CONFORMAL FIELD THEORY AND FROBENIUS ALGEBRAS IN … · IN MODULAR TENSOR CATEGORIES St.Petersburg 30 06 05 – p.1/15. CONFORMAL FIELD THEORY AND FROBENIUS ALGEBRAS IN MODULAR TENSOR

CONFORMAL FIELD THEORYAND FROBENIUS ALGEBRAS

IN MODULAR TENSOR CATEGORIES

St. Petersburg 30 06 05 – p.1/15

CONFORMAL FIELD THEORY

AND FROBENIUS ALGEBRAS


J

en Fu hsur cg


CONFORMAL FIELD THEORY

AND FROBENIUS ALGEBRAS


J

en Fu hsur cg


Plan

Aim:


Plan

Aim: Understand two-dimensional conformal field theory


Plan

Aim: Understand “holomorphic factorization” in RCFT


Plan


Step 1: Thicken the world sheet


Plan



Connecting manifold , ribbon graphs , modular tensor categories


Plan




Step 2: Triangulate the world sheet


Plan





Frobenius algebras in modular tensor categories


Plan






Step 3: Bulk fields as morphisms


Plan







Bimodules , alpha-induced bimodules , ...


Plan








Correlation functions , boundary fields , defect fields


Plan









Dictionary physical concepts ←→ mathematical structures


Plan










Some results


Plan










Some results


Holomorphic factorization

Free boson QFT in d=2 :




eq. of motion`∂2

∂x2− ∂2

∂t2

´φ(x, t) = 0




eq. of motion`∂2

∂x2− ∂2

∂t2

´φ(x, t) = 0 solved by φ = φ+(x+t) + φ−(x−t)




eq. of motion ∂ ∂ φ(z, z) = 0 solved by φ = φ1(z) + φ2(z)





“ left- and right-movers ”

“ holomorphic / antiholomorphic ”

“ chiral / antichiral ”





Observation: Same phenomenon for various aspects

of a large class of 2-d QFT models

on two-dimensional world sheets








— 2-d rational conformal field theories








— 2-d rational conformal field theories ≡ RCFT —









In particular: Bulk fields Φ of a RCFT

Φ










Φ

carry rep’s of both left and right world sheet symmetries Φ = Φij

ij










Φ

carry rep’s of both left and right world sheet symmetries Φ = Φij

ij

with multiplicities

α

α


Bulk fields

CFT =⇒ Conformal symmetry / extensions

2-d =⇒ Virasoro algebra / affine Lie algebras / W-algebras / ...

=⇒ conformal vertex algebra [ Borcherds 1986 , . . . ]


Bulk fields




RCFT =⇒ Rational conformal vertex algebra V [ Huang 2004 ]


Bulk fields





Rep category Rep(V) : a modular tensor category


Bulk fields






Tensor product U ⊗ V of V-modules and of intertwiners , V = 1 =: U0

Abelian,

�

-linear, semisimple, finite U ∼=M

i∈I

U⊕nii

Braiding

U V

TwistU

Duality

U U∨

U∨

U

Braiding maximally non-degenerate: det

Ui

Uj 6= 0


Bulk fields






Bulk field Φ = Φij

=⇒ two simple objects Ui , Uj ∈ C := Rep(V)


Bulk fields






Bulk field Φ = Φαij


and a morphism α ∈ Hom(Ui⊗Uj ,1)


Bulk fields






Bulk field Φ = Φαij


and a morphism α ∈ Hom(Ui⊗Uj ,1)

Recall: Free boson CFT: geometric separation of left- and right-movers


Bulk fields

Separate locally left- and right-movers :

double bX of world sheet X


Bulk fields



bX = orientation bundle over X modulo identification of points in fibers over ∂X

Φij 7−→

j

i


Bulk fields




To connect to X: Fatten the world sheet:

connecting manifold MX ∂MX = bX


Bulk fields





connecting manifold MX ∂MX = bXMX = interval bundle over X modulo identification over ∂X

= (bX× [−1, 1])/∼ with ([x, or2], t) ∼ ([x,−or2],−t)


Bulk fields






Field theory on MX non-dynamical

; 3-d TFT

; ribbon graphs

Φij 7−→

j

i


Bulk fields







; 3-d TFT

; ribbon graphs

j

i

α


Bulk fields







; 3-d TFT

; ribbon graphs

j

i

αX embedded as MX ⊃ X×{t=0}


Bulk fields

double bX`orientation bundle over X

´/∼

connecting manifold MX`interval bundle over X

´/∼

embedding X×{t=0} ⊂ MX

j

i

α


Bulk fields


´/∼


´/∼


j

i

α


Bulk fields


´/∼


´/∼


j

i

α

α ∈ Hom(Ui⊗Uj ,1)

= δi,�


Bulk fields


´/∼


´/∼


j

i

α

α ∈ Hom(Ui⊗Uj ,1)

= δi,�

too restrictive

?


