Factorization constraints and boundary conditions in rational CFT
aa r X i v : . [ h e p - t h ] J un Factorization constraints and boundary conditions in rational CFT
Carl Stigner
Teoretisk fysik, Karlstads UniversitetUniversitetsgatan 21, 651 88 Karlstad
June 30 2010
Abstract
Among (conformal) quantum field theories, the rational conformal field theories are sin-gled out by the fact that their correlators can be constructed from a modular tensor category C with a distinguished object, a symmetric special Frobenius algebra A in C , via the so-calledTFT-construction. These correlators satisfy in particular all factorization constraints, whichinvolve gluing homomorphisms relating correlators of world sheets of different topology.We review the action of the gluing homomorphisms and discuss the implications of thefactorization constraints for boundary conditions. The so-called classifying algebra A for aRCFT is a semisimple commutative associative complex algebra, which classifies the bound-ary conditions of the theory. We show that the annulus partition functions can be obtainedfrom the representation theory of A . There are various physical motivations to study quantum field theories on two-dimensional com-pact manifolds with a complex structure, possibly with non-empty boundary. Applications appeare.g. in condensed matter physics and in string theory. Such a surface is, by terminology inheritedfrom string theory, called a world sheet . The situation becomes particularly interesting for a (full,local) conformal field theory (CFT), i.e. a two-dimensional QFT with conformal symmetry definedon world sheets. In two dimensions, there are, apart from the global conformal transformations, aninfinite number of local conformal transformations giving rise to an infinite dimensional symmetryalgebra. In fact, as a consequence of the huge amount of symmetry, conformal field theories canbe studied in a fully non-perturbative manner. This is another reason to study 2d CFT.We denote by X c a world sheet, possibly with boundary, and a number of field insertions in thebulk or on the boundary. The correlation function Corr( X c ) for the world sheet X c associates to X c a map from the relevant space of fields to the complex numbers. Correlation functions are linearin the fields and satisfy a number of consistency conditions. Among them are the factorizationconstraints, which can be thought of as a concrete realization of the notion of inserting a completeset of states. Solving a CFT amounts to giving the correlation function for any world sheet X c .This paper concerns a special class of CFT’s, the so-called rational CFT’s (RCFT), for whichthere is a nice description of the construction in terms of modular tensor categories.An important issue in CFT is the classification of conformal boundary conditions. A priori thisis a difficult problem, except for some simple models. In e.g. the Ising model, a simple spin model,all boundary conditions can be described in terms of a fixed external magnetic field applied to thespin variables at the boundary. This gives rise to a one parameter family of boundary conditions,which renormalize to three different boundary conditions in the continuum model. Two of them,spin up and spin down, correspond to a non-zero external magnetic field, whereas the third one, the1ree boundary condition, corresponds to taking the external magnetic field to be zero. However,it is far from obvious that boundary conditions of this form exhaust the conformal boundaryconditions. E.g. in the three-states Potts model, there is one conformal boundary condition whichcan not be related in a simple way to the external magnetic field [3].In [12] it was conjectured that the conformal boundary conditions for a specific class of theoriesare classified by a semi-simple commutative associative complex algebra, the so-called classifyingalgebra A . In [13] we establish