aa r X i v : . [ m a t h - ph ] S e p MPP-2020-172
CARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS
MATTHIAS TRAUBE
Abstract.
In [SY19] it was shown how string-net spaces for the Cardy bulk algebra inthe Drinfeld center Z ( C ) of a modular tensor category C give rise to a consistent set ofcorrelators. We extend their results to include open-closed world sheets and allow for moregeneral field algebras, which come in the form of ( C | Z ( C ))-Cardy algebras. To be moreprecise, we show that a set of fundamental string-nets with input data from a ( C | Z ( C ))-Cardy algebra gives rise to a solution of the sewing constraints formulated in [KLR14] andthat any set of fundamental string-nets solving the sewing constraints determine a ( C | Z ( C ))-Cardy algebra up to isomorphism. Hence we give an alternative proof of the results in[KLR14] in terms of string-nets. Contents
1. Introduction 12. Categorical Preliminaries 32.1. Frobenius Algebras in Tensor Categories 72.2. Drinfeld Center 83. Cardy Algebras 94. String-Net Spaces 125. World Sheets, Sewing Constraints and the Block Functor 145.1. Generating Set and Sewing Constraint Relations 166. Consistent Correlators from String-Nets 176.1. Fundamental Correlators on Generating World Sheets 186.2. From Sewing Constraints on String-Net Spaces to Cardy Algebras 247. Conclusion 26Appendix A. Generating World Sheets and Sewing Relations 27A.1. Generating World Sheets 27Appendix B. Sewing Relations 27References 301.
Introduction
String-net spaces were originally introduced by Levin and Wen in [LW05] in order to describephenomena of topological phases of matter on surfaces. Roughly speaking a string-net is anequivalence class of an embedded graph on a surface S with boundary. Based on earlier works[KKR10][KMR10] a precise mathematical formulation of string-nets was given by Kirillov in[Kir11] and it can be seen as a higher genus enhancement of the graphical calculus for tensorcategories where the usual relations hold on any embedded disk. Similar discussions of categorieson surfaces have appeared in [HK19][Har19] for the case of a cylinder. In a series of papers[KB10][Bal10a][Bal10b][Kir11] it was shown how an appropriately defined vector space of formal Date : September 28, 2020. linear combinations of string-nets is equivalent to the state space of the Turaev-Viro threedimensional topological field theory (TFT) based on C or equivalently to the state space of theReshetikhin-Turaev (TFT) for Z ( C ).Two dimensional rational conformal field theory (RCFT) on the other hand has a cate-gorical description in the form of the FRS (Fuchs-Runkel-Schweigert)-formalism developed in[FRS02][FFRS06][FRS05][FRS04b][FRS04a], where the state space of the Reshetikhin-TuraevTFT features prominently. The monodromy data of an RCFT is described by a modular tensorcategory and the bulk field algebra is a Frobenius algebra H cl in Z ( C ). The space of closed con-formal blocks on a compact surface S gn | m of genus g with n incoming and m outgoing boundariesis Z RT, Z ( C ) ( H ∗ cl , . . . , H ∗ cl , H cl , . . . , H cl ) with n copies of H ∗ cl and m copies of H cl . A consistent setof correlators in the RCFT is an assignment of an element in Z RT, Z ( C ) ( H ∗ cl , . . . , H ∗ cl , H cl , . . . , H cl )for all surfaces S gn | m s.th. the vectors are invariant under the action of the mapping class groupand behave equivariant under sewings of surfaces. That is, there should exist a map for sewingcorrelators and sewn correlators should agree with the correlator on the sewn surface. Theequivalence between string-net spaces and spaces of conformal blocks was used in [SY19] to showthat a given set of genus zero string-nets with boundary value given by the Cardy-Frobeniusalgebra in Z ( C ) indeed give rise to consistent correlators.In this paper we generalize their result in two directions: Firstly we allow for world sheetswith open and closed field insertions. This leads to a formulation of sewing constraints interms of the category WS of open-closed world sheets given in [FFRS06]. String-nets on open-closed world sheets still give a symmetric monoidal functor BL : WS → Vect . Solving sewingconstraints for such a symmetric monoidal functor was reduced in [KLR14] to a set of 32relations, which have to be satisfied. The second generalization concerns the input data forthe construction. We allow arbitrary ( C | Z ( C ))-Cardy algebras as inputs. Based on an operadicformulation for vertex operator algebras (VOA), Cardy algebras were introduced in [Kon08a]and formulated entirely in terms of category theory in [KR09]. A ( C | Z ( C ))-Cardy algebraencodes the necessary data for an open-closed RCFT in genus one and zero and consists ofa triple ( H cl , H op , ι clop ), where H op , H cl are Frobenius algebras in C and Z ( C ) respectively,together with an algebra map ι cl − op : H cl → L ou ( H op ). Here L ou : C → Z ( C ) is the adjointfunctor to the forgetful functor. This data has to satisfy three conditions: modularity, the centerproperty and finally the Cardy condition. In [KLR14] it was shown, using the Reshetikhin-Turaev TFT, that Cardy algebras give rise to a consistent set of correlators and vice versa.The category WS is generated by a set(1.1) (cid:8) O m , O ∆ , O η , O ǫ , O prop , C m , C ∆ , C η , C ǫ , C prop , I, I † (cid:9) of fundamental world sheets. With the help of the ( C | Z ( C ))-Cardy algebra we define correlators(1.2) corr = (cid:8) corr opprop , corr opm corr op ∆ , corr opη , corr opǫ , corr clprop , corr clm , corr cl ∆ , corr clη , corr clǫ , corr I , corr I † (cid:9) in terms of string-nets and the first main result, which is Theorem 6.2, is Theorem I.
The set of correlators corr gives a solution to the sewing constraints for theconformal block functor
B l with boundary data determined by the ( C | Z ( C )) -Cardy algebra. Assuming a solution to the sewing constraints for
B l exists, we also show that the converseis true, which is our second main result (Theorem 6.9 in the text)
Theorem II.
An assignment of fundamental string-net correlators based on boundary data ( d H cl , d H op ) in C and Z ( C ) solving the sewing constraints determines a ( C | Z ( C )) -Cardy algebra ( H cl , H op , ι cl − op ) which is unique up to isomorphism. The proofs of the theorems very much use the fact that the graphical calculus in C carriesover to string-nets on surfaces. Hence the graphical representation of consistency relations forCardy algebras appear directly as string-nets on surfaces and can be manipulated accordingly.This renders the proof very tractable. As an example, figure 1 shows a correlator world sheet.Purple curves denote open insertions, orange ones correspond to closed insertion. Vertices standfor structure morphisms of the Frobenius algebras in The Cardy algebras and squares denote ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 3
Figure 1.
Example of an open-closed correlator on a genus 1 surface withtwo open insertions and one closed insertion.open-closed embedding maps. In addition there are circle decorations where world sheets areglued. All of these ingredients will be explained in the core of the text.The paper is organized as follows. In section 2 we give an account of the necessary categoricalpreliminaries, including modular tensor categories, the Drinfeld center and Frobenius algebrasin tensor categories. In section 3 we motivate Cardy algebras and recall their definition from[KR09]. In section 4 string-net spaces are discussed based on [Kir11]. In all of the paper weabusively use the term string-net to refer to an whole equivalence class of string-nets. Thedefinition of the category of world sheets WS from [FFRS06] and the formulation of sewingconstraints given in [KLR14] is recalled in section 5. Section 6 is the main part of the paper andcontains the precise formulation of the above theorems, as well as their proofs. In the appendixwe display the generating world sheets and the sewing relation for the readers convenience. Acknowledgement:
The author thanks Ralph Blumenhagen for valuable discussions andhaving a look on a first draft of this paper.2.
