DEFECT LINES IN CONFORMAL FIELD THEORY

Transcription

1 F Wien p. 1/23 DEFECT LINES IN CONFORML FIELD THEORY J urg hs c en Fu MEMORIL CONFERENCE FOR MIMILIN KREUZER JUNE 2011

2 F Wien p. 2/23 Plan Structures on the world sheet Why defect lines / defect fields? ulk fields as defect fields Working with defect fields classifying algebra for defects ppendix : Defect partition functions

3 F Wien p. 3/23 Defect lines and defect fields Defect lines and defect fields Motivation ulk fields as defect fields Working with defect fields The classifying algebra The defect partition function

4 F Wien p. 3/23 Structures on World sheet : smooth d -dimensional manifold may come with various additional structure : oundary dim d 1

5 F Wien p. 3/23 Structures on World sheet : smooth d -dimensional manifold may come with various additional structure : oundary dim d 1 Inner boundaries / domain walls / defects dim d 1 Phases separated by domain walls dim d Intersections of domain walls dim d Insertion points for fields / local defects dim 0

6 F Wien p. 3/23 Structures on World sheet : smooth d -dimensional manifold may come with various additional structure : oundary dim d 1 Inner boundaries / domain walls / defects dim d 1 Phases separated by domain walls dim d Intersections of domain walls dim d Insertion points for fields / local defects dim 0 Note : applies also to moduli spaces of theories Note : suggests d -categorical structure ( indeed works for d = 2 CFT )

7 F Wien p. 4/23 Structures on the world sheet World sheet of a CFT : conformal surface, comes with Phases CFT, CFT... Defect lines separating phases

8 F Wien p. 4/23 Structures on the world sheet World sheet of a CFT : conformal surface, comes with Phases CFT, CFT... Defect lines separating phases CFT CFT CFTC

9 F Wien p. 4/23 Structures on the world sheet World sheet of a CFT : conformal surface, comes with Phases CFT, CFT... Defect lines separating phases CFT CFT CFTC Defect fields changing the type of defect line Defect fields joining defect lines

10 F Wien p. 4/23 Structures on the world sheet World sheet of a CFT : conformal surface, comes with Phases CFT, CFT... Defect lines separating phases CFT CFT CFTC Defect fields changing the type of defect line Defect fields joining defect lines C Z α β Y

11 F Wien p. 5/23 Why defect lines / fields? Defect lines and defect fields Motivation ulk fields as defect fields Working with defect fields The classifying algebra The defect partition function

12 F Wien p. 5/23 Why defect lines / fields? Why defect lines? world sheet can contain regions with different phases present in concrete models ( e.g. antiferromagnetic frustration line in Ising model ) present in potential applications natural ingredient in a rep theoretic analysis of RCFT may be regarded as a kind of internal two-sided boundaries ( folding trick )

13 F Wien p. 5/23 Why defect lines / fields? Why defect lines? world sheet can contain regions with different phases present in concrete models ( e.g. antiferromagnetic frustration line in Ising model ) present in potential applications natural ingredient in a rep theoretic analysis of RCFT may be regarded as a kind of internal two-sided boundaries ( folding trick ) Why defect fields? can create / destroy defect lines ( disorder fields ) can join defect lines can change type of a defect line

14 F Wien p. 5/23 Why defect lines / fields? Why defect lines? world sheet can contain regions with different phases present in concrete models ( e.g. antiferromagnetic frustration line in Ising model ) present in potential applications natural ingredient in a rep theoretic analysis of RCFT may be regarded as a kind of internal two-sided boundaries ( folding trick ) Why defect fields? can create / destroy defect lines ( disorder fields ) can join defect lines can change type of a defect line generalize bulk fields help to understand bulk state state / torus partition function

15 F Wien p. 6/23 Torus partition functions Torus partition function of a RCFT : Z = Z(τ) = P i,j I Z ij χ i(τ) χ j (τ) χ i = character of irrep H i (i I) of the chiral symmetry algebra V τ = conformal structure of the torus Counts bulk states : Z = χ Hbulk H bulk = L i,j I Z ij H i H j Different bulk state spaces for the same chiral RCFT different phases