Frobenius algebras

Special case: meromorphic CFT


Frobenius algebras

Special case: meromorphic CFT

C = Vect �

2-d lattice topological CFT

corresponds to separable Frobenius algebra A in Vect �

properties of A ←→ triangulation independence


Frobenius algebras

Prescription for general RCFT:

triangulate the world sheet (trivalent vertices)

label edges (ribbons) by a Frobenius algebra A in C A A


Frobenius algebras




label vertices (coupons) in X\∂X

by product / coproduct morphisms

A

A A


Frobenius algebras






A

A A

connect to bulk field ribbons bysuitable morphisms in Hom(Ui⊗A⊗Uj , A)


Frobenius algebras






A

A A


More precisely: A a symmetric special Frobenius algebra


Frobenius algebras






A

A A



= = =


Frobenius algebras






A

A A


More precisely: A a symmetric special Frobenius co algebra

= = =


Frobenius algebras






A

A A



= =


Frobenius algebras






A

A A



=

A

A∨


Frobenius algebras






A

A A


More precisely: A a symmetric special = strongly separable Frobenius algebra

= =


Frobenius algebras






A

A A



In addition: reversion = isomorphism A↔ Aopp squaring to twist

when X is unoriented

prescription for vertices of triangulation on ∂X when ∂X 6= ∅


Bulk fields II

Prescription involves choices:

Triangulation

Insertion of ribbon graph fragment for edge (two possibilities)

Insertion of ribbon graph fragment for vertex (three possibilities)


Bulk fields II


Triangulation



Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices


Bulk fields II


Triangulation



Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choiceswithout field insertions


Bulk fields II


Triangulation



Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of

morphism spaces


Bulk fields II


Triangulation




morphism spaces

Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:

A is A-bimodule


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule left module properties:

= =


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule A as a left module:

=


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule A as a right module:

= left and right actions commuteby associativity


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule

Ui⊗A⊗Uj is A-bimodule


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule

Ui⊗A⊗Uj is A-bimodule Induced left module:

:=

A⊗U U


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule

Ui⊗A⊗Uj is A-bimodule ⊗+-induced left module:

Ui

( use braiding )


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule

Ui⊗A⊗Uj is A-bimodule ⊗−-induced right module:

Uj

cf alpha-induction of subfactors[ Longo-Rehren ]


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule


Ui⊗+A⊗−Uj ∈ Obj(CA|A)


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule



Subspace = space of bimodule morphisms


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule



Subspace = space of bimodule morphisms Left-module intertwiner:

=ρM

ρN


Bulk fields II


Triangulation




morphism spaces


A is A-bimodule



Subspace = space of bimodule morphisms

Prescription:

Space of bulk fields = HomA|A(Ui⊗+A⊗−Uj , A)with chiral labels i,j


Bulk fields II

Prescription: Space of bulk fields˘Φαij

¯= HomA|A(Ui⊗+A⊗−Uj , A)


Bulk fields II




Bulk fields II



copr.

unit

j

i

α


Correlation functions

Ribbon graphs in 3-manifolds vs morphisms in C :

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RCFT ; vertex algebra V

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; C = Rep(V) – modular tensor category

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; extended 3-d TFT [ Reshetikhin-Turaev 1991 , . . . ]

functor tftC : 3-CobC → Vect �

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Σ 7→ tftC(Σ) =: H(Σ)

H(∅) =

�

tftC(Hom(∅,Σ)) ∼= H(Σ)

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Conjecture: [ Witten 1989 , . . . ]

state spaces H(Σ) = spaces of conformal blocks of RCFT

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Correlation function C(X) for world sheet X with field insertions

= “ expectation value of product of field operators ”depending on position of insertions and on moduli of X

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Holomorphic factorization for correlation functions: C(X) ∈ H(bX)

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in words: a correlation function on X is a vector in the state space of the double bX

