TENSOR CATEGORIES IN RATIONAL AND NON-RATIONAL CFT

Transcription

1 TENSOR CATEGORIES IN RATIONAL AND NON-RATIONAL CFT J urg hs c en Fu STRASBOURG p.1/36

2 TENSOR CATEGORIES IN RATIONAL AND NON-RATIONAL CFT J urg hs c en Fu STRASBOURG p.1/36

3 TENSOR CATEGORIES IN RATIONAL AND NON-RATIONAL CFT J urg hs c en Fu STRASBOURG p.1/36

4 Plan Chiral sectors The category C Chiral vs full CFT Algebras in C Phases of R CFT A bicategory STRASBOURG p.2/36

5 Plan Chiral sectors The category C Chiral vs full CFT Algebras in C Phases of R CFT A bicategory Frobenius algebras in modular tensor categories STRASBOURG p.2/36

6 Plan Chiral sectors The category C Chiral vs full CFT Algebras in C Phases of R CFT A bicategory Frobenius algebras in modular tensor categories Nonrational / non-semisimple CFT Finite tensor categories STRASBOURG p.2/36

7 Plan Chiral sectors The category C Chiral vs full CFT Algebras in C Phases of R CFT A bicategory Frobenius algebras in modular tensor categories Nonrational / non-semisimple CFT Finite tensor categories A non-semisimple analogue of the Verlinde formula STRASBOURG p.2/36

8 Plan Chiral sectors The category C Chiral vs full CFT Algebras in C Phases of R CFT A bicategory Frobenius algebras in modular tensor categories Nonrational / non-semisimple CFT Finite tensor categories A non-semisimple analogue of the Verlinde formula Appendix : TFT construction of RCFT correlators ( knots and physics ) STRASBOURG p.2/36

9 The category of chiral sectors Basic concept in QFT : Space of states / fields carrying a representation of symmetries STRASBOURG p.3/36

10 The category of chiral sectors Basic concept in QFT : Space of states / fields carrying a representation of symmetries CFT : symmetries described by a conformal vertex algebra V STRASBOURG p.3/36

11 The category of chiral sectors Basic concept in QFT : Space of states / fields carrying a representation of symmetries CFT : Chiral symmetries described by a conformal vertex algebra V Rep(V) =: C called the category of chiral sectors STRASBOURG p.3/36

12 The category of chiral sectors Basic concept in QFT : Space of states / fields carrying a representation of symmetries CFT : Chiral symmetries described by a conformal vertex algebra V Rep(V) =: C called the category of chiral sectors Heuristic structures in CFT properties of C : STRASBOURG p.3/36

13 The category of chiral sectors Basic concept in QFT : Space of states / fields carrying a representation of symmetries CFT : Chiral symmetries described by a conformal vertex algebra V Rep(V) =: C called the category of chiral sectors Heuristic structures in CFT properties of C : Operator product tensor product C monoidal expansions of V-modules monoidal unit 1 = V Monodromy exchange symmetry of conformal blocks of intertwiners C braided Nondegeneracy of contragredient two-point blocks representation C rigid Scaling symmetry L 0 - grading C has twist / balancing STRASBOURG p.3/36

14 The category of chiral sectors Basic concept in QFT : Space of states / fields carrying a representation of symmetries CFT : Chiral symmetries described by a conformal vertex algebra V Rep(V) =: C called the category of chiral sectors Heuristic structures in CFT properties of C : Operator product tensor product C monoidal expansions of V-modules monoidal unit 1 = V Monodromy exchange symmetry of conformal blocks of intertwiners C braided Nondegeneracy of contragredient two-point blocks representation C rigid Scaling symmetry L 0 - grading C has twist / balancing Actually several different concepts of V-module available Properties of C also depend on additional technical properties of V STRASBOURG p.3/36

15 The category of chiral sectors Basic concept in QFT : Space of states / fields carrying a representation of symmetries CFT : Chiral symmetries described by a conformal vertex algebra V Rep(V) =: C called the category of chiral sectors Heuristic structures in CFT properties of C : Operator product tensor product C monoidal expansions of V-modules monoidal unit 1 = V Monodromy exchange symmetry of conformal blocks of intertwiners C braided Nondegeneracy of contragredient two-point blocks representation C rigid Scaling symmetry L 0 - grading C has twist / balancing Actually several different concepts of V-module available Properties of C also depend on additional technical properties of V Expect : To be relevant in CFT C must have most properties above STRASBOURG p.4/36

16 The category of chiral sectors Properties of Rep(V) = C : rigid balanced braided monoidal in short : C a ribbon category also : strict ( = strictified ) STRASBOURG p.5/36

17 The category of chiral sectors Properties of Rep(V) = C : rigid balanced braided monoidal in short : C a ribbon category also : strict ( = strictified ) Graphical notation : braiding isomorphisms c U,V : U V V U U V duality U U b U, d U U U twist θ U U STRASBOURG p.5/36

18 The category of chiral sectors Properties of Rep(V) = C : rigid balanced braided monoidal in short : C a ribbon category also : strict ( = strictified ) Graphical notation : braiding isomorphisms c U,V : U V V U U V duality U U b U, d U U U U U U U twist U θ U right-duality b U, d U sovereign :? =? U = U STRASBOURG p.5/36

19 The category of chiral sectors Axioms U U U U U U U U = = = = U U U U U U U U W f X g = W g X f W f U V = W U V f U V U V U V W U V W V f = V f U U = U U U V f = U V f like ribbons in S 3 U U U V U V STRASBOURG p.6/36

20 The category of chiral sectors Properties of Rep(V) = C : rigid balanced braided monoidal also : strict in short : C a ribbon category also : abelian and -linear STRASBOURG p.6/36

21 Rational CFT Rational CFT : conformal vertex algebra V STRASBOURG p.7/36

22 Rational CFT Rational CFT : conformal vertex algebra V satisfying V (n) = 0 for n < 0 V (0) = 1 V C 2 -cofinite V self-dual as a V-module every N-gradable weak V-module is fully reducible every simple V-module not isomorphic to V has > 0 STRASBOURG p.7/36

23 Rational CFT Rational CFT : conformal vertex algebra V satisfying V (n) = 0 for n < 0 V (0) = 1 unique vacuum state V C 2 -cofinite V self-dual as a V-module every N-gradable weak V-module is fully reducible every simple V-module not isomorphic to V has > 0 positive energy STRASBOURG p.7/36