the existence of the classifying algebra for any RCFT. The struc-ture constants of A are obtained by comparing bulk and boundary factorization of a disc withtwo bulk field insertions. The irreducible representations of A are the so-called reflection coeffi-cients. The reflection coefficients, which appear e.g. in [6, 12], are collected in so-called boundarystates. The boundary states contain essential physical information regarding boundary conditions,such as ground state degeneracies [1] and Ramond-Ramond charges of string compactifications[4]. Moreover it has been shown, for some special classes of models, see e.g. [2, 5, 14], that thereflection coefficients appears naturally in the annulus partition functions. In this paper we show,by applying bulk factorization, that essential information concerning the annulus partition func-tions for any RCFT is contained in A and its representation theory. Thus the appearance of thereflection coefficients in the annulus coefficients of a RCFT is a generic phenomenon.The symmetries of a CFT can be encoded in the mathematical structure of a conformal vertexalgebra V , by physicists often referred to as the chiral algebra. A rational CFT is distinguished bythe property that the strictification of the category R ep ( V ) of representations of V is a modulartensor category C . The correlation functions of a rational CFT satisfy holomorphic factorization,e.g. the correlation function Corr( X c ) is a vector in the space of conformal blocks on the double c X c . The double is obtained from X c by taking the orientation bundle over X c and pairwiseidentify points over the boundary ∂X c : c X c := or( X c ) (cid:14) ∼ , ( x, or) ∼ ( x, − or) ∀ x ∈ ∂X c . (1)The double is in particular a complex curve, thus we can study the space of conformal blocks on c X c .The solution of a rational conformal field theory, with given chiral algebra V , can be splitoff into two separate parts, a complex-analytic and a purely algebraic part. The first problemamounts to solving the chiral theory on c X c , i.e. to obtain the space of conformal blocks of V on c X c . The second problem amounts to selecting, from the space of conformal blocks, the particularvector Corr( X c ). This paper is concerned with the second problem. As a consequence, we will beable to restrict to topological world sheets. A topological world sheet X is obtained from X c bysuppressing the conformal structure.This paper is formulated in the framework of the TFT-construction. The TFT-constructionprovides all solutions to a rational CFT with given chiral algebra V . A rational CFT, withchiral algebra V , is constructed from the modular tensor category C , which is the strictification of R ep ( V ), and a distinguished object A in C , with the structure of a symmetric special Frobeniusalgebra. In fact, the rational CFT’s with chiral algebra V are classified by Morita classes ofsimple symmetric special Frobenius algebras in C . We will not discuss vertex algebras explicitly,we will rather work in the framework of an abstract modular tensor category. Thereby we coverall rational CFT’s simultaneously.A crucial tool in the TFT-construction is a topological field theory. A topological field theoryis a tensor functor tft C from the category 3- C ob ( C ) to the category V ect C of finite-dimensionalcomplex vector spaces. The morphisms of 3- C ob ( C ) are cobordisms, i.e. three-manifolds withembedded ribbon graph. The TFT-construction provides the correlator as the invariant of such2 cobordism. The correlator of a topological world sheet X is an element in a finite dimensionalvector space. This space can be identified with the space of conformal blocks on the world sheet X c , obtained by endowing X with a complex structure. Thus the structure constants of theexpansion of such a correlator are the same as the ones for the correlation function .In section 2 we review