Categorical Preliminaries
As stated in the introduction two dimensional (rational) conformal field theory can be conve-niently described in categorical terms. This section serves as a reminder of the necessary terms,together with the relevant graphical calculus. A classical source for the material presented hereis e.g. [EGNO16].Let C be an abelian K -linear category, i.e. for any morphism φ , Ker( φ ), Coker( φ ) and Im( φ )exist and moreover for A , B ∈ C , Hom C ( A, B ) is a K -vector space. A monoidal structureon C is a bilinear bifunctor ⊗ : C × C → C with associativity and unit natural isomorphismswhich are assumed to be identities in this paper. Thus we always consider strictly monoidalcategories. The unit object for ⊗ is denoted by . A braiding on ( C , ⊗ ) is a natural isomorphism β A,B : A ⊗ B → B ⊗ A . In the graphical calculus about to be introduced all diagrams run frombottom to top. Graphically β A,B is depicted by
MATTHIAS TRAUBE A B B A Figure 2. β A,B A B B A Figure 3. β − A,B .Besides a braiding, C is required to have dualities. For A ∈ C a right (left) dual is a triple( A ∗ , ev A , coev A ) ( ∗ A, g ev A , ^ coev A ) of an object A ∗ ( ∗ A ) with morphisms A A A A ev A coev A g ev A ^ coev A The morphisms satisfy straightening properties = with similar pictures for left duality. A monoidal category for which every object has rightand left duals is called rigid . In a rigid category there is the obvious isomorphism ∗ ( A ∗ ) ≃ A ≃ ( ∗ A ) ∗ . A pivotal structure in ( C , ⊗ ) is a natural isomorphism π : Id C → ( • ) ∗∗ . In fact anypivotal category is equivalent to a strict pivotal category, i.e. π = Id Id C . It easily follows thatfor a strict pivotal category left and right dualities coincide. In a rigid, strictly pivotal categorythere are left and right traces for endomorphisms f tr r ( f ) tr l ( f ). If tr( f ) = tr r ( f ) = tr l ( f ) holds in C , C is called spherical . We introduce the notation d A = tr(Id A ). A twist on ( C , ⊗ , β ) is a natural isomorphism θ : Id C ⇒ Id C satisfying θ A ⊗ B =( θ A ⊗ θ B ) ◦ β B,A ◦ β A,B . The twist isomorphism is depicted as.A rigid, pivotal, braided tensor category is ribbon if it has twist. Having defined all thenecessary structure on C we also want to control its size. Thus we require morphism vectorspaces to be finite dimensional. Recall that an object A ∈ C is simple if Hom C ( A, A ) ≃ K Id C .If the set of isomorphism classes of simple objects is finite and every object is isomorphic to a ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 5 direct sum of finitely many simple objects, the category is called semisimple . A ribbon category C is fusion if it is semisimple. Being semisimple has far reaching consequences, e.g. for anyobject there are maps b αi : A ≃ → L i ∈ I ( C ) U ⊕ n i i → U i , with α ∈ { , . . . , n i } , where the secondmap is the projection to the α ’s U i summand and a dual map b jβ : U i → L i ∈ I ( C ) U ⊕ n i i ≃ → A .These maps are dual in the following sense(2.1) X i ∈ I ( C ) n i X α =1 b iα ◦ b αi = Id A , b αi ◦ b jβ = δ ij δ αβ Id U i . Graphically we represent the duality as A = iAαα X i ∈ I ( C ) n i X α =1 A and = αβ iAj i δ ij δ αβ In addition for a semisimple category d i = tr ( U i ) = 0 and the global dimension is defined as D = P i ∈ I ( C ) d i . We state one more relation, which will be used in the proof of theorem 6.2.A proof can be found in e.g. [BK01, Corollary 3.1.11]. Lemma 2.1.
For l ∈ I ( C ) it holds X k ∈ I ( C ) d k D l k = lδ l, . One more ingredient for a modular tensor is needed, namely the S -isomorphism i j s ij = Definition 2.2. A modular tensor category is a spherical, ribbon fusion category s.th. s ij isinvertible for all isomorphism classes i = [ U i ], j = [ U j ] of simple objects. MATTHIAS TRAUBE In C there exists a pairing(2.2) Hom C ( A, B ) ⊗ K Hom C ( B, A ) → K f ⊗ g ( f, g ) ≡ tr( g ◦ f )which for C semisimple is non-degenerate. We are mainly interested in morphism spacesHom C ( , • ) for which we introduce some notation. Definition 2.3.
Let A , . . . , A n ∈ C , then(2.3) h A , . . . , A n i ≡ Hom C ( , A ⊗ · · · ⊗ A n ) Lemma 2.4.
There is a functorial isomorphism of vector spaces h A , . . . , A n i ≃ h A n , A , . . . , A n +1 i .Proof. The functorial isomorphism is given by A A n φ φA n which clearly has an inverse given by composing inverses of braiding and twist. (cid:3) As only the cyclic order of h A , . . . , A n i matters, instead of rectangular boxes for morphismswe introduce coupons in the graphical calculus φA A n ≡ φA A n For an arrow oriented towards the coupon with label A the respective element gets replaced by A ∗ . Coupons can be composed with the help of the evaluation morphisms. ≡ φ ψ φ ψ The following lemma can be found in [KB10]
Lemma 2.5.
For any A ∈ C there are isomorphismsa) = i A iφ ψ ( ψ,φ ) d i i b) For (cid:8) b iα (cid:9) a basis in h i, A ∗ n , . . . A ∗ i with dual basis { b αi } in h i, A , . . . A n i in the sensethat ( b iα , b βj ) = δ ij δ αβ it holds i P i ∈ I ( C ) d i A n A A n A = b b A A n where the b - b coupons indicate summation over basis and dual basis. The second relation is called completeness property and will be used several times.
ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 7
Frobenius Algebras in Tensor Categories.
Frobenius algebras are usually defined asassociative algebras on finite dimensional vector spaces having a non-degenerate bilinear formcompatible with the algebra multiplication. In this form Frobenius algebras correspond to twodimensional TFTs, see e.g. [LP08]. The notion of a Frobenius algebra has an enhancementto categories and the above notion correspond to a Frobenius algebra in
Vect , the category offinite dimensional vector spaces.
Definition 2.6.
Let ( A , ⊗ , ) be a tensor category. A Frobenius algebra in A with underlyingobject A consists of morphisms m : A ⊗ A → A η : → A ∆ : A → A ⊗ A ǫ : A → where all strands are colored with A . These have to satisfy:I) ( m, η ) define an associative algebra on A := = =II) (∆ , ǫ ) define a coassociative coalgebra on A := = =III) The Frobenius properties hold= =.If A is in addition pivotal, we can ask for ( A, m, η, ∆ , ǫ ) to be symmetric , i.e. there is an equalityof morphisms =For A braided, we can require ( A, m, η, ∆ , ǫ ) to be (co-)commutative, i.e. MATTHIAS TRAUBE = =We will use Frobenius algebras in modular tensor categories. If the category in question isthe representation category of a rational vertex operator algebra this corresponds to RCFTsvery much like Frobenius algebras in
Vect correspond to TFTs.2.2.
