log z, φ−(w)− iQ

]=∑n>0

z−n/lwn/l

n− 1

llog z = − log

(z1/l − w1/l

)(6.251)

where

iφ+(z) = −∑n>0

Jn/lnzn/l

iφ−(z) = −∑n<0

Jn/lnzn/l (6.252)

Define two operators

ψ∗(z) = z12l : eiφ(z) := z

12l eiφ−(z)eiφ+(z)eQzJ0/l

ψ(z) = z12l : e−iφ(z) := z

12l e−iφ−(z)e−iφ+(z)e−Qz−J0/l

(6.253)

with the OPE

ψ∗(z)ψ(w) =(zw)

12l

z1/l − w1/l: eiφ(z)−iφ(w) :=

=(zw)

12l

z1/l − w1/leiφ+(z)−iφ+(w)eiφ−(z)−iφ−(w)

( zw

)J0/l(6.254)

Then for the modes of their expansion

ψ∗(z) =∑k∈ 1

2+Z

ψ∗k/lzk/l

, ψ(z) =∑k∈ 1

2+Z

ψk/lzk/l (6.255)

one gets canonical anticommutation relations

ψ∗a, ψb = δa+b,0 (6.256)

Now one can express the l-component fermions in terms of a single twisted boson

ψ∗α(z) =1√lz−

12 ψ∗(e2πiαz), ψα(z) =

1√lz−

12 ψ(e2πiαz), α ∈ Z/lZ (6.257)

and it follows from (6.254), that their OPE is indeed

ψ∗α(z)ψβ(w) =z→w

δαβz − w

+ reg. (6.258)

The stress-energy tensor and U(1) current can be extracted from the expansion:

∑α∈Z/lZ

ψ∗α(z + t/2)ψα(z − t/2) =l

t+ J(z) + tT (z) + . . . (6.259)

197


Using (6.253), (6.254) and (6.257) one gets for the l.h.s.

∑α∈Z/lZ

1l(z + t

2)

1−l2l (z − t

2)

1−l2l

(z + t2)1/l − (z − t

2)1/l

: eiφ(e2πiα(z+t/2))−iφ(e2πiα(z−t/2)) : =

=∑

α∈Z/lZ

(1

t+ t

l2 − 1

24l2z2

)eit∂φ(e2πiαz) +O(t2) =

=l

t+∑

α∈Z/lZ

i∂φ(e2πiαz) +t

z2

l2 − 1

24l− t

2

∑α∈Z/lZ

: ∂φ(e2πiαz)2 : +O(t2)

(6.260)

One finds from here

J(z) =∑

α∈Z/lZ

i∂φ(e2πiαz) =∑k∈Z

Jnzn+1

T (z) =l2 − 1

24lz2+

1

l

∑k+n∈Z

: JnJk :

zn+k+2

(6.261)

which already have expansions over integer powers of z. Therefore

L0 =l2 − 1

24l+

1

2lJ2

0 +1

l

∑n>0

J−nJn (6.262)

and the character of this module is given by

tr qL0+rJ0 = ql2−124l

∑n∈Z

qn2

2l+rn

∞∏n=1

(1− q nl )(6.263)

l charged fermions – standard bosonization

From the modes (6.255) of the operators ψ(z), ψ∗(z) we can construct another lfermions

ψ(a)(z) =1√l

∑p∈Z+ 1

2

ψa+p

za+p+ 12

, ψ∗(a)(z) =1√l

∑p∈Z+ 1

2

ψ−a+p

z−a+p+ 12

(6.264)

where

a ∈ l − 1

2l,l − 3

2l, . . . ,

1− l2l (6.265)

These fermions can be bosonized in terms of l “normal”, untwisted, bosons

ψ∗(a)(z) = eiϕ(a),−(z)eiϕ(a),+(z)eQ(a)zJ(a),0(−1)

∑b<a

J(b),0

ψ(a)(z) = e−iϕ(a),−(z)e−iϕ(a),+(z)e−Q(a)z−J(a),0(−1)

∑b<a

J(b),0(6.266)

whereJ(a),0|0〉 = a|0〉 (6.267)

198

BIBLIOGRAPHY

Computation of character in this case gives us

tr qL0+r∑J(a),0 =

∑n0,...,nl−1

q

l−1∑k=0

( 1−l+2kl2l

+nk)2+rl−1∑k=0

nk

∞∏n=1

(1− qn)l

(6.268)

One can easily see that equality between (6.263) and (6.268) follows from particularcase of (6.223).

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· Contents 1 Introduction 1 1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.1.1 Conformal eld theory