some basic facts concerning modular tensor categories and the TFT-construction. Section 3 describes how the factorization constraints are implemented on a specificcorrelator. There are 2 types of factorization, bulk and boundary factorization. Boundary factor-ization is covered only briefly since we do not need it for the calculations in this paper. In section4 we use bulk factorization to show how A and its representation theory appear in the annuluspartition functions. tft C -functor A modular tensor category C is in particular an abelian, semisimple, C -linear, ribbon category.Thus any object is a finite direct sum of simple objects. Since the ground field of C is C thenotion of a simple object is the same as a ”scalar” object, meaning that End( U i ) = C . We chooserepresentatives of isomorphism classes of simple objects and label them by a finite index set I ,i.e. { U i | i ∈ I} , (2)where we take U = and ¯ k ∈ I such that U ¯ k ∼ = U ∨ k for all k ∈ I . Since C is ribbon we makeextensive use of graphical calculus, see e.g. [9, section 2]. Due to strictness lines labeled by areinvisible. Among the structures defining a ribbon category is the twist. We denote the twist of theobject U by θ U . The twist of a simple object U i , which is proportional to the identity morphism,is written as θ U i = θ i id U i , θ i ∈ C . (3)In a modular tensor category there is also a non-degenerate matrix S , c.f. [9, eqs. (2.21) & (2.27)],which is part of a representation of the modular group. We will also use the quantum dimensiondim( U ) of an object U , c.f. [9, eq. (2.17)], which for a simple object is related to the S -matrix:dim( U i ) := S i, S , . (4) An algebra in a modular tensor category is an object A , equipped with a product m ∈ Hom( A ⊗ A, A )and a unit η ∈ Hom( , A ) that satisfy associativity and unit constraints: m ◦ (id A ⊗ m ) = m ◦ ( m ⊗ id A ) and m ◦ ( η ⊗ id A ) = id A = m ◦ (id A ⊗ η ) . (5)Similarly a coalgebra A in C is an object A , together with a coproduct ∆ ∈ Hom(
A, A ⊗ A ) and acounit ε ∈ Hom( A, ) satisfying coassociativity and counit constraints:(id A ⊗ ∆) ◦ ∆ = (∆ ⊗ id A ) ◦ ∆ and ( ε ⊗ id A ) ◦ ∆ = id A = (id A ⊗ ε ) ◦ ∆ . (6) The correlation function depends in general on the metric on X c . However, a certain quotient of correlatorswill only depend on the conformal equivalence class of the metric, see [10, section 6.1.4]. It is these quotients thatcan be obtained via the TFT-construction. Frobenius algebra in a tensor category is an object A which is both an algebra and a coalgebra,such that the product and coproduct obey the following compatibility condition(id A ⊗ m ) ◦ (∆ ⊗ id A ) = ∆ ◦ m = ( m ⊗ id A ) ◦ (id A ⊗ ∆) . (7)A left-module M ≡ ( M, ρ ) over an algebra A in C is an object M , equipped with a represen-tation morphism ρ ∈ Hom( A ⊗ M, M ) satisfying ρ ◦ ( m ⊗ id M ) = ρ ◦ (id A ⊗ ρ ) and ρ ◦ ( η ⊗ id M ) = id M . (8)Similarly a right-module over A is an object M , together with a morphism ρ ∈ Hom( M ⊗ A, M ), satisfying analogous relations. For two algebras A and B in a tensor cate-gory, an A - B -bimodule X ≡ ( X, ρ L , ρ R ) is an object X , such that ( X, ρ L ) is a left A -module and( X, ρ R ) is a right B -module, such that the two actions commute. A simple module is a modulethat does not have a non-trivial subobject which is a module itself. For any two left A -modules M and N , we define the subspace of left A -module morphismsHom A ( M, N ) := { f ∈ Hom(
M, N ) | ρ N ◦ (id A ⊗ f ) = f ◦ ρ M } . (9)Similarly, for any two A - B -bimodules X and Y , the spaceHom A | B ( X, Y ) (10)consists of all morphisms in Hom(