24 Rational CFT Rational CFT : conformal vertex algebra V satisfying V (n) = 0 for n < 0 V (0) = 1 V C 2 -cofinite V self-dual as a V-module every N-gradable weak V-module is fully reducible every simple V-module not isomorphic to V has > 0 = finitely many simple V-modules {V i } i I up to isomorphism STRASBOURG p.7/36

25 Rational CFT Rational CFT : rational conformal vertex algebra V = C a modular tensor category [ Huang 2004 ] STRASBOURG p.8/36

26 Rational CFT Rational CFT : rational conformal vertex algebra V = C a modular tensor category abelian -linear semisimple ribbon, with simple 1 finite number of isomorphism classes of simple objects braiding maximally non-symmetric STRASBOURG p.8/36

27 Rational CFT Rational CFT : rational conformal vertex algebra V = C a modular tensor category abelian -linear semisimple ribbon, with simple 1 finite number of isomorphism classes of simple objects braiding maximally non-symmetric det i,j U i U j 0 no transparent objects besides 1 STRASBOURG p.8/36

28 Rational CFT Rational CFT : rational conformal vertex algebra V = C a modular tensor category abelian -linear semisimple ribbon, with simple 1 finite number of isomorphism classes of simple objects braiding maximally non-symmetric ( somewhat more restricted than Turaev s definition ) STRASBOURG p.8/36

29 Rational CFT Rational CFT : rational conformal vertex algebra V = C a modular tensor category abelian -linear semisimple ribbon, with simple 1 finite number of isomorphism classes of simple objects braiding maximally non-symmetric Application : C -decorated 3-d TFT [ R-T 1991 ] Arise also as : Rep(H) for H connected ribbon factorizable weak Hopf algebra with Haar integral over algebraically closed field [ N-T-V 2003 ] unitary rep s of double of connected C weak Hopf algebra [ N-T-V 2003 ] local sectors of strongly additive split finite-index net of von Neumann algebras on R [ K-L-M 2001 ] STRASBOURG p.8/36

30 Full CFT Distinguish : Chiral CFT ` CFT on a complex curve specified by a conformal vertex algebra V including its representation theory and its sheaves of conformal blocks STRASBOURG p.9/36

31 Full CFT Distinguish : Chiral CFT ` CFT on a complex curve specified by a conformal vertex algebra V including its representation theory and its sheaves of conformal blocks Full CFT ` CFT on a world sheet = conformal surface STRASBOURG p.9/36

32 Full CFT Distinguish : Chiral CFT ` CFT on a complex curve specified by a conformal vertex algebra V including its representation theory and its sheaves of conformal blocks Full CFT ` CFT on a world sheet in addition must combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = i,j I Zi,j V i Vj I finite for RCFT resp. the torus partition function Z = tr Hbulk q L 0 c 24 STRASBOURG p.9/36

33 Full CFT Distinguish : Chiral CFT ` CFT on a complex curve specified by a conformal vertex algebra V including its representation theory and its sheaves of conformal blocks Full CFT ` CFT on a world sheet in addition must combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = Early approach : Classify modular invariants i,j I Z `Z i,j Zi,j V i Vj I finite for RCFT [ Z, ρ χ (γ) ] = 0 for γ SL(2, Z) Z i,j Z 0 Z 0,0 = 1 ( V 0 V ) STRASBOURG p.9/36

34 Full CFT Distinguish : Chiral CFT ` CFT on a complex curve specified by a conformal vertex algebra V including its representation theory and its sheaves of conformal blocks Full CFT ` CFT on a world sheet in addition must combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = Early approach : Classify modular invariants i,j I Z `Z i,j Zi,j V i Vj I finite for RCFT e.g. A-D-E classification for b sl(2)-models [ C-I-Z 1987 ] STRASBOURG p.9/36

35 Full CFT Distinguish : Chiral CFT ` CFT on a complex curve specified by a conformal vertex algebra V including its representation theory and its sheaves of conformal blocks Full CFT ` CFT on a world sheet in addition must combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = Early approach : Classify modular invariants i,j I Z `Z i,j Zi,j V i Vj I finite for RCFT bad : no nice general tools no structural insight STRASBOURG p.9/36

36 Full CFT Distinguish : Chiral CFT ` CFT on a complex curve specified by a conformal vertex algebra V including its representation theory and its sheaves of conformal blocks Full CFT ` CFT on a world sheet in addition must combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = Early approach : Classify modular invariants i,j I Z `Z i,j Zi,j V i Vj I finite for RCFT bad : no nice general tools no structural insight worse : some solutions unphysical STRASBOURG p.9/36

37 Full CFT Distinguish : Chiral CFT ` CFT on a complex curve specified by a conformal vertex algebra V including its representation theory and its sheaves of conformal blocks Full CFT ` CFT on a world sheet in addition must combine left- and right-movers e.g. must specify the space of bulk fields H bulk = M i,j Z i,j V i Vj More recent : For R CFT, additional datum is a symmetric special Frobenius algebra A in C [ F-R-S 2001 ] STRASBOURG p.10/36

38 Full CFT Distinguish : Chiral CFT ` CFT on a complex curve specified by a conformal vertex algebra V including its representation theory and its sheaves of conformal blocks Full CFT ` CFT on a world sheet in addition must combine left- and right-movers e.g. must specify the space of bulk fields H bulk = M i,j Z i,j V i Vj More recent : For R CFT, additional datum is a symmetric special Frobenius algebra A in C [ F-R-S 2001 ] e.g. Z i,j (A) = dim Hom A A ( U i + A U j, A ) STRASBOURG p.10/36

39 Full CFT Distinguish : Chiral CFT ` CFT on a complex curve specified by a conformal vertex algebra V including its representation theory and its sheaves of conformal blocks Full CFT ` CFT on a world sheet in addition must combine left- and right-movers e.g. must specify the space of bulk fields H bulk = M i,j Z i,j V i Vj More recent : For R CFT, additional datum is a symmetric special Frobenius algebra A in C [ F-R-S 2001 ] e.g. Z i,j (A) = dim Hom A A ( U i + A U j, A ) More precisely : a Morita class of algebras A STRASBOURG p.10/36