Drinfeld Center.
For B a strictly monoidal category its Drinfeld center Z ( B ) has objects( B, β B, • ) where β B, • : B ⊗ • ⇒ • ⊗ B is a natural isomorphism called half braiding s.th.(2.4) β A,B ⊗ C = (Id B ⊗ β A,C ) ◦ ( β A,B ⊗ Id C ). Morphisms in Z ( B ) are morphisms f ∈ Hom B ( A, B ) s.th.(2.5) β B,C ◦ ( f ⊗ Id C ) = (Id C ⊗ f ) ◦ β A,C . The Drinfeld center becomes a monoidal category with tensor product(2.6) (
A, β A, • ) ⊗ ( B, β B, • ) = ( A ⊗ B, β A ⊗ B, • ) , β A ⊗ B,C = ( β A,C ⊗ Id B ) ◦ (Id A ⊗ β B,C )Note that B does not need to be braided. But Z ( B ) is naturally braided with braiding given by(2.7) β Z ( B )( A,β A, • ) , ( B,β B, • ) = β A,B . There exists an obvious forgetful functor F : Z ( B ) → B forgetting the half braiding. For B amodular tensor category it is shown in [KR09] that the adjoint of the forgetful functor reads(2.8) L ou : B → Z ( B ) B L ( B ) = M i ∈ I ( B ) B ⊗ U ∗ i ⊗ U i , β L ( B ) , • with over-under half braiding(2.9) β ouL ou ( B ) ,A = O i ∈ I ( C ) ( β B,A ⊗ Id U ∗ i ⊗ U i ) ◦ (Id B ⊗ β U ∗ i ,A ⊗ Id U i ) ◦ (Id B ⊗ U ∗ i ⊗ β − U i ,A )which has graphical representation β ouL ou ( B ) ,A = M i ∈ I ( C ) B i i A
On morphisms the functor is defined as L ou ( f ) = L i ∈ I ( C ) f ⊗ Id U ∗ i ⊗ U i . Note that this is afaithful functor [KR09, Lemma 2.22], but not a tensor functor since it doesn’t map the identityon B to the identity of Z ( B ). Nevertheless it transports Frobenius algebras from one categoryto the other preserving symmetry. Proposition 2.7. [KR09, Proposition 2.25]
For A a Frobenius algebra in B , the object L ou ( A ) has the structure of a Frobenius algebra in Z ( B ) . In addition, A is symmetric special, if andonly if L ou ( A ) is symmetric special. For finite categories A , B there exist a tensor product of categories. ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 9
Definition 2.8.
The
Deligne tensor product A ⊠ B has objects finite sums(2.10) M A i ⊠ B i , A i ∈ A , B i ∈ B and morphism spaces(2.11) Hom A ⊠ B ( A ⊠ B , A ⊠ B ) = Hom A ( A , A ) ⊗ K Hom B ( B , B ) . If A , B are fusion, their tensor product is also fusion with representatives for simple objectsgiven by U i ⊠ V j where U i ( V j ) are representatives of simple objects in A ( B ). For A , B braidedtensor categories, A ⊠ B is also a braided tensor category with tensor product(2.12) ( A ⊠ B ) ⊗ ( A ⊠ B ) ≡ ( A ⊗ A ) ⊠ ( B ⊗ B )and braiding(2.13) β A ⊠ B ( A ⊠ B ) , ( A ⊠ B ) = β A ,A ⊠ β B ,B For a finite ribbon category C let C be the category with the same objects and morphisms butwith inverse braiding and twist. In its most general form (even dropping semisimplicity) thefollowing theorem is proven in [Shi19]. Theorem 2.9. [Shi19, Theorem 3.3]
Let C be a finite ribbon category. If C is modular there isa braided equivalence (2.14) C ⊠ C → Z ( C )( A ⊠ B ) ( A ⊗ B, β ouA ⊗ B, • ) . Conversely if C ⊠ C ≃ Z ( C ) are braided equivalent, C is modular. This implies in particular that the finite set of simple objects in Z ( C ) is given by ( U i ⊗ U j , β ouij, • )for i, j ∈ I ( C ). 3. Cardy Algebras
In a series of paper [HK07][Kon08b][HK10][Hua91][Hua97a][Hua03][Hua05a][HK04] Huangand Kong gave a rigorous formulation of genus 0,1 two dimensional open-close conformal fieldtheory in the language of partial operads. A textbook account of the results appeared in[Hua97b]. The major outcome can be described in purely categorical terms and is given bythe notion of a Cardy algebra. The abstract formulation and its relation to sewing constraintswas developed in [KR09, KLR14]. Though we don’t need it in the core of the paper, we stillspend the next paragraph giving some intuition from CFT for the abstract formulation aboutto come.One way of formalizing two dimensional CFT is given by vertex operator algebras (VOA)which describe the chiral and antichiral symmetry algebras in full CFT. Roughly speaking aVOA encodes the operator-state correspondence and operator product expansions (OPEs). Ithas an underlying graded state space V and a vertex operator map Y : V → End( V )[[ z − , z ]],where z is a (formal) coordinate on the complex plane. There is a well studied notion ofrepresentations of VOAs including fully reducible and irreducible representations. Under theassumption that V is a rational VOA its representation category R V is a modular tensorcategory. Assuming that a CFT at hand has chiral symmetry algebra V L and antichiral sym-metry algebra V R , both of which are rational, the closed state space decomposes into a sum H cl = L ij N ij H Li × H Rj , where H Li , H Rj are the simple representations in R V L and R V R respectively. Hence it is naturally an object in R V L ⊠ R V R . From a physics perspective thecrucial object to compute are correlation functions, which in our situation split into productsof chiral and antichiral correlation functions. It is well know that chiral correlation functionshave an expansion in terms of so called conformal blocks which contain all the informationabout conformal weights and insertion points of the chiral insertions. Three point conformal For the precise definition see e.g. [HUA05b, Theorem 3.9]. blocks B ( v , v , v ) on the sphere for field insertions v i ∈ H Li at ( z , z , z ) = ( ∞ , z,
0) can bedescribed in terms of intertwining operators (3.1) Y ( z ) : H → Hom( H , H )[[ z − , z ]]satisfying(3.2) B ( v , v , v ) = h v , Y ( v , z ) v i H where h • , • i H is a well defined invariant inner product on H . The map Y ( z ) is said tobe of type (cid:0) H H H (cid:1) and the dimension of the vector space of intertwiners of type (cid:0) H k H i H j (cid:1) areprecisely the fusion rules N kij . Intertwining operators have an algebra structure, the so called intertwining operator algebra (IOA) . The upshot is that the OPE algebra in the CFT can beconveniently casted in the form of intertwining operators and the state space H cl inherits analgebra structure in R V L ⊠ R V R from the tensor product of chiral and antichiral IOAs. Underthe assumption that H cl carries a non degenerate invariant bilinear form (which we assume inthe presentation of intertwiners above already) H cl becomes a commutative Frobenius algebrawith trivial twist in R V L ⊠ R V R . The genus one enhancement is possible if it is in fact amodular invariant Frobenius algebra, a notion we discuss shortly in categorical terms. So farthe discussion was solely for closed states.Including boundaries one also has to consider open states, which due to their localization onsome boundary have only half of the symmetry of closed states. Hence a boundary CFT withsymmetry algebra V has an open state space H op = L i N i H i . Following the same lines as inthe closed case, the open state space becomes a symmetric Frobenius algebra in R V . Note thatit will in general not be commutative owing to the fact that field insertions on an interval can’tbe interchanged along the interval.Lastly boundary and bulk fields should interact, i.e. there are bulk-boundary OPEs for bulkfields approaching the boundary. This should correspond to a map ι cl − op : H cl × H op → H op satisfying certain compatibility relations. For this to work, we have to assume that left andright symmetry algebra of the closed theory agree. First of all, ι cl − op should be an algebra map,since taking first bulk OPEs and then approaching the boundary and taking bulk-boundaryOPEs better give the same as first approaching the boundary and taking bulk-boundary OPEsfollowed by taking boundary OPEs. Next it should be compatible with boundary OPEs andlastly it should commute with boundary OPEs as bulk fields can be transported along the bulkas shown in figure 4. Figure 4.