X, Y ) that commute with the left action of A and the rightaction of B . For any two objects U and V in C we define the A - A -bimodule U ⊗ + A ⊗ − V as U ⊗ + A ⊗ − V := (cid:16) U ⊗ A ⊗ V, (cid:2) (id U ⊗ m ⊗ id V ) ◦ ( c − U,A ⊗ id A ⊗ id V ) (cid:3) , (cid:2) (id U ⊗ m ⊗ id V ) ◦ (id U ⊗ id A ⊗ c − A,V ) (cid:3)(cid:17) . (11) We review some aspects concerning the TFT-construction. A detailed description can be foundin [10, section 3-4] or [15], see also [13, appendix A.1-A.5] for a shorter description.A modular tensor category C serves as a decoration of a geometric category 3- C ob ( C ). Theobjects of 3- C ob ( C ) are extended surfaces and the morphisms are cobordisms. An extended surface E is a compact closed oriented two-manifold, with marked points and a choice of Lagrangiansubspace λ ⊂ H ( E, R ). The data of a marked point contain in particular an object in C . A cobordism M : E → E ′ is a compact oriented three-manifold, with boundary ∂ M = ( − E ) ⊔ E ′ and an embedded ribbon graph with one ribbon ending at each marked point. The ribbon graphis colored by objects and morphisms in C .Given a modular tensor category C we can construct a three-dimensional topological field theory(3d TFT). A 3d TFT is a tensor functor from 3- C ob ( C ) to the category V ect C of finite dimensionalcomplex vector spaces. Thus tft C ( E ) ≡ H ( E ) is a vector space and tft C ( M ) ≡ Z ( M ) is a linearmap Z ( M ) : H ( E ) → H ( E ′ ) . (12)By projecting a ribbon graph locally to R in a non-singular manner we can consider it as amorphism in C and manipulate the ribbon graph locally by the rules of graphical calculus. Trans-formations of this kind leave the linear map Z ( M ) invariant. Furthermore, the linear map Z ( M )4s a topological invariant and we will refer to it as the invariant of M . A particular extendedsurface is the double b X of a topological world sheet X . For the purposes of this paper we canidentify the tft C -state space H ( b X ) with the space of conformal blocks on c X c .The tft C -functor is central in the TFT-construction of rational CFT. The TFT-constructiontakes as input a modular tensor category C and (a Morita class of) a symmetric special Frobeniusalgebra A in C . These data define a unique RCFT. The TFT-construction provides the correlator ofa world sheet X by giving the construction of a cobordism M X : ∅ → b X , the connecting manifold.As a three-manifold, M X is constructed by taking the interval bundle over X and identifyingpoints over the boundary: M X := X × [ − , (cid:14) ∼ , ( x, t ) ∼ ( x, − t ) ∀ x ∈ ∂X c and ∀ t ∈ [ − , . (13)Thus ∂ M X ∼ = b X , c.f. (1), and the world sheet is canonically embedded in M X as all points in( x, ∈ M X . Each field is indicated by a marked point on the world sheet X . A bulk fieldsgives rise to two marked points on b X , c.f. (13), whereas due to the identification of pointsover the boundary ∂X in (13), a boundary field gives rise to a single marked point on b X . Thestructure on the world sheet appears in M X as parts of the ribbon graph. The boundary conditionsare given by left A -modules and each boundary component appears as a ribbon, labeled by thecorresponding A -module. We refer to a boundary condition labeled by a simple A -module asan elementary boundary condition. Field insertions appear as coupons, labeled by morphismsin Hom A | A ( U ⊗ + A ⊗ − V, A ) and Hom A ( M ⊗ U, M ), with appropriate objects U and V , for bulkand boundary fields respectively. The correlator Corr( X ) is obtained from the invariant of theconnecting manifold: Corr( X ) = Z ( M X ) 1 ∈ H ( b X ) . (14)Since we identify H ( b X ) with the space of conformal blocks on c X c , (14) indeed defines a vectorin the space of conformal blocks on c X c . For the rest of this paper we can and will make theidentification Corr( X ) ≡ Z ( M X ) . (15) Factorization constraints relate correlators of world sheets of (possibly) different topology. Startingfrom one world sheet, we can cut it along an embedded circle S , which results in two holes in theworld sheet. A new world sheet X ′ is obtained by gluing a half sphere, with one primary bulkfield, to each hole. This describes bulk factorization. Boundary