Constructing modular categories from orbifold data
CConstructing modular categoriesfrom orbifold data
Vincentas Muleviˇcius Ingo Runkel [email protected] [email protected]
Fachbereich Mathematik, Universit¨at Hamburg, Germany
In Carqueville et al., arXiv:1809.01483 , the notion of an orbifolddatum A in a modular fusion category C was introduced as part of ageneralised orbifold construction for Reshetikhin-Turaev TQFTs. Inthis paper, given a simple orbifold datum A in C , we introduce a rib-bon category C A and show that it is again a modular fusion category.The definition of C A is motivated by properties of Wilson lines in thegeneralised orbifold. We analyse two examples in detail: (i) when A isgiven by a simple commutative ∆-separable Frobenius algebra A in C ;(ii) when A is an orbifold datum in C = Vect, built from a sphericalfusion category S . We show that in case (i), C A is ribbon-equivalentto the category of local modules of A , and in case (ii), to the Drinfeldcentre of S . The category C A thus unifies these two constructions intoa single algebraic setting. 1 a r X i v : . [ m a t h . QA ] F e b ontents
1. Introduction 2
2. Orbifold data 8
3. The category of Wilson lines 14 C A as linear category . . . . . . . . . . . . . . . . . . . 143.2. C A as ribbon category . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3. Semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4. Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4. Examples 32
A. Appendix 51
A.1. Useful identities for orbifold data and Wilson lines . . . . . . . . . . 51A.2. Monadicity for separable biadjunctions . . . . . . . . . . . . . . . . 53
References 57
1. Introduction
A modular fusion category (MFC) is a semisimple modular tensor category, that is,a fusion category which is equipped with a braiding and a ribbon twist, such thatthe braiding satisfies a non-degeneracy condition. Modular fusion categories areimportant ingredients in several constructions in mathematics and mathematicalphysics, such as link and three-manifold invariants, two-dimensional conformal fieldtheory, and topological phases of matter.Three “intrinsic” ways to produce examples of MFCs are:1. If S is a spherical fusion category, one can construct its Drinfeld centre Z ( S ),which is a modular fusion category (see [M¨u, Thm. 1.2]).2. If C is a MFC and if A ∈ C is a commutative simple special Frobenius algebra,then C loc A , the category of so-called local A -modules, is again a MFC (see [KO,2hm. 4.5]).3. If B = (cid:76) g ∈ G B g is a G -crossed ribbon category for some finite group G suchthat its neutral component B e is modular, then B G , its G -equivariantisation,aka. the gauging by G , is a MFC (see [Ki, Thm. 10.4], [DGNO, Prop. 4.56]).In [CRS3] it was found that all three cases produce examples of “orbifold data” in aMFC C , which means that they define so-called generalised orbifolds of Reshetikhin-Turaev 3d TQFTs.Below, we briefly recall the definition of an orbifold datum A in C and summarisehow it produces a new MFC C A . This is the main result of this paper. We thenmotivate the algebraic construction of C A from 3d TQFT and from the generalisedorbifold construction. Let C be a MFC over an algebraically closed field k . An orbifold datum in C is atuple A = ( A, T, α, ¯ α, ψ, φ ), where A : a ∆-separable symmetric Frobenius algebra in C (∆-separable means that µ ◦ ∆ = id A , where µ is the product and ∆ the coproduct of A ), T : an A - AA -bimodule in C , where we abbreviate A ⊗ A by AA , α : an A - AAA bimodule map T ⊗ A, T → T ⊗ A, T , where the additional indexindicates over which tensor factor of A ⊗ A the tensor product is taken,¯ α : up to normalisation the inverse of α , ψ, φ : normalisation factors ψ ∈ End AA ( A ) × and φ ∈ k × .These are subject to further conditions [CRS1, CRS3] which we recall in Section2.3. The three examples above give rise to orbifold data as follows [CRS3].1. In this case, C = Vect, the category of finite-dimensional k -vector spaces.Let I be a set of representatives of isomorphism classes of simple objects in S . Then A = (cid:76) i ∈I End S ( i ), T = (cid:76) i,j,l ∈I S ( l, i ⊗ j ), α , ¯ α are defined via theassociator of S , and ψ (id i ) = (dim i ) / · id i , φ = Dim( S ) − (see Section 4.2).2. Here, C is again an arbitrary MFC and we set T = A , α = ¯ α = ∆ ◦ µ and ψ = id A , φ = 1 (see Section 4.1).3. Choose a simple object m g ∈ B g for each g ∈ G . Then one can take A = (cid:76) g ∈ G m ∗ g ⊗ m g , T = (cid:76) g,h ∈ G m ∗ gh ⊗ m g ⊗ m h , and ψ (cid:12)(cid:12) m ∗ g ⊗ m g = (dim m g ) − / id, φ = 1 / | G | (see [CRS3, Sec. 5]). 3ix an orbifold datum A in C . We now turn to the main construction in thispaper, that of the category C A . Its objects are tuples ( M, τ , τ , τ , τ ), where M isan A - A -bimodule, τ i : M ⊗ A T → T ⊗ A,i M for i = 1 , A - AA -bimodule maps,and τ i is – up to normalisation – inverse to τ i . The τ i , τ i satisfy conditions listedin Section 3.1. Morphisms in C A are bimodule maps compatible with the τ i and τ i .We endow C A with the structure of a ribbon category (see Section 3.1). Forexample, the tensor product of two objects ( M, τ ’s) and (
N, τ ’s) has as underlyingbimodule simply M ⊗ A N . We call the orbifold datum A simple if the tensor unit C A := A is a simple object in C A . We stress that A can be simple in C A even if itis not simple as a bimodule over itself. We show (Theorem 3.17): Theorem 1.1.