The closed field insertion is moved along the half circle throughthe bulk past the open insertion (red dot).Dually we could consider a map ι ∗ cl − op : H op → H cl mapping open field insertions to boundarystates in the closed theory. Though only discussed heuristically we note that this can be madeentirely concrete. It is shown e.g. in [Kon08a, Proposition 2.8] that an appropriate rescaling of ι ∗ cl − op exactly constructs Ishibashi states. It is well known that Ishibashi states in general don’tcorrespond to true boundary states. Only linear combinations of Ishibashi states satisfying the Cardy condition are valid boundary states due to open-close duality.As stated in section 2 for a modular tensor category C , there is a braided equivalence C ⊠ C ≃ Z ( C ). As an open-closed CFT necessarily has coincident left and right symmetry algebras theappropriate representation category for the closed theory to take place is R V ⊗ V ≃ R V ⊠ R V . We again refer to e.g. [HK07, section 3] for the precise details of invariant bilinear forms on representationsof VOAs.
ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 11
To match the description for string-net spaces we formulate Cardy algebras in terms of Z ( R V )instead of R V ⊠ R V . Definition 3.1.
Let (
A, m A , η A , ∆ A , ǫ A ) and ( B, m B , η B , ∆ B , ǫ B ) be Frobenius algebras in amonoidal category C and f ∈ Hom C ( A, B ). The morphism f † ∈ Hom C ( B, A ) is defined as f † = f B A
Definition 3.2. [KR09, Definition 3.7] Let C be a modular tensor category. A ( C | Z ( C )) -Cardyalgebra ( H cl , H op , ι cl − op ) is the data ofA) a commutative symmetric Frobenius algebra ( H cl , m cl , η cl , ∆ cl , ǫ cl ) in Z ( C ).B) a symmetric Frobenius algebra ( H op , m op , η op , ∆ op , ǫ op ) in C .C) a morphism ι cl − op ∈ Hom Z ( C ) ( H cl , L ou ( H op )).This has to satisfy the following conditionsI) H cl has to be modular , i.e. there is the equality= H cl i ⊗ jd i d j D H cl i ⊗ j α H cl i ⊗ j α X α H cl II) ι cl − op is an algebra homomorphism.III) The center condition holds: = H cl L ou ( H op ) L ou ( H op ) ι ι H cl L ou ( H op ) L ou ( H op )IV) The Cardy condition holds: = L ou ( H op ) L ou ( H cl ) ι † ι L ou ( H op ) H cl Definition 3.3.
A morphism between ( C | Z ( C ))-Cardy algebras ( H cl , H op , ι cl − op ) and ( G cl , G op , ι ′ cl − op )is a pair of maps f cl ∈ Hom Z ( C ) ( H cl , G cl ), f op ∈ Hom C ( H op , G op ) s.th.:I) Both, f cl and f op , are homomorphisms of Frobenius algebras.II) The following diagram commutes H cl G cl L ou ( H op ) L ou ( G op ) . f cl ι cl − op ι ′ cl − op L ou ( f op ) Using the map ( • ) † , it is not hard to show that any morphism of Frobenius algebras hasan inverse (see [KR09, Lemma 2.18]). Thus any morphism of Cardy algebras is in fact anisomorphism. 4. String-Net Spaces
String-net spaces can be seen as a higher genus enhancement of graphical calculus for spher-ical categories which reduces to the usual graphical calculus on every embedded disk. For asurface S with boundary ∂S , let Γ ⊂ S be an embedded, finite, oriented graph, which we alwaysconsider up to isotopy. Γ is required to intersect the boundary in one valent vertices of Γ. Foran edge e the same edge with reversed orientation is denoted by e . Definition 4.1.
Let C be a spherical fusion category. A C -coloring of Γ is an assignment of anobject A ( e ) ∈ C to any edge e s.th. A ( e ) = A ( e ) ∗ . To a vertex v with incident edges e , . . . , e n ,taken in counterclockwise order, a C -coloring assigns an element(4.1) φ ∈ h A ( o ) , . . . , A ( o n ) i where o i is the edge e i oriented away from v . Hence if an edge is oriented towards a vertexand colored with A , the morphism corresponding to the vertex is an element in h· · · A ∗ · · ·i .An isomorphism of colorings is a collection of isomorphisms f e : A ( e ) → B ( e ) respectingorientations and mapping φ ( v ) = f ◦ φ ′ ( v ). The boundary value for a C -colored graph is thepair ( p , A ) with p = { Γ ∩ ∂S } and A is the list of colorings of edges incident to the boundaryvertices.Colored vertices have the graphical representation as coupons introduced in section 2. Let D ⊂ S be an embedded closed disk whose intersection with V (Γ) are only one valent vertices.Let { A ( e ) , . . . , A ( e n ) } be the colors of edges of Γ intersecting ∂D , taken in counterclockwiseorder. Then there is a unique surjective evaluation map [Kir11](4.2) h•i D : Γ ∩ D → h A ( o ) , . . . , A ( o n ) i ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 13 , where we again use the notation o i for the edge e i with orientation towards the boundary ∂D . The evaluation map satisfies some natural properties stated in [Kir11, Theorem 3.4,Corol-lary 3.5]. The properties include local relations Γ ∩ D = Γ ∩ D which have to be understoodin the sense that h Γ ∩ D i D = h Γ ∩ D i D . One of these relations is e.g.=coev A A A A and the fact that vertices lying in a disk can always be merged into a single coupon.
Definition 4.2.
To any surface S there is an associated vector space(4.3) V Graph( S, A ) = formal finite K -linear combinations of colored graphs withboundary value ADefinition 4.3.
Let Γ = P x i Γ i ∈ V Graph( S, A ) and D ⊂ S be an embedded disk intersecting Γ transversally. The graph Γ is a null graph if Γ i | S \ D = Γ j | S \ D and(4.4) h Γ i D = X x i h Γ i i D = 0The vector space of all null graphs with fixed boundary value will be denoted N Graph( S, A ). Definition 4.4.