16 F Wien p. 6/23 Torus partition functions Torus partition function of a RCFT : Z = Z(τ) = P i,j I Z ij χ i(τ) χ j (τ) Necessary conditions : Z ij Z ( dimensions ) Some examples : Z 00 = 1 ( unique vacuum ) Z(τ+1) = Z(τ) = Z( 1 ) ( same conformal structure ) τ Z ij = δ i,j charge conjugation for all RCFT Z ij = δ i,j diagonal for most RCFT -D-E classification for sl(2) WZW and classifications for other simple g Picard ( simple current ) constructions Galois constructions non-product solutions for sl(2) n [ JF-Klemm-Schmidt-Verstegen 1992 ] [ JF-Kreuzer 1994 ]

17 F Wien p. 6/23 Torus partition functions Torus partition function of a RCFT : Z = Z(τ) = P i,j I Z ij χ i(τ) χ j (τ) Necessary conditions : Z ij Z ( dimensions ) Z 00 = 1 ( unique vacuum ) Z(τ+1) = Z(τ) = Z( 1 ) τ ( same conformal structure ) Note : not sufficient search also reveals unphysical solutions sl(5) 5 [ Schellekens-Yankielowicz 1990 ] so(p 1 p 2 q) 2 [ JF-Schellekens-Schweigert 1996 ] doubles of finite groups [ Coste-Gannon-Ruelle 2000 ] Z ij = δ i,j for [u(1) p1 p 2 q/z 2 ] ω [ Schellekens-Sousa 2001 ]

18 F Wien p. 6/23 Torus partition functions Torus partition function of a RCFT : Z = Z(τ) = P i,j I Z ij χ i(τ) χ j (τ) Necessary conditions : Z ij Z ( dimensions ) Z 00 = 1 ( unique vacuum ) Z(τ+1) = Z(τ) = Z( 1 ) τ ( same conformal structure ) Note : not sufficient search also reveals unphysical solutions Necessary and sufficient : modular invariance at all genera + factorization / sewing constraints = φ φ φ

19 F Wien p. 6/23 Torus partition functions Torus partition function of a RCFT : Z = Z(τ) = P i,j I Z ij χ i(τ) χ j (τ) Necessary conditions : Z ij Z ( dimensions ) Z 00 = 1 ( unique vacuum ) Z(τ+1) = Z(τ) = Z( 1 ) τ ( same conformal structure ) Note : not sufficient search also reveals unphysical solutions Necessary and sufficient : modular invariance at all genera + factorization / sewing constraints Problem : infinitely many relations Solution : obtain Z as part of consistent collection of all correlators ( as elements in the relevant spaces of conformal blocks ) and develop methods for obtaining all such collections of correlators Tool : TFT construction relating full CFT on to chiral CFT on b

20 F Wien p. 7/23 Full vs chiral The double TFT construction : Holomorphic factorization : Full CFT = combine two chiral halves concretely : correlation function C of full CFT vector space () of conformal blocks () = space of conformal blocks on the double. b = orientation bundle over identif. on closed orientable = b = + e.g. for Z = C T; : (+T) = space of characters with basis {χ i } ( T) has basis {χ i }

21 F Wien p. 7/23 Full vs chiral The double TFT construction : Holomorphic factorization : Full CFT = combine two chiral halves concretely : correlation function C of full CFT vector space () of conformal blocks () = space of conformal blocks on the double. b = orientation bundle over identif. on closed orientable = b = + Problem : how to relate ( spectrum of ) bulk fields to chiral CFT on b? Solution : thicken the world sheet 3-manifold M M = b M U = U [ 1,1] = {0} M on M put a 3-d TFT naturally associated to the chiral CFT identify space of conformal blocks on b = state space of the 3-d TFT

22 F Wien p. 8/23 Full vs chiral the connecting manifold TFT construction : the double b of the world sheet i in vicinity of bulk field insertion : Φ ij α j

23 F Wien p. 8/23 Full vs chiral the connecting manifold TFT construction : the double b of the world sheet i in vicinity of bulk field insertion : Φ ij α j

24 F Wien p. 8/23 Full vs chiral the connecting manifold TFT construction : the double b of the world sheet i in vicinity of bulk field insertion : Φ ij α j

25 F Wien p. 8/23 Full vs chiral the connecting manifold TFT construction : the double b of the world sheet i in vicinity of bulk field insertion : Φ ij α j ribbon graph with ribbons starting / ending on b or on coupons in M \ b coupon for field insertion Φ ij α connected with arcs on b