40 Structures on the world sheet Basic ingredients of CFT : World sheet Y = smooth compact two-manifold with Riemannian metric may be oriented or not ( = two types of CFTs type I / II string theory ) may have non-empty boundary Boundary condition M on each segment of Y Boundary field Ψ: M M can change boundary condition STRASBOURG p.11/36

41 Structures on the world sheet Basic ingredients of CFT : World sheet Y = smooth compact two-manifold with Riemannian metric may be oriented or not ( = two types of CFTs type I / II string theory ) may have non-empty boundary Boundary condition M on each segment of Y Boundary field Ψ: M M can change boundary condition Different regions of Y can exist in different phases A ( = different full CFTs based on the same chiral CFT ) Defect line X provides separation between phases / regions A B X STRASBOURG p.11/36

42 Structures on the world sheet Basic ingredients of CFT : World sheet Y = smooth compact two-manifold with Riemannian metric may be oriented or not ( = two types of CFTs type I / II string theory ) may have non-empty boundary Boundary condition M on each segment of Y Boundary field Ψ: M M can change boundary condition Different regions of Y can exist in different phases A ( = different full CFTs based on the same chiral CFT ) Defect line X provides separation between phases / regions Defect field Θ: X X can change type of defect line (keeping phases) A X Θ B X STRASBOURG p.11/36

43 Structures on the world sheet Basic ingredients of CFT : World sheet Y = smooth compact two-manifold with Riemannian metric may be oriented or not ( = two types of CFTs type I / II string theory ) may have non-empty boundary Boundary condition M on each segment of Y Boundary field Ψ: M M can change boundary condition Different regions of Y can exist in different phases A ( = different full CFTs based on the same chiral CFT ) Defect line X provides separation between phases / regions Defect field Θ: X X can change type of defect line (keeping phases) Concrete physical realization : critical limit of spin model on 2-d lattice Boundary condition : e.g. prescribe values of outermost spin variables Defect line : e.g. change rule for interaction between neighbouring spins separated by the line ( e.g. ferromagnetic to antiferromagnetic ) ( frustration ) STRASBOURG p.11/36

44 A bicategory in CFT Interpretation : Categories M A : Categories D A B : objects = boundary conditions for phase A morphisms = boundary fields objects = defect lines between phases A and B morphisms = defect fields STRASBOURG p.12/36

45 A bicategory in CFT Interpretation : Categories M A : Categories D A B : objects = boundary conditions for phase A morphisms = boundary fields objects = defect lines between phases A and B morphisms = defect fields Defect line categories fit into a bicategory P : Obj(P) = phases A Hom P (A, B) = D A B STRASBOURG p.12/36

46 A bicategory in CFT Interpretation : Categories M A : Categories D A B : objects = boundary conditions for phase A morphisms = boundary fields objects = defect lines between phases A and B morphisms = defect fields Defect line categories fit into a bicategory P : Obj(P) = phases A Hom P (A, B) = D A B Includes also the boundary categories : have distinguished ( C -diagonal, Cardy ) CFT - phase A I s.t. D A I M A STRASBOURG p.12/36

47 A bicategory in CFT Interpretation : Categories M A : Categories D A B : objects = boundary conditions for phase A morphisms = boundary fields objects = defect lines between phases A and B morphisms = defect fields Defect line categories fit into a bicategory P : Obj(P) = phases A Hom P (A, B) = D A B Includes also the boundary categories : have distinguished ( C -diagonal, Cardy ) CFT - phase A I s.t. D A I M A related to chiral symmetries : D I I M I C Rep(V) STRASBOURG p.12/36

48 Structures on the world sheet II Ingredients : World sheet Y in phases A Boundary conditions M and boundary fields categories M A Defect lines X and defect fields categories D A B as well as : A - B - and B - C - defect lines can fuse to A - C - defect lines A B C X Y STRASBOURG p.13/36

49 Structures on the world sheet II Ingredients : World sheet Y in phases A Boundary conditions M and boundary fields categories M A Defect lines X and defect fields categories D A B as well as : A - B - and B - C - defect lines can fuse to A - C - defect lines A B C A C fuse X Y STRASBOURG p.13/36

50 Structures on the world sheet II Ingredients : World sheet Y in phases A Boundary conditions M and boundary fields categories M A Defect lines X and defect fields categories D A B as well as : A - B - and B - C - defect lines can fuse to A - C - defect lines tensor product functors B : D A B D B C D A C in particular : D A A monoidal ( but no natural notion of braiding of defect lines ) STRASBOURG p.13/36

51 Structures on the world sheet II Ingredients : World sheet Y in phases A Boundary conditions M and boundary fields categories M A Defect lines X and defect fields categories D A B as well as : A - B - and B - C - defect lines can fuse to A - C - defect lines tensor product functors B : D A B D B C D A C in particular : D A A monoidal analogously : A - B - defect lines can fuse with B - boundary conditions tensor product functors e B : D A B M B M A in particular : M A a (left-) module category over D A A STRASBOURG p.13/36

52 Structures on the world sheet II Ingredients : World sheet Y in phases A Boundary conditions M and boundary fields categories M A Defect lines X and defect fields categories D A B as well as : A - B - and B - C - defect lines can fuse to A - C - defect lines tensor product functors B : D A B D B C D A C in particular : D A A monoidal analogously : A - B - defect lines can fuse with B - boundary conditions tensor product functors e B : D A B M B M A in particular : M A a (left-) module category over D A A and Two possible orientations of boundary components M A as well as M A right- module category over D A A Local deformations of defect lines do not affect physical properties adjunctions D A B D B A in particular : D A A has dualities STRASBOURG p.13/36

53 A bicategory of algebras Equivalence of CFT phases : defect lines X, Y s.t. A X B Y A and isomorphisms X B Y = = ida & Y A X = = idb STRASBOURG p.14/36

54 A bicategory of algebras Equivalence of CFT phases : defect lines X, Y s.t. A X B Y A and isomorphisms X B Y = = ida & Y A X = = idb naturally associated to D D A A and M M A : Fun D (M, M) = category of module endofunctors of M STRASBOURG p.14/36