The string-net space on a surface with boundary value A is defined to be(4.5) H ( S, A ) = V Graph( S, A ) N Graph( S, A ) . So far boundary conditions are just sets of points on the boundary labeled by objects of C . Ofcourse boundary conditions should be subject to some natural relation as described in [Kir11,section 6], which turn them into a category of boundary conditions. Let N be an oriented onedimensional manifold and { p , . . . , p n } ⊂ N a finite subset of points. Let B ( N ) be the categorywith objects ( { p , . . . , p n } , { B , . . . , B n } ) ≡ B , where B i ∈ C and morphism spaces are givenby(4.6) Hom B ( N ) ( B , B ′ ) ≡ H ( N × I × ; B ∗ , B ) . The category of boundary values is defined to be the Karoubi envelope of B ( N ), which by abuseof notation will be denoted by the same symbol. This category has all the nice properties tobe expected, e.g. B ( N ) ≃ B ( N ′ ) for N ≃ N ′ and B ( N ⊔ N ′ ) ≃ B ( N ) ⊠ B ( N ′ ). Lemma 4.5. [Kir11, Theorem 6.4]
There are equivalences of categories B ( S ) ≃ Z ( C ) and B ( R ) ≃ C . Using this, one can give an enhancement of string-net spaces taking excited states on theboundary of a surface into account. Let S be a surface with boundary ∂S and B ∈ B ( ∂S ). Theextended string-net space is defined as the quotient(4.7) ˆ H s ( S, B ) = V Graph( S, B ) N Graph( S, B )where(4.8) V Graph( S, B ) = formal vector space of finite K -linear combinations of pairs ( f, Γ)with Γ a graph on S with boundary value A and f ∈ Hom B ( ∂S ) ( A , B )and(4.9) N Graph( S, B ) = subspace of null graphs under local relations as before plusrelation ( f γ, Γ) = ( f, γ
Γ) where γ ∈ Hom B ( ∂S ) ( B , B ′ ) ,f ∈ Hom B ( ∂S ) ( B ′ , A )The following is a result of a series of papers [KB10][Bal10b][Bal10b] and [Kir11, Theorem 7.3]. Theorem 4.6.
Let S be a compact oriented surface of genus g with boundary parameterizedcircles and objects A = (cid:8) A , . . . , A | π ( ∂S ) | = A n (cid:9) objects in Z ( C ) . Then there are isomorphisms (4.10) ˆ H s ( S, A ) ≃ Z T V, C ( S, A ) ≃ Z RT, Z ( C ) ( S, A ) ≃ Hom Z ( C ) ( , A ⊗ · · · ⊗ A n ⊗ ( L ) g ) where in the last vector space L = N i ∈ I ( Z ( C )) U i ⊗ U ∗ i . The following lemma is straightforward.
Lemma 4.7.
Let D be a closed disk and A ∈ C . Then ˆ H s ( D, L ou ( A )) = Hom C ( , A ) . Let P ∈ Hom B ( S ) ( A, A ) be the string-net= X k ∈ I ( C ) d k D k Figure 5.
Drinfeld center projectorwhich we call the projector and the circle winding around the circumference we call the projector circle . Graphically elements of the extended string-net spaces are represented bystring-nets with additional projectors P placed at each boundary component as shown in figure6. Figure 6.
Example of a projected string-net on a genus 1 surface.5.
World Sheets, Sewing Constraints and the Block Functor
In order to define a consistent system of correlators we have to give an appropriate categoryof open-closed world sheets for which our construction computes correlators. Since we are con-sidering open-closed world sheets, this needs a fair bit of data. Luckily an appropriate categoryis defined in [FFRS08, KLR14] and the first part of this section recalls this definition as well asthe notion of sewing constraints given in [KLR14, section 3.2]. Most of the problems concerningopen-closed world sheets is caused by properly disentangling open and closed boundaries, whichleads to the orientation double.
Definition 5.1. An open-closed world sheet is the data(5.1) ˆ S = ( ˜ S, ι S , B iS , B oS , or S , ord)whereA) ˜ S is an oriented topological surface with boundary ∂ ˜ S .B) ι S is an orientation reversing involution whose fixed point set is a submanifold. Thequotient S = ˜ S/ι S is a manifold and π S : ˜ S → S is a Z -bundle. ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 15 C) B iS , B oS is a disjoint partition of π ( ˜ S ) in incoming and outgoing boundary componentswhich is fixed by ι S . Fixed points of the induced map ι S : π ( ˜ S ) → π ( ˜ S ) are calledopen boundaries. The set of open boundaries is denoted B op and its complement in π ( ˜ S ) is B cl .D) or S : S → ˜ S is a global section.E) δ S : ∂ ˜ S → S is a boundary parameterization being a homeomorphism on every con-nected component s.th. δ S ◦ ι S = δ S , where • denotes complex conjugation. For a fixedpoint b ∈ π ( ˜ S ) of ι S , it has to hold δ | − b ( S ∩ H ) = Im(or) | b .F) ord : π ( ∂ ˜ S ) → n , . . . , | π ( ˜ S ) | o is an ordering function of boundary components forwhich we firstly demand that ord( o ) < ord( c ) for o ∈ B op and c ∈ B cl . Secondly,for a connected set P ⊂ ∂S having non-trivial intersection with a physical bound-ary (see the next paragraph) and ˜ P ⊂ ∂ ˜ S op with π S ( ˜ P ) ⊂ P there has to exist an n ∈ n , . . . , | π ( ˜ S ) | o s.th. ord( ˜ P ) = n n, n + 1 , . . . , n + | ˜ P | − o where the ordering ofcomponents is cyclically along the orientation of P . Definition 5.2. A sewing of a world sheet ˆ S has dataA) a subset S B ⊂ B iS × B oS s.th. if ( i, o ) ∈ S B there are no elements ( i, o ′ ) or ( i ′ , o ) in S B .B) for ( i, o ) ∈ S B it follows that ( ι S ( i ) , ι S ( o )) ∈ S B .C) either ( i, o ) ∈ B iop × B op or ( i, o ) ∈ B i ; l,rcl × B o ; l,rcl .The sewn world sheet [ S ( S ) hasI) ] S ( S ) = ˜ S/ ∼ , where δ | − i ( z ) ∼ δ − | o ( − z ). Let π S : ˜ S → ] S ( S ) be the projection.II) condition B) ensures that there is a well defined involution ι S ( S ) defined via ι S ( S ) ◦ π S = π S ◦ ι S .III) B i S ( S ) = (cid:8) i ∈ B iS | ( i, • ) / ∈ S B (cid:9) and B o S ( S ) = { o ∈ B oS | ( • , o ) / ∈ S B } IV) or S ( S ) is the unique section whose image in ] S ( S ) is π S ( S ) ◦ or S .There is an additional requirement on the ordering function. For the details we refer to[KLR14]. Note that the glueing defined above gives in addition a glueing projection S S : S → S ( S ) given by S S = π S ( S ) ◦ π S ◦ or S . We will be mainly concerned with S instead of its orientationdouble ˜ S . Its boundary components decompose into three different types: open, closed andphysical. Figure 7.
