TENSOR CATEGORIES AND HOPF ALGEBRAS IN CONFORMAL … · TENSOR CATEGORIES AND HOPF ALGEBRAS IN...

TENSOR CATEGORIES AND HOPF ALGEBRASIN CONFORMAL FIELD THEORY

Jen Fu

hsu

r g c

JF La Falda 03 09 09 – p. 1/28


J

en Fu

hsu

r g c


JF La Falda 03 09 09 – p. 1/28

Plan Categories and Hopf algebras in CFT

Rational CFT (semisimple) :

RCFT and 3-d TFT

Some ingredients

Sample results

Beyond rational CFT :

Verlinde-like formulas

Ribbon Hopf algebras

Coends

JF La Falda 03 09 09 – p. 2/28

CFT Categories and Hopf algebras in CFT

RCFT and 3-d TFT

Some ingredients

Sample results



Coends

#

JF La Falda 03 09 09 – p. 3/28

Correlators and conformal blocks Categories and Hopf algebras in CFT

CFT : Two-dimensional conformal quantum field theory

Central object of interest : correlators

Cor(Y) : MY ×~H → C multilinear in ~H = H1 × H2 × · · · × Hm

JF La Falda 03 09 09 – p. 3/28





Hℓ =

8

>

>

>

>

<

>

>

>

>

:

space of boundary fields– representation space of V

space of bulk fields– representation space of V⊕V

V: conformal vertex algebra ( ‘chiral algebra’ )

JF La Falda 03 09 09 – p. 3/28





Hℓ =

8

>

>

>

>

<

>

>

>

>

:

space of boundary fields– representation space of V

space of bulk fields– representation space of V⊕Vworld sheet Y ≡ (Y; ~τ, ~p , ...)

~τ moduli of conformal structure on Y

~p = p1, p2, ... , pm insertion points

m = 0p1(bulk)

p3 (bdy)

JF La Falda 03 09 09 – p. 3/28





determined by consistency conditions :

Ward identities :

Compatibility with action of V on ~H ; diff. equations

Sewing constraints :

Compatibility of correlators on different world sheets related by ‘ cutting and gluing ’

JF La Falda 03 09 09 – p. 3/28





determined by consistency conditions :

Ward identities :

Compatibility with action of V on ~H ; diff. equations

⊲ solutions for fixed p∈MY form finite-dim. vector space BY of conformal blocks

⊲ fit into vector bundle over moduli space of double cover bY of Y

⊲ carry action of mapping class group Map(bY) ⊃ Mapor(Y)

Sewing constraints :

⊲ include modular invariance : BY ∋ Cor(Y) invariant under action of Mapor(Y)

JF La Falda 03 09 09 – p. 3/28

Solution of the sewing constraints Categories and Hopf algebras in CFT

assumed for now : CFT “rational ” : Rep(V) a modular tensor category (semisimple) =⇒

For solving the sewing constraints ( and for other purposes )

combinatorial information sufficient :

⊲ regard BY as abstract vector space (finite-dim.)

⊲ Cor(Y) ∈ BY

⊲ encode symmetries in category C ≃ Rep(V) as abstract category

⊲ identify BY with state space tft C(bY) of a C-decorated 3-d topological field theory

JF La Falda 03 09 09 – p. 4/28






⊲ Cor(Y) ∈ BY



Actual solution of the sewing constraints

( infinitely many nonlinear equations in infinitely many variables ) :

⊲ Traditional approach : Find general solution to a specific small set of constraints

e.g. modular invariance of torus partition function Cor(T, τ, ∅)

= Z =`

Zi,j

´

[Z , ρχ(γ) ] = 0 for γ ∈ SL(2,Z) Zi,j ∈Z≥0 Z0,0 =1

JF La Falda 03 09 09 – p. 4/28






⊲ Cor(Y) ∈ BY



Actual solution of the sewing constraints

( infinitely many nonlinear equations in infinitely many variables ) :

⊲ Traditional approach : Find general solution to a specific small set of constraints

e.g. modular invariance of torus partition function Cor(T, τ, ∅)

giving the bulk state space Hbulk∼=L

i,j CZi,j ⊗CHi⊗CHj

JF La Falda 03 09 09 – p. 4/28






⊲ Cor(Y) ∈ BY



More recent : TFT construction :

Get one particular solution for all correlators as elements of the spaces tft C(∂MY) :

