CONFORMAL FIELD THEORYAND FROBENIUS ALGEBRAS
IN MODULAR TENSOR CATEGORIES
St. Petersburg 30 06 05 – p.1/15
CONFORMAL FIELD THEORY
AND FROBENIUS ALGEBRAS
IN MODULAR TENSOR CATEGORIES
J
en Fu hsur cg
St. Petersburg 30 06 05 – p.1/15
CONFORMAL FIELD THEORY
AND FROBENIUS ALGEBRAS
IN MODULAR TENSOR CATEGORIES
J
en Fu hsur cg
St. Petersburg 30 06 05 – p.1/15
Plan
Aim:
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand two-dimensional conformal field theory
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
Step 1: Thicken the world sheet
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
Step 1: Thicken the world sheet
Connecting manifold , ribbon graphs , modular tensor categories
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
Step 1: Thicken the world sheet
Connecting manifold , ribbon graphs , modular tensor categories
Step 2: Triangulate the world sheet
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
Step 1: Thicken the world sheet
Connecting manifold , ribbon graphs , modular tensor categories
Step 2: Triangulate the world sheet
Frobenius algebras in modular tensor categories
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
Step 1: Thicken the world sheet
Connecting manifold , ribbon graphs , modular tensor categories
Step 2: Triangulate the world sheet
Frobenius algebras in modular tensor categories
Step 3: Bulk fields as morphisms
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
Step 1: Thicken the world sheet
Connecting manifold , ribbon graphs , modular tensor categories
Step 2: Triangulate the world sheet
Frobenius algebras in modular tensor categories
Step 3: Bulk fields as morphisms
Bimodules , alpha-induced bimodules , ...
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
Step 1: Thicken the world sheet
Connecting manifold , ribbon graphs , modular tensor categories
Step 2: Triangulate the world sheet
Frobenius algebras in modular tensor categories
Step 3: Bulk fields as morphisms
Bimodules , alpha-induced bimodules , ...
Correlation functions , boundary fields , defect fields
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
Step 1: Thicken the world sheet
Connecting manifold , ribbon graphs , modular tensor categories
Step 2: Triangulate the world sheet
Frobenius algebras in modular tensor categories
Step 3: Bulk fields as morphisms
Bimodules , alpha-induced bimodules , ...
Correlation functions , boundary fields , defect fields
Dictionary physical concepts ←→ mathematical structures
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
Step 1: Thicken the world sheet
Connecting manifold , ribbon graphs , modular tensor categories
Step 2: Triangulate the world sheet
Frobenius algebras in modular tensor categories
Step 3: Bulk fields as morphisms
Bimodules , alpha-induced bimodules , ...
Correlation functions , boundary fields , defect fields
Dictionary physical concepts ←→ mathematical structures
Some results
St. Petersburg 30 06 05 – p.2/15
Plan
Aim: Understand “holomorphic factorization” in RCFT
Step 1: Thicken the world sheet
Connecting manifold , ribbon graphs , modular tensor categories
Step 2: Triangulate the world sheet
Frobenius algebras in modular tensor categories
Step 3: Bulk fields as morphisms
Bimodules , alpha-induced bimodules , ...
Correlation functions , boundary fields , defect fields
Dictionary physical concepts ←→ mathematical structures
Some results
St. Petersburg 30 06 05 – p.2/15
Holomorphic factorization
Free boson QFT in d=2 :
St. Petersburg 30 06 05 – p.3/15
Holomorphic factorization
Free boson QFT in d=2 :
eq. of motion`∂2
∂x2− ∂2
∂t2
´φ(x, t) = 0
St. Petersburg 30 06 05 – p.3/15
Holomorphic factorization
Free boson QFT in d=2 :
eq. of motion`∂2
∂x2− ∂2
∂t2
´φ(x, t) = 0 solved by φ = φ+(x+t) + φ−(x−t)
St. Petersburg 30 06 05 – p.3/15
Holomorphic factorization
Free boson QFT in d=2 :
eq. of motion ∂ ∂ φ(z, z) = 0 solved by φ = φ1(z) + φ2(z)