26 F Wien p. 8/23 Full vs chiral the connecting manifold TFT construction : the double b of the world sheet i in vicinity of bulk field insertion : Φ ij α α j ribbon graph with ribbons starting / ending on b or on coupons in M \ b coupon for field insertion Φ ij α connected with arcs on b

27 F Wien p. 8/23 Full vs chiral the connecting manifold TFT construction : the double b of the world sheet i in vicinity of bulk field insertion : Φ ij α α j ribbon graph with ribbons starting / ending on b or on coupons in M \ b coupon for field insertion Φ ij α connected with arcs on b

28 F Wien p. 8/23 Full vs chiral the connecting manifold TFT construction : the double b of the world sheet i in vicinity of bulk field insertion : Φ ij α α j ribbon graph with ribbons starting / ending on b or on coupons in M \ b coupon for field insertion Φ ij α connected with arcs on b Problem : works only when j = i i.e. Z = Z c.c.

29 F Wien p. 8/23 Full vs chiral the connecting manifold TFT construction : the double b of the world sheet i in vicinity of bulk field insertion : Φ ij α α j ribbon graph with ribbons starting / ending on b or on coupons in M \ b α coupon for field insertion Φ ij connected with arcs on b Problem : works only when j = i i.e. Z = Z c.c. Solution : realize that { bulk fields } { defect fields }

30 F Wien p. 9/23 ulk fields as defect fields Defect lines and defect fields Motivation ulk fields as defect fields Working with defect fields The classifying algebra The defect partition function

31 F Wien p. 9/23 Defect fields and defects i Picture for bulk fields generalizes to D Θ ij α D D D α j

32 F Wien p. 9/23 Defect fields and defects i Picture for bulk fields generalizes to D Θ ij α D D D α fter some work : j possible types of defects = certain V-rep s for each phase CFT distinguished defect line D 0 = = a symmetric special Frobenius algebra in Rep(V) [ Fjelstad-JF-Runkel-Schweigert 2008 ] [ Kapustin-Saulina 2010 ]

33 F Wien p. 9/23 Defect fields and defects i Picture for bulk fields generalizes to D Θ ij α D D D α fter some work : j possible types of defects = certain V-rep s for each phase CFT distinguished defect line D 0 = = a symmetric special Frobenius algebra in Rep(V) V-rep s D labeling topological defect lines = - -bimodules in Rep(V) possible couplings α = - -bimodule morphisms H i + D H j D -defect is invisible as a defect ( though visible as a V-rep )

34 F Wien p. 9/23 Defect fields and defects i Picture for bulk fields generalizes to D Θ ij α D D D α fter some work : j possible types of defects = certain V-rep s for each phase CFT distinguished defect line D 0 = = a symmetric special Frobenius algebra in Rep(V) V-rep s D labeling topological defect lines = - -bimodules in Rep(V) possible couplings α = - -bimodule morphisms H i + D H j D -defect is invisible as a defect ( though visible as a V-rep ) ulk field = defect field in CFT sending to

35 F Wien p. 10/23 Defect fields and defects i Picture for bulk fields α fter some work : j possible types of defects = certain V-rep s for each phase CFT distinguished defect line D 0 = = a symmetric special Frobenius algebra in Rep(V) V-rep s D labeling topological defect lines = - -bimodules in Rep(V) possible couplings α = - -bimodule morphisms H i + D H j D -defect is invisible as a defect ( though visible as a V-rep ) ulk field = defect field in CFT sending to

36 F Wien p. 10/23 Defect fields and defects i Picture for bulk fields α Thus : Z ij Z ij () = dimension of - -bimodule morphisms Hi + H j j

37 F Wien p. 10/23 Defect fields and defects i Picture for bulk fields α Thus : Z ij Z ij () = dimension of - -bimodule morphisms Hi + H j j symm. spec. Frobenius = Z ij Z Z 00 = 1 Z(τ+1) = Z(τ) = Z( 1 τ ) structure constants of = Z ij = m a bc a cb (c b j) ī G a k R (c b) a θ k F θ j a,b,c k I (c b j) ī k a ( mult. labels suppressed ) Classification of ( Morita classes of ) simple symm. special Frob. algebras in Rep(V) is finite problem has finite number of solutions for given chiral theory