The quotient surface of a genus 3 open-closed world sheet withclosed boundaries shown in purple. Open boundaries are colored green andphysical boundaries are red.i) Closed boundaries: A point p ∈ ∂S is on a closed boundary if its preimages under π S lieon different connected components of ∂ ˜ S . This implies that connected components ofclosed state boundaries are homeomorphic to S . Their preimages are pairs ( b cl , ι S ( b cl )) of connected components of boundaries in ˜ S and or S identifies them with one of thetwo boundaries.ii) Open boundaries: A point p ∈ ∂S is on an open boundary if its preimages under π S are on the same connected component in ∂ ˜ S . Hence its preimage is on a component b op ∈ B op . Since ι S was orientation reversing it acts on b op as a reflection. A reflectionon S has two fixed points and connected components of open boundaries map to oneof the open intervals stretching between the fixed points.iii) Physical boundaries: p ∈ ∂S is on a physical boundary if its preimage is on the fixedpoint set of ι S . In particular, preimages of physical boundaries aren’t boundary com-ponents of ˜ S except for the fixed points of ι S on open components of π ( ˜ S ). Ratherthey correspond to curves on ˜ S s.th. cutting ˜ S along the curves results in two copiesof S mapped to each other by the involution. Definition 5.3. A homeomorphism of world sheets is a homeomorphism F : ˜ S → ˜ T s.th.(5.2) F ◦ ι S = ι T ◦ F, δ T ◦ F = δ S , F ( B i ; oS ) = B i,oT , F ◦ Im(or S ) = Im(or T )The last point implies in particular, that f steps down to a homeomorphism f : S → T preserving all types of boundaries. Definition 5.4.
The category of world sheets WS has objects world sheets and morphismsHom WS ( ˆ S, ˆ T ) are given by pairs ( S , F ) where S is a sewing of ˆ S and F : ] S ( S ) → ˜ T is ahomeomorphism of world sheets. For the definition of the composition we refer to [KLR14]. WS is a symmetric monoidal category with the usual tensor product given by disjoint union.In addition two morphisms ( S , F ), ( S , F ) in WS are homotopic if S = S and F , F areisotopic maps. Definition 5.5.
Let
Fun ⊗ ( WS , Vect ) be the category of symmetric monoidal functors assigningthe same map to homotopic sewings. Morphisms are monoidal natural transformations.5.1.
Generating Set and Sewing Constraint Relations.
The category WS has an over-complete set of generating world sheets { S i | i ∈ G } which we give in appendix A. It is generat-ing in the sense that for any other world sheet S , there exists a list of generating world sheets S , . . . , S n and a morphism ( S , F ) : S ⊗· · ·⊗ S n → S . Non of this data needs to be unique. Thegenerating set allows to reduce the discussion of functors and natural transformations almostcompletely to the generating set and a set of relation among them. To be precise, considertriples of generating data ( S, { S i , } , ( S , F )) and functors Φ , Ψ ∈ Fun ⊗ ( WS , Vect ). In additionassume that Ψ(( S , F )) is an invertible linear map. Consider a collection of linear maps(5.3) g i : Ψ( S i ) → Φ( S i ) , i ∈ G defined for the generating set. To any world sheet S one can associate the map(5.4) G ( S ) ≡ Φ(( S , F ) ◦ ( g i ⊗ · · · ⊗ g i r ) ◦ Ψ(( S , F )) − where ( S , F ) : S i ⊗ · · · ⊗ S i r → S is the morphism from the generating property. The followingtheorem is the crucial simplification for the discussion of natural transformations. Theorem 5.6. [KLR14, Theorem 2.8]
Let Φ , Ψ ∈ Fun ⊗ ( WS , Vect ) , g i and G as above. Then G is a monoidal natural transformation if (5.5) G ( R i,l ) = G ( R i,r ) for { R i,l,r } the 32 fundamental world sheet sewings given in appendix B. There is an obvious symmetric monoidal functor : WS → Vect called the trivial functor .It maps any world sheet to K and any morphism to the identity on K . The following definitionof a solution to the sewing constraints is originally due to [FFRS08]. Definition 5.7.
A symmetric monoidal functor Θ ∈ Fun ⊗ ( WS , Vect ) satisfies the sewing con-straints if there is a monoidal natural transformation(5.6) ∆ : ⇒ Θ . ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 17
We briefly explain why this is a sensible definition for a solution of the sewing constraints.First of all the monoidal natural transformation ∆ picks a vector in Θ( ˆ S ) for any world sheet ˆ S .In physics terms one may call this the correlator of the surface. Recall that correlators in CFTon any surface should be invariant under the action of the mapping class group. An elementof the mapping class group gives a morphism ( ∅ , f ) and the functor assigns the identity toit. Thus Θ( ∅ , f ) has to map the correlator onto itself by naturality. By the same argument oftriviality for and naturality, correlators on lower genus surfaces are sewn to correlators onhigher genus surfaces. Hence this definition nicely captures all the features expected from aconsistent set of correlators.6. Consistent Correlators from String-Nets
For a consistent set of correlators we need a sensible functor of open-closed conformal blocks.This is achieved with the help of string-nets spaces. Let ( H cl , H op , ι cl − op ) be a ( C | Z ( C ))-Cardyalgebra. To a world sheet ˆ S i op ,i cl | o op ,o cl with i op / i cl incoming open/ closed boundaries and o op / o cl outgoing open / closed boundaries we associate the vector space(6.1) B l ( ˆ S ) = ˆ H s (cid:16) S, L ou ( H op ) , H cl (cid:17) where we introduced the notation L op ( H op ) = ^ L ou ( H op ) ⊗ · · · ⊗ ^ L ou ( H op ) i op + o op with(6.2) ^ L ou ( H op ) = ( L ou ( H op ) , for outgoing boundary L ou ( H op ) ∗ , for incoming boundary . and similar for H cl . Factors of incoming and outgoing insertions in the tensor product areinserted in the order given by the ordering function on the world sheet. Hence we assign toa world sheet the string-net space on its quotient surface decorated by the ingredients of theCardy algebra. To the sewing part of a morphism ( S , F ) in WS , B l assigns the map on string-net spaces given by stacking string-nets according to the induced sewing on S . Since the map F steps down to a homeomorphism on quotients, it gives a linear map on string-net spaces.Thus morphisms get mapped to the operation of stacking string-nets followed by a linear map.The following is straightforward. Proposition 6.1.
The map
B l : WS → Vect is a symmetric monoidal functor.
It is worthwhile analyzing which vector spaces the functor assigns to generating world sheets.I)
Open World Sheets:
The quotient surfaces of open generating world sheets are allhomeomorphic to a disk, though with different numbers of incoming and outgoing openboundaries. For a disk D ( n i , n o ) with n i incoming open boundaries and n o outgoingopen boundaries we get the vector space(6.3) B l ( D ( n i , n )) = ˆ H s ( D, L ou ( H op )) = Hom Z ( C ) ( , L ou ( H op )) ≃ Hom C ( , H op ) . II)
Closed World Sheets:
In this case the quotient surface is topologically a spherewith n i incoming closed boundaries and n o outgoing closed boundaries. With the samenotation as in the open case we get the vector space(6.4) B l ( S ( n i , n o )) = ˆ H s ( S , H cl ) = Hom Z ( C ) ( , H cl ) . III)
Open-Closed World Sheets:
Finally open-closed generating world sheets get mapped(6.5)
B l ( I ) = ˆ H s ( I, L op ( H op ) ⊗ H ∗ cl ) ≃ Hom Z ( C ) ( , L op ( H op ) ⊗ H ∗ cl ) B l ( I † ) = ˆ H s ( I † , L op ( H op ) ∗ ⊗ H cl ) = ≃ Hom Z ( C ) ( , L op ( H op ) ∗ ⊗ H cl ) . Thus
B l associates to generating world sheets the vector spaces expected from the categoricaldescription of RCFTs.
Fundamental Correlators on Generating World Sheets.