factorization amounts to cuttingthe world sheet along a line ℓ joining two boundary components, closing the gaps in the boundaryby gluing a half disc with a boundary field to each gap, and sum over all elementary boundaryfields.The correlators provided by the TFT-construction satisfy all factorization constraints [8]. Wewill restrict the discussion to orientable world sheets. The unorientable case works in a similarmanner. Factorization is described in detail in [8, section 2].A factorization introduces extra field insertions on the world sheet X ′ , obtained after factor-ization. As a consequence, if the double b X is marked by n points, the number of marked pointson the double c X ′ of the new world sheet will be n + 2 after boundary factorization and n + 4 afterbulk factorization. Thus H ( c X ′ ) ≇ H ( X ) and consequently, the correlator of the factorized world Morita equivalent algebras give rise to equivalent RCFT’s. X ′ is not in the same space as the correlator of the original world sheet. The factorizationconstraints, satisfied by the correlators of the TFT-construction, are stated in [8, theorem 2.9](boundary factorization) and [8, theorem 2.13] (bulk factorization). The theorems states first ofall that there exists a gluing homomorphism G : H ( c X ′ ) → H ( b X ) . (16)The composition G ◦ Corr( X ′ ) is thus in the same space as Corr( X ). Second, the two theoremsshow how these vectors are related. Schematically we can write this asCorr( X ) ∼ X fields G ◦ Corr( X ′ ) , (17)where the summation is over primary boundary fields or primary bulk fields depending on whatkind of factorization we are considering. For the purposes of this paper we do not need the gluinghomomorphism explicitly. We rather need the action of the gluing homomorphism on some specificcorrelator. Remember (14) that the correlators are given by invariants of cobordisms. The gluinghomomorphism G is also given as an invariant of a cobordism˜ G : c X ′ → b X. (18)Let M X ′ : ∅ → c X ′ be the connecting manifold of the factorized world sheet. The tft C -functorimplies that there exists a cobordism ˜ M X ′ = ˜ G ◦ M X ′ such that Z ( ˜ M X ′ ) = Z ( ˜ G ) ◦ Z ( M X ′ ) = G ◦ Corr( X ′ ) . (19)The proof of factorization is a local issue in the sense that it involves only the fibers over asmall neighborhood of the circle S or line ℓ along which the factorization is performed. Thus,for the proof, the explicit form of ˜ M X ′ is not needed. This is also a strength of the proof: Thefactorization constraints should be satisfied for any number of factorizations. Since the proof offactorization is a local consideration it treats an infinite number of factorizations simultaneously.On the other hand, for actual calculations of the correlator of the factorized world sheet we needto know ˜ M X ′ explicitly. Below we review how this manifold is constructed in the case of boundaryand bulk factorization. We refer the reader to [8] for the proof. Boundary factorization is a local issue also on the level of the connecting manifold. The cobor-dism ˜ M X ′ is obtained by applying an equality of morphisms in C to M X . Consider a strip ofthe world sheet with boundary conditions labeled by the left A -modules M and N . The ribbongraph in this neighborhood can be taken to be on a form that, when interpreted as a mor-phism in C , is a certain projector P M ∨ N ∈ End( M ∨ ⊗ N ), c.f. [15, eq. (4.7)]. The manifold M qγδ , playing the role of ˜ M X ′ in the case of boundary factorization, is then obtained by applying[8, eq. (4.22)] to P M ∨ N , c.f. [15, eq. (4.8)]. The labels γ and δ label the two boundary fields ψ γ ∈ Hom A ( N ⊗ U q , M ) and ψ δ ∈ Hom A ( M ⊗ U ¯ q , N ) respectively. The invariant of M qγδ is relatedto Z ( M X ) by Z ( M X ) = X q ∈I X γ,δ ( c bnd N,M,q ) − δγ Z ( M qγδ ) . (20)The elements of the matrix ( c bnd N,M,q ) are the structure constants of the correlator of