55 A bicategory of algebras Equivalence of CFT phases : defect lines X, Y s.t. A X B Y A and isomorphisms X B Y = = ida & Y A X = = idb naturally associated to D D A A and M M A : Fun D (M, M) = category of module endofunctors of M M left- module category over D and right- module category over Fun D (M, M) STRASBOURG p.14/36

56 A bicategory of algebras Equivalence of CFT phases : defect lines X, Y s.t. A X B Y A and isomorphisms X B Y = = ida & Y A X = = idb naturally associated to D D A A and M M A : Fun D (M, M) = category of module endofunctors of M Conjecture : Fun D (M, M) C STRASBOURG p.14/36

57 A bicategory of algebras Equivalence of CFT phases : defect lines X, Y s.t. A X B Y A and isomorphisms X B Y = = ida & Y A X = = idb naturally associated to D D A A and M M A : Fun D (M, M) = category of module endofunctors of M Conjecture : Fun D (M, M) C M (right-) module category over C Fun D (M, M) and M semisimple = there is a Morita class of algebras A in C ( obtainable as End s ) [ Ostrik 2003 ] s.t. M M A C A category of left A-modules in C M A C A category of right A-modules in C D D A A C A A category of A-A-bimodules in C STRASBOURG p.14/36

58 A bicategory of algebras Equivalence of CFT phases : defect lines X, Y s.t. A X B Y A and isomorphisms X B Y = = ida & Y A X = = idb naturally associated to D D A A and M M A : Fun D (M, M) = category of module endofunctors of M Conjecture : Fun D (M, M) C M (right-) module category over C Fun D (M, M) and M semisimple = there is a Morita class of algebras A in C ( obtainable as End s ) [ Ostrik 2003 ] s.t. M M A C A category of left A-modules in C M A C A category of right A-modules in C D D A A C A A category of A-A-bimodules in C thus Algebras A label the phases of the CFT Distinguished phase A I = monoidal unit 1 Morita equivalent algebras describe equivalent CFT phases STRASBOURG p.14/36

59 A bicategory of algebras Equivalence of CFT phases : defect lines X, Y s.t. A X B Y A and isomorphisms X B Y = = ida & Y A X = = idb naturally associated to D D A A and M M A : Fun D (M, M) = category of module endofunctors of M Conjecture : Fun D (M, M) C M (right-) module category over C Fun D (M, M) and M semisimple = there is a Morita class of algebras A in C ( obtainable as End s ) [ Ostrik 2003 ] s.t. M M A C A category of left A-modules in C M A C A category of right A-modules in C D D A A C A A category of A-A-bimodules in C Algebra ( monoid ) in C : A =,, s.t. = = = STRASBOURG p.14/36

60 A bicategory of algebras Equivalence of CFT phases : defect lines X, Y s.t. A X B Y A and isomorphisms X B Y = = ida & Y A X = = idb naturally associated to D D A A and M M A : Fun D (M, M) = category of module endofunctors of M Conjecture : Fun D (M, M) C M (right-) module category over C Fun D (M, M) and M semisimple = there is a Morita class of algebras A in C ( obtainable as End s ) [ Ostrik 2003 ] s.t. M M A C A category of left A-modules in C M A C A category of right A-modules in C D D A A C A A category of A-A-bimodules in C Left module : M =, s.t. = = STRASBOURG p.14/36

61 A bicategory of algebras Equivalence of CFT phases : defect lines X, Y s.t. A X B Y A and isomorphisms X B Y = = ida & Y A X = = idb naturally associated to D D A A and M M A : Fun D (M, M) = category of module endofunctors of M Conjecture : Fun D (M, M) C M (right-) module category over C Fun D (M, M) and M semisimple = there is a Morita class of algebras A in C ( obtainable as End s ) [ Ostrik 2003 ] s.t. M M A C A category of left A-modules in C M A C A category of right A-modules in C D D A A C A A category of A-A-bimodules in C Category of left A-modules in C : Objects M = `Ṁ, ρ M Morphisms f Hom(Ṁ, Ṅ) s.t. ρ M = f Ṁ Ṅ f ρ N STRASBOURG p.14/36

62 RCFT and Frobenius algebras Special situation : R CFT V a rational conformal vertex algebra C = Rep(V) a modular tensor category STRASBOURG p.15/36

63 RCFT and Frobenius algebras Special situation : R CFT V a rational conformal vertex algebra C = Rep(V) a modular tensor category invoking various other input from CFT : A a symmetric special Frobenius algebra STRASBOURG p.15/36

64 RCFT and Frobenius algebras Special situation : R CFT V a rational conformal vertex algebra C = Rep(V) a modular tensor category invoking various other input from CFT : A a symmetric special Frobenius algebra Frobenius algebra : also a co algebra = = = STRASBOURG p.15/36

65 RCFT and Frobenius algebras Special situation : R CFT V a rational conformal vertex algebra C = Rep(V) a modular tensor category invoking various other input from CFT : A a symmetric special Frobenius algebra Frobenius algebra : also a co algebra = = = with coproduct a bimodule morphism = = STRASBOURG p.15/36

66 RCFT and Frobenius algebras Special situation : R CFT V a rational conformal vertex algebra C = Rep(V) a modular tensor category invoking various other input from CFT : A a symmetric special Frobenius algebra symmetric Frobenius algebra : A = A STRASBOURG p.15/36

67 RCFT and Frobenius algebras Special situation : R CFT V a rational conformal vertex algebra C = Rep(V) a modular tensor category invoking various other input from CFT : A a symmetric special Frobenius algebra symmetric Frobenius algebra : A = A special Frobenius algebra : = = strongly separable STRASBOURG p.15/36

68 RCFT and Frobenius algebras RCFT with category C of chiral sectors bicategory SSFA C of symmetric special Frobenius algebras in C Relationship not yet under control e.g. concerning Fun D (M, M) C or specific properties of C and A STRASBOURG p.16/36

69 RCFT and Frobenius algebras RCFT with category C of chiral sectors bicategory SSFA C of symmetric special Frobenius algebras in C Relationship not yet under control Instead : start from C and a symmetric special Frobenius algebra A in C study the RCFT with the help of the categories C A and with tools from C -decorated 3-d TFT and C A A STRASBOURG p.16/36