For a consistent systemof correlators we have to give fundamental correlators on generating world sheets and showthe sewing constraints for this set of correlators. We start by defining fundamental correlatorsfor Cardy algebra ( H cl , H op , ι cl − op ). In the following purple curves always denote edges ofgraphs colored by L ou ( H op ). Orange curves are edges colored by H cl . Incoming and outgoingboundaries should be clear from the orientation of edges. Trivalent disk-shaped vertices eitherdenote multiplication or comultiplication in Frobenius algebras H cl , L ou ( H op ). Similarly one-valent disk-shaped vertices are either unit or counit. In both cases orientation of edges fix thekind of morphism assigned, so we suppress another graphical distinction between the two.I) Open World Sheets: corr opprop corr opm corr op ∆ corr opη corr opǫ Figure 8.
Open fundamental correlators.Note that for these world sheets the inserted projector is absent as it can be isotopedto a point and therefore vanishes.II)
Closed World Sheets: corr clprop corr clm corr cl ∆ corr clη corr clǫ Figure 9.
Closed fundamental correlators.III)
Open-Closed World Sheets:
World sheets I , I † are topologically cylinders and a single projector line is inserted.Boxes denote the morphisms ι cl − op or ι † cl − op , where again the orientation of edgesdisplayed fixes the type of morphism. ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 19 corr I corr I † Figure 10.
Open-closed fundamental correlators.We are now ready to state and prove the first main result of the paper.
Theorem 6.2.
The correlators (6.6) (cid:8) corr opprop , corr opm , corr op ∆ , corr opη , corr opǫ , corr clprop , corr clm , corr cl ∆ , corr clη , corr clǫ , corr I , corr I † (cid:9) satisfy the sewing constraints. In order to show the theorem we have to show that the correlators on both world sheets forall 32 relations in appendix B agree. We split the proof in several lemmas.
Lemma 6.3.
The correlators (6.7) (cid:8) corr opprop , corr opm , corr op ∆ , corr opη , corr opǫ (cid:9) satisfy all open relations.Proof. This is the easiest part of the theorem as string-nets on disks can be manipulatedaccording to the graphical calculus of its coloring category. It is immediate that the relationsdirectly follow from the fact that L ou ( H op ) is a symmetric Frobenius algebra in Z ( C ). RelationsR1)-R4) are unit and counit properties. R5) is satisfied as L ou ( H op ) is a symmetric Frobeniusalgebra. Relations R6) and R7) are (co-)associativity for (co-)multiplications. Next, R8) andR9) are the Frobenius property and finally the last four relations R10)-R13) are just the factthat composing with corr opprop leaves any morphism invariant in the graphical calculus. (cid:3) Lemma 6.4.
The correlators (6.8) (cid:8) corr clprop , corr clm , corr cl ∆ , corr clη , corr clǫ (cid:9) satisfy all closed relations.Proof. Again the first six relations R14)-R19) directly follow from (co-)unit properties andisotopy of the diagrams when composing with the propagator diagram. R20) and R21) followagain from (co-) associativity plus the fact that projector circles can be dragged from one sideto the other. That this is the case is already shown in [SY19, section 3.7]. Since the mainargument of dragging curves along projector circles will appear many times in the following,we refrain from displaying the graphical proof once again here.From the same argument and Frobenius property of H cl it follows that relations R22) andR23) are satisfied. More interesting relations are the Dehn twist R24) and the braid move R25).For the Dehn twist we have = X k,l ∈ I d k d l D α αkl = =In the first two equalities we used the completeness relation. The last equality holds as H cl hastrivial twist. The braid move follows again by the argument given in [SY19, section 3.6] and issimilar to the Dehn-twist. (cid:3) Lemma 6.5.
All open-closed relations are satisfied.Proof.
We start with relation R26). As the picture suggests this will follow from the centercondition of ι cl − op . = X i,k ∈ I d i d k D = = α αi k ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 21
Where we again use completeness followed by the center condition.Next we prove relation R27): =where we see the obvious equality from ι cl − op being an algebra homomorphism. Form the samereasoning it follows that relation R30) is satisfied. Relations R29) and R28) are consistencychecks for the definition of ι cl − op and ι † cl − op . We show R28), the other one goes exactly thesame. == =In the first picture red dashed lines indicate where we glued world sheets. In the secondequality we used the projector property and in the third equality we inserted the definition ofthe morphism ι † . For the Cardy condition R31) we again have to drag along projection circles: X i,k d i d k D = == kiα α (cid:3) Lemma 6.6.
The genus one one point correlator is invariant under the S -move.Proof. The proof of R32) is again graphical and given by the following steps.
ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 23 = X i,j ∈ I jiβ β = X i,j ∈ I d i d j D ji = X i,j,k,l ∈ I d i d j d k d l D jiα αlk = X i,j ∈ I d i d j D j i = X i,j,r ∈ I d i d j d r D j i rαα = X r,i ∈ I d i d r D ri = Figure 11.
A genus 3 surface S , with a-cycles respectively b-cycles shownin green and blue. Red circle show boundary generators in H ( S , ).The first equality uses the complete basis for elements in the Drinfeld center but in a differentnormalization. Graphically this is indicated by using squares instead of round coupons. Notethat there is no extra factor for the quantum dimension in this case. The second step is themodular property for H cl . In the third step we transported the projector circle along the torusand inserted the completeness relation in order to drag the H cl -colored curve along the circlein the fourth step. Using again completeness and finally lemma 2.1 yields the result. (cid:3) This completes the proof of Theorem 6.2.6.2.
From Sewing Constraints on String-Net Spaces to Cardy Algebras.
In the previ-ous section we defined a fixed set of fundamental correlators and showed that the properties of aCardy algebra leads to a solution of the sewing constraints for these fundamental correlators. Inthis section we go the other way round, i.e. we assume that a solution to the sewing constraintsfor the functor
B l exists and show that this gives in fact a ( C | Z ( C ))-Cardy algebra. A string-net on a surface of genus g with n boundary components can have finitely many connectedcomponents winding non-contractible 1-cycles on the surface. However, on projector decoratedsurfaces things simplify considerably. Recall that the first homology group of a compact surface S g,n of genus g with n boundary components has 2 g + n generators. The first 2 g -generatorsare the usual a - and b -cycle running around holes of tori. The other n generators are simpleclosed curves homotopic to the boundary (see figure 11.). Proposition 6.7.
Let S g,n be a compact surface of genus g with n boundary components and A , . . . , A n ∈ Z ( C ) . Any element in ˆ H ( S g,n , A , . . . , A n ) is equivalent to a string-net withtrivial winding around boundary generators of H ( S g,n ) . Corollary 6.8.