St. Petersburg 30 06 05 – p.3/15
Holomorphic factorization
Free boson QFT in d=2 :
eq. of motion ∂ ∂ φ(z, z) = 0 solved by φ = φ1(z) + φ2(z)
“ left- and right-movers ”
“ holomorphic / antiholomorphic ”
“ chiral / antichiral ”
St. Petersburg 30 06 05 – p.3/15
Holomorphic factorization
Free boson QFT in d=2 :
eq. of motion ∂ ∂ φ(z, z) = 0 solved by φ = φ1(z) + φ2(z)
Observation: Same phenomenon for various aspects
of a large class of 2-d QFT models
on two-dimensional world sheets
St. Petersburg 30 06 05 – p.3/15
Holomorphic factorization
Free boson QFT in d=2 :
eq. of motion ∂ ∂ φ(z, z) = 0 solved by φ = φ1(z) + φ2(z)
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
St. Petersburg 30 06 05 – p.3/15
Holomorphic factorization
Free boson QFT in d=2 :
eq. of motion ∂ ∂ φ(z, z) = 0 solved by φ = φ1(z) + φ2(z)
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 ≡ RCFT —
St. Petersburg 30 06 05 – p.3/15
Holomorphic factorization
Free boson QFT in d=2 :
eq. of motion ∂ ∂ φ(z, z) = 0 solved by φ = φ1(z) + φ2(z)
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 ≡ RCFT —
In particular: Bulk fields Φ of a RCFT
Φ
St. Petersburg 30 06 05 – p.3/15
Holomorphic factorization
Free boson QFT in d=2 :
eq. of motion ∂ ∂ φ(z, z) = 0 solved by φ = φ1(z) + φ2(z)
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 ≡ RCFT —
In particular: Bulk fields Φ of a RCFT
Φ
carry rep’s of both left and right world sheet symmetries Φ = Φij
ij
St. Petersburg 30 06 05 – p.3/15
Holomorphic factorization
Free boson QFT in d=2 :
eq. of motion ∂ ∂ φ(z, z) = 0 solved by φ = φ1(z) + φ2(z)
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 ≡ RCFT —
In particular: Bulk fields Φ of a RCFT
Φ
carry rep’s of both left and right world sheet symmetries Φ = Φij
ij
with multiplicities
α
α
St. Petersburg 30 06 05 – p.3/15
Bulk fields
CFT =⇒ Conformal symmetry / extensions
2-d =⇒ Virasoro algebra / affine Lie algebras / W-algebras / ...
=⇒ conformal vertex algebra [ Borcherds 1986 , . . . ]
St. Petersburg 30 06 05 – p.4/15
Bulk fields
CFT =⇒ Conformal symmetry / extensions
2-d =⇒ Virasoro algebra / affine Lie algebras / W-algebras / ...
=⇒ conformal vertex algebra [ Borcherds 1986 , . . . ]
RCFT =⇒ Rational conformal vertex algebra V [ Huang 2004 ]
St. Petersburg 30 06 05 – p.4/15
Bulk fields
CFT =⇒ Conformal symmetry / extensions
2-d =⇒ Virasoro algebra / affine Lie algebras / W-algebras / ...
=⇒ conformal vertex algebra [ Borcherds 1986 , . . . ]
RCFT =⇒ Rational conformal vertex algebra V [ Huang 2004 ]
Rep category Rep(V) : a modular tensor category
St. Petersburg 30 06 05 – p.4/15
Bulk fields
CFT =⇒ Conformal symmetry / extensions
2-d =⇒ Virasoro algebra / affine Lie algebras / W-algebras / ...
=⇒ conformal vertex algebra [ Borcherds 1986 , . . . ]
RCFT =⇒ Rational conformal vertex algebra V [ Huang 2004 ]
Rep category Rep(V) : a modular tensor category
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
St. Petersburg 30 06 05 – p.4/15
Bulk fields
CFT =⇒ Conformal symmetry / extensions
2-d =⇒ Virasoro algebra / affine Lie algebras / W-algebras / ...
=⇒ conformal vertex algebra [ Borcherds 1986 , . . . ]
RCFT =⇒ Rational conformal vertex algebra V [ Huang 2004 ]
Rep category Rep(V) : a modular tensor category
Bulk field Φ = Φij
=⇒ two simple objects Ui , Uj ∈ C := Rep(V)
St. Petersburg 30 06 05 – p.4/15
Bulk fields
CFT =⇒ Conformal symmetry / extensions
2-d =⇒ Virasoro algebra / affine Lie algebras / W-algebras / ...
=⇒ conformal vertex algebra [ Borcherds 1986 , . . . ]
RCFT =⇒ Rational conformal vertex algebra V [ Huang 2004 ]
Rep category Rep(V) : a modular tensor category
Bulk field Φ = Φαij
=⇒ two simple objects Ui , Uj ∈ C := Rep(V)
and a morphism α ∈ Hom(Ui⊗Uj ,1)
St. Petersburg 30 06 05 – p.4/15
Bulk fields
CFT =⇒ Conformal symmetry / extensions
2-d =⇒ Virasoro algebra / affine Lie algebras / W-algebras / ...