the disc withtwo boundary fields ψ γ and ψ δ , see [8, eq. (2.27)].6 .2 Bulk factorization Bulk factorization is a more involved issue. The reason is that the construction of ˜ M X ′ is a non-local problem. Bulk factorization is performed along an embedding ι ( S ) of a circle S in X . Wewill be interested in a millstone-shaped neighborhood of N X ⊂ M X obtained as the fibers over atubular neighborhood of ι ( S ). The preimage Y S := π − X ( ι ( S )) ∈ M X , (21)of ι ( S ) under the canonical projection π X from M X to X (c.f. (13)) separates N X into two disjointparts. Y S is an annulus whose two boundary components are contained in the boundary of M X .Removing Y S from M X and taking the closure results in a manifold with corners, M ◦ X . Theboundary of M ◦ X contains two copies Y S and Y S of Y S .The manifold ˜ M X ′ in (19) is constructed by composing M ◦ X with another manifold with corners.This manifold which we denote by T q q γδ is as a three-manifold D × S : T q q γδ = q ¯ q ¯ q q φ γ φ δ Y T Y T (22)Here S is running vertically with top and bottom identified. We use black board framing forribbon graphs, i.e. we depict ribbons as lines, see [13, appendix. A.4] for details. The two spacesof bulk fields Hom A | A ( U q ⊗ + A ⊗ − U q , A ) and Hom A | A ( U ¯ q ⊗ + A ⊗ − U ¯ q , A ) are labeled by φ γ and φ δ respectively. The boundary of T q q γδ contains two copies of Y S as well. We denote them by Y T and Y T . See [13] for more details on T q q γδ .The manifold M X ; q q γδ , playing the role of ˜ M X ′ in (19), is obtained by identifying Y S with Y T and Y S with Y T . There is a unique way to make this identification such that the orientations ofthe A -ribbons as well as the boundary components agree. The invariant of M X ; q q γδ is related to M X according to Z ( M X ) = X q ,q ∈I X γ,δ dim( U q ) dim( U q ) ( c bulk q ,q ) − δγ Z ( M X ; q q γδ ) . (23)Here, ( c bulk q ,q ) is a non-degenerate matrix whose elements are the structure constants of the twopoints function on the sphere, c.f. [8, eq. (2.42)] This is the precise form of (17) in the case ofbulk factorization. 7 The annulus partition functions
Let the world sheet be an annulus with no field insertions, and with the boundary conditions at thetwo boundary components given by the simple A -modules M and N respectively. The correlatorof this world sheet is the annulus partition function A NM , see [9, section 5.8]. The double of theworld sheet is a torus, and the connecting manifold M A NM is a full torus with embedded ribbongraph, see [9, eq. (5.117)]. Consequently, the annulus partition function is an element in the spaceof conformal zero-point blocks on the torus. We choose a basis {| χ k ; T i| k ∈ I} for this space with | χ k ; T i = Z ( M χ ; k ) , k ∈ I . (24)The cobordism M χ ; k is a full torus with an annular ribbon labeled by U k inserted along the non-contractible cycle, c.f. [9, eq. (5.15)]. The dual basis {h χ k ; T | | k ∈ I} is given in [9, eq. (5.18)].The elements h χ k ; T | of the dual basis are obtained as h χ k ; T | = Z ( M ∗ χ ; k ) , k ∈ I . (25)The manifold M ∗ χ ; k differs from M χ ; k by reversion of the three-orientation and the orientation ofthe ribbon core, c.f. [9, eq. (5.18)]. We wish to expand A NM as A NM = X k ∈I A NkM | χ k ; T i . (26)The duality of the bases implies that the annulus coefficients A NkM are obtained by composing M ∗ χ ; k with M A NM . This yields a ribbon graph in S × S . A NkM is obtained by applying the tft C -functor to this ribbon graph, c.f. [9, section 5.8]. We investigate a bulk factorization along a circle S , embedded between and aligned with thetwo boundary components of the annulus. Using the prescription for bulk factorization, we firstconstruct M ◦ A NM by decomposing M A NM into a disjoint sum of the following two components: M ◦ , A NM = Y SM (27)and M ◦ , A NM = Y S N (28)Each component is a full torus