CATEGORICAL STRUCTURES IN CONFORMAL FIELD THEORY

Transcription

1 CATEGORICAL STRUCTURES IN CONFORMAL FIELD THEORY J urg hs c en Fu p.1/32

2 CATEGORICAL STRUCTURES IN CONFORMAL FIELD THEORY J urg hs c en Fu p.1/32

3 CATEGORICAL STRUCTURES IN CONFORMAL FIELD THEORY J urg hs c en Fu p.1/32

4 Plan Aim : Field-theoretic structures on a world sheet ( C F T ) categorical / combinatorial structures which are under control ( C F T ) p.2/32

5 Plan Aim : Field-theoretic structures on a world sheet ( C F T ) categorical / combinatorial structures which are under control ( C F T ) Present : Rational CFT Frobenius algebras in modular tensor categories construct correlators with tools from 3-d TFT p.2/32

6 Plan Aim : Field-theoretic structures on a world sheet ( C F T ) categorical / combinatorial structures which are under control ( C F T ) Present : Rational CFT Frobenius algebras in modular tensor categories construct correlators with tools from 3-d TFT Future : General CFT One approach : study phases of CFT and defect lines between them Bicategory P p.2/32

7 Plan Aim : Field-theoretic structures on a world sheet ( C F T ) categorical / combinatorial structures which are under control ( C F T ) Present : Rational CFT Frobenius algebras in modular tensor categories construct correlators with tools from 3-d TFT Future : General CFT One approach : study phases of CFT and defect lines between them Bicategory P Appendix : TFT construction of RCFT correlators p.2/32

8 Structures on the world sheet Geometry : World sheet = smooth compact two-manifold Y with Riemannian metric may be oriented or not (= two types of CFTs type I / II string theory ) may have non-empty boundary Field theory on the world sheet = decorations : in particular possible types of fields / state insertions at points (arcs) in Y e.g. describing quasiparticle excitations in condensed matter systems p.3/32

9 Structures on the world sheet Geometry : World sheet = smooth compact two-manifold Y with Riemannian metric may be oriented or not (= two types of CFTs type I / II string theory ) may have non-empty boundary Field theory on the world sheet = decorations : Boundary condition M on each segment of Y ( D-branes in string theory ) p.3/32

10 Structures on the world sheet Geometry : World sheet = smooth compact two-manifold Y with Riemannian metric may be oriented or not (= two types of CFTs type I / II string theory ) may have non-empty boundary Field theory on the world sheet = decorations : Boundary condition M on each segment of Y ( D-branes in string theory ) Boundary field Ψ on Y can change boundary condition M M ( open strings ) p.3/32

11 Structures on the world sheet Geometry : World sheet = smooth compact two-manifold Y may be oriented or not may have non-empty boundary with Riemannian metric Field theory on the world sheet = decorations : Boundary condition M on each segment of Y Boundary field Ψ on Y can change boundary condition M M Different regions of Y can exist in different phases A ( different full CFTs based on the same chiral CFT ) p.4/32

12 Structures on the world sheet Geometry : World sheet = smooth compact two-manifold Y may be oriented or not may have non-empty boundary with Riemannian metric Field theory on the world sheet = decorations : Boundary condition M on each segment of Y Boundary field Ψ on Y can change boundary condition M M 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 p.4/32

13 Structures on the world sheet Geometry : World sheet = smooth compact two-manifold Y may be oriented or not may have non-empty boundary with Riemannian metric Field theory on the world sheet = decorations : Boundary condition M on each segment of Y Boundary field Ψ on Y can change boundary condition M M 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 Defect field Θ can change type of defect line X X (keeping phases) X Θ X p.4/32

14 Structures on the world sheet Geometry : World sheet = smooth compact two-manifold Y may be oriented or not may have non-empty boundary with Riemannian metric Field theory on the world sheet = decorations : Boundary condition M on each segment of Y Boundary field Ψ on Y can change boundary condition M M 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 Defect field Θ can change type of defect line X X includes bulk fields ( closed strings ) and disorder fields (keeping phases) X X Θ p.4/32

15 Structures on the world sheet Geometry : World sheet = smooth compact two-manifold Y may be oriented or not may have non-empty boundary with Riemannian metric Field theory on the world sheet = decorations : Boundary condition M on each segment of Y Boundary field Ψ on Y can change boundary condition M M 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 can change type of defect line X X Concrete realization : critical limit of spin model on 2-d lattice, e.g. Ising model Boundary condition : prescribe values of outermost spin variables Defect line : change rule for interaction between neighbouring spins separated by the line ( e.g. frustration : ferromagnetic to antiferromagnetic ) p.4/32

16 Chiral and full CFT Insight : can formalize boundary conditions, defect lines and bulk / boundary / defect field insertions as nice mathematical structures can analyze these by standard methods thus can make precise statements and establish proofs as well as concretely calculate quantities of interest in specific models p.5/32