=⇒ conformal vertex algebra [ Borcherds 1986 , . . . ]
RCFT =⇒ Rational conformal vertex algebra V [ Huang 2004 ]
Rep category Rep(V) : a modular tensor category
Bulk field Φ = Φαij
=⇒ two simple objects Ui , Uj ∈ C := Rep(V)
and a morphism α ∈ Hom(Ui⊗Uj ,1)
Recall: Free boson CFT: geometric separation of left- and right-movers
St. Petersburg 30 06 05 – p.4/15
Bulk fields
Separate locally left- and right-movers :
double bX of world sheet X
St. Petersburg 30 06 05 – p.5/15
Bulk fields
Separate locally left- and right-movers :
double bX of world sheet X
bX = orientation bundle over X modulo identification of points in fibers over ∂X
Φij 7−→
j
i
St. Petersburg 30 06 05 – p.5/15
Bulk fields
Separate locally left- and right-movers :
double bX of world sheet X
bX = orientation bundle over X modulo identification of points in fibers over ∂X
To connect to X: Fatten the world sheet:
connecting manifold MX ∂MX = bX
St. Petersburg 30 06 05 – p.5/15
Bulk fields
Separate locally left- and right-movers :
double bX of world sheet X
bX = orientation bundle over X modulo identification of points in fibers over ∂X
To connect to X: Fatten the world sheet:
connecting manifold MX ∂MX = bXMX = interval bundle over X modulo identification over ∂X
= (bX× [−1, 1])/∼ with ([x, or2], t) ∼ ([x,−or2],−t)
St. Petersburg 30 06 05 – p.5/15
Bulk fields
Separate locally left- and right-movers :
double bX of world sheet X
bX = orientation bundle over X modulo identification of points in fibers over ∂X
To connect to X: Fatten the world sheet:
connecting manifold MX ∂MX = bXMX = interval bundle over X modulo identification over ∂X
Field theory on MX non-dynamical
; 3-d TFT
; ribbon graphs
Φij 7−→
j
i
St. Petersburg 30 06 05 – p.5/15
Bulk fields
Separate locally left- and right-movers :
double bX of world sheet X
bX = orientation bundle over X modulo identification of points in fibers over ∂X
To connect to X: Fatten the world sheet:
connecting manifold MX ∂MX = bXMX = interval bundle over X modulo identification over ∂X
Field theory on MX non-dynamical
; 3-d TFT
; ribbon graphs
j
i
α
St. Petersburg 30 06 05 – p.5/15
Bulk fields
Separate locally left- and right-movers :
double bX of world sheet X
bX = orientation bundle over X modulo identification of points in fibers over ∂X
To connect to X: Fatten the world sheet:
connecting manifold MX ∂MX = bXMX = interval bundle over X modulo identification over ∂X
Field theory on MX non-dynamical
; 3-d TFT
; ribbon graphs
j
i
αX embedded as MX ⊃ X×{t=0}
St. Petersburg 30 06 05 – p.5/15
Bulk fields
double bX`orientation bundle over X
´/∼
connecting manifold MX`interval bundle over X
´/∼
embedding X×{t=0} ⊂ MX
j
i
α
St. Petersburg 30 06 05 – p.6/15
Bulk fields
double bX`orientation bundle over X
´/∼
connecting manifold MX`interval bundle over X
´/∼
embedding X×{t=0} ⊂ MX
j
i
α
St. Petersburg 30 06 05 – p.6/15
Bulk fields
double bX`orientation bundle over X
´/∼
connecting manifold MX`interval bundle over X
´/∼
embedding X×{t=0} ⊂ MX
j
i
α
α ∈ Hom(Ui⊗Uj ,1)
= δi,�
St. Petersburg 30 06 05 – p.6/15
Bulk fields
double bX`orientation bundle over X
´/∼
connecting manifold MX`interval bundle over X
´/∼
embedding X×{t=0} ⊂ MX
j
i
α
α ∈ Hom(Ui⊗Uj ,1)
= δi,�
too restrictive
?
St. Petersburg 30 06 05 – p.6/15
Frobenius algebras
Special case: meromorphic CFT
St. Petersburg 30 06 05 – p.7/15
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
St. Petersburg 30 06 05 – p.7/15
Frobenius algebras
Prescription for general RCFT:
triangulate the world sheet (trivalent vertices)
label edges (ribbons) by a Frobenius algebra A in C A A
St. Petersburg 30 06 05 – p.7/15
Frobenius algebras
Prescription for general RCFT:
triangulate the world sheet (trivalent vertices)
label edges (ribbons) by a Frobenius algebra A in C A A
label vertices (coupons) in X\∂X
by product / coproduct morphisms
A
A A
St. Petersburg 30 06 05 – p.7/15
Frobenius algebras
Prescription for general RCFT:
triangulate the world sheet (trivalent vertices)
label edges (ribbons) by a Frobenius algebra A in C A A
label vertices (coupons) in X\∂X
by product / coproduct morphisms
A
A A
connect to bulk field ribbons bysuitable morphisms in Hom(Ui⊗A⊗Uj , A)
St. Petersburg 30 06 05 – p.7/15
Frobenius algebras
Prescription for general RCFT:
triangulate the world sheet (trivalent vertices)
label edges (ribbons) by a Frobenius algebra A in C A A
label vertices (coupons) in X\∂X
by product / coproduct morphisms
A
A A
connect to bulk field ribbons bysuitable morphisms in Hom(Ui⊗A⊗Uj , A)
More precisely: A a symmetric special Frobenius algebra
St. Petersburg 30 06 05 – p.7/15