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Idea: Specify the vector C(X) as a concrete cobordism ∅ → bX( 3-manifold + ribbon graph )

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Prescription for cobordism C(X) :

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3-manifold: Connecting manifold MX

∂MX = bX X embedded as X×{t=0}

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Ribbon graph: Triangulation of X×{t=0} labeled by A

connected to bulk field ribbons labeled by Uis & Ujsby coupons labeled by HomA|A(Uis⊗

+A⊗−Ujs , A)

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+A⊗−Ujs , A)

Example : Three bulk fields on the sphere S2 = R2∪{∞} :

p1

p2

p3Φ1

Φ2

Φ3

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+A⊗−Ujs , A)


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+A⊗−Ujs , A)


yields OPE coefficientswhen expressed as linear combination of standard basis blocks

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+A⊗−Ujs , A)



Additional ingredients for ribbon graph needed in general:

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+A⊗−Ujs , A)




For ∂X 6= ∅ : annular ribbon for each boundary componentlabeled by boundary condition

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+A⊗−Ujs , A)





Boundary fields – can change boundary condition

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+A⊗−Ujs , A)






Defect lines and defect fields

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+A⊗−Ujs , A)






Defect lines and defect fields

Be careful with 1- and 2-orientations of ribbons , . . .St. Petersburg 30 06 05 – p.11/15


DICTIONARY

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DICTIONARY

chiral labels ←→ C = Rep(V) , simple objects ∼= Ui , i ∈ I

full non-diagonal CFT ←→ symmetric special Frobenius algebra A in C

actually: Morita class [A]

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DICTIONARY



bulk fields Φij ←→ bimodule morphisms HomA|A(Ui⊗+A⊗− Uj , A)

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DICTIONARY




boundary conditions ←→ A-modules M ∈ Obj(CA)

boundary fields ΨMNi ←→ module morphisms HomA(M ⊗Ui, N)

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DICTIONARY






defect lines ←→ A-bimodules Y ∈ Obj(CA|A)

defect fields ΘY Zij ←→ bimodule morphisms HomA|A(Ui⊗+Y ⊗− Uj , Z)

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DICTIONARY






defect lines ←→ A-B-bimodules Y ∈ Obj(CA|B)

defect fields ΘY Zij ←→ bimodule morphisms HomA|B(Ui⊗+Y ⊗− Uj , Z)

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DICTIONARY








CFT on unoriented ←→ Jandl algebraworld sheet

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DICTIONARY









simple current models ←→ Schellekens algebra A ∼=Lg∈G Lg for G ≤ Pic(C)

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DICTIONARY









simple current models ←→ Schellekens algebra A ∼=Lg∈G Lg for G ≤ Pic(C)

internal symmetries ←→ Picard group Pic(CA|A)

Kramers-Wannier ←→ duality bimodules Y : Y ∨⊗AY ∈ Pic(CA|A)like dualities

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Results

See:

JF , Ingo Runkel , Christoph Schweigert :

TFT construction of RCFT correlators

I: Partition functions Nucl. Phys. B 646 (2002) 353–497 hep-th/0204148

II: Unoriented world sheets Nucl. Phys. B 678 (2004) 511–637 hep-th/0306164

III: Simple currents Nucl. Phys. B 694 (2004) 277–353 hep-th/0403157

IV: Structure constantsand correlation functions Nucl. Phys. B 715 (2005) 539–638 hep-th/0412290

& Jens Fjelstad :

V: Proof of modular invariance and factorisation hep-th/0503194

& Jürg Fröhlich :

Correspondences of ribbon categories Adv. Math. ... (2005) ... math.CT/0309465

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Results

Finding the possible structures of symmetric special Frobenius algebra

(if any) on an object of a modular tensor category C is a finite problem

( and only one of the equations to be solved is nonlinear )

For any symmetric special Frobenius algebra A in C

constructing CA and CA|A is a finite problem

In any modular tensor category C there is only a finite number

of Morita classes of simple symmetric special Frobenius algebras

( simple: A a simple A-bimodule )

A symmetric special Frobenius algebra can be reconstructed

from the operator product of boundary fields ΨMMi

( for any full CFT with at least one consistent boundary condition M )

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Results

For A a symmetric special Frobenius algebra in a modular tensor category C :

C(X) is independent of the choices involved in the prescription

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Results


C(X) is independent of the choices involved in the prescription

Choices to be made:

Triangulation TX Prop. V: 3.1, V: 3.7 ( fusion and bubble moves )