For C a MFC and A a simple orbifold datum in C , C A is also aMFC.Recall that the dimension of a MFC is defined as the sum over the squares ofthe quantum dimensions of simple objects. The dimension of C A can be directlyexpressed in terms of the constituents of the orbifold datum A . Namely, we showthat tr C ψ is non-zero and thatDim C A = Dim C φ · (tr C ψ ) . (1.1)Let us also illustrate this in the first two examples. For example 1, Dim C = 1, φ = Dim( S ) − and tr C ψ = Dim( S ) and so Dim C A = (Dim S ) . For example 2 wehave φ = 1 and tr C ψ = dim( A ), so that Dim C A = Dim C / dim( A ) . Both resultsare to be expected in light of the following theorem (Theorems 4.1 and 4.2). Theorem 1.2.
For the orbifold datum in example 1 we have the equivalence ofribbon categories C A ∼ = Z ( S ), and for that in example 2 we have C A ∼ = C loc A .This provides a unified proof of modularity of Z ( S ) and of C loc A . In the thirdexample, the evident conjecture is that C A ∼ = B G , but we do not treat this here.Indeed, in this case φ = 1 / | G | , tr C ψ = | G | and the dimension formula (1.1) givesDim C A = Dim( B e ) | G | , as expected. Further support is given in part 2 of the nextremark. Remark 1.3.
1. Examples 2 and 3 can also be obtained by a construction using Hopf monadsdeveloped in [CZW], but example 1 is in general not covered by that construc-tion. 4. In [KZ] an enriched version of the Drinfeld centre was introduced. Let C , D be fusion categories and let C be in addition braided. Let F : C → Z ( D ) be abraided functor. In the case that C is a MFC, as we consider here, the enrichedcentre is F ( C ) (cid:48) , i.e. the commutant of the image of C in Z ( D ). The constructions1–3 mentioned in the beginning are all instances of enriched centres. For case1 this is trivial, and for cases 2 and 3 this is established by the followingfactorisations of Drinfeld centres Z ( C A ) ∼ = C rev (cid:2) C loc A , Z ( B ) ∼ = ( B e ) rev (cid:2) B G , (1.2)where ( − ) rev refers to the category with inverse braiding and twist, see [DMNO,Cor. 3.30] and [CGPW, Thm. 2]. It is therefore expected that C A is an enrichedDrinfeld centre in general. Conjecturally, the relevant category D which sat-isfies Z ( D ) ∼ = C rev (cid:2) C A is defined similarly to C A , but objects now are triples( M, τ , τ ) which correspondingly satisfy fewer conditions, cf. Remark 3.9. Inparticular, D itself is no longer braided. This will be elaborated in a separatepaper [Mu].Note that proving this equivalence would give as a corollary that C and C A areWitt equivalent [DMNO]. Given a MFC C , the Reshetikhin-Turaev construction [RT, Tu] provides a 3d TQFT Z RT C , i.e. a symmetric monoidal functor Z RT C : (cid:91) Bord ( C ) → Vect k . (1.3)The source category is that of three-dimensional bordisms with embedded C -coloured ribbon graphs, and the hat denotes a certain extension needed to absorba glueing anomaly, see [Tu] for details.One can extend Z RT C to a larger bordism category (cid:91) Bord def3 ( C ) of stratified bor-disms [CRS2], where the various strata are labelled by algebraic data in C : 3-strata are unlabelled, or, equivalently, all labelled by the MFC C ; 2-strata, aka.surface defects, are labelled by ∆-separable symmetric Frobenius algebras in C [KS, FSV, CRS2]; 1-strata are labelled by (bi)modules over an appropriate ten-sor product of these algebras and 0-strata by the corresponding intertwiners, see[CRS2] for details.Starting from such a TQFT on stratified manifolds one can introduce the gen-eralised orbifold construction [CRS1, CRS3]. The idea is to carry out a state sum We are grateful to David Penneys and David Reutter for bringing the enriched centre to ourattention and for explaining the relation to C A . A one picks a simplicial decomposition of a bordism, passes to the Poincar´edual cell decomposition, and decorates each 2-stratum by A , each 1-stratum by T and each 0-stratum by α or ¯ α , depending on orientations. Finally, in each 