70 RCFT and Frobenius algebras RCFT with category C of chiral sectors bicategory SSFA C of symmetric special Frobenius algebras in C Relationship not yet under control Instead : start from C and a symmetric special Frobenius algebra A in C study the RCFT with the help of the categories C A and with tools from C -decorated 3-d TFT and C A A = TFT construction of RCFT correlators [ F-R-S 2002 ] for details see Appendix here: Dictionary STRASBOURG p.16/36

71 Dictionary chiral sectors objects U Obj(C) C = Rep(V) CFT phases symmetric special Frobenius algebras A in C boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) defect lines A-B-bimodules X Obj(C A B ) defect fields Θ XX ij bimodule morphisms Hom A B (U i + X U j, X ) STRASBOURG p.17/36

72 Dictionary chiral sectors objects U Obj(C) C = Rep(V) CFT phases symmetric special Frobenius algebras A in C boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) defect lines A-B-bimodules X Obj(C A B ) defect fields Θ XX ij bimodule morphisms Hom A B (U i + X U j, X ) Braiding on C = -induced left A-module : `U A, (idu m) (c A,U id A ) U STRASBOURG p.17/36

73 Dictionary chiral sectors objects U Obj(C) C = Rep(V) CFT phases symmetric special Frobenius algebras A in C boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) defect lines A-B-bimodules X Obj(C A B ) defect fields Θ XX ij bimodule morphisms Hom A B (U i + X U j, X ) Braiding on C = + -induced left A-module : `U A, (idu m) (c 1 U,A id A) U STRASBOURG p.17/36

74 Dictionary chiral sectors objects U Obj(C) C = Rep(V) CFT phases symmetric special Frobenius algebras A in C boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) defect lines A-B-bimodules X Obj(C A B ) defect fields Θ XX ij bimodule morphisms Hom A B (U i + X U j, X ) Braiding on C = ± -induced bimodules : Analogously : U + A V U + X V U V U X V STRASBOURG p.17/36

75 Dictionary chiral sectors objects U Obj(C) C = Rep(V) CFT phases symmetric special Frobenius algebras A in C boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) defect lines A-B-bimodules X Obj(C A B ) defect fields Θ XX ij bimodule morphisms Hom A B (U i + X U j, X ) bulk fields Φ ij bimodule morphisms Hom A A (U i + A U j, A) i.e. bulk fields are special defect fields, attached to invisible defect lines STRASBOURG p.17/36

76 Dictionary chiral sectors objects U Obj(C) C = Rep(V) CFT phases symmetric special Frobenius algebras A in C boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) defect lines A-B-bimodules X Obj(C A B ) defect fields Θ XX ij bimodule morphisms Hom A B (U i + X U j, X ) bulk fields Φ ij bimodule morphisms Hom A A (U i + A U j, A) CFT on unoriented Jandl algebra ( braided version of algebra with involution ) world sheet simple current models Schellekens algebra A = L g G L g for G Pic(C) internal symmetries Picard group Pic(C A A ) Kramers-Wannier duality bimodules X: X A X Pic(C A A ) like dualities STRASBOURG p.17/36

77 Sample results For A a symmetric special Frobenius algebra in a modular tensor category C : A Azumaya dim HomA A`Ui +A Uj, A ij a permutation matrix A Azumaya = exact sequence 1 Inn(A) Aut(A) Pic(C) [ VO-Z 1998 ] Theorem [ S:7 ] : exact sequence 1 Inn(A) Aut(A) Pic(C A A ) Theorem [ C:4.8 ] : C rigid monoidal, A special Frobenius algebra in C = every M Obj(C A ) is a submodule of Ind A (U) for a suitable U Obj(C) Theorem [ D:4.10 ] : C modular, A simple symmetric special Frobenius algebra in C = every X Obj(C A A ) is a sub-bimodule of U + A V for suitable U, V Obj(C) Theorem [ III:3.6 ] : The number of Morita classes of simple symmetric special Frobenius algebras in a modular tensor category C is finite STRASBOURG p.18/36

78 References J F, Ingo Runkel, Christoph Schweigert : TFT construction of RCFT correlators I: Partition functions Nucl. Phys. B 646 (2002) hep-th/ II: Unoriented world sheets Nucl. Phys. B 678 (2004) hep-th/ III: Simple currents Nucl. Phys. B 694 (2004) hep-th/ IV: Structure constants and correlation functions Nucl. Phys. B 715 (2005) hep-th/ Categorification and... Proceedings ICM 2006 math.ct/ & Jens Fjelstad : V: Proof of modular invariance and factorisation hep-th/ & Jürg Fröhlich : Kramers-Wannier duality from conformal defects Phys. Rev. Lett. 93 (2004) cond-mat/ Correspondences of ribbon categories Adv. Math. 199 (2006) math.ct/ STRASBOURG p.19/36

79 Non-rational CFT To be understood : Details of the relevant vertex algebras and their representation theory Properties of the categories C, M A, D A B Properties of the relevant algebras in C STRASBOURG p.20/36

80 Non-rational CFT To be understood : Details of the relevant vertex algebras and their representation theory Properties of the categories C, M A, D A B Properties of the relevant algebras in C So far only very restricted understanding e.g. in representation theory must deal with weak / generalized modules on which L 0 can act non-semisimply [ Milas 2002, H-L-Z 2003 ] STRASBOURG p.20/36

81 Non-rational CFT To be understood : Details of the relevant vertex algebras and their representation theory Properties of the categories C, M A, D A B Properties of the relevant algebras in C Expected properties of the categories : C still monoidal ( operator product expansions ) braided ( monodromies of conformal blocks (?) ) rigid ( non-degeneracy of 2-point blocks on Riemann sphere (?) ) STRASBOURG p.20/36

82 Non-rational CFT To be understood : Details of the relevant vertex algebras and their representation theory Properties of the categories C, M A, D A B Properties of the relevant algebras in C Expected properties of the categories : C still monoidal ( operator product expansions ) braided ( monodromies of conformal blocks (?) ) rigid ( non-degeneracy of 2-point blocks on Riemann sphere (?) ) D A B still rigid monoidal ( topological defects ) M A still module category over D A B ( consistent conformal boundary conditions ) thus still bicategory P STRASBOURG p.20/36