17 Chiral and full CFT Insight : can formalize boundary conditions, defect lines and bulk / boundary / defect field insertions as nice mathematical structures can analyze these by standard methods thus can make precise statements and establish proofs as well as concretely calculate quantities of interest in specific models One way to start : formalize the symmetries : chiral symmetries conformal vertex algebra V aspects of fields representation category Rep(V) p.5/32

18 Chiral and full CFT Insight : can formalize boundary conditions, defect lines and bulk / boundary / defect field insertions as nice mathematical structures can analyze these by standard methods thus can make precise statements and establish proofs as well as concretely calculate quantities of interest in specific models One way to start : formalize the symmetries : chiral symmetries conformal vertex algebra V aspects of fields representation category Rep(V) Chiral : free boson field φ in d = 2 : φ = 0 thus left- and right-movers φ ± : φ + = 0 = φ φ ± also called chiral fields p.5/32

19 Chiral and full CFT One way to proceed : concentrate on combinatorial aspects p.6/32

20 Chiral and full CFT One way to proceed : concentrate on combinatorial aspects Symmetries : forget about V, keep Rep(V) p.6/32

21 Chiral and full CFT One way to proceed : concentrate on combinatorial aspects Symmetries : forget about V, keep Rep(V) Geometry : world sheet as topological manifold ( do not specify conformal structure / metric ) p.6/32

22 Chiral and full CFT One way to proceed : concentrate on combinatorial aspects = allows for neat separation of Chiral CFT ` CFT on complex curves ingredients a conformal vertex algebra V and a class of V-modules V i chiral vertex operators and sheaves of conformal blocks Full CFT ` CFT on world sheets = real curves / conformal surfaces p.6/32

23 Chiral and full CFT One way to proceed : concentrate on combinatorial aspects = allows for neat separation of Chiral CFT ` CFT on complex curves ingredients a conformal vertex algebra V and a class of V-modules V i chiral vertex operators and sheaves of conformal blocks Full CFT ` CFT on world sheets describe fields, boundary conditions, defect lines = must in addition combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = Zi,j V i V j i,j I p.6/32

24 Chiral and full CFT One way to proceed : concentrate on combinatorial aspects = allows for neat separation of Chiral CFT ` CFT on complex curves ingredients a conformal vertex algebra V and a class of V-modules V i chiral vertex operators and sheaves of conformal blocks Full CFT ` CFT on world sheets describe fields, boundary conditions, defect lines = must in addition combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = Zi,j V i V j i,j I p.6/32

25 Chiral and full CFT One way to proceed : concentrate on combinatorial aspects = allows for neat separation of Chiral CFT ` CFT on complex curves ingredients a conformal vertex algebra V and a class of V-modules V i chiral vertex operators and sheaves of conformal blocks Full CFT ` CFT on world sheets describe fields, boundary conditions, defect lines = must in addition combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = Zi,j V i V j i,j I Traditional approach : Classify modular invariants Z `Z i,j 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) p.6/32

26 Chiral and full CFT One way to proceed : concentrate on combinatorial aspects = allows for neat separation of Chiral CFT ` CFT on complex curves ingredients a conformal vertex algebra V and a class of V-modules V i chiral vertex operators and sheaves of conformal blocks Full CFT ` CFT on world sheets describe fields, boundary conditions, defect lines = must in addition combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = Zi,j V i V j i,j I Traditional approach : Classify modular invariants Z `Z i,j I finite for RCFT e.g. A-D-E classification for sl(2)-models b [ C-I-Z 1987 ] p.6/32

27 Chiral and full CFT One way to proceed : concentrate on combinatorial aspects = allows for neat separation of Chiral CFT ` CFT on complex curves ingredients a conformal vertex algebra V and a class of V-modules V i chiral vertex operators and sheaves of conformal blocks Full CFT ` CFT on world sheets describe fields, boundary conditions, defect lines = must in addition combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = Zi,j V i V j i,j I Traditional approach : Classify modular invariants Z `Z i,j I finite for RCFT bad : no nice general tools no structural insight p.6/32

28 Chiral and full CFT One way to proceed : concentrate on combinatorial aspects = allows for neat separation of Chiral CFT ` CFT on complex curves ingredients a conformal vertex algebra V and a class of V-modules V i chiral vertex operators and sheaves of conformal blocks Full CFT ` CFT on world sheets describe fields, boundary conditions, defect lines = must in addition combine left- and right-movers M Z e.g. specify the space of bulk fields H bulk = Zi,j V i V j i,j I Traditional approach : Classify modular invariants Z `Z i,j I finite for RCFT bad : no nice general tools no structural insight worse : some solutions unphysical: do not satisfy further constraints p.6/32

29 Rational CFT Restriction for now : Rational CFT p.7/32

30 Rational CFT Restriction for now : Rational CFT rational conformal vertex algebra V p.7/32

31 Rational CFT Restriction for now : Rational CFT rational conformal vertex algebra V = C Rep(V) a modular tensor category [ Huang 2004 ] p.7/32

32 Rational CFT Restriction for now : Rational CFT rational conformal vertex algebra V = C Rep(V) a modular tensor category abelian -linear semisimple ribbon, with simple 1 finitely many simple objects U i up to isomorphism braiding maximally non-symmetric p.7/32