2-cellone inserts ψ and in each 3-cell one inserts φ . The conditions on A ensure that thevalue of Z RT C on such a stratified bordism is independent of the choice of simplicialdecomposition. It is shown in [CRS3] that one obtains a new 3d TQFT, called the generalised orbifold , Z orb , A C : (cid:91) Bord → Vect k , (1.4)which, at least at this point, is only defined on bordisms without embedded ribbongraphs.The name “generalised orbifold” derives from the observation that example 3 isan actual orbifold by G , but that the same construction also covers examples 1and 2. This is similar to the use of the term “generalised symmetries” in [CZW].Two natural questions are now whether Z orb , A C is equivalent to a Reshetikhin-Turaev type TQFT Z RT C (cid:48) for some other MFC C (cid:48) , and, if so, how to obtain C (cid:48) from C and A . The second question was the motivation to undertake the researchpresented here. Indeed, we conjecture that C (cid:48) = C A , and will return to this questionin a future publication.The idea to extract C (cid:48) is to investigate how to describe ribbon graphs – or inother words Wilson lines and their junctions – in the generalised orbifold TQFT.We will think of such ribbon graphs as line defects embedded in the 2-strata oforbifold stratification. By [CRS2], such line defects are given by A - A -bimodules(Figure 1.1 a). The line defects need to be able to cross T -labelled 1-strata intoadjacent 2-strata. Such junctions are described precisely by the data τ i , τ i (Fig-ure 1.1 b). To achieve independence of the initial simplicial decomposition, onehas to be able to slide line defects across junction points of T -defects in variousways, for example as in Figure 1.1 c. The tensor product is given by placing twoline defects parallel to each other, and the T -crossings are described in this way,too (Figure 1.1 d). Finally, the braiding is obtained by inserting a bubble on a2-stratum and using this to make one line-defect pass over another (Figure 1.1 e).In the main text we will not use the connection between the algebraic definitionof C A and line defects in the generalised orbifold TQFT, but the entire constructionwas found by exploiting this relation.This paper is organised as follows. In Section 2 we recall some basic definitionsand facts about algebras and modules, and then give the definition of an orbifolddatum A . Section 3 contains the definition of the category C A and of its ribbonstructure, as well as our main theorem that C A is a modular fusion category. InSection 4 we present two examples in detail: local modules and the Drinfeld centre.6 a) (b) = (c)(d) · φ (e) Figure 1.1: (a) Line defects in a surface defect are labelled by bimodules M . (b) T -crossings τ , τ into adjacent surface defects. (c) Example of a com-patibility condition between the T -crossings τ i and α . (d) Example ofthe T -crossing for the tensor product M ⊗ A N of Wilson line defects.(e) Braiding of two Wilson line defects via a surface defect bubble.The various insertions of ψ and φ are result from the orbifold construc-tion and can be ignored at first.7ome useful identities to work with the category C A and a variant of the monadicitytheorem and are collected in the Appendix. Acknowledgements
We would like to thank Nils Carqueville, David Penneys, David Reutter, IordanisRomaidis, Gregor Schaumann, Daniel Scherl, Christoph Schweigert, and ZhenghanWang for helpful discussions. We are grateful to Nils Carqueville for his carefulreading of a draft of this paper. VM is supported by the Deutsche Forschungs-gemeinschaft (DFG) via the Research Training Group RTG 1670. IR is partiallysupported by the DFG via the RTG 1670 and the Cluster of Excellence EXC 2121.
2. Orbifold data
In this section we will recall the definition of an orbifold datum from [CRS3]. Todo so, we first list our conventions for spherical and modular fusion categoriesand summarise the definition of Frobenius algebras and their modules in a tensorcategory.