38 F Wien p. 10/23 Defect fields and defects i Picture for bulk fields α Thus : Z ij Z ij () = dimension of - -bimodule morphisms Hi + H j j Examples : Z c.c. is physical [ Felder-Fröhlich-JF-Schweigert 1999 ] -D-E classification for sl(2) WZW [ Kirillov-Ostrik 2001 ] simple current invariants are physical [ JF-Runkel-Schweigert 2004 ] permutation invariants are physical [ armeier-f-r-s 2008 ]

39 F Wien p. 11/23 Working with defect fields Defect lines and defect fields Motivation ulk fields as defect fields Working with defect fields The classifying algebra The defect partition function

40 F Wien p. 11/23 Topological defects Type of defect assigned to a defect line encoded in transition conditions : Conformal defect : T T continuous across the defect line Totally reflective defect : T = T on either side of the defect line Totally transmissive defect : T and T individually continuous tensionless : can be deformed without affecting value of a correlator ( as long as not taken through field insertion or other defect line )

41 F Wien p. 11/23 Topological defects Type of defect assigned to a defect line encoded in transition conditions : Conformal defect : T T continuous across the defect line Totally reflective defect : T = T on either side of the defect line Totally transmissive defect : T and T individually continuous tensionless : can be deformed without affecting value of a correlator ( as long as not taken through field insertion or other defect line ) Topological defect : preserve chiral symmetries of a rational chiral algebra already assumed partly above to be assumed from now on topological - -defects = objects of the category C of --bimodules in C = Y in C and Y label physically equivalent - -defects

42 F Wien p. 11/23 Topological defects Type of defect assigned to a defect line encoded in transition conditions : Conformal defect : T T continuous across the defect line Totally reflective defect : T = T on either side of the defect line Totally transmissive defect : T and T individually continuous tensionless : can be deformed without affecting value of a correlator ( as long as not taken through field insertion or other defect line ) Topological defect : preserve chiral symmetries of a rational chiral algebra already assumed partly above to be assumed from now on topological - -defects = objects of the category C of --bimodules in C = Y in C and Y label physically equivalent - -defects R CFT : C semisimple every defect type is finite superposition of elementary defects ( finite direct sum of simple objects )

43 F Wien p. 11/23 Topological defects Type of defect assigned to a defect line encoded in transition conditions : Conformal defect : T T continuous across the defect line Totally reflective defect : T = T on either side of the defect line Totally transmissive defect : T and T individually continuous tensionless : can be deformed without affecting value of a correlator ( as long as not taken through field insertion or other defect line ) Topological defect : preserve chiral symmetries of a rational chiral algebra already assumed partly above to be assumed from now on topological - -defects = objects of the category C of --bimodules in C = Y in C and Y label physically equivalent - -defects R CFT : C semisimple every defect type is finite superposition of elementary defects number of elementary --defect types is finite for any two phases and

44 F Wien p. 11/23 Topological defects Type of defect assigned to a defect line encoded in transition conditions : Conformal defect : T T continuous across the defect line Totally reflective defect : T = T on either side of the defect line Totally transmissive defect : T and T individually continuous tensionless : can be deformed without affecting value of a correlator ( as long as not taken through field insertion or other defect line ) Topological defect : preserve chiral symmetries of a rational chiral algebra already assumed partly above to be assumed from now on topological - -defects = objects of the category C of --bimodules in C = Y in C and Y label physically equivalent - -defects R CFT : C semisimple every defect type is finite superposition of elementary defects number of elementary --defect types is finite for any two phases and amenable to TFT construction [ Fröhlich-JF-Runkel-Schweigert 2004, 2007 ]

45 F Wien p. 12/23 Fusing defects Tensor product over of - - and -C -bimodule defect lines can be fused C C ( smooth limit of vanishing distance ) Y fuse Y relate defects between different phases associative up to equivalence of defects

46 F Wien p. 12/23 Fusing defects Tensor product over of - - and -C -bimodule defect lines can be fused C C ( smooth limit of vanishing distance ) Y fuse Y Tensor product over with left -module M fuse M relate boundary conditions adjecent to different phases defect lines can be fused to boundaries

47 F Wien p. 12/23 Fusing defects Tensor product over of - - and -C -bimodule defect lines can be fused C C ( smooth limit of vanishing distance ) Y fuse Y Tensor product over with left -module M fuse M defect lines can be fused to boundaries Tensor unit : each phase has an invisible defect : as bimodule over itself Duality : change of orientation of - -defect results in - -defect Fusion ring of - -defects with distinguished basis consisting of simple defect types ( in general not commutative no braiding of defect lines )