33 Rational CFT Restriction for now : Rational CFT rational conformal vertex algebra V = C Rep(V) a modular tensor category abelian -linear semisimple ribbon, with simple 1 finitely many simple objects U i up to isomorphism braiding maximally non-symmetric det i,j U i U j 0 strict monoidal, rigid, braided, balanced no transparent objects besides p.7/32

34 Rational CFT Restriction for now : Rational CFT rational conformal vertex algebra V = C Rep(V) a modular tensor category abelian -linear semisimple ribbon, with simple 1 finitely many simple objects U i up to isomorphism braiding maximally non-symmetric ( somewhat more restricted than Turaev s definition ) p.7/32

35 Rational CFT Restriction for now : Rational CFT rational conformal vertex algebra V = C Rep(V) a modular tensor category = C-decorated 3-d TFT functor tft C : 3-Cob C Vect assigning vector spaces to extended surfaces and linear maps to cobordisms [ Reshetikhin - Turaev 1990 ] p.8/32

36 Rational CFT Restriction for now : Rational CFT rational conformal vertex algebra V = C Rep(V) a modular tensor category Combinatorial aspects of rational chiral CFT ( CFT on complex curves ) encoded in C as an abstract category p.8/32

37 Rational CFT Restriction for now : Rational CFT rational conformal vertex algebra V = C Rep(V) a modular tensor category Combinatorial aspects of rational chiral CFT ( CFT on complex curves ) encoded in C as an abstract category More recent : Combinatorial aspects of rational full CFT ( CFT on world sheets ) encoded in C together with one additional datum : p.8/32

38 Rational CFT Restriction for now : Rational CFT rational conformal vertex algebra V = C Rep(V) a modular tensor category Combinatorial aspects of rational chiral CFT ( CFT on complex curves ) encoded in C as an abstract category More recent : Combinatorial aspects of rational full CFT ( CFT on world sheets ) encoded in C together with one additional datum : a symmetric special Frobenius algebra A in C ( more precisely: a Morita class of such algebras ) [ F-R-S 2001 ] p.8/32

39 Rational CFT Restriction for now : Rational CFT rational conformal vertex algebra V = C Rep(V) a modular tensor category Combinatorial aspects of rational chiral CFT ( CFT on complex curves ) encoded in C as an abstract category More recent : Combinatorial aspects of rational full CFT ( CFT on world sheets ) encoded in C together with one additional datum : a symmetric special Frobenius algebra A in C ( more precisely: a Morita class of such algebras ) [ F-R-S 2001 ] e.g. Z i,j (A) = dim Hom A A( U i + A U j, A) p.8/32

40 Algebras in monoidal categories Algebra ( monoid ) in C : A =,, s.t. = = = p.9/32

41 Algebras in monoidal categories Algebra ( monoid ) in C : A =,, s.t. = = = Frobenius algebra : also a co algebra = = = p.9/32

42 Algebras in monoidal categories Algebra ( monoid ) in C : A =,, s.t. = = = Frobenius algebra : also a co algebra = = = with coproduct a bimodule morphism : = = p.9/32

43 Algebras in monoidal categories Algebra ( monoid ) in C : A =,, s.t. = = = symmetric Frobenius algebra : A = for C rigid A p.9/32

44 Algebras in monoidal categories Algebra ( monoid ) in C : A =,, s.t. = = = symmetric Frobenius algebra : A = for C rigid A special Frobenius algebra : = = strongly separable p.9/32

45 RCFT and Frobenius algebras algebras A label the phases of a CFT with given C Morita equivalent algebras describe equivalent CFT phases p.10/32

46 RCFT and Frobenius algebras algebras A label the phases of a CFT with given C Morita equivalent algebras describe equivalent CFT phases Indeed : Phases of RCFT with category C of chiral sectors bicategory SSFA C of symmetric special Frobenius algebras in C p.10/32

47 RCFT and Frobenius algebras algebras A label the phases of a CFT with given C Morita equivalent algebras describe equivalent CFT phases Indeed : Phases of RCFT with category C of chiral sectors bicategory SSFA C of symmetric special Frobenius algebras in C to be discussed later on ( details still to be explored ) Instead : select a phase = a symmetric special Frobenius algebra A in C study the RCFT phase with the help of the categories C A (A-modules) and C A A (A-bimodules) and with tools from C -decorated 3-d TFT p.10/32

48 RCFT and Frobenius algebras algebras A label the phases of a CFT with given C Morita equivalent algebras describe equivalent CFT phases Indeed : Phases of RCFT with category C of chiral sectors bicategory SSFA C of symmetric special Frobenius algebras in C to be discussed later on ( details still to be explored ) Instead : select phases = symmetric special Frobenius algebras A,B,... in C study the RCFT phases with the help of the categories C A (A-modules) and C A B (A-B-bimodules) and with tools from C -decorated 3-d TFT p.10/32