Let k be an algebraically closed field. For a spherical fusion category S over k ,an object X ∈ S and a morphism f : X → X , denote by tr S f ∈ k the trace of f and by | X | S = tr S id X the categorical dimension of X . Irr S will always denotea set of representatives of isomorphism classes of simple objects of S and we willassume that the tensor unit ∈ S is in Irr S . The dimension of S is the sumDim S := (cid:80) k ∈ Irr S ( | k | S ) . For k ∈ Irr S we have | k | S (cid:54) = 0, see [EGNO, Prop. 4.8.4].If in addition the characteristic of k is zero, then automatically also Dim S (cid:54) = 0[ENO, Thm. 2.3], but in non-zero characteristic it can happen that Dim S = 0[ENO, Sec. 9.1].Throughout the entire paper we will extensively use string diagram notation.Diagrams are to be read from bottom to top, and a downwards directed strandrepresents the dual of an object. The evaluation/coevaluation morphisms of anobject X ∈ S will be denoted by:[ev X : X ∗ ⊗ X → ] = , [coev X : → X ⊗ X ∗ ] = , [ (cid:101) ev X : X ⊗ X ∗ → ] = , [ (cid:103) coev X : → X ∗ ⊗ X ] = . (2.1)8abels for objects and morphisms will be omitted whenever they are clear fromthe context.Let C be a braided fusion category. For all X, Y ∈ C , the braiding morphisms c X,Y : X ⊗ Y → Y ⊗ X , and their inverses will be depicted by: c X,Y = , c − X,Y = . (2.2)A braided spherical fusion category C is automatically ribbon (see e.g. [TV,Lem.4.5]), with the twist morphism of an object X ∈ C built out of braiding andduality morphisms as θ X := = , θ − X := = . (2.3)A ribbon fusion category is also called premodular . A premodular category C iscalled modular if the matrix s i,j := tr S ( c j,i ◦ c i,j ) = , i, j ∈ Irr C (2.4)is invertible.We fix a modular fusion category C over k . For notational simplicity, C will beassumed to have strict monoidal and pivotal structures (without loss of generality[ML, NS]), and the symbol ⊗ for the monoidal product will sometimes be omitted. In this section we briefly recall the notion of Frobenius algebras and their modulesin C , more details can be found e.g. in [FRS].A Frobenius algebra in C is a tuple A ∈ C , µ : A ⊗ A → A, η : → A, ∆ : A → A ⊗ A, ε : A → , (2.5)where ( A, µ, η ) is an associative unital algebra and ( A, ∆ , ε ) is a coassociative9ounital coalgebra, such that = = . (2.6)The unit (resp. counit) will be denoted by (resp. ). A Frobenius algebrais called symmetric if = and ∆- separable if = . (2.7)A ( left- ) module of a Frobenius algebra A is a module ( M ∈ C , λ : A ⊗ M → M )of the underlying algebra. It is simultaneously a comodule with the coaction givenby [ M (∆ ◦ η ) ⊗ id M −−−−−−→ A ⊗ A ⊗ M id A ⊗ λ −−−−→ A ⊗ M ], or in graphical notation:= . (2.8)One easily generalises this to right modules and bimodules of a Frobenius algebra.Let A C A be the category of A - A -bimodules. As usual, it is a tensor categorywith the monoidal product M ⊗ A N of M, N ∈ A C A given by M ⊗ A N := coker[ M ⊗ A ⊗ N ρ M ⊗ id N − id M ⊗ λ N −−−−−−−−−−−→ M ⊗ N ] , (2.9)where ρ M and λ N are the corresponding right and left actions. By [FS, Prop. 5.24,Rem. 5.25] we have: Proposition 2.1.
For A ∈ C a ∆-separable Frobenius algebra, A C A is a finitelysemisimple monoidal category. In the online version, A -coloured strands are drawn in green. [FS] uses a special Frobenius algebra A , which is the same as ∆-separable, but with the extraassumption dim C A (cid:54) = 0; for this particular result, this assumption is not necessary. By “finitely semisimple” we mean that there are finitely many isomorphism classes of simpleobjects, and that each object is isomorphic to a finite direct sum of simple objects. A is a ∆-separable Frobenius algebra, M ⊗ A N is isomorphic to the image ofthe idempotent := . (2.10)The projection π : M ⊗ N → M ⊗ A N has a section ı in A C A , and π and ı thensplit the idempotent in (2.10). The graphical notation we will use is π = and ı = . (2.11)For any object X ∈ C , a morphism f : X → M ⊗ A N or g : M ⊗ A N → X can beuniquely given by morphisms (cid:98) f : X → M ⊗ N , (cid:98) g : M ⊗ N → X , such that= , = , (2.12)namely (cid:98) f := ı ◦ f and (cid:98) g := g ◦ π . By abuse of notation, the overhats like in (2.12)will be omitted in the following.If A, B ∈ C are ∆-separable Frobenius algebras, so is A ⊗ B , where we equip itwithproduct: , coproduct: , unit: , counit: . (2.13)An A ⊗ B module M is the same as a simultaneous A and B module such that= . (2.14)11ere and for the rest of the paper we indicate the action of the first and secondtensor factor by indices 1 , M be a right AB -module and K , L be left A - and B -modules respectively.We define the following partial tensor products M ⊗ K := im , M ⊗ L := im , (2.15)where the horizontal lines denote the idempotents as in (2.10). Note that M ⊗ K (resp. M ⊗ L ) is a right B -module (resp. right A -module).We will often encounter A - AA -bimodules and their partial tensor products. Inthis case, the left action will be indicated by 0 whenever it is necessary to avoidambiguity. The two right actions will be distinguished by indices 1, 2, like in (2.14).For an A - AA -bimodule M , the dual M ∗ is an AA - A -bimodule, M ⊗ M , M ⊗ M are A - AAA bimodules, M ∗ ⊗ M is an AA - AA -bimodule, etc. Here we recall the definition of an orbifold datum as given in [CRS3, Def. 3.4](where it is called a “special orbifold datum”).
