ALGEBRAS IN MONOIDAL CATEGORIES

Transcription

1 F CRM p. 1/24 ALGEBRAS IN MONOIDAL CATEGORIES J urg hs c en Fu

2 F CRM p. 2/24 Motivation MESSAGE : algebras in monoidal categories are natural and nice

3 F CRM p. 2/24 Motivation POSSIBLE MOTIVATIONS : Z express similarities : traditionally : Hopf algebras as generalizations of group algebras in fact : group = Hopf algebra in Set Z disclose fake peculiarities : e.g. non-associativity : z octonions = associative algebra in Z 3 2 -Vect ALBUQUERQUE-MAJID 1999 Z increase degree of familiarity : z Beilinson-Drinfeld chiral algebra = Lie algebra in some category of D-modules z vertex algebra (= B-D chiral algebra on the formal disk ) = singular commutative associative algebra in certain functor category SOIBELMAN 1997 BORCHERDS 1998, 2001

4 F CRM p. 2/24 Motivation POSSIBLE MOTIVATIONS : Z express similarities : traditionally : Hopf algebras as generalizations of group algebras in fact : group = Hopf algebra in Set Z disclose fake peculiarities : e.g. non-associativity : z octonions = associative algebra in Z 3 2 -Vect ALBUQUERQUE-MAJID 1999 Z present in applications : z coend Hopf algebras for 3-mf invariants and MCG-reps LYUBASHENKO 1995 z coalgebras for cloning operations in quantum mechanics COECKE-DUNCAN 2011 z Frobenius algebras for classifying full CFTs associated with given chiral CFT SPECIFIC MOTIVATION : Z bulk state space of a full CFT as CLAIM : JF-RUNKEL-SCHWEIGERT 2002 Frobenius algebra in Z(C) C C rev allows to construct bulk fields correlation functions on any closed world sheet

5 F CRM p. 3/24 Plan PLAN Z algebras Frobenius algebras Hopf algebras weak Hopf algebras Lie algebras Z sample results Z handle Hopf algebras Z conformal field theory Z bulk state spaces in CFT Z bulk field correlation functions Z appendix

6 F CRM p. 4/24 k-algebras Z k-algebra = vector space with multiplication and unit element Z category Vect k : z objects = vector spaces over k z morphisms = linear maps z monoidal structure : tensor product k of vector spaces with tensor unit 1 = k Z interpret z multiplication bi linear map A A A linear map m: A k A A z unit element 1 A A linear map η: k A η(c)=c1 = k-algebra = algebra object (A,m,η) Vect k

7 F CRM p. 5/24 Algebras Z associative algebra = object A + morphism m : A A A A A A

8 F CRM p. 5/24 Algebras Z associative algebra = object A + morphism m : A A A such that m (m id) = m (id m) =

9 F CRM p. 5/24 Algebras Z associative algebra = object A + morphism m : A A A Z unital algebra (A, m, η) : η : 1 A m (η id) = id = m (id η) = = 1 Z natural setting : strict monoidal categories (C,, 1)

10 F CRM p. 6/24 Coalgebras Z coassociative coalgebra (C, ) : : A A A ( id) = (id ) = Z co-unital coalgebra (C,, ε) : ε : A 1 (ε id) = id = (id ε) = = Z natural setting : strict monoidal categories (C,, 1)

11 F CRM p. 7/24 Frobenius algebras Z Algebra : = = = Z Coalgebra : = = = Z Frobenius algebra : algebra and coalgebra and coproduct a bimodule morphism : = =

12 F CRM p. 7/24 Frobenius algebras Z Algebra : = = = Z Coalgebra : = = = Z Frobenius algebra : algebra and coalgebra and coproduct a bimodule morphism : = = equivalent to existence of non-degenerate invariant form if C is rigid