49 Frobenius algebras: Sample results For A a symmetric special Frobenius algebra in a modular tensor category C : A Azumaya C A A C A A Azumaya = exact sequence 1 Inn(A) Aut(A) Pic(C) [ Van Oystaeyen - Zhang 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 p.11/32

50 Frobenius algebras: Sample results For A a symmetric special Frobenius algebra in a modular tensor category C : A Azumaya C A A C A A Azumaya = exact sequence 1 Inn(A) Aut(A) Pic(C) [ Van Oystaeyen - Zhang 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 Convenient tool : graphical presentation of morphisms p.11/32

51 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent p.12/32

52 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A Proof : in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent P P = (id d) (c 1 A,A id) (id m id) ( b ) (id d) (c 1 A,A id) (id m id) ( b ) = = (id d) (c 1 A,A id) (id m id) (id m ) ( b id id d) (c 1 A,A c 1 A,A id) ( b ) = = (id d) (c 1 d id) (id c 1 A,A A,A id id ) (id id m m id ) (id b d b) = = (id d) (c 1 A,A id) (id m id) ( b id m) ( id) = = = P (c 1 A,A c 1 A,A id) ( b ) p.12/32

53 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = p.12/32

54 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = p.12/32

55 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = = p.12/32

56 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = = p.12/32

57 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = = p.12/32

58 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = = p.12/32

59 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = = p.12/32

60 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = = p.12/32

61 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = = p.12/32

62 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = = = p.12/32

63 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = = = = p.12/32

64 Graphical proofs: An illustration Lemma [ I: 5.2 ] : For any symmetric special Frobenius algebra A in a ribbon category C the morphism P := (id d) (c 1 A,A id) (id m id) ( b ) is an idempotent = Proof : P P = = = = = = P p.12/32

65 RCFT and Frobenius algebras II algebras A label the phases of a CFT with given C Morita equivalent algebras describe equivalent CFT phases Indeed : Phases of RCFT with category C of chiral sectors bicategory SSFA C of symmetric special Frobenius algebras in C to be discussed later on Instead : select phases = symmetric special Frobenius algebras A,B,... in C study the RCFT phases with the help of the categories C A (A-modules) and C A B (A-B-bimodules) and with tools from C -decorated 3-d TFT p.13/32

66 RCFT and Frobenius algebras II algebras A label the phases of a CFT with given C Morita equivalent algebras describe equivalent CFT phases Indeed : Phases of RCFT with category C of chiral sectors bicategory SSFA C of symmetric special Frobenius algebras in C to be discussed later on Instead : select phases = symmetric special Frobenius algebras A,B,... in C study the RCFT phases with the help of the categories C A (A-modules) and C A B (A-B-bimodules) and with tools from C -decorated 3-d TFT concretely : correlators as invariants of ribbon graphs in three-manifolds p.13/32

67 CFT correlators Correlator ( correlation function, amplitude ) Cor(Y) for a world sheet Y multilinear map from appropriate product of state spaces to depending on insertion points, moduli of Y and chiral data of field insertions map from world sheets to elements of spaces of conformal blocks satisfying consistency conditions ( factorization / sewing, modular invariance ˆ= extension to stable curves / action of mapping class group ) section in appropriate sheaf of conformal blocks p.14/32

68 CFT correlators Correlator ( correlation function, amplitude ) Cor(Y) for a world sheet Y multilinear map from appropriate product of state spaces to depending on insertion points, moduli of Y and chiral data of field insertions map from world sheets to elements of spaces of conformal blocks satisfying consistency conditions ( factorization / sewing, modular invariance ˆ= extension to stable curves / action of mapping class group ) section in appropriate sheaf of conformal blocks Combinatorial part : space of conformal blocks as state space of decorated 3-d TFT tft C Cor(Y) as specific vector in this state space consistency conditions system of correlators as monoidal natural transformation p.14/32

69 CFT correlators Correlator ( correlation function, amplitude ) Cor(Y) for a world sheet Y multilinear map from appropriate product of state spaces to depending on insertion points, moduli of Y and chiral data of field insertions map from world sheets to elements of spaces of conformal blocks satisfying consistency conditions ( factorization / sewing, modular invariance ˆ= extension to stable curves / action of mapping class group ) section in appropriate sheaf of conformal blocks Combinatorial part : space of conformal blocks as state space of decorated 3-d TFT tft C Cor(Y) as specific vector in this state space consistency conditions system of correlators as monoidal natural transformation I. Runkel s talk Strategy for computing Cor(Y) : associate to Y a three-manifold M Y with embedded ribbon graph regarded as cobordism M Y : M Y use 3-d TFT to assign to M Y an element of the vector space tft C ( M Y ) : Cor(Y) = tft C (M Y ) p.14/32

70 TFT construction of CFT correlators Construction of Cor(Y) = tft C (M Y )1: slightly involved p.15/32

71 TFT construction of CFT correlators Construction of Cor(Y) = tft C (M Y )1: here : Basic ingredients of the ribbon graph Dictionary CFT C, C A, C A B ( more ingredients ) Example: torus partition function and Klein bottle partition function References p.15/32