Definition 2.2. An orbifold datum in C is a tuple A = ( A, T, α, α, ψ, φ ), where • A is a symmetric ∆-separable Frobenius algebra in C ; • T is an A - AA bimodule in C ; • α : T ⊗ T → T ⊗ T , α : T ⊗ T → T ⊗ T are A - AAA bimodule morphisms; • ψ : A → A is an invertible A - A -bimodule morphism; • φ ∈ k × ;These are subject to conditions (O1)–(O8) in Figure 2.1, where the following no-tation is used::= , := , := , i = 1 , . (2.16)As mentioned in the introduction, [CRS3] provides three examples of orbifolddata. We will look at two of them in detail in Section 4.12 (O1)= (O2) = (O3)= (O4) = (O5)= (O6) = (O7)= = = · φ − (O8)Figure 2.1: Identities an orbifold datum has to satisfy. All these string diagramsare drawn in C , but by the comment below (2.12) they induce identitiesalso between the appropriate tensor products over A in A C A .To manipulate expressions involving orbifold data it is best to firstignore all the actions of ψ , which is why we draw them in grey.13 . The category of Wilson lines In this section, we give the main definition of this paper and prove our main result,namely we define the category C A of Wilson lines and show that it is a modularfusion category.For the remainder of this section, we fix a MFC C and an orbifold datum A =( A, T, α, α, ψ, φ ) in C . C A as linear category Definition 3.1.
Define the category C A to have: • Objects : tuples (
M, τ , τ , τ , τ ), where – M is an A - A bimodule; – τ : M ⊗ T → T ⊗ M , τ : M ⊗ T → T ⊗ M ), τ : T ⊗ M → M ⊗ T , τ : T ⊗ M → M ⊗ T are A - AAA -bimodule morphisms, denoted by τ i = , τ i := , i = 1 , , (3.1)such that the identities in Figure 3.1 are satisfied. (Recall from the endof Section 2.2 that the notation ⊗ refers to the left- A -action on T .) • Morphisms : A morphism f : ( M, τ M , τ M , τ M , τ M ) → ( N, τ N , τ N , τ N , τ N )is an A - A -bimodule morphism f : M → N , such that τ Ni ◦ ( f ⊗ id T ) =(id T ⊗ i f ) ◦ τ Mi , i = 1 ,
2, or graphically= , i = 1 , . (M)Note that (T4) and (T5) imply that τ i is uniquely determined by τ i . We will referto the morphisms τ , τ as T -crossings and to τ , τ as their pseudo-inverses .14 (T1) = (T2)= (T3)= (T4) = (T5) i = 1 ,
2= (T6) = (T7) i = 1 , C A .15 xample 3.2. For each λ ∈ Z , ( A, τ λ , τ λ , τ λ , τ λ ) is an object of C A , where the T -crossings are τ λi := , τ λi := , i = 1 , . (3.2)Moreover, all these objects are isomorphic in C A . Indeed, define a morphism f :( A, τ λ , τ λ , τ λ , τ λ ) → ( A, τ µ , τ µ , τ µ , τ µ ) to be the bimodule map f = ψ µ − λ . For i = 1 , , i.e. f is an invertible morphism in C A .For M ∈ A C A , the following notation will be used to handle ψ -insertions::= , := , := . (3.3)By abuse of notation, the label M in ψ Ml , ψ Mr , ω M will be omitted whenever it isclear from the context.Definitions 2.2 and 3.1 imply a large number of additional algebraic identities,which will be used in computations below. We list several of them in AppendixA.1. C A as ribbon category We equip C A with the following monoidal structure: • product: ( M, τ M , τ M , τ M , τ M ) ⊗ ( N, τ N , τ N , τ N , τ N ) := ( M ⊗ A N, τ
M,N , τ M,N , τ M,N , τ M,N ) , where the T -crossings are: τ M,Ni := , τ M,Ni := , i = 1 , unit: C A = ( A, τ , τ , τ , τ ) from Example 3.2 (with the choice λ = 1), i.e.with the T -crossings:= , := , i = 1 , • associators and unitors: as in A C A .One checks that τ M,N satisfies the conditions (T1)–(T7) of a T -crossing. For ex-ample, to check (T1) for τ M,N , one first applies (T1) for τ M and τ N separately.This generates two insertions of ψ , one of which combines with ω in (3.4) intothe two insertions of ω required for the two copies of τ M,N . Note that (T1) wouldfail without the ω i in (3.4).The remaining conditions for C A to be a monoidal category follow from those in A C A . Definition 3.3.