Cor(Y) = tft C(MY) 1

JF La Falda 03 09 09 – p. 4/28






⊲ Cor(Y) ∈ BY



More recent : TFT construction :

Get one particular solution for all correlators as elements of the spaces tft C(∂MY) :

Cor(Y) = tft C(MY) 1

1 ∈ C = tft C(∅)

connecting 3-manifold ∅MY−−−→ bY

JF La Falda 03 09 09 – p. 4/28

TFT construction of CFT correlators Categories and Hopf algebras in CFT

In short :

Correlator Cor(Y) as invariant of 3-manifold MY ( with embedded ribbon graph )

Input data for construction of MY :

⊲ a modular tensor category C

⊲ a symmetric special Frobenius algebra A in C

JF La Falda 03 09 09 – p. 5/28

TFT construction of CFT correlators Categories and Hopf algebras in CFT

In short :

Correlator Cor(Y) as invariant of 3-manifold MY ( with embedded ribbon graph )

Input data for construction of MY :

⊲ a modular tensor category C

⊲ a symmetric special Frobenius algebra A in C

Result : Data C and A necessary and sufficient :

Theorem : Given a simple symmetric special Frobenius algebra A in C

the TFT construction gives a solution to the sewing constraints :

bulk and boundary state spaces and family {Cor(Y) } of correlators

satisfying all constraints [ JF-Runkel-Schweigert 2002, 2005 ]

Theorem : Every solution to the sewing constraints of a ( non-degenerate ) CFT

can be obtained via the TFT construction with a simple symmetric

special Frobenius algebra A determined uniquely up to isomorphism

[ Fjelstad- JF-Runkel-Schweigert 2008 ]

JF La Falda 03 09 09 – p. 5/28

Ingredients Categories and Hopf algebras in CFT

RCFT and 3-d TFT

Some ingredients

Sample results



Coends

" #

JF La Falda 03 09 09 – p. 6/28

Modular tensor categories Categories and Hopf algebras in CFT

Rational CFT ! rational conformal vertex algebra V

=⇒ C ≃ Rep(V) a modular tensor category

⊲ abelian C-linear

⊲ semisimple

⊲ ribbon , with simple 1

⊲ finitely many simple objects Si up to isomorphism

⊲ braiding maximally non-symmetric

JF La Falda 03 09 09 – p. 6/28





⊲ semisimple




deti,j

Si

Sj 6= 0

JF La Falda 03 09 09 – p. 6/28





⊲ semisimple




deti,j

Si

Sj 6= 0

strict monoidal ; graphical notation for morphisms

JF La Falda 03 09 09 – p. 6/28

Ribbon categories Categories and Hopf algebras in CFT

Ribbon category :

⊲ monoidal : tensor product bifunctor ⊗ : C×C → C and tensor unit 1

⊲ strict monoidal

⊲ braiding

U V

cU,V

⊲ duality

U

U∨

U∨

UbU , dU

⊲ twist

U

θU

right-duality bU , dU

∨U U

U ∨U

satisfying relations like ribbons in S3

=⇒ ∨? = ?∨ ?∨∨∼= IdC ⊗ exact

JF La Falda 03 09 09 – p. 7/28

Decorated 3-d TFT Categories and Hopf algebras in CFT

3-d topological field theory = projective monoidal functor X → VectC

JF La Falda 03 09 09 – p. 8/28


3-d topological field theory = projective monoidal functor X → VectCC-decorated 3-d topological field theory tft C :

objects / morphisms of cobordism category X decorated by objects / morphisms of C

object E 7−→ vector space tft C(E)

morphism M: E→E′ 7−→ linear map tft C(M) : tft C(E)→ tft C(E′)

JF La Falda 03 09 09 – p. 8/28






⊲ Extended surface E = closed oriented compact 2-manifold with

finitely many disjoint decorated marked arcs and . . .