with corners, with the boundary torus divided into two parts. Y S and Y S constitutes the ”outer” and ”inner” part of the boundary of M ◦ , A NM and M ◦ , A NM respectively.8he remaining boundary parts constitute the boundary of M A NM . The manifold M X ; q q γδ isobtained by composing M ◦ A NM with T q q γδ . Following the prescription of the previous sectionwe glue M ◦ , A NM and M ◦ , A NM to T q q γδ . The component M ◦ , A NM is readily composed with T q q γδ byidentifying Y S and Y T .The composition of M ◦ , A NM with T q q γδ is straightforward as well but needs a bit explanation.First of all it has to be glued with the black side of the ribbon graph facing upwards in orderto match the A -ribbon in T q q γδ . Second, think of M ◦ , A NM as a cylinder with the two oppositeboundary discs identified, i.e. as D × [ − ,
1] with the discs D × {− } and D × { } identified. Thecomposition is then performed by first identifying Y S with Y T and afterwards identifying D ×{− } with D × { } . The result is a cobordism ( M A NM ) q q ,γδ , which is a ribbon graph in D × S :( M A NM ) q q ,γδ = ¯ q q q N φ γ q ¯ q φ δ M (29)Again S is running vertically with top and bottom identified. Here we have also deformed theribbon graph by a π rotation of the part of the ribbon graph that shows its black side. The upperhalf of the ribbon graph can be interpreted as a morphism in Hom( U ¯ q , U q ). This morphism canbe non-zero only if ¯ q = q . Consequently, the invariant is non-zero only if q = ¯ q . Thus, applying(23) the annulus partition function can be written as A NM = X q ∈I Z q ¯ q X γ,δ =1 dim( U q ) ( c bulk − q, ¯ q ) γδ Z (( M A NM ) q ¯ q,γδ ) . (30) When an extended surface E appears as the boundary of a three-manifold M , there is a canonicalchoice of Lagrangian subspace given by the kernel of the inclusion map H ( E, R ) → H ( M, R ).The purpose of the Lagrangian subspace is to define the surface unambiguously. Let E be a torusand denote by the A -cycle the cycle that does not become contractible when E appears as theboundary of a full torus, and let the B -cycle be the other one. The canonical choice of Lagrangian A ribbon with its preferred orientation is displayed as a solid line, whereas a dashed line, like the upper A -ribbon in (22), indicates that the ribbon is endowed with the opposite orientation. We refer to these to orientationsas that the ribbon is showing its ”white side” and its ”black side” respectively. H ( ∂ M A NM , R ) and H ( ∂ M ∗ χ ; k , R ) is spanned by the B -cycle. When we extract structureconstants in (26) by composing M A NM and M ∗ χ ; k we do this with the two B-cycles aligned.The relation (23) involves cutting out a full torus and gluing it back after an S -transformation.As a consequence, the factorization procedure exchanges the A - and B -cycles on ∂ ( M A NM ) q ¯ q,γδ compared to ∂ M A NM . Therefore, also the Lagrangian subspace is changed, such that in H ( ∂ ( M A NM ) q ¯ q,γδ , R ) it is spanned by the A -cycle. Thus, when extracting the annulus coefficient A NkM , after factorization, the manifold M ∗ χ ; k has to be glued to ( M A NM ) q ¯ q,γδ with the B -cycle on ∂ M ∗ χ ; k aligned with the A -cycle on ∂ ( M A NM ) q ¯ q,γδ . The resulting cobordism ( A NkM ) qγδ is a ribbongraph in S : ( A NkM ) qγδ = ¯ qq ¯ q qN φ γ kφ δ M (31)Combining with (30), the annulus coefficients can be written as A NkM = X q ∈I Z q ¯ q X γ,δ =1 dim( U q ) ( c bulk − q, ¯ q ) γδ Z (( A NkM ) qγδ ) . (32)In general, the choice of Lagrangian subspaces is related to an anomaly of the tft C -functor undergluing. However, in the case at hand the extended surfaces are doubles, which come with anorientation reversing involution. In this case this anomaly factor is unity, see [7, Lemma 2.2].Next we evaluate Z (( A NkM ) qγδ ). The invariant of a ribbon graph in S is calculated as follows:First we project the ribbon graph to R and