We call an orbifold datum A in C simple if dim End C A ( C A ) = 1.From now on, we will omit the T -crossings when referring to an object of C A . If M is an object of C A , so is the dual bimodule M ∗ , where the T -crossings are:= , := i = 1 , ψ M ∗ l = ( ψ Mr ) ∗ and ψ M ∗ r = ( ψ Ml ) ∗ ). M ∗ is a left and right dual of M simultaneously, with evaluation/coevaluation morphisms given by:[ev M : M ∗ ⊗ A M → A ] := , [coev M : A → M ⊗ A M ∗ ] := , [ (cid:101) ev M : M ⊗ A M ∗ → A ] := , [ (cid:103) coev M : A → M ∗ ⊗ A M ] := . (3.7)The various insertions of ψ l and ψ r are necessary to make the dualities into mor-phisms in C A . The four zig-zag identities follow from those in A C A , the ψ -insertions17ancel each other in each case. The above duality morphisms equip C A with pivotalstructure (for that it is enough to check that the identities in [CR, Lem. 2.12] hold).For a pair of objects M, N ∈ C A , define the morphisms c M,N : M ⊗ A N → N ⊗ A M and c − M,N : N ⊗ A M → M ⊗ A N in A C A as follows: c M,N := · φ , c − M,N := · φ . (3.8)That the notation c − M,N is indeed justified is part of the claim in Proposition 3.5below.
Lemma 3.4.
For all
M, N ∈ C A , the following identities hold:= , = . (3.9) Proof.
Using also the identities in Appendix A.1 (and which are denoted by aprime, e.g. (T16 (cid:48) )) for the first equality one has = · φ (T4) = (T1) · φ As already mentioned in Figure 2.1, in this and the following computations it is helpful toignore the φ - and ψ -insertions at first and only verify that these also work out as a secondstep. To make this easier, all ψ ’s are shown in grey in string diagrams. T3) = · φ (T16 (cid:48) ) = · φ (T14 (cid:48) ) = · φ (T6) = · φ (O6) = · φ (O8) = . Similarly one can show the second identity.
Proposition 3.5. { c M,N } M,N ∈C A defines a braiding on C A . Proof.
One must check that for
M, N ∈ C A , c M,N and c − M,N are natural in M , N ,satisfy the hexagon identities, are inverses of each other and that the identity (M)holds. All of this can be done by repeatedly applying (O1)-(O8), (T1)-(T7) and(M); we only show one of the hexagon identities for c M,N . Using Lemma 3.4 onegets: = · φ (3.9) = · φ T5) = · φ = · φ = . Proposition 3.6. C A is spherical. Proof.
One needs to check that for all M ∈ C A and f ∈ End C A , the left and righttraces of f are equal, i.e. (cid:101) ev M ◦ ( f ⊗ id M ∗ ) ◦ coev M = ev M ◦ (id M ∗ ⊗ f ) ◦ (cid:103) coev M . Wehave: (3.7) = (O8) = · φ (T5) = (M) · φ (T6) = · φ = · φ = · φ ( ∗ ) = · φ = · φ ( ∗∗ ) = = , where in step ( ∗ ) one uses that C is ribbon to flip the rightmost part of the M -ribbon around the back of the circular T -ribbon (recall that all string diagrams inthis section are drawn in C ). For step ( ∗∗ ) one basically runs the first four stepsin reverse order. 20 emark 3.7.