Local orientations at vertices of TX Sec. II: 3.1, IV: 3.2, IV: 3.3for unoriented X

Insertion of ribbon graph fragmentsfor vertices and edges of TX Sec. I: 5.1, II: 3.1

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Results


Thm. I: 5.1 (Torus partition function)

The coefficients Zij of C(T; ∅) =Pi,j∈I Zij |χi,T〉⊗ |χj ,−T〉

satisfy [ Γ, Z ] = 0 for Γ∈SL(2,Z)

and Zij = dimHomA|A(Ui⊗+A⊗− Uj , A) ∈ Z≥0

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Results






C(T; ∅) =

T×[−1,1]

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Results






C(T; ∅) =

T×[−1,1]

=⇒ Zij =

i A j

A

A

A

S2×S1

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Results






Lemma I: 5.2 :

A

A U

A U

A

is an idempotent

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Results






Proof : = = =

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Results






Lemma I: 5.2 :

A

A U

A U

A

is an idempotent

Composition with the idempotent projects End(A⊗U)→ EndA|A(A⊗− U)

Analogously: End(V ⊗A)→ EndA|A(V ⊗+A)

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Results






Proof of [T,Z] = 0 :

= =

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Results






Prop. I: 5.3 (Torus partition function)

ZA⊕B = ZA + ZB eZA⊗B = eZA eZB ZAopp

=`ZA

´t

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Results








=`ZA

´t

eZA⊗Bij =

i A B j

eZij ≡ Zıj

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Results








=`ZA

´t

ZAopp

ij =

i A

mopp

j

=

i

m

A j

=

i

m

A j

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Results








=`ZA

´t

Thm. II: 3.7 (Klein bottle partition function)

The coefficients Kj of C(K; ∅) =Pj∈I Kj |χj ,T〉

satisfy Kj ∈ Z Kj = K12

(Zjj +Kj) ∈ {0, 1, ... , Zjj}

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Results








=`ZA

´t

Thm. II: 3.7 (Klein bottle partition function)Kj ∈ Z Kj = K

12

(Zjj +Kj) ∈ {0, 1, ... , Zjj}

C(K; ∅) =A

σ

I×S1×I/∼

(r,φ)top∼ ( 1

r,−φ)bottom

=⇒ Kj =

j

A

σ

j

S2×I/∼

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Results

For A a simple symmetric special Frobenius algebra in a modular tensor category C :

Thm. I: 5.1 & Prop. I: 5.3 (Torus)

[ Γ, Z ] = 0 (Γ∈SL(2,Z)) Zij = dimHomA|A(Ui⊗+A⊗− Uj , A)

Z00 = 1 ZA⊕B = ZA + ZB eZA⊗B = eZA eZB ZAopp

=`ZA

´t

Thm. II: 3.7 (Klein bottle) Kj ∈ Z Kj = K(Zjj+Kj)

2∈ {0, ... , Zjj}

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Results





=`ZA

´t


2∈ {0, ... , Zjj}

Special case: A = 1 =⇒ Zij = δi Kj =

(±1 if j=

0 else( FS indicator )

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Results





=`ZA

´t


2∈ {0, ... , Zjj}

OPE coefficients for the fundamental correlation functions

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Results





=`ZA

´t


2∈ {0, ... , Zjj}


Three bulk fields on the sphereSec. IV: 4.4

C =Pµ,ν c(Φ1Φ2Φ3)µν B(p1, p2, p3)µν

p1

p2

p3Φ1

Φ2

Φ3

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Results





=`ZA

´t


2∈ {0, ... , Zjj}



c(Φ1Φ2Φ3)µν =1

S0,0 i k

m m

µ

φ1 φ2 φ3

A

ν

S3n.

j l

n n

Φ1=(i,j,φ1,p1,[γ1],or2)

Φ2=(k,l,φ2,p2,[γ2],or2)

Φ3=(m,n,φ3,p3,[γ3],or2)

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Results





=`ZA

´t


2∈ {0, ... , Zjj}



C =Pµ,ν c(Φ1Φ2Φ3)µν B(p1, p2, p3)µν

c(Φ1Φ2Φ3)µν =dim(A)