interpret it as a morphism in C . The result is anendomorphism of the tensor unit, i.e. a complex number. The invariant of the cobordism in S isthis number multiplied by S , . Thus, we obtain after some manipulations A NkM = dim( M )dim( N ) X q ∈I S k,q θ q Z q ¯ q X γ,δ =1 ( c bulk q ¯ q ) − δγ b q,γN b ¯ q,δM . (33)The number b q,γN , with q ∈ I and N a simple A -module, is a so-called reflection coefficients,c.f. [15, eqs. (3.24) & (3.26)]. b q,γN is related to the single structure constant, c (Φ γ ; N ), of theone-point correlator of the disc with boundary condition N and a single bulk field, labeled by φ γ ∈ Hom A | A ( U q ⊗ + A ⊗ − U ¯ q , A ), by c (Φ γ ; N ) = dim( N ) b q,γN . To arrive at the expression (33) wefirst remove the annular U k -ribbon, which yields a factor S k,q S , . Second, we use dominance inEnd( U ¯ q ⊗ U q ), which separates the morphism into two morphisms, each of them proportional to areflection coefficient. Simplifying the results by braiding and fusion moves we arrive at (33).10he reflection coefficients can be calculated by evaluating the morphisms[15, eq. (3.24)] in C , corresponding to the one-point functions. However, they also appear asrepresentation matrices of a semisimple associative complex algebra A , the classifying algebra[13]. As a vector space, A is given as the space of primary bulk fields with non-zero correlator onthe disc: A := M q ∈I Hom A | A ( U q ⊗ + A ⊗ − U ¯ q , A ) . (34)The irreducible representations of A are all one-dimensional and are labeled by simple modulesover A in C . Choosing a basis { φ q,α | α = 1 , ..., Z q ¯ q } of Hom A | A ( U q ⊗ + A ⊗ − U ¯ q , A ), the representationmatrices are ρ M ( φ q,α ) = b q,αM . Furthermore, A is equipped with a non-degenerate bilinear form, ω .In the basis { φ q,α } , the bilinear form is given by ω ( φ p,α , φ q,β ) = ω pα,qβ where ω pα,qβ = [ θ p dim( U p ) c bulk00 ] − δ ¯ q,p c bulk p ¯ p,αβ , (35)c.f. [13, eq. (4.26)]. ω is a dim( A ) × dim( A ) block matrix, where each block, labeled by p ∈ I , isproportional to c bulk p ¯ p . Combining (33) and (35), we can rewrite the annulus coefficient A NkM as A NkM = dim( M )dim( N ) S , dim( A ) X q ∈I S k,q S ,q Z q ¯ q X γ,δ =1 ( ω − ) ¯ qδ,qγ b q,γN b ¯ q,δM . (36)Thus, much of the significant information concerning the annulus partition functions is containedin A and its representation theory.We conclude by comparing (33) with some previous results. In the Cardy case, i.e. when A is Morita equivalent to the tensor unit, the irreducible boundary conditions are labeled by simpleobjects M = U m and N = U n in C . The matrix ( c bulk q ¯ q ) − δγ is a scalar given by S q, θ q , and the reflectioncoefficients are S n, ¯ q dim( U m ) S q, and S m,q dim( U m ) S q, respectively. Combining this with the Verlinde formulawe obtain from (33) A nkm = N nkm . (37)This result was established already in [5], and it also follows directly from e.g.[2, eq. (2.16)] or [9, eq. (5.119)].In [9, Theorem 5.20] some more results on the annulus coefficients are listed. The result (36)corresponds to point (iv) in that list, with the difference that (36) is written in a more symmetricalmanner. Furthermore, using S ¯ q, ¯ k = S q,k and ( c bulk q ¯ q ) − δγ = ( c bulk¯ qq ) − γδ , the result [9, Theorem 5.20 (iii)] A NkM = A M ¯ kN (38)is reproduced. Finally, combining [9, Theorem 5.20 (ii)], which states A N M = δ M,N , with (36), weobtain an orthogonality relation for the representations of the classifying algebra: S , X q ∈I Z q ¯ q X γ,δ =1 µ − qδ,qγ b q,γN b ¯ q,δM = dim( A )dim( M )dim( N ) δ M,N . (39) Acknowledgements:
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A classifying algebra for CFT boundary con-ditions licentiate thesis, Karlstad University Press 2009, [http://kau.diva-portal.org/smash/record.jsf?searchId=1&pid=diva2:276168][http://kau.diva-portal.org/smash/record.jsf?searchId=1&pid=diva2:276168]