13 F CRM p. 7/24 Frobenius algebras Z Frobenius algebra : algebra and coalgebra and coproduct a bimodule morphism : = = natural setting : monoidal categories (C,, 1) Z symmetric Frobenius algebra : A = A special Frobenius algebra : = = 0 natural setting : sovereign rigid monoidal categories

14 F CRM p. 8/24 Hopf algebras Z bi algebra : algebra and coalgebra and coproduct and counit algebra morphisms : = = Z Hopf algebra : bialgebra with antipode s = s s = = natural setting : braided monoidal categories

15 F CRM p. 8/24 Hopf algebras Z bi algebra : algebra and coalgebra and coproduct and counit algebra morphisms : = = Z Hopf algebra : bialgebra with antipode s = s s = = Z left integral on H : Λ Hom(1,H) s.t. m (id A Λ) = Λ ε Λ = Λ

16 F CRM p. 8/24 Hopf algebras Z bi algebra : algebra and coalgebra and coproduct and counit algebra morphisms : = = Z Hopf algebra : bialgebra with antipode s = s s = = Z right cointegral on H : λ Hom(H,1) s.t. (λ id H ) = η λ λ = λ

17 F CRM p. 9/24 NB : weak Hopf algebras Z Weak Hopf algebra (H,m,η,,ε,s) : weak antipode ε not nec. not nec. algebra morphism unital S = = = = S S = S = = S

18 F CRM p. 10/24 Lie algebras Z Lie algebra : object L with morphism l Hom C (L L,L) s.t. z antisymmetry τ = z Jacobi identity + + = 0 GUREVICH 1986 Z natural setting : symmetric monoidal additive categories (C,, 1, τ) Z examples : z Lie superalgebra = Lie algebra in SVect = Z 2 -Vect z color Lie algebra = Lie algebra in Γ-Vect (Γ abelian group with skew bicharacter )

19 F CRM p. 10/24 Lie algebras Z Lie algebra : object L with morphism l Hom C (L L,L) s.t. z antisymmetry τ = z Jacobi identity + + = 0 GUREVICH 1986 Z natural setting : symmetric monoidal additive categories (C,, 1, τ) Z examples : z Hom-Lie algebra in C ( Jacobi identity modified by an automorphism ) = Lie algebra in HC CAENEPEEL-GOYVAERTS 2011

20 F CRM p. 11/24 Lie algebras Z Lie algebra : object L with morphism l Hom C (L L,L) s.t. z antisymmetry c = = c 1 z two Jacobi identities + + = = 0 Z natural setting : braided monoidal additive categories (C,, 1, c) Z example : commutator algebras satisfying antisymmetry ZHANG-ZHANG 2004

21 F CRM p. 11/24 Lie algebras Z Lie algebra : object L with morphism l Hom C (L L,L) s.t. z antisymmetry c = = c 1 z Jacobi identity / ies (C symmetric / braided ) Z more conceptual approaches for braided C : z quantum Lie bracket related to adjoint action for Hopf algebras in C compatible with a coalgebra structure of L MAJID 1994 satisfies a generalized Jacobi identity

22 F CRM p. 11/24 Lie algebras Z Lie algebra : object L with morphism l Hom C (L L,L) s.t. z antisymmetry c = = c 1 z Jacobi identity / ies (C symmetric / braided ) Z more conceptual approaches for braided C : z quantum Lie bracket related to adjoint action for Hopf algebras in C z family of n-ary products in Γ-Vect ( Γ abelian group with bicharacter ) PAREIGIS 1997 KHARCHENKO 1998

23 F CRM p. 11/24 Lie algebras Z Lie algebra : object L with morphism l Hom C (L L,L) s.t. z antisymmetry c = = c 1 z Jacobi identity / ies (C symmetric / braided ) Z more conceptual approaches for braided C : z quantum Lie bracket related to adjoint action for Hopf algebras in C z family of n-ary products in Γ-Vect ( Γ abelian group with bicharacter ) e.g. primitive elements of a Hopf algebra in Γ-Vect e.g. derivations of an algebra in Γ-Vect reduces to color Lie algebras for skew bicharacter PAREIGIS 1997