48 F Wien p. 13/23 Shrinking defects Shrink defect loop around bulk field : φ = D (φ) map bulk fields of CFT bulk fields of CFT

49 F Wien p. 13/23 Shrinking defects Shrink defect loop around bulk field : φ = D (φ) map bulk fields of CFT bulk fields of CFT Shrink defect around bulk field : Y α φ = Y D,Y,α (φ) map bulk fields of CFT disorder fields of CFT

50 F Wien p. 14/23 Inflating defects Inflate defect loop on a world sheet : = dim() dim(y ) Y = dim() dim(y ) relate correlators in different phases

51 F Wien p. 14/23 Inflating defects Inflate defect loop on a world sheet : = dim() dim(y ) Y = dim() dim(y ) relate correlators in different phases pplication : Y a - -duality defect = relate Z() and Z g h () g = 1 S r (Y ) g,h S r (Y ) gh=hg h h g auto-orbifold relation [ Ruelle 2005 ]

52 F Wien p. 15/23 Pushing defects Push defect through a bulk field : Y φ = µ,γ θ µγ µ Y γ Y

53 F Wien p. 15/23 Pushing defects Push defect through a bulk field : Y φ = µ,γ θ µγ µ Y γ Y Special case : invertible defects symmetries g g = ± g g g φ g = = ± g g φ g non-chiral internal symmetries of a full CFT in phase in bijection with equivalence classes of invertible - -defects form the Picard group Pic of the phase

54 F Wien p. 15/23 Pushing defects Push defect through a bulk field : Y φ = µ,γ θ µγ µ Y γ Y Special case : invertible defects symmetries g g = ± g g g φ g = = ± g g φ g non-chiral internal symmetries of a full CFT in phase in bijection with equivalence classes of invertible - -defects form the Picard group Pic of the phase can be non-abelian e.g. Pic = S 3 for critical three-states Potts model (c =4/5, =1 J ) associativity constraint of C endows Pic with a class in H 3 (Pic, ) similarly as in theory of K S bihomomorphisms [ Runkel-Suszek 2009 ]

55 F Wien p. 15/23 Pushing defects Push defect through a bulk field : Y φ = µ,γ θ µγ µ Y γ Y Special case : invertible defects symmetries g g = ± g g g φ g = = ± g g φ g Special case : duality defects dualities Y Y = L h Pic h = Y φ dim() dim(y ) h S l (Y ) = Y φ h Y

56 F Wien p. 15/23 Pushing defects Push defect through a bulk field : Y φ = µ,γ θ µγ µ Y γ Y Special case : invertible defects symmetries g g = ± g g g φ g = = ± g g φ g Special case : duality defects dualities Y Y = L h Pic h = Y φ dim() dim(y ) h S l (Y ) = Y φ h Y

57 F Wien p. 16/23 classifying algebra for defects Defect lines and defect fields Motivation ulk fields as defect fields Working with defect fields The classifying algebra The defect partition function

58 F Wien p. 16/23 Transmission through defects Transmission of bulk field Φ α through a defect : fuse parallel pieces of defect line and shrink circular piece create of disorder fields Θ γ with same chiral labels as Φ α Φ α = = Y τ Y = Y,τ γ d αγ,,;y,τ Y Θ γ = defect transmission coefficients d ıj,αγ,,;y,τ independent of insertion point of Φ α

59 F Wien p. 16/23 Transmission through defects TFT construction three-dimensional picture of transmission : i φ α i j = Y τ τ Y τ φ α i j = Y,τ γ d ij,αγ,,;y,τ τ Y θ γ j

60 F Wien p. 16/23 Transmission through defects Defect transmission coefficients d ıj,αγ,,;y,τ play multiple role : transmitting bulk field Φ α from phase to phase through defect

61 F Wien p. 16/23 Transmission through defects Defect transmission coefficients d ıj,αγ,,;y,τ play multiple role : transmitting bulk field Φ α from phase to phase through defect appear naturally in expansion of partition function on a torus with defect lines into characters : Z Y T, ij = S 2 0,0 dim() dim(y ) S i,p Sj,q p,q I (c bulk 1 β 1,β 2,β 3,β 4 bulk 1 ;p,q ) (c β 2 β 1 ;p,q ) d pq,β 1β 4 β 4 β 3 d pq,β 3β 2 Y