Frobenius algebras
Prescription for general RCFT:
triangulate the world sheet (trivalent vertices)
label edges (ribbons) by a Frobenius algebra A in C A A
label vertices (coupons) in X\∂X
by product / coproduct morphisms
A
A A
connect to bulk field ribbons bysuitable morphisms in Hom(Ui⊗A⊗Uj , A)
More precisely: A a symmetric special Frobenius algebra
= = =
St. Petersburg 30 06 05 – p.7/15
Frobenius algebras
Prescription for general RCFT:
triangulate the world sheet (trivalent vertices)
label edges (ribbons) by a Frobenius algebra A in C A A
label vertices (coupons) in X\∂X
by product / coproduct morphisms
A
A A
connect to bulk field ribbons bysuitable morphisms in Hom(Ui⊗A⊗Uj , A)
More precisely: A a symmetric special Frobenius co algebra
= = =
St. Petersburg 30 06 05 – p.7/15
Frobenius algebras
Prescription for general RCFT:
triangulate the world sheet (trivalent vertices)
label edges (ribbons) by a Frobenius algebra A in C A A
label vertices (coupons) in X\∂X
by product / coproduct morphisms
A
A A
connect to bulk field ribbons bysuitable morphisms in Hom(Ui⊗A⊗Uj , A)
More precisely: A a symmetric special Frobenius algebra
= =
St. Petersburg 30 06 05 – p.7/15
Frobenius algebras
Prescription for general RCFT:
triangulate the world sheet (trivalent vertices)
label edges (ribbons) by a Frobenius algebra A in C A A
label vertices (coupons) in X\∂X
by product / coproduct morphisms
A
A A
connect to bulk field ribbons bysuitable morphisms in Hom(Ui⊗A⊗Uj , A)
More precisely: A a symmetric special Frobenius algebra
=
A
A∨
St. Petersburg 30 06 05 – p.7/15
Frobenius algebras
Prescription for general RCFT:
triangulate the world sheet (trivalent vertices)
label edges (ribbons) by a Frobenius algebra A in C A A
label vertices (coupons) in X\∂X
by product / coproduct morphisms
A
A A
connect to bulk field ribbons bysuitable morphisms in Hom(Ui⊗A⊗Uj , A)
More precisely: A a symmetric special = strongly separable Frobenius algebra
= =
St. Petersburg 30 06 05 – p.7/15
Frobenius algebras
Prescription for general RCFT:
triangulate the world sheet (trivalent vertices)
label edges (ribbons) by a Frobenius algebra A in C A A
label vertices (coupons) in X\∂X
by product / coproduct morphisms
A
A A
connect to bulk field ribbons bysuitable morphisms in Hom(Ui⊗A⊗Uj , A)
More precisely: A a symmetric special Frobenius algebra
In addition: reversion = isomorphism A↔ Aopp squaring to twist
when X is unoriented
prescription for vertices of triangulation on ∂X when ∂X 6= ∅
St. Petersburg 30 06 05 – p.7/15
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)
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choiceswithout field insertions
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule left module properties:
= =
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule A as a left module:
=
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule A as a right module:
= left and right actions commuteby associativity
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule
Ui⊗A⊗Uj is A-bimodule
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule
Ui⊗A⊗Uj is A-bimodule Induced left module:
:=
A⊗U U
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule
Ui⊗A⊗Uj is A-bimodule ⊗+-induced left module:
Ui
( use braiding )
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule
Ui⊗A⊗Uj is A-bimodule ⊗−-induced right module:
Uj
cf alpha-induction of subfactors[ Longo-Rehren ]
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule
Ui⊗A⊗Uj is A-bimodule
Ui⊗+A⊗−Uj ∈ Obj(CA|A)
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule
Ui⊗A⊗Uj is A-bimodule
Ui⊗+A⊗−Uj ∈ Obj(CA|A)
Subspace = space of bimodule morphisms
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule
Ui⊗A⊗Uj is A-bimodule
Ui⊗+A⊗−Uj ∈ Obj(CA|A)
Subspace = space of bimodule morphisms Left-module intertwiner:
=ρM
ρN
St. Petersburg 30 06 05 – p.8/15
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)
Properties of algebra A ⇐⇒ correlation functions (see below)independent of all choices+ restriction of
morphism spaces
Restriction of Hom(Ui⊗A⊗Uj , A) to a subspace:
A is A-bimodule
Ui⊗A⊗Uj is A-bimodule
Ui⊗+A⊗−Uj ∈ Obj(CA|A)
Subspace = space of bimodule morphisms
Prescription:
Space of bulk fields = HomA|A(Ui⊗+A⊗−Uj , A)with chiral labels i,j
St. Petersburg 30 06 05 – p.8/15
Bulk fields II
Prescription: Space of bulk fields˘Φαij
¯= HomA|A(Ui⊗+A⊗−Uj , A)
St. Petersburg 30 06 05 – p.9/15
Bulk fields II
Prescription: Space of bulk fields˘Φαij
¯= HomA|A(Ui⊗+A⊗−Uj , A)
St. Petersburg 30 06 05 – p.9/15
Bulk fields II
Prescription: Space of bulk fields˘Φαij
¯= HomA|A(Ui⊗+A⊗−Uj , A)
copr.