⊲ Extended cobordism EM−→ E′ = compact oriented 3-manifold with

embedded oriented decorated ribbon graph

or Homeomomorphim EM−→ E

JF La Falda 03 09 09 – p. 8/28






JF La Falda 03 09 09 – p. 8/28






JF La Falda 03 09 09 – p. 8/28






Decoration data from category C :

⊲ ribbons labeled by U ∈ Obj(C)

⊲ marked arcs labeled (U,±) with U ∈Obj(C)

⊲ coupons labeled by f ∈ HomC(·, ·)

(U,+) (V,−)

(W,−)

U

V

W

X

f

E

E′

JF La Falda 03 09 09 – p. 8/28

The connecting manifold Categories and Hopf algebras in CFT

Detailed prescription for MY with its embedded ribbon graph : a bit lengthy

Some aspects :

Relevant for Cor(Y) is space BY of conformal blocks on double bY

bY =“

Y × {−1, 1}”.

identif. on ∂Y

e.g. ddisk = S2

Y closed orientable =⇒ bY = +Y ⊔ −Y

JF La Falda 03 09 09 – p. 9/28

The connecting manifold Categories and Hopf algebras in CFT

Detailed prescription for MY with its embedded ribbon graph : a bit lengthy

Some aspects :

Relevant for Cor(Y) is space BY of conformal blocks on double bY

bY =“

Y × {−1, 1}”.

identif. on ∂Y

e.g. ddisk = S2

Y closed orientable =⇒ bY = +Y ⊔ −Y

Principle : No dynamical information in 3-d , just fatten the world sheet

; 3-manifold MY as interval bundle over Y modulo identification over ∂Y

∂MY = bY e.g. Mdisk = three-ball

Crucial further ingredient : Dual triangulation Γ of Y

Prescription : Cover Γ with ribbons labeled by A and morphisms . . .

JF La Falda 03 09 09 – p. 9/28

Frobenius algebras Categories and Hopf algebras in CFT

Algebra (≡ monoid ) in monoidal category C :

= = =A =“

, ,”

s.t.

Frobenius algebra : also a co algebra

= = =

with coproduct a bimodule morphism :

= =

JF La Falda 03 09 09 – p. 10/28


Algebra (≡ monoid ) in monoidal category C :

= = =A =“

, ,”

s.t.

symmetric Frobenius algebra :

=

A

A∨

for C rigid

special Frobenius algebra :

= =

6= 0

simple Frobenius algebra : simple as A-bimodule

JF La Falda 03 09 09 – p. 10/28


Examples of simple symmetric special Frobenius algebras :

⊲ A = (1, id1, id1, id1, id1, id1) for any C

⊲ A = (X∨⊗X, idX∨⊗dX⊗idX , bX , idX∨⊗bX⊗idX , dX) for any object X of C

⊲ bsl(2) A-D-E classification :

A(A) = S0 ≡ 1 A(D) = S0 ⊕ Sk A(E6) = S0 ⊕ S6

A(E7) = S0 ⊕ S8 ⊕ S16 A(E8) = S0 ⊕ S10 ⊕ S18 ⊕ S28

⊲ A(H,ω) =M

g∈H

Sg ( ‘ Schellekens algebra ’ )

Sg invertible , H ≤ {g ∈Pic(C) | θNgg = 1} , dω = ψC |H

⊲ A =M

i1,i2,...,im

(Si1 ×Si2 × . . . ×Sim ) ∈ Obj(C⊠m)

JF La Falda 03 09 09 – p. 11/28

Sample results Categories and Hopf algebras in CFT

RCFT and 3-d TFT

Some ingredients

Sample results



Coends

" #

JF La Falda 03 09 09 – p. 12/28

Sample results – CFT Categories and Hopf algebras in CFT

TFT construction =⇒ universal formulas involving the structural data of C and A

Example : Torus partition function

Cor(T) =

T×[−1,1]

=⇒ Zi,j =

i A j

A

A

A

S2×S1

JF La Falda 03 09 09 – p. 12/28




i.e. Zi,j = tft C

“

i A j

”

(S2×S1)

In particular : Recover bsl(2) A-D-E classification

JF La Falda 03 09 09 – p. 12/28




i.e. Zi,j = tft C

“

i A j

”

(S2×S1)

In particular : Recover bsl(2) A-D-E classification

Theorem :[ I:5.1 ]

The coefficients Zi,j of Cor(T) satisfy

[ Γ, Z ] = 0 for Γ∈ SL(2,Z)

and Zi,j = dimCHomA|A(Si⊗+A⊗− Sj , A) ∈ Z≥0

JF La Falda 03 09 09 – p. 12/28


Example : Klein bottle partition function

Cor(K) =A

σ

I×S1×I/∼

(r,φ)top∼ ( 1

r,−φ)bottom

=⇒ K =

A

σ

S2×I/∼

Theorem [ II:3.7 ] :