1. For
M, N ∈ C A let Ω (cid:48) : M ⊗ A N ⊗ T → N ⊗ A M ⊗ T be themorphism on the left hand side of the first identity in Lemma 3.4 and denoteΩ := (id N ⊗ A id M ⊗ ( ψ ◦ ψ ))) ◦ Ω (cid:48) · φ (3.10)Then one has c M,N = tr C ,T Ω, where tr C ,T denotes the partial trace withrespect to T , taken in C .2. In what follows we show that C A is multifusion (see [EGNO] for the definition;a multifusion category with simple tensor unit is fusion). Proposition 3.6 thenimplies that C A is ribbon (see Section 2.1). The twist of an object M ∈ C A isobtained as in (2.3), or explicitly: θ M (2.3) := (3.7) (3.8) = · φ. (3.11) We will show the semisimplicity of C A by exploiting a sequence of adjunctions.These will involve some auxiliary categories D , D , which we now define. Definition 3.8.
Define the categories D i , i = 1 , • objects of D i are triples ( M, τ i , τ i ), where M ∈ A C A and τ i : M ⊗ T → T ⊗ i M is a T -crossing with pseudo-inverse τ i , i.e. it satisfies the identities (T1) and(T4)-(T7) (for i = 1) and (T3)-(T7) (for i = 2); • morphisms of D i are bimodule morphisms, satisfying the identity (M) (forthe given value of i only). Remark 3.9.
The categories D , D can be interpreted in the orbifold TQFTsetting briefly outlined in Section 1.2. Namely, they describe Wilson lines that liveon an interface between the TQFT described by C and the TQFT given by theorbifold datum A . D , D are also candidates for the category D alluded to inRemark 1.3. This will be further elaborated in [Mu].21e will now define four functors: A C A D D C A H H H H . (3.12)Namely, for any bimodule M ∈ A C A , define two bimodules H ( M ) and H ( M )together with T -crossings τ H ( M )1 , τ H ( M )2 and their pseudo-inverses by H ( M ) := im , H ( M ) := im , (3.13) τ H ( M )1 := , τ H ( M )2 := , (3.14) τ H ( M )1 := , τ H ( M )2 := . (3.15)It is easy to check that H ( M ), H ( M ) are objects of D and D , respectively.Next, for K ∈ D define H ( K ) to have the same underlying bimodule as H ( K ), and set τ H ( K )1 = τ H ( K )1 , τ H ( K )1 = τ H ( K )1 . (3.16)The T -crossings τ are defined as follows: τ H ( K )2 := , τ H ( K )2 := . (3.17)22 a) (b) Figure 3.2: Stratifications corresponding to the compositions (a) H ◦ H ( M ) and(b) H ◦ H ( M ).For H one proceeds analogously. Given L ∈ D , H ( L ) has the same bimoduleas H ( L ), the T -crossings τ agree with those of H , while the T -crossings τ aregiven by τ H ( L )1 := , τ H ( L )1 := . (3.18)One verifies that this makes H ( K ), H ( L ) into objects of C A .On a morphism f each functor acts as π ◦ (id T ⊗ f ⊗ id T ∗ ) ◦ ι , with π , ι thecorresponding projection and embedding. Remark 3.10.
Let M ∈ A C A . As Wilson lines in the orbifold TQFT from Sec-tion 1.2, H ◦ H ( M ) and H ◦ H ( M ) correspond to the stratifications in Fig-ure 3.2. For this reason, we call these functors “pipe functors”.As explained in Appendix A.2, for any pair of k -linear categories A , B , a bi-adjunction between functors X : A → B , Y : B → A is called separable , if thenatural transformation ε ◦ (cid:101) η : Id B ⇒ Id B is invertible (here (cid:101) η : Id B ⇒ XY is theunit of the adjunction Y (cid:97) X and ε : XY ⇒ Id B is the counit of the adjunction X (cid:97) Y ). Suppose now that the category A is finitely semisimple, B is idempotentcomplete, and that there exists a separable biadjunction between A and B . Thenit is shown in Proposition A.3 that B is finitely semisimple as well.23onsider the following commuting square of forgetful functors: A C A D D C A U U U U . (3.19)We have: Proposition 3.11. ( H , U ), ( H , U ), ( H , U ), ( H , U ) are pairs of biadjointfunctors and in each case the biadjunction is separable. Proof.