unit
j
i
α
St. Petersburg 30 06 05 – p.9/15
Correlation functions
Ribbon graphs in 3-manifolds vs morphisms in C :
St. Petersburg 30 06 05 – p.10/15
Correlation functions
Ribbon graphs in 3-manifolds vs morphisms in C :
RCFT ; vertex algebra V
St. Petersburg 30 06 05 – p.10/15
Correlation functions
Ribbon graphs in 3-manifolds vs morphisms in C :
RCFT ; vertex algebra V
; C = Rep(V) – modular tensor category
St. Petersburg 30 06 05 – p.10/15
Correlation functions
Ribbon graphs in 3-manifolds vs morphisms in C :
RCFT ; vertex algebra V
; C = Rep(V) – modular tensor category
; extended 3-d TFT [ Reshetikhin-Turaev 1991 , . . . ]
functor tftC : 3-CobC → Vect �
St. Petersburg 30 06 05 – p.10/15
Correlation functions
Ribbon graphs in 3-manifolds vs morphisms in C :
RCFT ; vertex algebra V
; C = Rep(V) – modular tensor category
; extended 3-d TFT [ Reshetikhin-Turaev 1991 , . . . ]
functor tftC : 3-CobC → Vect �
Σ 7→ tftC(Σ) =: H(Σ)
H(∅) =
�
tftC(Hom(∅,Σ)) ∼= H(Σ)
St. Petersburg 30 06 05 – p.10/15
Correlation functions
Ribbon graphs in 3-manifolds vs morphisms in C :
RCFT ; vertex algebra V
; C = Rep(V) – modular tensor category
; extended 3-d TFT [ Reshetikhin-Turaev 1991 , . . . ]
functor tftC : 3-CobC → Vect �
Conjecture: [ Witten 1989 , . . . ]
state spaces H(Σ) = spaces of conformal blocks of RCFT
St. Petersburg 30 06 05 – p.10/15
Correlation functions
Ribbon graphs in 3-manifolds vs morphisms in C :
RCFT ; vertex algebra V
; C = Rep(V) – modular tensor category
; extended 3-d TFT [ Reshetikhin-Turaev 1991 , . . . ]
functor tftC : 3-CobC → Vect �
Conjecture: [ Witten 1989 , . . . ]
state spaces H(Σ) = spaces of conformal blocks of RCFT
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
St. Petersburg 30 06 05 – p.10/15
Correlation functions
Ribbon graphs in 3-manifolds vs morphisms in C :
RCFT ; vertex algebra V
; C = Rep(V) – modular tensor category
; extended 3-d TFT [ Reshetikhin-Turaev 1991 , . . . ]
functor tftC : 3-CobC → Vect �
Conjecture: [ Witten 1989 , . . . ]
state spaces H(Σ) = spaces of conformal blocks of RCFT
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
Holomorphic factorization for correlation functions: C(X) ∈ H(bX)
St. Petersburg 30 06 05 – p.10/15
Correlation functions
Ribbon graphs in 3-manifolds vs morphisms in C :
RCFT ; vertex algebra V
; C = Rep(V) – modular tensor category
; extended 3-d TFT [ Reshetikhin-Turaev 1991 , . . . ]
functor tftC : 3-CobC → Vect �
Conjecture: [ Witten 1989 , . . . ]
state spaces H(Σ) = spaces of conformal blocks of RCFT
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
Holomorphic factorization for correlation functions: C(X) ∈ H(bX)
in words: a correlation function on X is a vector in the state space of the double bX
St. Petersburg 30 06 05 – p.10/15
Correlation functions
Ribbon graphs in 3-manifolds vs morphisms in C :
RCFT ; vertex algebra V
; C = Rep(V) – modular tensor category
; extended 3-d TFT [ Reshetikhin-Turaev 1991 , . . . ]
functor tftC : 3-CobC → Vect �
Conjecture: [ Witten 1989 , . . . ]
state spaces H(Σ) = spaces of conformal blocks of RCFT
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
Holomorphic factorization for correlation functions: C(X) ∈ H(bX)
Idea: Specify the vector C(X) as a concrete cobordism ∅ → bX( 3-manifold + ribbon graph )
St. Petersburg 30 06 05 – p.10/15
Correlation functions
Prescription for cobordism C(X) :
St. Petersburg 30 06 05 – p.11/15
Correlation functions
Prescription for cobordism C(X) :
3-manifold: Connecting manifold MX
∂MX = bX X embedded as X×{t=0}
St. Petersburg 30 06 05 – p.11/15
Correlation functions
Prescription for cobordism C(X) :
3-manifold: Connecting manifold MX
∂MX = bX X embedded as X×{t=0}
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)
St. Petersburg 30 06 05 – p.11/15
Correlation functions
Prescription for cobordism C(X) :
3-manifold: Connecting manifold MX
∂MX = bX X embedded as X×{t=0}
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)
Example : Three bulk fields on the sphere S2 = R2∪{∞} :
p1
p2
p3Φ1
Φ2
Φ3
St. Petersburg 30 06 05 – p.11/15
Correlation functions
Prescription for cobordism C(X) :
3-manifold: Connecting manifold MX
∂MX = bX X embedded as X×{t=0}
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)
Example : Three bulk fields on the sphere S2 = R2∪{∞} :
St. Petersburg 30 06 05 – p.11/15
Correlation functions
Prescription for cobordism C(X) :
3-manifold: Connecting manifold MX
∂MX = bX X embedded as X×{t=0}
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)
Example : Three bulk fields on the sphere S2 = R2∪{∞} :
yields OPE coefficientswhen expressed as linear combination of standard basis blocks
St. Petersburg 30 06 05 – p.11/15
Correlation functions
Prescription for cobordism C(X) :
3-manifold: Connecting manifold MX
∂MX = bX X embedded as X×{t=0}
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)
