tensor categories and hopf algebras in conformal … · tensor categories and hopf algebras in...
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TENSOR CATEGORIES AND HOPF ALGEBRASIN CONFORMAL FIELD THEORY
Jen Fu
hsu
r g c
JF La Falda 03 09 09 – p. 1/28
TENSOR CATEGORIES AND HOPF ALGEBRASIN CONFORMAL FIELD THEORY
J
en Fu
hsu
r g c
TENSOR CATEGORIES AND HOPF ALGEBRASIN CONFORMAL FIELD THEORY
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
Verlinde-like formulas
Ribbon Hopf algebras
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Verlinde-like formulas
Ribbon Hopf algebras
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
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
deti,j
Si
Sj 6= 0
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
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
Decorated 3-d TFT Categories and Hopf algebras in CFT
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
Decorated 3-d TFT Categories and Hopf algebras in CFT
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′)
⊲ 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
Decorated 3-d TFT Categories and Hopf algebras in CFT
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
Decorated 3-d TFT Categories and Hopf algebras in CFT
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
Decorated 3-d TFT Categories and Hopf algebras in CFT
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′)
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
Frobenius algebras Categories and Hopf algebras in CFT
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
Frobenius algebras Categories and Hopf algebras in CFT
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
Verlinde-like formulas
Ribbon Hopf algebras
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
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
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
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
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
Sample results – CFT Categories and Hopf algebras in CFT
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
Sample results – CFT Categories and Hopf algebras in CFT
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
Verlinde-like formulas
Ribbon Hopf algebras
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
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
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
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
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
RCFT Verlinde formula Categories and Hopf algebras in CFT
Theorem : C modular tensor category
=⇒ S⊗ = S◦◦
[ Witten 1989 ][ Moore-Seiberg 1989 ]
[ 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
RCFT Verlinde formula Categories and Hopf algebras in CFT
Theorem : C modular tensor category
=⇒ S⊗ = S◦◦
[ Witten 1989 ][ Moore-Seiberg 1989 ]
[ 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
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
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
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
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
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 ]
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
Two fusion algebras Categories and Hopf algebras in CFT
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
[ Feigin-Gainutdinov-Semikhatov-Tipunin 2005 ]
[ JF-Hwang-Semikhatov-Tipunin 2004 ]
JF La Falda 03 09 09 – p. 21/28
Two fusion algebras Categories and Hopf algebras in CFT
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
[ Feigin-Gainutdinov-Semikhatov-Tipunin 2005 ]
[ 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
Verlinde-like formulas
Ribbon Hopf algebras
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 )
[ Feigin-Gainutdinov-Semikhatov-Tipunin 2005 ]
[ 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
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 )
[ Feigin-Gainutdinov-Semikhatov-Tipunin 2005 ]
[ Nagatomo-Tsuchiya 2009 ]
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)
[ Feigin-Gainutdinov-Semikhatov-Tipunin 2005 ]
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 )
[ Feigin-Gainutdinov-Semikhatov-Tipunin 2005 ]
[ Nagatomo-Tsuchiya 2009 ]
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)
[ Feigin-Gainutdinov-Semikhatov-Tipunin 2005 ]
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
Ribbon Hopf algebras Categories and Hopf algebras in CFT
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
Ribbon Hopf algebras Categories and Hopf algebras in CFT
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 )
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
Ribbon Hopf algebras Categories and Hopf algebras in CFT
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 )
Then already known : Verlinde-like formula for the Higman ideal of H
purely categorical : concerns S⊗! S◦◦ [ Cohen-Westreich 2008 ]
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
Ribbon Hopf algebras Categories and Hopf algebras in CFT
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 )
Then already known : Verlinde-like formula for the Higman ideal of H
purely categorical : concerns S⊗! S◦◦ [ Cohen-Westreich 2008 ]
Basic ingredient : Chains of subalgebras in the center and in the space of central forms
Z0(H) ⊆ Hig(H) ⊆ Rey(H) ⊆ Z(H)
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
Ribbon Hopf algebras Categories and Hopf algebras in CFT
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 )
Then already known : Verlinde-like formula for the Higman ideal of H
purely categorical : concerns S⊗! S◦◦ [ Cohen-Westreich 2008 ]
Basic ingredient : Chains of subalgebras in the center and in the space of central forms
Z0(H) ⊆ Hig(H) ⊆ Rey(H) ⊆ Z(H)
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
Ribbon Hopf algebras Categories and Hopf algebras in CFT
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 )
Then already known : Verlinde-like formula for the Higman ideal of H
purely categorical : concerns S⊗! S◦◦ [ Cohen-Westreich 2008 ]
Basic ingredient : Chains of subalgebras in the center and in the space of central forms
Z0(H) ⊆ Hig(H) ⊆ Rey(H) ⊆ Z(H)
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
Verlinde-like formulas
Ribbon Hopf algebras
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
Coends Categories and Hopf algebras in CFT
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
Coends Categories and Hopf algebras in CFT
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
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
Coends Categories and Hopf algebras in CFT
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
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∫
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
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∫
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
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∫
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
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∫
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
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∫
X∨⊗X Categories and Hopf algebras in CFT
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∫
X∨⊗X Categories and Hopf algebras in CFT
H =RXX∨⊗X
Further structure :
⊲ Left integral ; 3-manifold invariants
⊲ 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∫
X∨⊗X Categories and Hopf algebras in CFT
H =RXX∨⊗X
Further structure :
⊲ Left integral ; 3-manifold invariants
⊲ 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