72 TFT construction of CFT correlators Construction of Cor(Y) = tft C (M Y )1: Construction of the three-manifold M Y : connecting manifold = interval bundle over Y modulo identification over Y Y embedded as M Y Y {t=0} M Y = b Y = double of Y = orientation bundle over Y modulo identification over Y Example : Y oriented, Y = = b Y = Y Y p.15/32

73 TFT construction of CFT correlators Construction of Cor(Y) = tft C (M Y )1: Construction of the three-manifold M Y : connecting manifold = interval bundle over Y modulo identification over Y Y embedded as M Y Y {t=0} M Y = b Y = double of Y Construction of ribbon graph in M Y : cotriangulate the world sheet ( trivalent vertices ) label edges ( ribbons ) on Y\ Y by A label vertices ( coupons ) in Y\ Y by product / coproduct morphisms p.15/32

74 TFT construction of CFT correlators Construction of Cor(Y) = tft C (M Y )1: Construction of the three-manifold M Y : connecting manifold = interval bundle over Y modulo identification over Y Y embedded as M Y Y {t=0} M Y = Y b = double of Y Construction of ribbon graph in M Y : cotriangulate the world sheet ( trivalent vertices ) label edges ( ribbons ) on Y\ Y by A A A label vertices ( coupons ) in Y\ Y by product / coproduct morphisms A A A p.15/32

75 TFT construction of CFT correlators Construction of Cor(Y) = tft C (M Y )1: Construction of the three-manifold M Y : connecting manifold = interval bundle over Y modulo identification over Y Y embedded as M Y Y {t=0} M Y = Y b = double of Y Construction of ribbon graph in M Y : cotriangulate the world sheet ( trivalent vertices ) label edges ( ribbons ) on Y\ Y by A A A label vertices ( coupons ) in Y\ Y by product / coproduct morphisms A A Properties of A ( symmetric, special, Frobenius ) correlators Cor(Y) ( without boundary / without field insertions ) independent of all choices and satisfy all consistency constraints A p.15/32

76 Dictionary CFT phases symmetric special Frobenius algebras A in C chiral sectors objects U Obj(C) boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) topol. 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 ) p.16/32

77 Dictionary CFT phases symmetric special Frobenius algebras A in C chiral sectors objects U Obj(C) boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) topol. 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 ) Left module : M =, s.t. = = p.16/32

78 Dictionary CFT phases symmetric special Frobenius algebras A in C chiral sectors objects U Obj(C) boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) topol. 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 ) 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 p.16/32

79 Dictionary CFT phases symmetric special Frobenius algebras A in C chiral sectors objects U Obj(C) boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) topol. 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 p.16/32

80 Dictionary CFT phases symmetric special Frobenius algebras A in C chiral sectors objects U Obj(C) boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) topol. 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 p.16/32

81 Dictionary CFT phases symmetric special Frobenius algebras A in C chiral sectors objects U Obj(C) boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) topol. 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 ) ± -induced bimodules : Analogously : U + A V U + X V U V U X V p.16/32

82 Dictionary CFT phases symmetric special Frobenius algebras A in C chiral sectors objects U Obj(C) boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) topol. 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 p.16/32

83 Dictionary CFT phases symmetric special Frobenius algebras A in C chiral sectors objects U Obj(C) boundary conditions A-modules M Obj(C A ) boundary fields Ψ MM i module morphisms Hom A (M U i, M ) topol. 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 CFT on unoriented Jandl algebra ( braided version of algebra with involution ) world sheet p.16/32

84 The torus partition function Partition functions : correlators without field insertions p.17/32

85 The torus partition function Torus partition function T [ 1,1] Cor(T; ) = also e.g. Klein bottle partition function I S 1 I/ (r,φ) top ( 1 r, φ) bottom Cor(K; ) = A σ p.17/32

86 The torus partition function Torus partition function S 2 S 1 T [ 1,1] A Cor(T; ) = = Z i,j = A A i A j also e.g. Klein bottle partition function I S 1 I/ (r,φ) top ( 1 r, φ) bottom S 2 I/ Cor(K; ) = A σ = K j = A σ j j p.17/32

87 The torus partition function For A a symmetric special Frobenius algebra in a modular tensor category C : Theorem [ I:5.1 ] : The coefficients Z i,j of Cor(T; ) = P i,j I Z i,j χ i,t χ j, T satisfy [Γ, Z ] = 0 for Γ SL(2, Z) and Z i,j dim = Hom A A(U i + A U j, A) Z p.17/32

88 The torus partition function For A a symmetric special Frobenius algebra in a modular tensor category C : Theorem [ I:5.1 ] : The coefficients Z i,j of Cor(T; ) = P i,j I Z i,j χ i,t χ j, T satisfy [Γ, Z ] = 0 for Γ SL(2, Z) and Z i,j dim = Hom A A(U i + A U j, A) Z 0 recall : is an idempotent p.17/32