83 Finite tensor categories To be understood : Details of the relevant vertex algebras and their representation theory Properties of the categories C, M A, D A B Properties of the relevant algebras in C Modest first step : consider models still close to rational CFT STRASBOURG p.21/36

84 Finite tensor categories To be understood : Details of the relevant vertex algebras and their representation theory Properties of the categories C, M A, D A B Properties of the relevant algebras in C Modest first step : consider models still close to rational CFT = interesting class of categories : finite tensor categories [ Etingof-Ostrik 2004 ] abelian, k-linear ( k algebraically closed for CFT : ) finite-dimensional morphism spaces all objects having finite-length Jordan-Hölder series every simple object having a projective cover finite number of isomorphism classes of simple objects rigid monoidal, with simple monoidal unit ( for C also : braided ) STRASBOURG p.21/36

85 Finite tensor categories To be understood : Details of the relevant vertex algebras and their representation theory Properties of the categories C, M A, D A B Properties of the relevant algebras in C Modest first step : consider models still close to rational CFT = interesting class of categories : finite tensor categories [ Etingof-Ostrik 2004 ] abelian, k-linear ( k algebraically closed for CFT : ) finite-dimensional morphism spaces all objects having finite-length Jordan-Hölder series every simple object having a projective cover finite number of isomorphism classes of simple objects rigid monoidal, with simple monoidal unit ( for C also : braided ) indeed seem to fit certain non-rational CFTs such as the (1, p) minimal models STRASBOURG p.21/36

86 Finite tensor categories To be understood : Details of the relevant vertex algebras and their representation theory Properties of the categories C, M A, D A B Properties of the relevant algebras in C Modest first step : consider models still close to rational CFT = interesting class of categories : finite tensor categories [ Etingof-Ostrik 2004 ] abelian, k-linear ( k algebraically closed for CFT : ) finite-dimensional morphism spaces all objects having finite-length Jordan-Hölder series every simple object having a projective cover finite number of isomorphism classes of simple objects rigid monoidal, with simple monoidal unit ( for C also : braided ) indeed seem to fit certain non-rational CFTs such as the (1, p) minimal models similar to highest weight categories of modular representations of algebraic groups STRASBOURG p.21/36

87 Finite tensor categories To be understood : Details of the relevant vertex algebras and their representation theory Properties of the categories C, M A, D A B Properties of the relevant algebras in C Modest first step : consider models still close to rational CFT = interesting class of categories : finite tensor categories Some properties : any projective object is also injective and vice versa 1 is projective iff C is semisimple if P is projective, then also P an indecomposable projective object has a unique simple subobject up to isomorphism C not necessarily sovereign, but L = L for every invertible object L simple objects are absolutely simple STRASBOURG p.22/36

88 Finite tensor categories To be understood : Details of the relevant vertex algebras and their representation theory Properties of the categories C, M A, D A B Properties of the relevant algebras in C Modest first step : consider models still close to rational CFT = interesting class of categories : finite tensor categories Some properties : any projective object is also injective and vice versa 1 is projective iff C is semisimple if P is projective, then also P an indecomposable projective object has a unique simple subobject up to isomorphism C not necessarily sovereign, but L = L for every invertible object L simple objects are absolutely simple rigidity = tensor product functor exact = ring structure on K 0 (C) : fusion product [U] [V ] := [U V ] STRASBOURG p.22/36

89 Fusion rules Grothendieck group K 0 (C) = free abelian group generated by isomorphism classes [U] modulo [U] = [V ] + [W ] for 0 V U W 0 STRASBOURG p.23/36

90 Fusion rules Grothendieck group K 0 (C) inherits properties from C: Exact = unital ring [U] [V ] := [U V ] [1] [U] = [U] Dualities = dual classes [U] [U ] = [1] +... [ U] [U] = [1] +... Braiding = commutative Finiteness = finite Z-basis [U i ] i I [U i ] [U j ] = X i I N ij k U k STRASBOURG p.23/36

91 Fusion rules Grothendieck group K 0 (C) inherits properties from C: Exact = unital ring [U] [V ] := [U V ] [1] [U] = [U] Dualities = dual classes [U] [U ] = [1] +... [ U] [U] = [1] +... Braiding = commutative Finiteness = finite Z-basis [U i ] i I [U i ] [U j ] = X i I N ij k U k Describes immediately tensor products between the simple objects U i and their indecomposable projective covers P i : M P i U j = N kj i P k k I U j P i = M while P i P j k I N jk i P k M = B k ij P k with some B k ij Z 0 k I STRASBOURG p.23/36

92 Fusion algebras in RCFT Study K 0 (C) via fusion algebra F := K 0 (C) Z STRASBOURG p.24/36

93 Fusion algebras in RCFT Study K 0 (C) via fusion algebra F := K 0 (C) Z Properties of F for R CFT : commutative associative unital evaluation at unit is involutive automorphism = semisimple STRASBOURG p.24/36

94 Fusion algebras in RCFT Study K 0 (C) via fusion algebra F := K 0 (C) Z Properties of F for R CFT : commutative associative unital evaluation at unit is involutive automorphism = semisimple = Basis {e l } of idempotents F M = l e l e l e m = δ lm e m X e l = [1] l STRASBOURG p.24/36

95 Fusion algebras in RCFT Study K 0 (C) via fusion algebra F := K 0 (C) Z Properties of F for R CFT : commutative associative unital evaluation at unit is involutive automorphism = semisimple = Basis {e l } of idempotents F M = Basis transformation : [U i ] = P l Q i,l e l l e l e l e m = δ lm e m X e l = [1] l = invertible matrix Q ( unitary up to normalization ) = structure constants N ij k = X l Q i,l Q j,l Q 1 l,k Diagonalization of the fusion rules STRASBOURG p.24/36

96 Fusion algebras in RCFT Study K 0 (C) via fusion algebra F := K 0 (C) Z Properties of F for R CFT : commutative associative unital evaluation at unit is involutive automorphism = semisimple = Basis {e l } of idempotents F M = Basis transformation : [U i ] = P l Q i,l e l l e l e l e m = δ lm e m X e l = [1] l = invertible matrix Q ( unitary up to normalization ) = structure constants N ij k = X l Q i,l Q j,l Q 1 l,k Diagonalization of the fusion rules i.e. N ij k = X l S i,l S j,l S l,k S 0,l with unitary matrix S S i,l = ξ l Q i,l ξ l := ` P i I Q i,l 2 1/2 = S 0,l 0 STRASBOURG p.24/36