The coefficients K satisfy K = K , K + Z ∈ 2Z , |K| ≤12Z

JF La Falda 03 09 09 – p. 13/28


Example : Klein bottle partition function

Cor(K) =A

σ

I×S1×I/∼

(r,φ)top∼ ( 1

r,−φ)bottom

=⇒ K =

A

σ

S2×I/∼

Theorem [ II:3.7 ] :

The coefficients K satisfy K = K , K + Z ∈ 2Z , |K| ≤12Z

Special case : A ≃ 1 [ Felder-Fr ohlich- JF-Schweigert 2000 ]

=⇒ Zı, = δı, and K =

(

FS() if = ( Frobenius-Schur0 else indicator )

JF La Falda 03 09 09 – p. 13/28

Dictionary Φ ↔ M Categories and Hopf algebras in CFT

DICTIONARY

CFT phases ←→ symmetric special Frobenius algebras A in C

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

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

boundary fields ←→ module morphisms HomA(M ⊗U,M ′)

bulk fields ←→ bimodule morphisms HomA|A(U ⊗+A⊗−V ,A)

defect fields ←→ bimodule morphisms HomA|B(U ⊗+X ⊗−V ,X′)

simple current model ←→ Schellekens algebra

CFT on unoriented ←→ Jandl algebraworld sheet ( braided version of algebra with involution )

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

JF La Falda 03 09 09 – p. 14/28

Sample results – Frobenius algebrasCategories and Hopf algebras in CFT

Theorem :[ S:7 ]

C modular tensor category , A ∈Obj(C) symmetric special Frobenius

=⇒ exact sequence 1 → Inn(A) → Aut(A) → Pic(CA|A)

Theorem :[ B:O′ ]

C modular , A simple symmetric special Frobenius

=⇒ bimodule fusion rules G0(CA|A)⊗ZC isomorphic as C-algebra

toL

i,j∈I EndC`HomA|A(A⊗+ Ui, A⊗− Uj)

Theorem :[ III:3.6 ]

The number of Morita classes of simple symmetric special

Frobenius algebras in a modular tensor category C is finite

JF La Falda 03 09 09 – p. 15/28

Sample results – Frobenius algebrasCategories and Hopf algebras in CFT

Theorem :[ S:7 ]

C modular tensor category , A ∈Obj(C) symmetric special Frobenius

=⇒ exact sequence 1 → Inn(A) → Aut(A) → Pic(CA|A)

Theorem :[ B:O′ ]

C modular , A simple symmetric special Frobenius

=⇒ bimodule fusion rules G0(CA|A)⊗ZC isomorphic as C-algebra

toL

i,j∈I EndC`HomA|A(A⊗+ Ui, A⊗− Uj)

Theorem :[ III:3.6 ]

The number of Morita classes of simple symmetric special

Frobenius algebras in a modular tensor category C is finite

Corollary : In RCFT finitely many different torus partition functions for given V

JF La Falda 03 09 09 – p. 15/28

Verlinde-like formulas Categories and Hopf algebras in CFT

RCFT and 3-d TFT

Some ingredients

Sample results



Coends

" #

JF La Falda 03 09 09 – p. 16/28

Beyond rational CFT Categories and Hopf algebras in CFT

Challenge : Generalize as much as possible to some class of non-rational CFTs

Problem : Proper mathematical setting still largely unknown

⊲ e.g. not enough information about vertex algebra V

⊲ several different concepts of V-module in use

JF La Falda 03 09 09 – p. 16/28




In any case : desired features of CFT ; properties of V ; properties of C

e.g.⊲ Operator product expansions ; C monoidal

⊲ Monodromy of conformal blocks?; C braided

⊲ Nondegeneracy of two-point blocks ; C rigid

⊲ Scaling symmetry?; C has twist / balancing

In particular : Fusion rules G0(C)⊗Z

C

JF La Falda 03 09 09 – p. 16/28




In any case : desired features of CFT ; properties of V ; properties of C

e.g.⊲ Operator product expansions ; C monoidal

⊲ Monodromy of conformal blocks?; C braided

⊲ Nondegeneracy of two-point blocks ; C rigid

⊲ Scaling symmetry?; C has twist / balancing

In particular : Fusion rules G0(C)⊗Z

CHope : Get some ideas from Verlinde-like formulas found in specific classes of models