24 F CRM p. 12/24 Lie algebras in symmetric C consider C symmetric rigid monoidal idempotent-completek-linear abelian Z FACTS : z trivial abelian Lie algebra (1, 0) z commutator algebras (A,m m τ) z Frobenius algebra (U U, id U d U id U ) Morita equivalent to (1,id 1 ) z U U = 1 R and dim(u) 0 = R inherits Lie algebra structure e.g. adjoint representation sl n (k)-mod e.g. R 2024 M 23 -mod or M 24 -mod e.g. for U = st gl(m n)-mod for m n z L simple as object ( and non-trivial ) = simple as Lie algebra z U U = 1 R and dim(u) 0 and U generating C = every V C carries natural R-action ( but no handle on all of R-mod )

25 F CRM p. 12/24 Lie algebras in symmetric C consider C symmetric rigid monoidal idempotent-completek-linear abelian Z FACTS : z trivial abelian Lie algebra (1, 0) z commutator algebras (A,m m τ) z Frobenius algebra (U U, id U d U id U ) Morita equivalent to (1,id 1 ) z U U = 1 R and dim(u) 0 = R inherits Lie algebra structure e.g. adjoint representation sl n (k)-mod e.g. R 2024 M 23 -mod or M 24 -mod e.g. for U = st gl(m n)-mod for m n z L simple as object ( and non-trivial ) = simple as Lie algebra z U U = 1 R and dim(u) 0 and U generating C = every V C carries natural R-action ( but no handle on all of R-mod ) Z desirable application : group-theoretical coefficients for Feynman diagrams CVITANOVIĆ 1976

26 F CRM p. 13/24 Sample results Z C fusion category and M semisimple indecomposable C-module category = M A-mod C for algebra A C OSTRIK 2003 Z M in addition endowed with module trace = A Frobenius SCHAUMANN 2013 Z C modular tensor category = finite number of Morita classes of simple symmetric special Frobenius algebras Z C ribbon and A commutative symmetric special Frobenius = z category A-mod loc C of local A-modules ribbon z C semisimple = A-mod loc C semisimple z C modular and A simple = A-mod loc C modular Z C rigid monoidal and A special Frobenius JF-RUNKEL-SCHWEIGERT 2004 = every M A-mod C is submodule of an induced module (A U,m id U ) Z C modular and A symmetric special Frobenius : = every X A-bimod C is sub-bimodule of a braided-induced module U + A V

27 F CRM p. 13/24 Sample results Z C fusion category and M semisimple indecomposable C-module category = M A-mod C for algebra A C OSTRIK 2003 Z M in addition endowed with module trace = A Frobenius SCHAUMANN 2013 Z C modular tensor category = finite number of Morita classes of simple symmetric special Frobenius algebras JF-RUNKEL-SCHWEIGERT 2004 Z C ribbon and A Azumaya = 1 Inn(A) Aut(A) Pic(C) exact VAN OYSTAEYEN-ZHANG 1998 Z C modular and A symmetric special Frobenius : ( A Azumaya dim C Hom A A Ui + A U j,a ) permutation matrix ij = exact sequence 1 Inn(A) Aut(A) Pic(C) Z C modular and A simple symmetric special Frobenius = bimodule fusion rules K 0 (A-bimod C ) Z C isomorphic as C-algebra to i,j I End ( C HomA A (U i + A U j,a) )