97 S = S Define s i,j := (d Uj d Ui ) [id U j (c Ui,U j c Uj,U i ) id U i ] ( b Uj b Ui ) STRASBOURG p.25/36

98 S = S Define s i,j := (d Uj d Ui ) [id U j (c Ui,U j c Uj,U i ) id U i ] ( b Uj b Ui ) = U i U j STRASBOURG p.25/36

99 S = S Define s i,j := (d Uj d Ui ) [id U j (c Ui,U j c Uj,U i ) id U i ] ( b Uj b Ui ) = U i U j Then s i,k s 0,k s j,k = s i,k s 0,k j k = j i k = X p X α p α α k i j = X p X α p k i α j α = X p N ij p s p,k STRASBOURG p.25/36

100 S = S Define s i,j := (d Uj d Ui ) [id U j (c Ui,U j c Uj,U i ) id U i ] ( b Uj b Ui ) = U i U j Then s i,k s 0,k s j,k = s i,k s 0,k j k = j i k = X p X α p α α k i j = X p X α p k i α j α = X p N ij p s p,k = S = S with S := S 0,0 s STRASBOURG p.25/36

101 S = S Define s i,j := (d Uj d Ui ) [id U j (c Ui,U j c Uj,U i ) id U i ] ( b Uj b Ui ) = U i U j Then s i,k s 0,k s j,k = s i,k s 0,k j k = j i k = X p X α p α α k i j = X p X α p k i α j α = X p N ij p s p,k = S = S with S := S 0,0 s i.e. topological part of Verlinde formula STRASBOURG p.25/36

102 S = S χ Analytic / algebro-geometric part of Verlinde formula : S = S χ STRASBOURG p.26/36

103 S = S χ Analytic / algebro-geometric part of Verlinde formula : S = S χ involves non-categorical information : Characters = graded dimensions of V i : χ i (q) = X dim(v (n) i ) q n+ i c/24 n 0 ( more generally: m -point conformal blocks on the torus ) STRASBOURG p.26/36

104 S = S χ Analytic / algebro-geometric part of Verlinde formula : S = S χ involves non-categorical information : Characters = graded dimensions of V i : χ i (q) = X dim(v (n) i ) q n+ i c/24 n 0 R CFT : span i I {χ i } carries a unitary representation ρ of SL(2, Z) [ Zhu 1996 ] SL(2, Z) a b c d = γ ρ(γ) χ i (γτ,...) = X j ρ(γ) ij χ j (τ,...) S χ := ρ q = e 2πiτ γτ = aτ+b cτ+d STRASBOURG p.26/36

105 S = S χ Analytic / algebro-geometric part of Verlinde formula : S = S χ involves non-categorical information : Characters = graded dimensions of V i : χ i (q) = X dim(v (n) i ) q n+ i c/24 n 0 R CFT : span i I {χ i } carries a unitary representation ρ of SL(2, Z) [ Zhu 1996 ] SL(2, Z) a b c d = γ ρ(γ) χ i (γτ,...) = X j ρ(γ) ij χ j (τ,...) S χ := ρ q = e 2πiτ γτ = aτ+b cτ+d But in non-rational CFT : no longer SL(2, Z)-representation on span i I {χ i } = a priori no longer a matrix S χ STRASBOURG p.26/36

106 S = S χ Analytic / algebro-geometric part of Verlinde formula : S = S χ involves non-categorical information : Characters = graded dimensions of V i : χ i (q) = X dim(v (n) i ) q n+ i c/24 n 0 R CFT : span i I {χ i } carries a unitary representation ρ of SL(2, Z) [ Zhu 1996 ] SL(2, Z) a b c d = γ ρ(γ) χ i (γτ,...) = X j ρ(γ) ij χ j (τ,...) S χ := ρ q = e 2πiτ γτ = aτ+b cτ+d But in non-rational CFT : no longer SL(2, Z)-representation on span i I {χ i } = a priori no longer a matrix S χ Still : prescriptions for extracting an SL(2, Z)-representation at least when V is C 2 -cofinite STRASBOURG p.26/36

107 Fusion algebras for braided finite tensor categories Fusion algebra of a braided finite tensor category : in short : F not necessarily semisimple = not diagonalizable = not directly a matrix S but still block-diagonalizable Expected example : (1, p) minimal models STRASBOURG p.27/36

108 Fusion algebras for braided finite tensor categories Properties of F : commutative associative unital = F sum of Jacobson radical R (ideal) and semisimple part = basis {y l } = {e a } {w a,l } of idempotents e a and nilpotent elements w a,l STRASBOURG p.27/36

109 Fusion algebras for braided finite tensor categories Properties of F : commutative associative unital = F sum of Jacobson radical R (ideal) and semisimple part = basis {y l } = {e a } {w a,l } of idempotents e a and nilpotent elements w a,l F = R M a e a e a e b = δ ab e b P a e a = [1] STRASBOURG p.27/36

110 Fusion algebras for braided finite tensor categories Properties of F : commutative associative unital = F sum of Jacobson radical R (ideal) and semisimple part = basis {y l } = {e a } {w a,l } of idempotents e a and nilpotent elements w a,l F = R M a e a R = M a ν a 1 M w a,l l=1 e a e b = δ ab e b P a e a = [1] w a,l w b,l = 0 w a,l e b = δ ab w a,l for a b STRASBOURG p.27/36

111 Fusion algebras for braided finite tensor categories Properties of F : commutative associative unital = F sum of Jacobson radical R (ideal) and semisimple part = basis {y l } = {e a } {w a,l } of idempotents e a and nilpotent elements w a,l F = R M a e a R = M a ν a 1 M w a,l l=1 e a e b = δ ab e b P a e a = [1] Structure constants : `Ni k j w a,l w b,l = 0 for a b w a,l e b = δ ab w a,l = N k ij in [U i ]-basis Basis transformation: [U i ] = X l Q i,l y l STRASBOURG p.27/36