62 F Wien p. 16/23 Transmission through defects Defect transmission coefficients d ıj,αγ,,;y,τ play multiple role : transmitting bulk field Φ α from phase to phase through defect appear naturally in expansion of partition function on a torus with defect lines into characters describe operator product of two bulk fields Φ α and Φ β give matrix elements of the action of on bulk fields describe scattering of bulk fields in the background of the defect in phases separated by

63 F Wien p. 16/23 Transmission through defects Defect transmission coefficients d ıj,αγ,,;y,τ play multiple role : transmitting bulk field Φ α from phase to phase through defect appear naturally in expansion of partition function on a torus with defect lines into characters describe operator product of two bulk fields Φ α and Φ β give matrix elements of the action of on bulk fields describe scattering of bulk fields in the background of the defect in phases separated by TFT construction express d ıj,αγ,,;y,τ as invariant of ribbon graph in S 3 ( thus as morphism in End(1) = ) dim() d ıj,αβ = φ α φ β ı ı j j

64 F Wien p. 17/23 Reflection coefficients nalogue on boundary : Reflection coefficients b ı,α M satisfy bı,α M bk,β M = P q I P Zq q γ=1 C qγ ıα,kβ bq,γ M

65 F Wien p. 17/23 Reflection coefficients nalogue on boundary : Reflection coefficients b ı,α M satisfy bı,α M bk,β M = P q I P Zq q γ=1 C qγ ıα,kβ bq,γ M Classifying algebra of boundary conditions obtained by comparing boundary and bulk factorization for correlator of one left-right symmetric bulk field on disk with boundary condition M in the vacuum channel

66 F Wien p. 17/23 Reflection coefficients nalogue on boundary : Reflection coefficients b ı,α M satisfy bı,α M bk,β M = P q I P Zq q γ=1 C qγ ıα,kβ bq,γ M Classifying algebra of boundary conditions Schematically : boundary factorization bulk factorization projection to vacuum channel projection to vacuum channel

67 F Wien p. 18/23 Defect transmission coefficients Defect transmission coefficients satisfy d ij,αβ d kl,γδ = P p,q I Pµ,ν Cij,αβ;kl,γδ pq,µν d pq,µν

68 F Wien p. 18/23 Defect transmission coefficients Defect transmission coefficients satisfy d ij,αβ d kl,γδ = P p,q I Pµ,ν Cij,αβ;kl,γδ pq,µν d pq,µν with complex numbers C i,jα;β,klγ δ,pqµ ν that do not depend on the simple defect Classifying algebra of defect lines obtained by comparing double bulk factorization and bulk factorization across the defect line for correlator of two bulk fields on sphere with separating defect line in the vacuum channel

69 F Wien p. 18/23 Defect transmission coefficients Defect transmission coefficients satisfy d ij,αβ Schematically : d kl,γδ = P p,q I Pµ,ν Cij,αβ;kl,γδ pq,µν d pq,µν defect-crossing factoriz. double bulk factorization

70 F Wien p. 18/23 Defect transmission coefficients Defect transmission coefficients satisfy d ij,αβ Schematically : d kl,γδ = P p,q I Pµ,ν Cij,αβ;kl,γδ pq,µν d pq,µν defect-crossing factoriz. double bulk factorization plus projection to vacuum channel

71 Defect transmission coefficients defect-crossing factorization TFT description : double bulk factorization projection to vacuum channel projection to vacuum channel F Wien p. 19/23

72 Defect transmission coefficients defect-crossing factorization TFT description : double bulk factorization three-manifold : S 2 S 1 minus two four-punctured three-balls ( entire correlator ) projection to vacuum channel projection to vacuum channel F Wien p. 19/23

73 F Wien p. 20/23 The classifying algebra Result : C ıj,αβ;kl,γδ pq,µν = S 2 0,0 dim(u p)dim(u q ) θ j θ l θ q κ,λ (cbulk 1 ;pq ) κµ (c bulk 1 ; p q ) Z(K αγκ;βδλ ) λν ı kp;j lq with ribbon graph K α 3 α 4 β 2;α 1 α 2 β 4 ı 1 ı 2 p;j 1 j 2 q = j 1 j q 2 φ α3 φ φ α4 β2 ı 1 ı p 2 ı 1 p ı 2 φ α1 φ α2 φ β4 j 1 j 2 q in S 2 S 1