28 F CRM p. 13/24 Sample results Z C fusion category and M semisimple indecomposable C-module category = M A-mod C for algebra A C OSTRIK 2003 Z M in addition endowed with module trace = A Frobenius SCHAUMANN 2013 Z C modular tensor category = finite number of Morita classes of simple symmetric special Frobenius algebras Z H Hopf algebra with s invertible and (co)integral s.t. λ Λ k = H Frobenius with same algebra structure ε F = λ F = s = JF-RUNKEL-SCHWEIGERT 2004 PAREIGIS 1971 KADISON-STOLIN 2000 JF-SCHWEIGERT Λ Z Hopf algebras have symmetric self-braiding : SCHAUENBURG 1998 =

29 F CRM p. 14/24 Handle Hopf algebras Z example : 1-holed torus in category of 3-cobordisms with corners ( 1-morphism in bicategory ) YETTER 1995 CRANE-YETTER 1999

30 F CRM p. 14/24 Handle Hopf algebras Z example : 1-holed torus in category of 3-cobordisms with corners Z example : coend L = U D U U in any finite abelian k-linear ribbon category D z exists and carries a natural structure of a Hopf algebra in D z endowed with left integral and Hopf pairing L MAJID, LYUBASHENKO, KERLER, m L (ι U ι V ) := ι V U (γ U,V id V U ) (id U c U,V V ) L ι U := (ι U ι U ) (id U b U id U ) η L := ι 1 ε L ι U := d U s L ι U := (d U ι U ) (id U c U,U id U ) (b U c U,U) L (ι U ι V ) := (d U d V ) [ id U (c V,U c U,V id V ) ] z special case : D modular = L = i I S i S i

31 F CRM p. 14/24 Handle Hopf algebras Z example : 1-holed torus in category of 3-cobordisms with corners Z example : coend L = L U D U U L in any finite abelian k-linear ribbon category D L L L L ι U m L ι V = γ U,V c id V U c L = U U V V U U V V U U L U U L L η L = L ε L U U = U U s L = U U c c U U U U

32 F CRM p. 14/24 Handle Hopf algebras Z example : 1-holed torus in category of 3-cobordisms with corners Z example : coend L = U D U U in any finite abelian k-linear ribbon category D L c = c U U V V U U V V

33 F CRM p. 14/24 Handle Hopf algebras Z example : 1-holed torus in category of 3-cobordisms with corners Z example : coend L = U D U U in any finite abelian k-linear ribbon category D Z for D factorizable : ( e.g. D = H-mod for factorizable ribbon Hopf algebra H ) z integral two-sided z Hopf pairing non-degenerate z projective rep π g,m of Map Σ on Hom D (L g,u 1 U m ) LYUBASHENKO 1995

34 F CRM p. 14/24 Handle Hopf algebras Z example : 1-holed torus in category of 3-cobordisms with corners Z example : coend L = U D U U in any finite abelian k-linear ribbon category D Z for D factorizable : ( e.g. D = H-mod for factorizable ribbon Hopf algebra H ) z integral two-sided z Hopf pairing non-degenerate z projective rep π g,m of Map Σ on Hom D (L g,u 1 U m ) z e.g. modular S-transformation = composition with automorphism S L := (ε L id L ) Q L,L (id L Λ L ) L L L L ı L U ı L V Q L,L = c ı L U ı L V c U U V V U U V V

35 F CRM p. 14/24 Handle Hopf algebras Z example : 1-holed torus in category of 3-cobordisms with corners Z example : coend L = U D U U in any finite abelian k-linear ribbon category D Z for D factorizable : ( e.g. D = H-mod for factorizable ribbon Hopf algebra H ) z integral two-sided z Hopf pairing non-degenerate z projective rep π g,m of Map Σ on Hom D (L g,u 1 U m ) Z every V D has natural L-module structure (V,κ L V ) κ L V := (ε L id V ) Q L V L V L V ı L U Q L V := c ı L U c JF-SCHWEIGERT-STIGNER 2013 U U V U U V

36 F CRM p. 15/24 Conformal field theory 2d conformal quantum field theory ( CFT ) in a nutshell : Z chiral CFT : z chiral symmetry algebra conformal vertex algebra V ( say ) z category C of V-representations z for sufficiently nice V : C factorizable finite ribbon category assumed below z conformal blocks sheaves on moduli spaces of curves with marked points [ cf. Belkale s talk ]