89 The torus partition function For A a symmetric special Frobenius algebra in a modular tensor category C : Theorem [ I:5.1 ] : The coefficients Z i,j of Cor(T; ) = P i,j I Z i,j χ i,t χ j, T satisfy [Γ, Z ] = 0 for Γ SL(2, Z) and Z i,j dim = Hom A A(U i + A U j, A) Z 0 A U Lemma [ I:5.2 ] : A is an idempotent A A U p.17/32

90 The torus partition function For A a symmetric special Frobenius algebra in a modular tensor category C : Theorem [ I:5.1 ] : The coefficients Z i,j of Cor(T; ) = P i,j I Z i,j χ i,t χ j, T satisfy [Γ, Z ] = 0 for Γ SL(2, Z) and Z i,j dim = Hom A A(U i + A U j, A) Z 0 Propos. [ I:5.3 ] : Z A B = Z A + Z B, e Z A B = e Z A e Z B, Z Aopp = `Z A t Theorem [ II:3.7 ] : The coefficients K j of Cor(K; ) = P j I K j χ j,t satisfy K j Z, K j = K j, 1 2 (Z jj + K j ) {0,1,..., Z jj } p.17/32

91 The torus partition function For A a symmetric special Frobenius algebra in a modular tensor category C : Theorem [ I:5.1 ] : The coefficients Z i,j of Cor(T; ) = P i,j I Z i,j χ i,t χ j, T satisfy [Γ, Z ] = 0 for Γ SL(2, Z) and Z i,j dim = Hom A A(U i + A U j, A) Z 0 Propos. [ I:5.3 ] : Z A B = Z A + Z B, e Z A B = e Z A e Z B, Z Aopp = `Z A t Special case : Theorem [ II:3.7 ] : The coefficients K j of Cor(K; ) = P j I K j χ j,t satisfy K j Z, K j = K j, 1 2 (Z jj + K j ) {0,1,..., Z jj } C-diagonal CFT A = 1 = Z i,j = δ i,j [ Felder - F - F - S 2002 ] ( ±1 if j = j K j = ( F-S indicator ) 0 else p.17/32

92 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... in CFT Proceedings ICM math.ct/ & Jens Fjelstad : V: Proof of modular invariance and factorisation Theo. Appl. Cat. 16 (2006) hep-th/ Uniqueness of open/closed rational CFT... & Jürg Fröhlich : hep-th/ Correspondences of ribbon categories Adv. Math. 199 (2006) math.ct/ Duality and defects in RCFT Nucl. Phys. B 763 (2007) hep-th/ p.18/32

93 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... in CFT Proceedings ICM math.ct/ & Jens Fjelstad : V: Proof of modular invariance and factorisation Theo. Appl. Cat. 16 (2006) hep-th/ Uniqueness of open/closed rational CFT... & Jürg Fröhlich : hep-th/ Correspondences of ribbon categories Adv. Math. 199 (2006) math.ct/ Duality and defects in RCFT Nucl. Phys. B 763 (2007) hep-th/ p.19/32

94 Non-rational CFT Problem : representation theory of non-rational vertex algebras complicated much recent progress e.g. [ Huang - Lepowsky - Zhang 2003/2006 ] but still far from having a good characterization of Rep(V) for any class of non-rational CFTs Idea : formalize aspects of general CFTs directly at combinatorial level ( forget about A, keep C A and C A B ) p.20/32

95 The bicategory SSFA C Tensor product of C = bifunctor C A C C A ( C monoidal ) Tensor product A = bifunctor C A A C A C A ( C abelian ) ( A algebra in C ) p.21/32

96 The bicategory SSFA C Tensor product of C = bifunctor C A C C A ( C monoidal ) Tensor product A = bifunctor C A A C A C A ( C abelian ) ( A algebra in C ) = C A right module category over C and left module category over C A A p.21/32

97 The bicategory SSFA C Tensor product of C = bifunctor C A C C A Tensor product A = bifunctor C A A C A C A More generally : Bifunctors C A B C B C C A C ( C A C A 1, C C 1 1 ) p.22/32

98 The bicategory SSFA C Tensor product of C = bifunctor C A C C A Tensor product A = bifunctor C A A C A C A More generally : Bifunctors C A B C B C C A C furnish horizontal composition for a bicategory with objects = algebras A in C 1-cells = bimodules Obj(C A B ) ( C small ) 2-cells = bimodule morphisms p.23/32

99 The bicategory SSFA C Tensor product of C = bifunctor C A C C A Tensor product A = bifunctor C A A C A C A More generally : Bifunctors C A B C B C C A C furnish horizontal composition for bicategory SSFA C with objects = symmetric special Frobenius algebras A in C 1-cells = bimodules Obj(C A B ) ( C small ) 2-cells = bimodule morphisms p.23/32

100 The bicategory SSFA C Tensor product of C = bifunctor C A C C A Tensor product A = bifunctor C A A C A C A More generally : Bifunctors C A B C B C C A C furnish horizontal composition for bicategory SSFA C with objects = symmetric special Frobenius algebras A in C 1-cells = bimodules Obj(C A B ) ( C small ) 2-cells = bimodule morphisms Conversely : M semisimple indecomposable right module category over modular tensor category C = M C A for an algebra A in C ( unique up to Morita equivalence, obtainable as End ) [ Ostrik 2003 ] p.23/32