Example : Three bulk fields on the sphere S2 = R2∪{∞} :
yields OPE coefficientswhen expressed as linear combination of standard basis blocks
Additional ingredients for ribbon graph needed in general:
St. Petersburg 30 06 05 – p.11/15
Correlation functions
Prescription for cobordism C(X) :
3-manifold: Connecting manifold MX
∂MX = bX X embedded as X×{t=0}
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)
Example : Three bulk fields on the sphere S2 = R2∪{∞} :
yields OPE coefficientswhen expressed as linear combination of standard basis blocks
Additional ingredients for ribbon graph needed in general:
For ∂X 6= ∅ : annular ribbon for each boundary componentlabeled by boundary condition
St. Petersburg 30 06 05 – p.11/15
Correlation functions
Prescription for cobordism C(X) :
3-manifold: Connecting manifold MX
∂MX = bX X embedded as X×{t=0}
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)
Example : Three bulk fields on the sphere S2 = R2∪{∞} :
yields OPE coefficientswhen expressed as linear combination of standard basis blocks
Additional ingredients for ribbon graph needed in general:
For ∂X 6= ∅ : annular ribbon for each boundary componentlabeled by boundary condition
Boundary fields – can change boundary condition
St. Petersburg 30 06 05 – p.11/15
Correlation functions
Prescription for cobordism C(X) :
3-manifold: Connecting manifold MX
∂MX = bX X embedded as X×{t=0}
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)
Example : Three bulk fields on the sphere S2 = R2∪{∞} :
yields OPE coefficientswhen expressed as linear combination of standard basis blocks
Additional ingredients for ribbon graph needed in general:
For ∂X 6= ∅ : annular ribbon for each boundary componentlabeled by boundary condition
Boundary fields – can change boundary condition
Defect lines and defect fields
St. Petersburg 30 06 05 – p.11/15
Correlation functions
Prescription for cobordism C(X) :
3-manifold: Connecting manifold MX
∂MX = bX X embedded as X×{t=0}
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)
Example : Three bulk fields on the sphere S2 = R2∪{∞} :
yields OPE coefficientswhen expressed as linear combination of standard basis blocks
Additional ingredients for ribbon graph needed in general:
For ∂X 6= ∅ : annular ribbon for each boundary componentlabeled by boundary condition
Boundary fields – can change boundary condition
Defect lines and defect fields
Be careful with 1- and 2-orientations of ribbons , . . .St. Petersburg 30 06 05 – p.11/15
Correlation functions
DICTIONARY
St. Petersburg 30 06 05 – p.12/15
Correlation functions
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]
St. Petersburg 30 06 05 – p.12/15
Correlation functions
DICTIONARY
chiral labels ←→ C = Rep(V) , simple objects ∼= Ui , i ∈ I
full non-diagonal CFT ←→ symmetric special Frobenius algebra A in C
bulk fields Φij ←→ bimodule morphisms HomA|A(Ui⊗+A⊗− Uj , A)
St. Petersburg 30 06 05 – p.12/15
Correlation functions
DICTIONARY
chiral labels ←→ C = Rep(V) , simple objects ∼= Ui , i ∈ I
full non-diagonal CFT ←→ symmetric special Frobenius algebra A in C
bulk fields Φij ←→ bimodule morphisms HomA|A(Ui⊗+A⊗− Uj , A)
boundary conditions ←→ A-modules M ∈ Obj(CA)
boundary fields ΨMNi ←→ module morphisms HomA(M ⊗Ui, N)
St. Petersburg 30 06 05 – p.12/15
Correlation functions
DICTIONARY
chiral labels ←→ C = Rep(V) , simple objects ∼= Ui , i ∈ I
full non-diagonal CFT ←→ symmetric special Frobenius algebra A in C
bulk fields Φij ←→ bimodule morphisms HomA|A(Ui⊗+A⊗− Uj , A)
boundary conditions ←→ A-modules M ∈ Obj(CA)
boundary fields ΨMNi ←→ module morphisms HomA(M ⊗Ui, N)
defect lines ←→ A-bimodules Y ∈ Obj(CA|A)
defect fields ΘY Zij ←→ bimodule morphisms HomA|A(Ui⊗+Y ⊗− Uj , Z)
St. Petersburg 30 06 05 – p.12/15
Correlation functions
DICTIONARY
chiral labels ←→ C = Rep(V) , simple objects ∼= Ui , i ∈ I
full non-diagonal CFT ←→ symmetric special Frobenius algebra A in C
bulk fields Φij ←→ bimodule morphisms HomA|A(Ui⊗+A⊗− Uj , A)
boundary conditions ←→ A-modules M ∈ Obj(CA)
boundary fields ΨMNi ←→ module morphisms HomA(M ⊗Ui, N)
defect lines ←→ A-B-bimodules Y ∈ Obj(CA|B)
defect fields ΘY Zij ←→ bimodule morphisms HomA|B(Ui⊗+Y ⊗− Uj , Z)
St. Petersburg 30 06 05 – p.12/15
Correlation functions
DICTIONARY
chiral labels ←→ C = Rep(V) , simple objects ∼= Ui , i ∈ I
full non-diagonal CFT ←→ symmetric special Frobenius algebra A in C
bulk fields Φij ←→ bimodule morphisms HomA|A(Ui⊗+A⊗− Uj , A)
boundary conditions ←→ A-modules M ∈ Obj(CA)
boundary fields ΨMNi ←→ module morphisms HomA(M ⊗Ui, N)
defect lines ←→ A-B-bimodules Y ∈ Obj(CA|B)
defect fields ΘY Zij ←→ bimodule morphisms HomA|B(Ui⊗+Y ⊗− Uj , Z)
CFT on unoriented ←→ Jandl algebraworld sheet
St. Petersburg 30 06 05 – p.12/15
Correlation functions
DICTIONARY
chiral labels ←→ C = Rep(V) , simple objects ∼= Ui , i ∈ I