37 F CRM p. 15/24 Conformal field theory 2d CFT in a nutshell : Z chiral CFT : z chiral symmetry algebra conformal vertex algebra V ( say ) z category C of V-representations z for sufficiently nice V : C factorizable finite ribbon category z conformal blocks sheaves on moduli spaces of curves with marked points Z full local CFT : z conformal world sheet possibly with boundary, possibly non-orientable z holomorphic factorization : bulk fields C C rev WITTEN 1992 z local correlation functions : specific sections in the chiral blocks for complex double of the world sheet

38 F CRM p. 15/24 Conformal field theory 2d CFT in a nutshell : Z chiral CFT : z chiral symmetry algebra conformal vertex algebra V ( say ) z category C of V-representations z for sufficiently nice V : C factorizable finite ribbon category z conformal blocks sheaves on moduli spaces of curves with marked points Z full local CFT : z conformal world sheet possibly with boundary, possibly non-orientable z holomorphic factorization : bulk fields C C rev WITTEN 1992 z local correlation functions : specific sections in the chiral blocks for complex double of the world sheet z consistent boundary conditions bulk fields Z(C) = take C factorizable : C C rev Z(C) z in particular : bulk state space F C C rev z torus partition function Z T (τ) = Virasoro character of F

39 F CRM p. 16/24 The bulk state space F Bulk state space F C C rev Z simplest situation : C modular tensor category and F charge conjugation bulk state space F = F rat. C i I S i S i Z T = i I χ ī χ i z indeed appears as part of a consistent full CFT for any MTC C FELDER-FRÖHLICH-JF-SCHWEIGERT 2000

40 F CRM p. 16/24 The bulk state space F Bulk state space F C C rev Z simplest situation : C modular tensor category and F charge conjugation bulk state space F = F rat. C i I S i S i Z T = i I χ ī χ i Z interpretation : combine every object of C with its dual ( = charge conjugate ) accounting for all relations among objects Z proper concept : coend F = U C U U C C rev

41 F CRM p. 16/24 The bulk state space F Bulk state space F C C rev Z simplest situation : C modular tensor category and F charge conjugation bulk state space F = F rat. C i I S i S i Z T = i I χ ī χ i Z interpretation : combine every object of C with its dual ( = charge conjugate ) accounting for all relations among objects Z proper concept : coend F = U C U U C C rev z object together with dinatural family of morphisms ı F U : U U F z defined by universal property z C semisimple ( thus modular ) : F = F rat. C z C = H-mod for factorizable ribbon Hopf C-algebra : F = coregular bimodule H H-bimod C C rev

42 F CRM p. 17/24 The bulk state space F Z C semisimple ( RCFT ) : z F rat. C = Z(1) full center

43 F CRM p. 17/24 The bulk state space F Z C semisimple ( RCFT ) : z any bulk state space is of the form Z(A) z Z(A) = i,j I ( Si S j for A C simple symmetric special Frobenius algebra ) zij (A) with z ij (A) = dim C ( HomA A (S i + A S j,a)) commutative cocommutative symmetric Frobenius algebra in C C rev FJELSTAD-JF-RUNKEL-SCHWEIGERT 2008

44 F CRM p. 17/24 The bulk state space F Z C semisimple ( RCFT ) : z any bulk state space is of the form Z(A) z Z(A) = i,j I Z physics terminology : ( Si S j for A C simple symmetric special Frobenius algebra ) zij (A) with z ij (A) = dim C ( HomA A (S i + A S j,a)) commutative cocommutative symmetric Frobenius algebra in C C rev z multiplication on F formalizes operator product of bulk fields z Frobenius pairing formalizes non-degeneracy of two-point correlators of bulk fields on the sphere z (co-) commutativity implements monodromy invariance of correlators z