101 The bicategory SSFA C Tensor product of C = bifunctor C A C C A Tensor product A = bifunctor C A A C A C A More generally : Bifunctors C A B C B C C A C furnish horizontal composition for bicategory SSFA C with objects = symmetric special Frobenius algebras A in C 1-cells = bimodules Obj(C A B ) ( C small ) 2-cells = bimodule morphisms Conversely : M semisimple indecomposable right module category over modular tensor category C = M C A for an algebra A in C ( unique up to Morita equivalence, obtainable as End ) [ Ostrik 2003 ] C A A Fun C (M, M) = category of module endofunctors of M A symmetric special Frobenius ( ˆ= some obscure property of M ) K 0 (C A A ) M Z = End `Hom A A (U i + A U j, A) i,j I recall : dim Hom A A(U i + A U j, A) = Z i,j p.23/32

102 The bicategory P C In RCFT : Bicategory P C SSFA C with objects = phases A of the CFT 1-cells = topol. defect lines Obj(C A B ) 2-cells = defect fields Boundary conditions form module categories C A C A A A C A C p.24/32

103 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Boundary conditions form module categories M A D A A M A p.25/32

104 The bicategory P In CFT expect : objects = topol. defect lines joining phases A and B morphisms = defect fields Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Boundary conditions form module categories M A D A A M A objects = boundary conditions adjacent to phase A morphisms = boundary fields p.25/32

105 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Boundary conditions form module categories M A D A A M A naturally associated to D D A A and M M A : Fun D (M, M) also expect : special phase I s.t. Fun D (M, M) D I I Rep(V) p.25/32

106 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Boundary conditions form module categories M A D A A M A D I I naturally associated to D D A A and M M A : Fun D (M, M) also expect : special phase I s.t. Fun D (M, M) D I I Rep(V) M A D A I p.25/32

107 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Boundary conditions form module categories M A D A A M A D I I naturally associated to D D A A and M M A : Fun D (M, M) also expect : special phase I s.t. Fun D (M, M) D I I Rep(V) M A D A I Equivalence of CFT phases adjunction in P: defect lines X, Y s.t. A X B Y A and X Y = = ida, Y X = = idb p.25/32

108 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Indeed : A - B - and B - C - defect lines can fuse to A - C - defect lines A B C X Y p.26/32

109 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Indeed : A - B - and B - C - defect lines can fuse to A - C - defect lines A B C A C X Y fuse X Y p.26/32

110 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Indeed : A - B - and B - C - defect lines can fuse to A - C - defect lines horizontal composition of 1-cells p.26/32

111 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Indeed : A - B - and B - C - defect lines can fuse to A - C - defect lines horizontal composition of 1-cells existence of operator products of defect fields horizontal and vertical composition of 2-cells p.26/32

112 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Indeed : A - B - and B - C - defect lines can fuse to A - C - defect lines horizontal composition of 1-cells existence of operator products of defect fields horizontal and vertical composition of 2-cells associativity of operator product expansion associativity of composition ( in particular : D A A strict monoidal ) p.26/32

113 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Indeed : A - B - and B - C - defect lines can fuse to A - C - defect lines horizontal composition of 1-cells existence of operator products of defect fields horizontal and vertical composition of 2-cells associativity of operator product expansion associativity of composition ( in particular : D A A strict monoidal ) A - B - defect lines can fuse with B - boundary conditions M A module category over D A A p.26/32

114 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields Indeed : A - B - and B - C - defect lines can fuse to A - C - defect lines horizontal composition of 1-cells existence of operator products of defect fields horizontal and vertical composition of 2-cells associativity of operator product expansion associativity of composition ( in particular : D A A strict monoidal ) A - B - defect lines can fuse with B - boundary conditions M A module category over D A A Local deformations of defect lines do not affect correlators adjunctions D A B D B A ( in particular : D A A rigid ) p.26/32

115 The bicategory P In CFT expect : Bicategory P with objects = phases A of the CFT 1-cells = topol. defect lines Obj(D A B ) 2-cells = defect fields However : most details still to be worked out so far no new insight p.27/32

116 Outlook BETTER TRY TO WORK FROM BOTH SIDES p.28/32

117 Outlook BETTER TRY TO WORK FROM BOTH SIDES p.28/32

118 Outlook BETTER TRY TO WORK FROM BOTH SIDES p.28/32

119 TFT construction of CFT correlators Construction of M Y : = connecting manifold = interval bundle over Y modulo identification over Y Y embedded as M Y Y {t=0} M Y = Y b = double of Y Construction of ribbon graph in M Y : ( no fields or defect lines, for now ) cotriangulate the world sheet ( trivalent vertices ) label edges ( ribbons ) on Y\ Y by a symmetric special Frobenius algebra A in C A A label edges on Y by A-modules M label vertices ( coupons ) in Y\ Y by product / coproduct morphisms A A label vertices on Y by representation morphisms A p.29/32