Dream : Generalize various features with the help of coends

JF La Falda 03 09 09 – p. 16/28

RCFT fusion rules Categories and Hopf algebras in CFT

Grothendieck group G0(C) of a modular tensor category C

inherits properties from C :

commutative unital ring [U ] ∗ [V ] = [U ⊗V ] with involution

C finite semisimple =⇒ basis˘

[Si]¯

i ∈I [Si] ∗ [Sj ] =X

i∈I

Nijk Sk

Fusion algebra F := G0(C)⊗Z

C of C

is commutative, associative, unital, evaluation at unit is involutive automorphism

=⇒ semisimple =⇒ basis {el} of idempotents

JF La Falda 03 09 09 – p. 17/28

RCFT fusion rules Categories and Hopf algebras in CFT

Grothendieck group G0(C) of a modular tensor category C

inherits properties from C :

commutative unital ring [U ] ∗ [V ] = [U ⊗V ] with involution

C finite semisimple =⇒ basis˘

[Si]¯

i ∈I [Si] ∗ [Sj ] =X

i∈I

Nijk Sk

Fusion algebra F := G0(C)⊗Z

C of C

is commutative, associative, unital, evaluation at unit is involutive automorphism

=⇒ semisimple =⇒ basis {el} of idempotents

{el} ↔˘

[Si]¯

=⇒ unitary matrix S⊗ S⊗

0l 6= 0

Diagonalization of the fusion rules : Nijk =

X

l

S⊗

il S⊗

jl S⊗∗

lk

S⊗

0l

JF La Falda 03 09 09 – p. 17/28

RCFT Verlinde formula Categories and Hopf algebras in CFT

Theorem : C modular tensor category

=⇒ S⊗ = S◦◦

[ Witten 1989 ][ Moore-Seiberg 1989 ]

[ Cardy 1989 ]

with S◦◦= S⊗0,0 s◦◦ s◦◦i,j :=

Si S

j

JF La Falda 03 09 09 – p. 18/28



=⇒ S⊗ = S◦◦


[ Cardy 1989 ]

Proof :

s◦◦i,k

s◦◦0,k

s◦◦j,k =s◦◦i,k

s◦◦0,k

j k = j i k

=X

p

X

α

pαα k

i∨ j∨

=X

p

X

α

p ki∨ j∨

α

α

=X

p

Nijps◦◦p,k

JF La Falda 03 09 09 – p. 18/28



=⇒ S⊗ = S◦◦


[ Cardy 1989 ]

Theorem : C ≃ Rep(V) for a rational conformal vertex algebra V

=⇒ S⊗ = Sχ [ Verlinde 1988 ]

[ Tsuchiya-Ueno-Yamada 1989 ][ ...... ]

[ Huang 2004 ]

with Sχ implementing τ 7→− 1τ

on characters of simple V-modules

Thus Nijk =

X

l

Sχil Sχ

jl Sχ ∗

lk

Sχ0l

JF La Falda 03 09 09 – p. 18/28

Logarithmic minimal models Categories and Hopf algebras in CFT

Verlinde-like relations also found for specific class of logarithmic minimal models L1,p

Features :

V(L1,p) ( or rather, corresponding W-algebra ) sufficiently well known

In particular : still finitely many ( 2p ) simple objects Si up to isomorphism

JF La Falda 03 09 09 – p. 19/28



Features :



More precisely : C C-linear abelian rigid monoidal, braided and finite :

⊲ finitely many isomorphism classes of simple objects

⊲ every object of finite length

⊲ every object has projective cover

JF La Falda 03 09 09 – p. 19/28



Features :



More precisely : C C-linear abelian rigid monoidal, braided and finite :

⊲ finitely many isomorphism classes of simple objects

⊲ every object of finite length

⊲ every object has projective cover

C not semisimple

; no finite-dim. SL(2,Z)-representation on characters χ of V-modules

But 3p−1 -dimensional SL(2,Z)-representation bρ on characters together with

pseudo-characters ψa(τ) = i τP

j∈IaCaj χj(τ) (Ia = linkage classes )

(C suitable matrix )

[ Feigin-Gainutdinov-Semikhatov-Tipunin 2006 ]

JF La Falda 03 09 09 – p. 19/28

Two fusion algebras Categories and Hopf algebras in CFT

Commutative unital associative extended fusion algebra bF

spanned by { [Si] , [Pi] } [ Pearce-Rasmussen-Ruelle 2008 ]

Pi = P (Si) : 2p−2 non-simples =⇒ dimC(bF) = 4p−2

Matrices bN• of structure constants of bF brought simultaneously to Jordan form

by similarity transformation : bN• = bQ bNJ•bQ−1

with entries of both bQ and bNJ• expressed through entries of bSχ = bρ

`

“

0 −1

1 0

”