112 Fusion algebras for braided finite tensor categories Properties of F : commutative associative unital = F sum of Jacobson radical R (ideal) and semisimple part = basis {y l } = {e a } {w a,l } of idempotents e a and nilpotent elements w a,l F = R M a e a R = M a ν a 1 M w a,l l=1 e a e b = δ ab e b P a e a = [1] Structure constants : `Ni k j w a,l w b,l = 0 for a b w a,l e b = δ ab w a,l = N k ij in [U i ]-basis Basis transformation: [U i ] = X l Q i,l y l = M i := Q 1 N i Q = R reg ([U i ]) in y l -basis are block-diagonal with blocks of size ν a STRASBOURG p.27/36

113 Fusion algebras for braided finite tensor categories Properties of F : commutative associative unital = F sum of Jacobson radical R (ideal) and semisimple part = basis {y l } = {e a } {w a,l } of idempotents e a and nilpotent elements w a,l = M i := Q 1 N i Q block-diagonal Semisimple case ( all ν a = 1 ) : all M i diagonal obtain S S uniquely determined by normalization condition Q = S K with K l,m := δ lm /S 0,m STRASBOURG p.28/36

114 Fusion algebras for braided finite tensor categories Properties of F : commutative associative unital = F sum of Jacobson radical R (ideal) and semisimple part = basis {y l } = {e a } {w a,l } of idempotents e a and nilpotent elements w a,l = M i := Q 1 N i Q block-diagonal Semisimple case ( all ν a = 1 ) : all M i diagonal obtain S S uniquely determined by normalization condition Q = S K with K l,m := δ lm /S 0,m General case : again rewrite Q =: S K = N i = S `KM i K 1 S 1 STRASBOURG p.28/36

115 Fusion algebras for braided finite tensor categories Properties of F : commutative associative unital = F sum of Jacobson radical R (ideal) and semisimple part = basis {y l } = {e a } {w a,l } of idempotents e a and nilpotent elements w a,l = M i := Q 1 N i Q block-diagonal Semisimple case ( all ν a = 1 ) : all M i diagonal obtain S S uniquely determined by normalization condition Q = S K with K l,m := δ lm /S 0,m General case : again rewrite Q =: S K = N i = S `KM i K 1 S 1 i.e. N i = S bs i S 1 with S S bs i KM i K 1 STRASBOURG p.28/36

116 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ Obtain suitably normalized S from block-diagonalisation with blocks of matching size Insert S χ for S S in N i = S ` Si b S 1 STRASBOURG p.29/36

117 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ involving only characters of simple modules ` since 0 U W V 0 = χ W = χ U + χ V STRASBOURG p.29/36

118 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ Three approaches : SL(2, Z)-representation on space of zero-point blocks on the torus spanned by {χ i } together with pseudo-trace functions [ Miyamoto 2002 ] linear combinations of characters with coefficients in no distinguished basis [τ] STRASBOURG p.29/36

119 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ Three approaches : SL(2, Z)-representation on space of zero-point blocks on the torus spanned by {χ i } together with pseudo-trace functions [ Miyamoto 2002 ] Perform a contraction procedure [ F-G-S-T 2005 ] to obtain an SL(2, Z)-representation of the desired dimension STRASBOURG p.29/36

120 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ Three approaches : SL(2, Z)-representation on space of zero-point blocks on the torus spanned by {χ i } together with pseudo-trace functions [ Miyamoto 2002 ] Perform a contraction procedure [ F-G-S-T 2005 ] to obtain an SL(2, Z)-representation of the desired dimension Find appropriate restricted quantum group U q and contract [ F-G-S-T 2005 ] SL(2, Z) -representation on the center of U q [ L, M, K 1995 ] STRASBOURG p.29/36

121 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ Three approaches : SL(2, Z)-representation on space of zero-point blocks on the torus spanned by {χ i } together with pseudo-trace functions [ Miyamoto 2002 ] Perform a contraction procedure [ F-G-S-T 2005 ] to obtain an SL(2, Z)-representation of the desired dimension Find appropriate restricted quantum group U q and contract [ F-G-S-T 2005 ] SL(2, Z) -representation on the center of U q [ L, M, K 1995 ] Separate matrix-valued automorphy factor [ F-H-S-T 2004 ] from modular transformations of characters analogous to scalar automorphy factor ζ 1 c,d (cτ+d) 1/2 e iπcν2 /(cτ+d) making ϑ(τ, ν) a scalar under Γ 1,2 SL(2, Z) STRASBOURG p.29/36

122 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ Three approaches : SL(2, Z)-representation on space of zero-point blocks on the torus spanned by {χ i } together with pseudo-trace functions [ Miyamoto 2002 ] Perform a contraction procedure [ F-G-S-T 2005 ] to obtain an SL(2, Z)-representation of the desired dimension Find appropriate restricted quantum group U q and contract [ F-G-S-T 2005 ] SL(2, Z) -representation on the center of U q [ L, M, K 1995 ] Separate matrix-valued automorphy factor [ F-H-S-T 2004 ] from modular transformations of characters Methods work and give identical results for the (1, p) minimal models STRASBOURG p.29/36

123 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ Obtain suitably normalized S from block-diagonalisation with blocks of matching size STRASBOURG p.30/36

124 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ Obtain suitably normalized S from block-diagonalisation with blocks of matching size Prescription : Choose a block-diagonal K = L a K a X m such that S 0,m K m,l = Q 0l ` {z } {z } ν 1 ν 2 Block structure the same as present in S χ ( and in exact sequences of V-modules ) Each block K a depending only on S 0,l y l {e a, w a,1,..., w a,νa } STRASBOURG p.30/36

125 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ Obtain suitably normalized S from block-diagonalisation with blocks of matching size Prescription : Choose a block-diagonal K = L a K a X m such that S 0,m K m,l = Q 0l ` {z } {z } ν 1 ν 2 Block structure the same as present in S χ ( and in exact sequences of V-modules ) Each block K a depending only on S 0,l y l {e a, w a,1,..., w a,νa } leaves ν a 1 normalizations of w a,l and (ν a 1) 2 further parameters to be adjusted ad hoc STRASBOURG p.30/36

126 A generalized Verlinde relation Strategy : Extract S χ := ρ` from V via a finite-dimensional SL(2, Z)-representation ρ Obtain suitably normalized S from block-diagonalisation with blocks of matching size Insert S χ for S S in N i = S ` bs i S 1 STRASBOURG p.30/36