120 TFT construction of CFT correlators Prescription involves choices : Triangulation Insertion of ribbon graph fragment for edges ( two possibilities per edge ) Insertion of ribbon graph fragment for vertices ( three possibilities per vertex ) With field insertions, in addition certain local orientations p.30/32

121 TFT construction of CFT correlators Prescription involves choices : Triangulation Insertion of ribbon graph fragment for edges ( two possibilities per edge ) Insertion of ribbon graph fragment for vertices ( three possibilities per vertex ) With field insertions, in addition certain local orientations Properties of A ( symmetric, special, Frobenius ) correlators Cor(Y) = tft C (M Y )1 independent of all choices p.30/32

122 TFT construction of CFT correlators Prescription involves choices : Triangulation Insertion of ribbon graph fragment for edges ( two possibilities per edge ) Insertion of ribbon graph fragment for vertices ( three possibilities per vertex ) With field insertions, in addition certain local orientations Properties of A ( symmetric, special, Frobenius ) correlators Cor(Y) = tft C (M Y )1 independent of all choices and satisfy all consistency constraints ( without field insertions ) p.30/32

123 TFT construction of CFT correlators Prescription involves choices : Triangulation Insertion of ribbon graph fragment for edges ( two possibilities per edge ) Insertion of ribbon graph fragment for vertices ( three possibilities per vertex ) With field insertions, in addition certain local orientations Properties of A ( symmetric, special, Frobenius ) correlators Cor(Y) = tft C (M Y )1 independent of all choices and satisfy all consistency constraints Construction of ribbon graph in M Y when Y has field insertions : for each field have a coupon in ı(y) M Y labelled by a morphism in C coupons connected to the triangulation and to M Y p.30/32

124 TFT construction of CFT correlators Prescription involves choices : Triangulation Insertion of ribbon graph fragment for edges ( two possibilities per edge ) Insertion of ribbon graph fragment for vertices ( three possibilities per vertex ) With field insertions, in addition certain local orientations Properties of A ( symmetric, special, Frobenius ) correlators Cor(Y) = tft C (M Y )1 independent of all choices and satisfy all consistency constraints Construction of ribbon graph in M Y when Y has field insertions : for each field have a coupon in ı(y) M Y labelled by a morphism in C coupons connected to the triangulation and to M Y Morphisms for fields module / bimodule morphisms correlators still independent of all choices and still satisfy all consistency constraints p.30/32

125 Example: Bulk fields Bulk field Φ : special defect field : separating phase A from phase A changing the trivial defect A to the trivial defect A p.31/32

126 Example: Bulk fields Bulk field Φ : coupon Φ p.31/32

127 Example: Bulk fields Bulk field Φ : coupon connect to triangulation p.31/32

128 Example: Bulk fields Bulk field Φ : carries rep s of left and right chiral world sheet symmetries coupon connect to triangulation p.31/32

129 Example: Bulk fields Bulk field Φ : coupon connect to triangulation connect ribbons labelled by objects U, V say U i, U j ` R CFT p.31/32

130 Example: Bulk fields Bulk field Φ : coupon connect to triangulation connect ribbons labelled by objects U, V say U i, U j coupon labelled by morphism α Hom A A (U i A U j, A) p.31/32

131 Example: Bulk fields Bulk field Φ : coupon connect to triangulation connect ribbons labelled by objects U, V say U i, U j coupon labelled by morphism α Hom A A (U i + A U j, A) Hom A A (U i A U j, A) p.31/32

132 Example: Bulk fields Bulk field Φ : coupon connect to triangulation connect ribbons labelled by objects U, V say U i, U j coupon labelled by morphism α Hom A A (U i + A U j, A) i locally, double b Y looks as Φ ij j p.31/32

133 Example: Bulk fields Bulk field Φ : coupon connect to triangulation connect ribbons labelled by objects U, V say U i, U j coupon labelled by morphism α Hom A A (U i + A U j, A) i locally, double b Y looks as j p.31/32

134 Example: Bulk fields Bulk field Φ : coupon connect to triangulation connect ribbons labelled by objects U, V say U i, U j coupon labelled by morphism α Hom A A (U i + A U j, A) i connect coupon to M Y α j p.31/32

135 Example: Bulk fields Bulk field Φ : coupon connect to triangulation connect ribbons labelled by objects U, V say U i, U j coupon labelled by morphism α Hom A A (U i + A U j, A) i connect coupon to M Y α j p.31/32

136 Example: Bulk fields Bulk field Φ : coupon connect to triangulation connect ribbons labelled by objects U, V say U i, U j coupon labelled by morphism in Hom A A (U i + A U j, A) connect coupon to M Y i α j p.32/32

137 Example: Bulk fields Bulk field Φ Hom A A (U i + A U j, A) 1 2 i α j p.32/32

138 Example: Bulk fields Bulk field Φ Hom A A (U i + A U j, A) 1 2 i α j p.32/32

139 Example: Bulk fields Bulk field Φ Hom A A (U i + A U j, A) 1 2 i copr. α unit j p.32/32