45 F CRM p. 17/24 The bulk state space F Z C semisimple ( RCFT ) : z any bulk state space is of the form Z(A) z Z(A) = Z general case : i,j I ( Si S j for A C simple symmetric special Frobenius algebra ) zij (A) with z ij (A) = dim C ( HomA A (S i + A S j,a)) commutative cocommutative symmetric Frobenius algebra in C C rev z set m F (ı F U ıf V ) := ıf V U (γ U,V c U,V ) γ U,V : U V = (V U) η F := ı F 1 z (F,m F,η F ) is commutative associative unital algebra in C C rev JF-SCHWEIGERT-STIGNER 2013

46 F CRM p. 17/24 The bulk state space F Z C semisimple ( RCFT ) : z any bulk state space is of the form Z(A) z Z(A) = Z general case : i,j I ( Si S j for A C simple symmetric special Frobenius algebra ) zij (A) with z ij (A) = dim C ( HomA A (S i + A S j,a)) commutative cocommutative symmetric Frobenius algebra in C C rev z set m F (ı F U ıf V ) := ıf V U (γ U,V c U,V ) γ U,V : U V = (V U) η F := ı F 1 z (F,m F,η F ) is commutative associative unital algebra in C C rev Z C = H-mod ( with H not necessarily semisimple ) z also symmetric Frobenius and cocommutative JF-SCHWEIGERT-STIGNER 2012

47 F CRM p. 17/24 The bulk state space F Z C semisimple ( RCFT ) : z any bulk state space is of the form Z(A) z Z(A) = Z general case : i,j I ( Si S j for A C simple symmetric special Frobenius algebra ) zij (A) with z ij (A) = dim C ( HomA A (S i + A S j,a)) commutative cocommutative symmetric Frobenius algebra in C C rev z set m F (ı F U ıf V ) := ıf V U (γ U,V c U,V ) γ U,V : U V = (V U) η F := ı F 1 z (F,m F,η F ) is commutative associative unital algebra in C C rev H Z C = H-mod ( with H not necessarily semisimple ) H λ H H m F = η F = F = ε F = Λ H H H H

48 F CRM p. 18/24 Bulk correlation functions Z recall : handle Hopf algebra L = U D U U D recall : partial monodromy action κ L U of L on U D recall : composition with automorphism S L gives modular S-transformation

49 F CRM p. 18/24 Bulk correlation functions Z input : bulk handle Hopf algebra K = X C C rev X X C C rev input : partial monodromy action κ K X

50 F CRM p. 18/24 Bulk correlation functions Z input : bulk handle Hopf algebra K = X C C rev X X C C rev input : partial monodromy action κ K X Z further input : (F,m F,η F, F,ε F ) algebra - coalgebra in C C rev

51 F CRM p. 18/24 Bulk correlation functions Z input : bulk handle Hopf algebra K = X C C rev X X input : partial monodromy action κ K X Z further input : (F,m F,η F, F,ε F ) algebra - coalgebra Z define family of morphisms Cor g;p,q = Cor g;p,q (F): Cor 0;1,1 := id F Cor 1;1,1 := m F (ρ K F id F) (id K F ) F m F κ F K K F F Hom C C rev(k F,F)

52 F CRM p. 18/24 Bulk correlation functions Z input : bulk handle Hopf algebra K = X C C rev X X input : partial monodromy action κ K X Z further input : (F,m F,η F, F,ε F ) algebra - coalgebra Z define family of morphisms Cor g;p,q = Cor g;p,q (F): Cor 0;1,1 := id F Cor 1;1,1 := m F (ρ K F id F) (id K F ) Cor g;1,1 := Cor 1;1,1 (id K Cor g 1;1,1 ) for g>1 Cor 2;1,1 =