full non-diagonal CFT ←→ symmetric special Frobenius algebra A in C
bulk fields Φij ←→ bimodule morphisms HomA|A(Ui⊗+A⊗− Uj , A)
boundary conditions ←→ A-modules M ∈ Obj(CA)
boundary fields ΨMNi ←→ module morphisms HomA(M ⊗Ui, N)
defect lines ←→ A-B-bimodules Y ∈ Obj(CA|B)
defect fields ΘY Zij ←→ bimodule morphisms HomA|B(Ui⊗+Y ⊗− Uj , Z)
CFT on unoriented ←→ Jandl algebraworld sheet
simple current models ←→ Schellekens algebra A ∼=Lg∈G Lg for G ≤ Pic(C)
St. Petersburg 30 06 05 – p.12/15
Correlation functions
DICTIONARY
chiral labels ←→ C = Rep(V) , simple objects ∼= Ui , i ∈ I
full non-diagonal CFT ←→ symmetric special Frobenius algebra A in C
bulk fields Φij ←→ bimodule morphisms HomA|A(Ui⊗+A⊗− Uj , A)
boundary conditions ←→ A-modules M ∈ Obj(CA)
boundary fields ΨMNi ←→ module morphisms HomA(M ⊗Ui, N)
defect lines ←→ A-B-bimodules Y ∈ Obj(CA|B)
defect fields ΘY Zij ←→ bimodule morphisms HomA|B(Ui⊗+Y ⊗− Uj , Z)
CFT on unoriented ←→ Jandl algebraworld sheet
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
St. Petersburg 30 06 05 – p.12/15
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
St. Petersburg 30 06 05 – p.13/15
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 )
St. Petersburg 30 06 05 – p.13/15
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
St. Petersburg 30 06 05 – p.13/15
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
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
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
C(T; ∅) =
T×[−1,1]
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
C(T; ∅) =
T×[−1,1]
=⇒ Zij =
i A j
A
A
A
S2×S1
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
Lemma I: 5.2 :
A
A U
A U
A
is an idempotent
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
Proof : = = =
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
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)
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
Proof of [T,Z] = 0 :
= =
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
Prop. I: 5.3 (Torus partition function)
ZA⊕B = ZA + ZB eZA⊗B = eZA eZB ZAopp
=`ZA
´t
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
Prop. I: 5.3 (Torus partition function)
ZA⊕B = ZA + ZB eZA⊗B = eZA eZB ZAopp
=`ZA
´t
eZA⊗Bij =
i A B j
eZij ≡ Zıj
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
Prop. I: 5.3 (Torus partition function)
ZA⊕B = ZA + ZB eZA⊗B = eZA eZB ZAopp
=`ZA
´t
ZAopp
ij =
i A
mopp
j
=
i
m
A j
=
i
m
A j
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
Prop. I: 5.3 (Torus partition function)
ZA⊕B = ZA + ZB eZA⊗B = eZA eZB ZAopp
=`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}
St. Petersburg 30 06 05 – p.13/15
Results
For A a symmetric special Frobenius algebra in a modular tensor category C :
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
Prop. I: 5.3 (Torus partition function)
ZA⊕B = ZA + ZB eZA⊗B = eZA eZB ZAopp
=`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/∼
St. Petersburg 30 06 05 – p.13/15
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}
St. Petersburg 30 06 05 – p.14/15
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}
Special case: A = 1 =⇒ Zij = δi Kj =
(±1 if j=
0 else( FS indicator )
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
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
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
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)
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
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
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
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
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
Three defect fields on the sphereSec. IV: 4.5
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)
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
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
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
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
Three defect fields on the sphereSec. IV: 4.5
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
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
Three defect fields on the sphereSec. IV: 4.5
Three boundary fields on the diskSec. IV: 4.2
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])
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
Three defect fields on the sphereSec. IV: 4.5
Three boundary fields on the diskSec. IV: 4.2
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
St. Petersburg 30 06 05 – p.14/15
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
Three defect fields on the sphereSec. IV: 4.5
Three boundary fields on the diskSec. IV: 4.2
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
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
Three defect fields on the sphereSec. IV: 4.5
Three boundary fields on the diskSec. IV: 4.2