´

; Verlinde-like formula [ Rasmussen 2009 ]

JF La Falda 03 09 09 – p. 20/28


Commutative unital associative extended fusion algebra bF

spanned by { [Si] , [Pi] } [ Pearce-Rasmussen-Ruelle 2008 ]

Pi = P (Si) : 2p−2 non-simples =⇒ dimC(bF) = 4p−2

Matrices bN• of structure constants of bF brought simultaneously to Jordan form

by similarity transformation : bN• = bQ bNJ•bQ−1

with entries of both bQ and bNJ• expressed through entries of bSχ = bρ

`

“

0 −1

1 0

”

´

; Verlinde-like formula [ Rasmussen 2009 ]

Fusion algebra F = G0(C)⊗Z

C spanned by { [Si] }

Matrices Ni of structure constants of F brought simultaneously to Jordan form :

Ni = QNJi Q

−1 with entries of Q and NJi expressed through entries of bSχ

; Verlinde-like formula [ Gaberdiel-Runkel 2008 ][ Pearce-Rasmussen-Ruelle 2009 ]

JF La Falda 03 09 09 – p. 20/28


Can also obtain 2p -dimensional SL(2,Z)-representation ρ

such that S := ρ`

“

0 −1

1 0

”

´

= Sχχ − SχψC with bSχ =

Sχχ Sχψ

Sψχ 0

!

; another Verlinde-like formula for F

with entries of Q and NJi expressed through entries of S


[ JF-Hwang-Semikhatov-Tipunin 2004 ]

JF La Falda 03 09 09 – p. 21/28


Can also obtain 2p -dimensional SL(2,Z)-representation ρ

such that S := ρ`

“

0 −1

1 0

”

´

= Sχχ − SχψC with bSχ =

Sχχ Sχψ

Sψχ 0

!

; another Verlinde-like formula for F

with entries of Q and NJi expressed through entries of S


[ JF-Hwang-Semikhatov-Tipunin 2004 ]

None of the explicit expressions particularly illuminating

JF La Falda 03 09 09 – p. 21/28

Ribbon Hopf algebras Categories and Hopf algebras in CFT

RCFT and 3-d TFT

Some ingredients

Sample results



Coends

" #

JF La Falda 03 09 09 – p. 22/28

Uq(sl2) Categories and Hopf algebras in CFT

Interpretation of the L1,p results :

Rep(V(L1,p)) ≃ Rep(Uq(sl2)) equivalent as abelian categories ( q=eiπ/p )


[ Nagatomo-Tsuchiya 2009 ]

Restricted quantum group Uq(sl2) :

⊲ algebra : freely generated by {E,F,H } modulo

Ep = 0 = F p K2p = 1 KEK−1 = q2 E K F K−1 = q−2 F

E F − F E =K −K−1

q − q−1dimC(Uq(sl2)) = 2p3

⊲ coalgebra : ∆(E) = 1⊗E + E⊗K ∆(F ) = K−1⊗F + F ∆(K) = K ⊗K

ε(E) = 0 = ε(F ) ε(K) = 1

⊲ Hopf algebra : antipode S(E) = −EK−1 S(F ) = −K F S(K) = K−1

⊲ reduced form (basic algebra) : C ⊕ C ⊕ hP`•=⇒⇐=•

´

/∼

i⊕p−1

∼= C8

JF La Falda 03 09 09 – p. 22/28






bρL1,p equivalent to SL(2,Z)-representation on the center Z(Uq(sl2))

obtained by composing Frobenius and Drinfeld maps between Uq(sl2) and its dual

and get ρ from bρ via multiplicative Jordan decomposition of ribbon element of Uq(sl2)


JF La Falda 03 09 09 – p. 22/28






bρL1,p equivalent to SL(2,Z)-representation on the center Z(Uq(sl2))

obtained by composing Frobenius and Drinfeld maps between Uq(sl2) and its dual

and get ρ from bρ via multiplicative Jordan decomposition of ribbon element of Uq(sl2)


Analogue of extended fusion algebra

with pseudo-characters involving specific linear maps ( not Uq(sl2)-morphisms )

Soc(Pi) ∼= Si −→ Si∼= Top(Pi) [ Gainutdinov-Tipunin 2008 ]