For S ,n a sphere with n boundary components any string-net in ˆ H s ( S ,n , A , . . . , A n ) is equivalent to a string-net with a single coupon.Proof. This is the same argument we already used over and over again when manipulatingstring-nets. Assume a string-net has non-trivial winding along a boundary component. In anannular neighborhood of the boundary the string-net can be manipulated as follows
ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 25 = X k,l d k d l D α αkl =. Orientation of curves is chosen arbitrarily in the picture. For any other orientation thecomputation is exactly the same. (cid:3) The proposition and its corollary imply that any string-net on a generating world sheet is ofthe form shown in Figures 8, 9, 10 where disk-shaped vertices are now fixed morphisms of theright type. Assume a boundary coloring by the closed object c G cl and open object L ou ( d G op ),i.e. closed boundaries of a world sheet have boundary value c G cl and open ones L ou ( d G op ). Forexample, on the world sheet C m the vertex corresponds to a morphism b m cl : c G cl ⊗ c G cl → c G cl in Z ( C ). On C ∆ the vertex is a morphism b ∆ cl : c G cl → c G cl ⊗ c G cl and so on. We have to takespecial care of world sheets O prop and C prop . Those give rise to maps(6.9) ˆ p op ∈ Hom Z ( C ) ( L ou ( d G op ) , L ou ( d G op )) , ˆ p cl ∈ Hom Z ( C ) ( c G cl , c G cl ) . Assuming the sewing constraints hold it readily follows that ˆ p op and ˆ p cl are idempotent maps.In the previous discussion these maps were fixed to be the identity maps. Thus we may makethe additional assumption that ˆ p op , ˆ p cl are invertible, which by finiteness of the morphismspaces implies that ˆ p op = Id and ˆ p cl = Id. But we don’t have to. Since Z ( C ) is abelian we canchoose a retract ( G cl , e cl , r cl ) for ˆ p op , ˆ p cl , i.e.(6.10) e cl : G cl → c G cl , r cl : c G cl → G cl e cl ◦ r cl = ˆ p cl , r cl ◦ e cl = Id G cl . Since L ou , F are faithful functors there is a unique idempotent p ∈ Hom C ( d G op , d G op ) s.th. L ou ( p o ) = ˆ p o . We can choose a retract ( G op , e o , r o ) for p o in C . This gives a retract( L ou ( G op ) , L ou ( e o ) , L ou ( r o )) for ˆ p o in Z ( C ). On their images G cl , G op the propagator mor-phisms act as the identity and sewing constraints realize a ( C | Z ( C ))-Cardy algebra on G cl , G op rather than on c G cl , L ou ( d G op ). Theorem 6.9.
Given any set of fundamental string-nets on generating world sheets with closedboundary values c G cl and open boundary values L ou ( d G op ) which satisfy the sewing constraintsdefines a ( C | Z ( C )) -Cardy algebra ( G cl , G op , ι cl − op ) , which is unique up to isomorphism.Proof. As discussed above from fundamental world sheets we get the following ten maps O m O ∆ O η O ǫ C m C ∆ C η C η I I † m op ∆ op η op ǫ op m cl ∆ cl η cl ǫ cl ι ι † Table 1.
The first row states the type of world sheet and the second row thecorresponding maps.where the morphisms are(6.11) m op = r o ◦ d m op ◦ ( e o ⊗ e o ) : G op ⊗ G op → G op ∆ op = ( r o ⊗ r o ) ◦ d ∆ op ◦ e o : G op → G op ⊗ G op η op = r o ◦ c η op : → G op ǫ op = c ǫ op ◦ e o : G op → m cl = r cl ◦ d m cl ◦ ( e cl ⊗ e cl ) : G cl ⊗ G cl → G cl ∆ cl = ( r cl ⊗ r cl ) ◦ d ∆ cl ◦ e cl : G cl → G cl ⊗ G cl η cl = r cl ◦ c η cl : → G cl ǫ cl = c ǫ cl ◦ e cl : G cl → ι cl − op = L ou ( r o ) ◦ \ ι cl − op ◦ e cl : G cl → L ou ( G op ) ι † cl − op = r cl ◦ \ ι † cl − op ◦ L ou ( e o ) : L ou ( G op ) → G cl . The hatted morphisms are the maps appearing in the coupons for the string-nets. Since thegraphical representation of fundamental correlators stays the same, the proofs in section 6.1can be just run backwards giving the defining relations of a ( C | Z ( C ))-Cardy algebra for thesemorphisms. For the uniqueness part suppose we have chosen another set of retracts ( G ′ cl , e ′ cl , r ′ cl )and ( L ou ( G ′ op ) , L ou ( e ′ o ) , L ou ( r ′ o )), then it is easy to see that f o = r o ◦ e ′ o : G ′ op → G op and f cl = r cl ◦ e ′ cl : G ′ cl → G cl are isomorphisms of Frobenius algebras and in addition the diagram G ′ cl G cl L ou ( G ′ op ) L ou ( G op ) f cl ι ′ cl − op ι cl − op L ou ( f o ) commutes. (cid:3) Conclusion
In this paper we have shown how string-nets on topological surfaces generate solutions toopen-closed sewing relations. The major advantage of string-nets is the transportation of cat-egorical graphical calculus onto surfaces, which allows to use the defining conditions for Cardyalgebras directly when solving the sewing constraints. There are some open ends related to thiswork. First of all, as noted in [SY19] one could further generalize the results including defects.This seems likely to be possible using the description of defect world sheets given in [FFS12].Furthermore the qualifyer ”rational” may be given up, leading to a more general notion ofmodular tensor categories, which are not fusion. As shown in [FS17][FGSS18] many of the cat-egorical description can be transported to this situation by replacing sums over simple objectsby coends. Since dragging curves along projector circles was the crucial point in manipulatingstring-nets for fusion categories there should be an appropriate procedure for string-nets withnon-fusion colorings.Constructions of open-close interaction using curves on surfaces have appeared in [KLP03][KP06]in the form of the
A r c -operad. Since the graphical representation of the construction very muchresembles string-nets, there may be a connection between the two approaches. In general one
ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 27 may wonder about a (wheeled) PROP-description of string-nets since null graphs give a pastingscheme for string-net diagrams. We plan to address some of these questions in future work.
Appendix A. Generating World Sheets and Sewing Relations
In this appendix we give all the generating world sheets in WS and relations for sewingconstraints in theorem (5.6).A.1. Generating World Sheets.
The following figures display the quotients of the orienta-tion double for generating world sheets.I)
Open World Sheets: O prop O m O ∆ O η O ǫ Purple colored parts of the boundary correspond to open boundaries. Black bound-aries are physical boundaries.II)
Closed World Sheets: C prop C m C ∆ C η C ǫ III)
Open-Closed World Sheets:
I I † Appendix B. Sewing Relations
In the following figures red curves indicate how the world sheet displayed is glued from easierparts. The blue flag on glueing curves indicate the direction of glueing. For the part containingthe flag an incoming boundary is glued. In the figures blue boundaries denote in-boundariesand green ones correspond to out-boundaries.I)
Open Relations: ←→ ←→
R1) R2) ←→ ←→
R3) R4) ←→ R5) ←→ ←→
R6) R7) ←→ ←→
R8) R9) ←→ ←→
R10) R11) ←→ ←→
R12) R13)II)
Closed Relations: ←→ ←→
R14) R15) ←→ ←→
R16) R17) ←→ ←→
R18) R19)
ARDY ALGEBRAS, SEWING CONSTRAINTS AND STRING-NETS 29 ←→ R20) ←→ R21) ←→ ←→
R22) R23)In the picture of the Dehn-twist and braid move the red dashed lines are not glueinglines, but auxiliary curves to display the action of the elements of the mapping classgroup corresponding to the moves. ←→ ←→
R24) R25)III)
Open-Closed Relations ←→ ←→
R26) R27) ←→ ←→
R28) R29) ←→ ←→
R30) R31)IV)
Genus 1 Relation:
The genus one move takes place on a torus with one boundarycomponent and interchanges a - and b -cycle of the torus as indicated by the colors. ←→ R32)
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Matthias Traube: Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), F¨ohringer Ring6, 80805 M¨unchen, Germany
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