One bulk and one boundary field on the diskSec. IV: 4.3
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
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}
OPE coefficients for the fundamental correlation functions
Three bulk fields on the sphereSec. IV: 4.4
Three defect fields on the sphereSec. IV: 4.5
Three boundary fields on the diskSec. IV: 4.2
One bulk and one boundary field on the diskSec. IV: 4.3
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
For A a simple symmetric special Frobenius algebra in a modular tensor category C :
Thm. I: 5.1 Torus Thm. I: 5.20 Annulus
Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip
OPE coefficients for the fundamental correlation functions
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
For A a simple symmetric special Frobenius algebra in a modular tensor category C :
Thm. I: 5.1 Torus Thm. I: 5.20 Annulus
Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip
OPE coefficients for the fundamental correlation functions
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
For A a simple symmetric special Frobenius algebra in a modular tensor category C :
Thm. I: 5.1 Torus Thm. I: 5.20 Annulus
Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip
OPE coefficients for the fundamental correlation functions
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))´
� � � � � � � � � � � � � �
� � � � � � � � � � � � � �
� � � � � � � � � � � � � �
� � � � � � � � � � � � � �
� � � � � � � � � � � � � �
� � � � � � � � � � � � � �
� � � � � � � � � � � � � �
� � � � � � � � � � � � � �
� � � � � � � � � � � � � �
� � � � � � � � � � � � � �
� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �
� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �
f(annulus)↓
7−→ 7−→
(U,−)
(U,+)
St. Petersburg 30 06 05 – p.15/15
Results
For A a simple symmetric special Frobenius algebra in a modular tensor category C :
Thm. I: 5.1 Torus Thm. I: 5.20 Annulus
Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip
OPE coefficients for the fundamental correlation functions
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))´
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
For A a simple symmetric special Frobenius algebra in a modular tensor category C :
Thm. I: 5.1 Torus Thm. I: 5.20 Annulus
Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip
OPE coefficients for the fundamental correlation functions
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))´
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))´
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
For A a simple symmetric special Frobenius algebra in a modular tensor category C :
Thm. I: 5.1 Torus Thm. I: 5.20 Annulus
Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip
OPE coefficients for the fundamental correlation functions
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))´
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))´
Cor. V: 2.2 Modular invariance:
C(X) = f](C(X))`f ∈Mapor(X)
´
St. Petersburg 30 06 05 – p.15/15
Results
For A a simple symmetric special Frobenius algebra in a modular tensor category C :
Thm. I: 5.1 Torus Thm. I: 5.20 Annulus
Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip
OPE coefficients for the fundamental correlation functions
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))´
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))´
Cor. V: 2.2 Modular invariance:
C(X) = f](C(X))`f ∈Mapor(X)
´
St. Petersburg 30 06 05 – p.15/15
Results
For A a simple symmetric special Frobenius algebra in a modular tensor category C :
Thm. I: 5.1 Torus Thm. I: 5.20 Annulus
Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip
OPE coefficients for the fundamental correlation functions
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))´
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))´
Cor. V: 2.2 Modular invariance:
C(X) = f](C(X))`f ∈Mapor(X)
´
St. Petersburg 30 06 05 – p.15/15
Results
For A a simple symmetric special Frobenius algebra in a modular tensor category C :
Thm. I: 5.1 Torus Thm. I: 5.20 Annulus
Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip
OPE coefficients for the fundamental correlation functions
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))´
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))´
Cor. V: 2.2 Modular invariance:
C(X) = f](C(X))`f ∈Mapor(X)
´
St. Petersburg 30 06 05 – p.15/15
Results
For A a simple symmetric special Frobenius algebra in a modular tensor category C :
Thm. I: 5.1 Torus Thm. I: 5.20 Annulus
Thm. II: 3.7 Klein bottle Thm. II: 3.5 Möbius strip
OPE coefficients for the fundamental correlation functions
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))´
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))´
Cor. V: 2.2 Modular invariance:
C(X) = f](C(X))`f ∈Mapor(X)
´
St. Petersburg 30 06 05 – p.15/15