74 F Wien p. 20/23 The classifying algebra Result : C ıj,αβ;kl,γδ pq,µν = S 2 0,0 dim(u p)dim(u q ) θ j θ l θ q κ,λ (cbulk 1 ;pq ) κµ (c bulk 1 ; p q ) Z(K αγκ;βδλ ) λν ı kp;j lq Recall : d ij,αβ d kl,γδ Natural interpretation : C ij,αβ;kl,γδ pq,µν = P p,q I Pµ,ν Cij,αβ;kl,γδ pq,µν d pq,µν structure constants of multiplication on vector space D := L p,q I Hom (U p + U q, ) Hom (U p + U q, )

75 F Wien p. 20/23 The classifying algebra Result : C ıj,αβ;kl,γδ pq,µν = S 2 0,0 dim(u p)dim(u q ) θ j θ l θ q κ,λ (cbulk 1 ;pq ) κµ (c bulk 1 ; p q ) Z(K αγκ;βδλ ) λν ı kp;j lq Recall : d ij,αβ d kl,γδ Natural interpretation : C ij,αβ;kl,γδ pq,µν = P p,q I Pµ,ν Cij,αβ;kl,γδ pq,µν d pq,µν structure constants of multiplication on vector space D := L p,q I Hom (U p + U q, ) Hom (U p + U q, ) in basis {φ pq,αβ } = {φ pq,α φ p q,β p, q I, α =1,2,..., Z pq(), β =1,2,..., Z pq ()} ( pairs of bulk fields in phase with chiral labels p, q and in phase with p, q ) dim (D ) = tr(z()z() t ) = number of isom. classes of simple --bimodules number of types of simple --defects

76 F Wien p. 20/23 The classifying algebra Result : C ıj,αβ;kl,γδ pq,µν = S 2 0,0 dim(u p)dim(u q ) θ j θ l θ q κ,λ (cbulk 1 ;pq ) κµ (c bulk 1 ; p q ) Z(K αγκ;βδλ ) λν ı kp;j lq Recall : d ij,αβ d kl,γδ Natural interpretation : C ij,αβ;kl,γδ pq,µν = P p,q I Pµ,ν Cij,αβ;kl,γδ pq,µν d pq,µν structure constants of multiplication on vector space D := L p,q I Hom (U p + U q, ) Hom (U p + U q, ) in basis {φ pq,αβ } = {φ pq,α φ p q,β p, q I, α =1,2,..., Z pq(), β =1,2,..., Z pq ()} Theorem : [ JF-Schweigert-Stigner 2011 ] D with this product is a semisimple commutative unital associative algebra in Vect The ( one-dimensional ) irreducible representations of D are in bijection with the types of simple topological defects separating the phases and ; the representation matrices are furnished by the defect transmission coefficients

77 F Wien p. 21/23 The classifying algebra Comments : Cardy case : C ıı, ;jj, p p, = θ ı θ j θ p dim(u p ) 2 dim(u ı ) 2 dim(u j ) 2 N ıj p Note : multiple role of chiral fusion rules only in Cardy case fusion algebra

78 F Wien p. 21/23 The classifying algebra Comments : Cardy case : C ıı, ;jj, p p, = θ ı θ j θ p dim(u p ) 2 dim(u ı ) 2 dim(u j ) 2 N ıj p Note : multiple role of chiral fusion rules only in Cardy case fusion algebra Proof of associativity uses higher products : show that D can be endowed with an n-ary product for any n 2 with structure constants expressed through ribbon graphs similar to n = 2 show that all n-ary products are totally commutative a commutative algebra with totally commutative ternary product is associative if at least one bracketing of a twofold binary product equals the ternary product

79 F Wien p. 21/23 The classifying algebra Comments : Cardy case : C ıı, ;jj, p p, = θ ı θ j θ p dim(u p ) 2 dim(u ı ) 2 dim(u j ) 2 N ıj p Note : multiple role of chiral fusion rules only in Cardy case fusion algebra Proof of associativity uses higher products Proof of semisimplicity relies on associativity : the dim (D ) dim (D ) -matrix of defect transmission coefficients ( rows labeled by simple - -bimodules, columns by pairs of bulk fields ) is non-degenerate [ Fröhlich-JF-Runkel-Schweigert 2007 ] = n simp (D ) dim (D ) basic structure theory of associative algebras = dim (D ) n simp (D )