S0,0

X

β

F[A|A](ki0jl)0α10α2 , βmnµν

F[A|A](mm0nn)0β0α3 , ·00··

φ1 = ξα1(i0j)0

φ2 = ξα2(k0l)0

φ3 = ξα3(m0n)0

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Results





=`ZA

´t


2∈ {0, ... , Zjj}



Three defect fields on the sphereSec. IV: 4.5

C =Pµ,ν c(X,Θ1, Y,Θ2, Z,Θ3,X)µν B(p1, p2, p3)µν

p1

p2

p3Θ1

Θ2

Θ3

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Results





=`ZA

´t


2∈ {0, ... , Zjj}




c(X,Θ1, Y,Θ2, Z,Θ3,X)µν =1

S0,0 i k

m m

µ

ϑ1 ϑ2 ϑ3Y Z

X

ν

S3n.

j l

n n

Θ1=(X,or2,Y,or2,i,j,ϑ1,p1,[γ1],or2)

Θ2=(Y,or2,Z,or2,k,l,ϑ2,p2,[γ2],or2)

Θ3=(Z,or2,X,or2,m,n,ϑ3,p3,[γ3],or2)

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Results





=`ZA

´t


2∈ {0, ... , Zjj}




C =Pµ,ν c(X,Θ1, Y,Θ2, Z,Θ3,X)µν B(p1, p2, p3)µν

c(X,Θ1, Y,Θ2, Z,Θ3,X)εϕ =dim(Xµ)

S0,0

X

β

F[A|A](kiµjl)κα1να2 , βmnεϕ

F[A|A](mmµnn)µβκα3 , ·00··

X = Xµ Y = Xν Z = Xκ ϑ1 = ξα1(iµj)ν

ϑ2 = ξα2(kνl)κ

ϑ3 = ξα3(mκn)µ

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Results





=`ZA

´t


2∈ {0, ... , Zjj}




Three boundary fields on the diskSec. IV: 4.2

C =PNij

k

δ=1 c(MΨ1NΨ2KΨ3M)δ B(x1, x2, x3)δ

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �Mx1 x2 x3

Ψ1 Ψ2 Ψ3 MKN

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Results





=`ZA

´t


2∈ {0, ... , Zjj}





c(MΨ1NΨ2KΨ3M)δ =

N

M

K

ij

k

k

δ

ψ1ψ2ψ3

S3n.

Ψ1=(N,M,Ui,ψ1,x1,[γ1])

Ψ2=(K,N,Uj ,ψ2,x2,[γ2])

Ψ3=(M,K,Uk,ψ3,x3,[γ3])

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Results





=`ZA

´t


2∈ {0, ... , Zjj}





c(MΨ1NΨ2KΨ3M)δ = dim(Mµ)PAkMκ

Mµ

β=1 G[A](κji)µ

α2να1 , βkδG[A]

(µkk)µα3κβ , ·0·

M = Mµ N = Mν K = Mκ ψ1 = ψα1(νi)µ

ψ2 = ψα2(κj)ν

ψ3 = ψα3(µk)κ

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Results





=`ZA

´t


2∈ {0, ... , Zjj}





One bulk and one boundary field on the diskSec. IV: 4.3

C =Pδ c(Φ;MΨ)δ B(x, y, s)δ � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �M MΨ

Φ

s

(x,y)

St. Petersburg 30 06 05 – p.14/15

Results





=`ZA

´t


2∈ {0, ... , Zjj}






c(Φ;MΨ)δ =

i j

k k

ψM

AAA

M

φ

δ

S3n.

Φ=(i,j,φ,p,[γ],or2)

Ψ=(M,M,Uk,ψ,s,[γ′])

St. Petersburg 30 06 05 – p.14/15

Results





=`ZA

´t


2∈ {0, ... , Zjj}






c(Φ;MΨ)• =

M=Mµ ψ=ψα(µk)µ

φ=ξβ

(i0j)0

Nijk ∈{0,1} 〈Ui,A〉 ∈ {0,1}

X

m,n,a,p,ρ,σ

[ξβ(i0j)0

]0jaρMµ (mρ)

a (nσ)[ψα(µk)µ]mnρσ R

(na)m dim(Um) θnθi

θpG

(mkk)mn0 F

(n i j)map G

(n i j)m

p k

St. Petersburg 30 06 05 – p.14/15

Results


Thm. I: 5.1 Torus Thm. I: 5.20 Annulus

Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip


Sec. IV: 4.4 Three bulk fields on the sphere

Sec. IV: 4.5 Three defect fields on the sphere

Sec. IV: 4.2 Three boundary fields on the disk

Sec. IV: 4.3 One bulk and one boundary field on the disk

Sec. IV: 4.6 / 7 One bulk / defect field on the cross cap

St. Petersburg 30 06 05 – p.15/15

Results





Thm. V: 2.9 Bulk factorisation:

C(X) =Pi,j∈I

Pα,β dim(Ui)dim(Uj) (cbulk

i,j−1

)βαGbulkf,ij

`C(Γbulk

f,ij,αβ(X))´

with α, β bases of HomA|A(Ui⊗+A⊗−Uj , A) and HomA|A(Uı⊗+A⊗−U, A)

`f injective continuous orientation preserving map open annulus → X

no defect field insertions´

St. Petersburg 30 06 05 – p.15/15

Results






C(X) =Pi,j∈I


i,j−1

)βαGbulkf,ij

`C(Γbulk

f,ij,αβ(X))´

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � ��

� � � � � � � � � � � � ��

f(annulus)↓

7−→ 7−→

(U,−)

(U,+)

St. Petersburg 30 06 05 – p.15/15

Results






C(X) =Pi,j∈I


i,j−1

)βαGbulkf,ij

`C(Γbulk

f,ij,αβ(X))´

Thm. V: 2.6 Boundary factorisation:

C(X) =Pk∈I

Pα,β dim(Uk) (cbnd −1

Ml,Mr ,k)βα

Gbndf,k

`C(Γbnd

f,k,αβ(X))´

with α, β bases of HomA(Ml⊗Uk,Mr) and HomA(Mr ⊗Uk,Ml)`f injective, continuous 2-orientation preserving map strip Rε → X

f(∂Rε∩Rε)⊂ ∂X Ml/r boundary conditions at left/right end´

St. Petersburg 30 06 05 – p.15/15

Results






C(X) =Pi,j∈I


i,j−1

)βαGbulkf,ij

`C(Γbulk

f,ij,αβ(X))´


C(X) =Pk∈I


Ml,Mr ,k)βα

Gbndf,k

`C(Γbnd

f,k,αβ(X))´

Thm. V: 2.1 Covariance:

C(Y) = f](C(X))`[f ]∈Map(X,Y) orientation preserving

f] = tftC`X× [−1, 0]tY× [0, 1]

´/∼

´

Proof : consequence of triangulation independence

St. Petersburg 30 06 05 – p.15/15

Results






C(X) =Pi,j∈I


i,j−1

)βαGbulkf,ij

`C(Γbulk

f,ij,αβ(X))´


C(X) =Pk∈I


Ml,Mr ,k)βα

Gbndf,k

`C(Γbnd

f,k,αβ(X))´

Cor. V: 2.2 Modular invariance:

C(X) = f](C(X))`f ∈Mapor(X)

´

St. Petersburg 30 06 05 – p.15/15

Results






C(X) =Pi,j∈I


i,j−1

)βαGbulkf,ij

`C(Γbulk

f,ij,αβ(X))´


C(X) =Pk∈I


Ml,Mr ,k)βα

Gbndf,k

`C(Γbnd

f,k,αβ(X))´



´

St. Petersburg 30 06 05 – p.15/15

Results






C(X) =Pi,j∈I


i,j−1

)βαGbulkf,ij

`C(Γbulk

f,ij,αβ(X))´


C(X) =Pk∈I


Ml,Mr ,k)βα

Gbndf,k

`C(Γbnd

f,k,αβ(X))´



´

St. Petersburg 30 06 05 – p.15/15

Results






C(X) =Pi,j∈I


i,j−1

)βαGbulkf,ij

`C(Γbulk

f,ij,αβ(X))´


C(X) =Pk∈I


Ml,Mr ,k)βα

Gbndf,k

`C(Γbnd

f,k,αβ(X))´



´

St. Petersburg 30 06 05 – p.15/15

Results






C(X) =Pi,j∈I


i,j−1

)βαGbulkf,ij

`C(Γbulk

f,ij,αβ(X))´


C(X) =Pk∈I


Ml,Mr ,k)βα

Gbndf,k

`C(Γbnd

f,k,αβ(X))´



´

St. Petersburg 30 06 05 – p.15/15

Documents

CONFORMAL FIELD THEORY AND FROBENIUS ALGEBRAS IN … · IN MODULAR TENSOR CATEGORIES St.Petersburg 30 06 05 – p.1/15. CONFORMAL FIELD THEORY AND FROBENIUS ALGEBRAS IN MODULAR TENSOR