Warning : H = Uq(sl2) only almost quasitriangular :

monodromy matrix in H ⊗H , R-matrix in eH ⊗ eH

JF La Falda 03 09 09 – p. 22/28


Hope : Results generalize to CFT models with C ≃ H-mod

for suitable class of Hopf algebras H

Specifically : H finite-dimensional factorizable ribbon Hopf k-algebra( k algebraically closed, char(k)= 0 )

JF La Falda 03 09 09 – p. 23/28





Then already known : Verlinde-like formula for the Higman ideal of H

purely categorical : concerns S⊗! S◦◦ [ Cohen-Westreich 2008 ]

JF La Falda 03 09 09 – p. 23/28







Basic ingredient : Chains of subalgebras in the center and in the space of central forms

Z0(H) ⊆ Hig(H) ⊆ Rey(H) ⊆ Z(H)

⊲ Reynolds ideal Rey(H) = Soc(H) ∩ Z(H)

⊲ Higman ideal / projective center Hig(H) = im(τ)

⊲ Z0(H) = span of those central primitive idempotents e for which He is simple

JF La Falda 03 09 09 – p. 23/28









C0(H) ⊆ I(H) ⊆ R(H) ⊆ C(H)

⊲ C(H) = { x∈H⋆ |x ◦m=x ◦m ◦ cH,H } central forms / class functions

⊲ R(H) = span of characters of all H-modules

⊲ I(H) = span of characters of all projective H-modules

⊲ C0(H) = span of characters of all simple projective H-modules

JF La Falda 03 09 09 – p. 23/28









C0(H) ⊆ I(H) ⊆ R(H) ⊆ C(H)

any of these inclusions an equality =⇒ H semisimple

Frobenius map and Drinfeld map furnish algebra isomorphisms Z(H) ∼= C(H) etc.

For H = Uq(sl2) : dimensions 2 < p+1 < 2p < 3p− 1

JF La Falda 03 09 09 – p. 23/28









C0(H) ⊆ I(H) ⊆ R(H) ⊆ C(H)

any of these inclusions an equality =⇒ H semisimple

Frobenius map and Drinfeld map furnish algebra isomorphisms Z(H) ∼= C(H) etc.

Open problem : General prescription for finding pseudo-characters

JF La Falda 03 09 09 – p. 23/28

Coends Categories and Hopf algebras in CFT

RCFT and 3-d TFT

Some ingredients

Sample results



Coends

"

JF La Falda 03 09 09 – p. 24/28

Three-manifold invariants Categories and Hopf algebras in CFT

Modular tensor category C ; three-manifold invariants ( and tft C )

[ Reshetikhin-Turaev 1990 ]

Finite-dimensional ribbon Hopf algebra ; three-manifold invariants

[ Kauffman-Radford 1995 ]

[ Hennings 1996 ]

Certain coends in ribbon categories are Hopf algebras

and give three-manifold invariants [ Lyubashenko 1995 ]

JF La Falda 03 09 09 – p. 24/28


Dinatural transformation F ⇒B from functor F : Cop×C→D to object B ∈D :

family of morphisms

ϕX : F (X,X)→B such thatF (Y,X)

F (idY,f)−→ F (Y, Y )

F(f

∨,idX

)−→

ϕY−→

F (X,X)ϕX −→ B

commutes

for all f : X→Y

JF La Falda 03 09 09 – p. 25/28



family of morphisms


F (idY,f)−→ F (Y, Y )

F(f

∨,idX

)−→

ϕY−→

F (X,X)ϕX −→ B

commutes

for all f : X→Y

Coend (A, i) of F :

dinatural transformation

that is universal :

F (Y,X)F (idY ,f)

−→ F (Y, Y )F

(f∨,idX

)−→

iY−→ ϕY

−→

F (X,X)iX −→ A

ϕX

−→

∃!

−−−→

B

JF La Falda 03 09 09 – p. 25/28



family of morphisms


F (idY,f)−→ F (Y, Y )

F(f

∨,idX

)−→

ϕY−→

F (X,X)ϕX −→ B

commutes

for all f : X→Y

Coend (A, i) of F :

dinatural transformation

that is universal :

F (Y,X)F (idY ,f)

−→ F (Y, Y )F

(f∨,idX

)−→

iY−→ ϕY

−→

F (X,X)iX −→ A

ϕX

−→

∃!