80 F Wien p. 21/23 The classifying algebra Comments : Cardy case : C ıı, ;jj, p p, = θ ı θ j θ p dim(u p ) 2 dim(u ı ) 2 dim(u j ) 2 N ıj p fusion algebra Note : multiple role of chiral fusion rules only in Cardy case Proof of associativity uses higher products Proof of semisimplicity relies on associativity dim (D ) = n simp (D ) For = : comparison with algebra D PZ [ Petkova-Zuber 2001 ] known to be of same dimension as D and to share other properties some assumptions involving complex conjugation = D PZ = D ut : within TFT approach no natural place for complex conjugation

81 F Wien p. 21/23 The classifying algebra Comments : Cardy case : C ıı, ;jj, p p, = θ ı θ j θ p dim(u p ) 2 dim(u ı ) 2 dim(u j ) 2 N ıj p fusion algebra Note : multiple role of chiral fusion rules only in Cardy case Proof of associativity uses higher products Proof of semisimplicity relies on associativity dim (D ) = n simp (D ) For = : comparison with algebra D PZ [ Petkova-Zuber 2001 ] known to be of same dimension as D and to share other properties some assumptions involving complex conjugation = D PZ = D ut : within TFT approach no natural place for complex conjugation Motivation : study aspects of defects with the help of classical algebra possible generalization to non-semisimple CFT

82 F Wien p. 22/23 ppendix : Defect partition functions Defect lines and defect fields Motivation ulk fields as defect fields Working with defect fields The classifying algebra The defect partition function

83 F Wien p. 22/23 ppendix : Defect partition functions Recall : T, ij = S 2 0,0 dim() dim(y ) S i,p Sj,q Z Y p,q I (c bulk 1 β 1,β 2,β 3,β 4 bulk 1 ;p,q ) (c β 2 β 1 ;p,q ) d pq,β 1β 4 β 4 β 3 d pq,β 3β 2 Y

84 F Wien p. 22/23 ppendix : Defect partition functions Recall : T, ij = S 2 0,0 dim() dim(y ) S i,p Sj,q Z Y p,q I (c bulk 1 β 1,β 2,β 3,β 4 bulk 1 ;p,q ) (c β 2 β 1 ;p,q ) d pq,β 1β 4 β 4 β 3 d pq,β 3β 2 Y nalogue : nnulus coefficients k,m N = dim(m) dim(n) q I Z q q S k,q θ q γ,δ=1 (c bulk 1 ;q q ) δγ b q,γ N b q,δ M [ Stigner 2011 ]

85 F Wien p. 22/23 ppendix : Defect partition functions Recall : T, ij = S 2 0,0 dim() dim(y ) S i,p Sj,q Z Y p,q I (c bulk 1 β 1,β 2,β 3,β 4 bulk 1 ;p,q ) (c β 2 β 1 ;p,q ) d pq,β 1β 4 β 4 β 3 d pq,β 3β 2 Y nalogue : nnulus coefficients k,m N = dim(m) dim(n) q I Z q q S k,q θ q γ,δ=1 (c bulk 1 ;q q ) δγ b q,γ N b q,δ M [ Stigner 2011 ] Obtained by double bulk factorization ( cutting circles parallel to the defects lines ) : Z Y T = dim(u p )dim(u q ) dim(u r ) dim(u s ) p,q,r,s I 1 ;p,q ) bulk 1 (c β 2 β 1 ;r,s ) Z(M T,Y β 4 β 3 β 1,β 2,β 3,β 4 (c bulk pqβ 1 β 2,rsβ 3 β 4 )

86 F Wien p. 23/23 ppendix : Defect partition functions Z Y T = p,q,r,s dim(u p ) dim(u q )dim(u r )dim(u s ) (c bulk 1 β 1,β 2,β 3,β 4 ;p,q ) β 2 β 1 (c bulk 1 ;r,s ) β 4 β 3 Z(M T,Y pqβ 1 β 2,rsβ 3 β 4 ) q φ β1 s M T,Y pqβ 1 β 2,rsβ 3 β 4 ) = p p φ β4 r φ β2 φ β3 r Y q s