53 F CRM p. 18/24 Bulk correlation functions Z input : bulk handle Hopf algebra K = X C C rev X X F F F input : partial monodromy action κ K X Z further input : (F,m F,η F, F,ε F ) algebra - coalgebra F Z define family of morphisms Cor g;p,q = Cor g;p,q (F): m F Cor 0;1,1 := id F κ F K Cor 1;1,1 := m F (ρ K F id F) (id K F ) Cor g;1,1 := Cor 1;1,1 (id K Cor g 1;1,1 ) for g>1 Cor g;p,q := (p) F Cor g;1,1 ( id K g m (q) ) F Cor g;0,n = η F K K K

54 F CRM p. 18/24 Bulk correlation functions Z input : bulk handle Hopf algebra K = X C C rev X X input : partial monodromy action κ K X Z further input : (F,m F,η F, F,ε F ) algebra - coalgebra Z define family of morphisms Cor g;p,q = Cor g;p,q (F): Cor 0;1,1 := id F Cor 1;1,1 := m F (ρ K F id F) (id K F ) Cor g;1,1 := Cor 1;1,1 (id K Cor g 1;1,1 ) for g>1 Cor g;p,q := (p) F Cor g;1,1 ( id K g m (q) ) F Z goal : various desirable properties of Cor(F) from properties of F = Cor candidate for bulk correlation functions of a full CFT

55 F CRM p. 19/24 Bulk correlation functions PROPERTIES OF Cor(F) Z action of mapping class group : z F Frobenius ( and associative and coassociative ) = Cor(F) invariant under fusing move z F commutative and cocommutative ( and trivial twist ) = Cor(F) invariant under braiding move z F = Cor(F) also invariant under rotation move

56 F CRM p. 19/24 Bulk correlation functions PROPERTIES OF Cor(F) Z action of mapping class group : z F Frobenius ( and associative and coassociative ) = Cor(F) invariant under fusing move z F commutative and cocommutative ( and trivial twist ) = Cor(F) invariant under braiding move z F = Cor(F) also invariant under rotation move call F S-invariant iff Cor 1;1,0 S K = Cor 1;1,0 z F S-invariant = Cor(F) invariant under S-move z F symmetric = can exchange incoming and outgoing field insertions

57 F CRM p. 19/24 Bulk correlation functions PROPERTIES OF Cor(F) Z action of mapping class group : z F Frobenius ( and associative and coassociative ) = Cor(F) invariant under fusing move z F commutative and cocommutative ( and trivial twist ) = Cor(F) invariant under braiding move z F = Cor(F) also invariant under rotation move call F S-invariant iff Cor 1;1,0 S K = Cor 1;1,0 z F S-invariant = Cor(F) invariant under S-move z F symmetric = can exchange incoming and outgoing field insertions Z Lego-Teichmüller game : systematic description of diffeomorphisms via connected simply connected CW complex with F / B / R / S moves as 1-cells = invariance of Cor(F) under moves implies invariance of Cor(F) under whole mapping class group BAKALOV-KIRILLOV 2000 BARMEIER 2007

58 F CRM p. 19/24 Bulk correlation functions PROPERTIES OF Cor(F) Z action of mapping class group : z F Frobenius ( and associative and coassociative ) = Cor(F) invariant under fusing move z F commutative and cocommutative ( and trivial twist ) = Cor(F) invariant under braiding move z F = Cor(F) also invariant under rotation move call F S-invariant iff Cor 1;1,0 S K = Cor 1;1,0 z F S-invariant = Cor(F) invariant under S-move z F symmetric = can exchange incoming and outgoing field insertions Z STATUS : z general C : arguments straightforward but not written up z C = H-mod : F = U C U ω(u ) is S-invariant Frobenius JF-SCHWEIGERT-STIGNER 2012