−−−→

B

Notation : (A, i) =

Z X

F (X,X)

unique up to isomorphism ( if exists )

JF La Falda 03 09 09 – p. 25/28

The Hopf algebra∫

X∨⊗X Categories and Hopf algebras in CFT

Theorem : C finite tensor category =⇒ the coend H =RXX∨⊗X of

F : Cop×C ∋ (X,Y ) 7−→ X∨⊗Y ∈ C

exists and is a Hopf algebra in C [ Lyubashenko, Kerler 1995 ]

[ Virelizier 2006 ]

JF La Falda 03 09 09 – p. 26/28

The Hopf algebra∫



F : Cop×C ∋ (X,Y ) 7−→ X∨⊗Y ∈ C

exists and is a Hopf algebra in C

Structure morphisms :

m ◦ (iX ⊗ iY ) := iY ⊗X ◦ (γX,Y ⊗ idY ⊗X) ◦ (idX∨ ⊗ cX,Y ∨⊗Y )

η := i1

∆ ◦ iX := (iX ⊗ iX) ◦ (idX∨ ⊗ bX ⊗ idX)

ε ◦ iX := dX

S ◦ iX := (dX ⊗ iX∨ ) ◦ (idX∨ ⊗ cX∨∨,X ⊗ idX∨ ) ◦ (bX∨ ⊗ cX∨,X)

JF La Falda 03 09 09 – p. 26/28

The Hopf algebra∫



F : Cop×C ∋ (X,Y ) 7−→ X∨⊗Y ∈ C


X∨X

H

m

Y∨Y

iX iY

=

X∨X

H

Y∨Y

(Y⊗X)∨Y⊗X

H H

X∨X

∆ =

H H

X∨ X

H

η

=

H

ε

X∨ X

=

X∨ X

S

H

=

H

X∨∨ X∨

JF La Falda 03 09 09 – p. 26/28

The Hopf algebra∫



F : Cop×C ∋ (X,Y ) 7−→ X∨⊗Y ∈ C


Examples :

⊲ C semisimple : H ∼=L

i∈I S∨i ⊗Si

⊲ C ≃ H-mod for finite-dimensional ribbon Hopf algebra H :

H = H∗ with coadjoint H-action

and iX : X∨⊗X ∋ x⊗x 7−→`

h 7→ 〈x, h.x〉 )

JF La Falda 03 09 09 – p. 26/28

The Hopf algebra∫



F : Cop×C ∋ (X,Y ) 7−→ X∨⊗Y ∈ C


Examples :

⊲ C semisimple : H ∼=L

i∈I S∨i ⊗Si

⊲ C ≃ H-mod for finite-dimensional ribbon Hopf algebra H :

H = H∗ with coadjoint H-action

X∨

iX

H∗

X

:= ρX

X∨ X

H∗

JF La Falda 03 09 09 – p. 26/28

The Hopf algebra∫


H =RXX∨⊗X

Further structure :

⊲ Left integral ; 3-manifold invariants

⊲ Hopf pairing ω��

��

X∨X Y∨Y

ω

:=

X∨X Y∨Y

JF La Falda 03 09 09 – p. 27/28

The Hopf algebra∫


H =RXX∨⊗X

Further structure :


⊲ Hopf pairing ω

ω non-degenerate

=⇒ analogues of conformal blocks ( ‘ 2-d part of a 3-d TFT ’ )

e.g. projective representation of SL(2,Z) on Hom(1,H)

[ Lyubashenko 1995 ]C semisimple : Hom(1,H) = tft C(T)

and get the usual SL(2,Z)-representation on tft C(T)

JF La Falda 03 09 09 – p. 27/28

The Hopf algebra∫


H =RXX∨⊗X

Further structure :


⊲ Hopf pairing ω

ω non-degenerate

=⇒ analogues of conformal blocks ( ‘ 2-d part of a 3-d TFT ’ )

e.g. projective representation of SL(2,Z) on Hom(1,H)

[ Lyubashenko 1995 ]C semisimple : Hom(1,H) = tft C(T)

and get the usual SL(2,Z)-representation on tft C(T)

Generalized characters : Obj(C) −→ Hom(1,H)

X 7−→ iX ◦ bX

. . . . . .

JF La Falda 03 09 09 – p. 27/28

Outlook Categories and Hopf algebras in CFT

· · · · · · work in progress · · · · · ·

JF La Falda 03 09 09 – p. 28/28

Outlook Categories and Hopf algebras in CFT

· · · · · · work in progress · · · · · ·

JF La Falda 03 09 09 – p. 28/28

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