59 F CRM p. 19/24 Bulk correlation functions PROPERTIES OF Cor(F) Z action of mapping class group : z F Frobenius ( and associative and coassociative ) = Cor(F) invariant under fusing move z F commutative and cocommutative ( and trivial twist ) = Cor(F) invariant under braiding move z F = Cor(F) also invariant under rotation move call F S-invariant iff Cor 1;1,0 S K = Cor 1;1,0 z F S-invariant = Cor(F) invariant under S-move z F symmetric = can exchange incoming and outgoing field insertions Z STATUS : z general C : arguments straightforward but not written up z C = H-mod : F = U C U ω(u ) is S-invariant Frobenius and Cor(F) is mapping class group invariant JF-SCHWEIGERT-STIGNER 2014 by brute force : invariance under action of set of generators of Map(Σ)

60 F CRM p. 20/24 Bulk correlation functions PROPERTIES OF Cor(F) Z behavior under sewing Σ Σ of world sheets : known : sewing / factorization relations among spaces of conformal blocks desired : sewing relations map Cor(Σ;F) to Cor(Σ ;F)

61 F CRM p. 20/24 Bulk correlation functions PROPERTIES OF Cor(F) Z behavior under sewing Σ Σ of world sheets : known : sewing / factorization relations among spaces of conformal blocks desired : sewing relations map Cor(Σ;F) to Cor(Σ ;F) Z two types of sewings : z non-handle creating : result follows from simple form of sewing relations X C C rev e.g. Hom C C rev(k g 1,X) k Hom C C rev(k g 2 X,Y) = Hom C C rev(k g 1+g 2,Y) with dinatural family f g g (id K g 2 f)

62 F CRM p. 20/24 Bulk correlation functions PROPERTIES OF Cor(F) Z behavior under sewing Σ Σ of world sheets : known : sewing / factorization relations among spaces of conformal blocks desired : sewing relations map Cor(Σ;F) to Cor(Σ ;F) Z two types of sewings : z non-handle creating : result follows from simple form of sewing relations X C C rev e.g. Hom C C rev(k g 1,X) k Hom C C rev(k g 2 X,Y) = Hom C C rev(k g 1+g 2,Y) with dinatural family f g g (id K g 2 f) z handle creating : must form coend in category of left exact functors LYUBASHENKO 1995 = technically involved

63 F CRM p. 20/24 Bulk correlation functions PROPERTIES OF Cor(F) Z behavior under sewing Σ Σ of world sheets : known : sewing / factorization relations among spaces of conformal blocks desired : sewing relations map Cor(Σ;F) to Cor(Σ ;F) Z two types of sewings : z non-handle creating : result follows from simple form of sewing relations X C C rev e.g. Hom C C rev(k g 1,X) k Hom C C rev(k g 2 X,Y) = Hom C C rev(k g 1+g 2,Y) with dinatural family f g g (id K g 2 f) z handle creating : must form coend in category of left exact functors LYUBASHENKO 1995 Z OUTLOOK : to be written up quite soon

64 THANK YOU F CRM p. 21/24

65 APPENDIX F CRM p. 22/24

66 F CRM p. 23/24 Appendix: Generators of Map Σ Z exact sequence 1 B g;n Map g;n Map g;0 1 Z thus generated by permutations of holes and Dehn twists U 1 d m a m b 1 b m 1 b m b l b k b g e m S l t j,k U j U j+1 U n ( relations not needed )

67 F CRM p. 24/24 Appendix: Coends Z given a functor G: D op D E and an object B E a dinatural transformation G B is a family of morphisms ϕ X : G(X,X) B s.t. G(Y, X) G(id Y,f) G(Y,Y) G(f,id X ) G(X, X) ϕ X B ϕ Y commutes for all f: X Y Z coend (A,ι) for G : initial object in category of dinatural transformations G : z notation : (A,ι) = X G(X,X) G(f,id X ) G(Y, X) G(X, X) z unique up to unique isomorphism ( if exists ) G(id Y,f) ι X ϕ X G(Y,Y) A ι Y ϕ Y B