We show this for ( H , U ) only, the proofs for other cases are similar. Thebiadjunction is given by maps D ( H M, N ) → A C A ( M, U N ) ϕ M,N : (cid:55)→ · φϕ − M,N : ← (cid:91) , D ( K, H L ) → A C A ( U K, L ) χ K,L : (cid:55)→ · φχ − K,L : ← (cid:91) for all M, L ∈ A C A , N, K ∈ D . As an example, we will check that ϕ M,N and ϕ − M,N are indeed inverses of each other: Let f ∈ D ( H M, N ), g ∈ A C A ( M, U N ). Then ϕ M,N ◦ ϕ − M,N ( g ) = g follows by applying (T5) and (O8) to remove the T -loop whilethe other composition needs the following more elaborate computation: ϕ − M,N ◦ ϕ M,N ( f ) = · φ (O4) = · φ T14 (cid:48) ) = (T4) · φ deform = · φ (3.15) = · φ (O3) = · φ (O3) = · φ (O8) = f . Similarly, χ K,L and χ − K,L are inverses of each other. One also checks that ϕ − M,N ( g ), χ − K,L ( g ) are morphisms in D .As always, the counit ε : H U ⇒ Id D is given by { ε N = ϕ − U N,N (id U N ) } N ∈D and the unit (cid:101) η : Id D ⇒ H U by { (cid:102) η N = χ − N,U N (id U N ) } N ∈D . For all N ∈ D onehas: ε N ◦ (cid:102) η N = (T4) = (O8) = · φ − , i.e. ε N ◦ (cid:102) η N is invertible and hence the biadjunction is separable. Remark 3.12.
Note that the diagram in (3.19) commutes with identity naturalisomorphism, U := U ◦ U = U ◦ U : C A → A C A , as each path sends an object in C A to its underlying bimodule. By Proposition 3.11, both H ◦ H and H ◦ H arebiadjoint to U , and hence in particular naturally isomorphic. Thus the diagramin (3.12) commutes as well. In view of the stratifications in Figure 3.2 this is notsurprising, and an explicit natural isomorphism can be build from α and α . Belowwe will work exclusively with the the composition P := H ◦ H : A C A → C A , (3.20)where ‘ P ’ stands for ‘pipe functor’. Proposition 3.13.
The categories D , D and C A are finitely semisimple.25 roof. From Proposition 2.1 we already know that A C A is finitely semisimple.Therefore by Proposition 3.11 and the argument preceding it, it is enough to showthat D , D and C A are idempotent complete. We show this for D only, since theother cases are analogous. Let p : M → M be an idempotent in D . Then it isalso an idempotent in A C A and hence has a retract ( S, e, r ) in A C A . Equip S withthe morphisms: τ S := , τ S := . (3.21)They satisfy the axioms of T -crossings, e.g.= = = ( ∗ ) == = , where in step ( ∗ ) we used that p is a morphism in D . The argument that e and r are morphisms in D is similar. ( S, e, r ) is therefore a retract in D .Combining Propositions 3.6 and 3.13 with Remark 3.7, we get: Corollary 3.14. C A is a ribbon multifusion category. In this section we will in addition assume that A is a simple orbifold datum (seeDefinition 3.3), so that by Corollary 3.14, C A is a ribbon fusion category. We willshow that C A is in fact modular. 26et Ind A : C → A C A , X (cid:55)→ A ⊗ X ⊗ A be the induced bimodule functor. It isbiadjoint to the forgetful functor U AA : A C A → C (e.g. apply the adjunction for leftmodules in [FS, Prop. 4.10, 4.11] to the algebra A ⊗ A op ).We will use the pipe functor P = H ◦ H : A C A → C A , which is biadjoint to theforgetful functor U : C A → A C A , cf. Remark 3.12. It will prove useful to note thate.g. for M ∈ A C A , the braiding of P ( M ) with any object N ∈ C A can be simplifiedas follows: c P M,N = , c − N,P M = . (3.22)Let M, N ∈ C A and f : M → N be a morphism in A C A . Define the averagedmorphism f : M → N to be the A - A -bimodule morphism f := · φ . (3.23)One can check that f is a morphism in C A . Moreover, if f ∈ C A , then f = f , i.e.averaging is an idempotent on the morphism spaces of A C A , projecting onto themorphism spaces of C A .Recall the notations | X | C , tr C f , Irr C and Dim C from Section 2.1. For the re-mainder of the section, a thick loop (red in the online version) in a string diagram D will mean then sum (cid:80) k ∈ Irr C | k | C D k , where D k denotes the string diagram inwhich the red loop is labelled by k ∈ Irr C . Lemma 3.15. (cf. [KO], Lemma 4.6) A premodular category E is modular iff for27ll i ∈ Irr E = c · δ i, · (3.24)for some c (cid:54) = 0. Moreover, in this case one necessarily has c = Dim E .A useful corollary of Lemma 3.15 is the following identity, which holds for anymodular fusion category C and an object X ∈ C := Dim C · (cid:88) α , (3.25)where α and α run over a basis of C ( , X ) and its dual (with respect to thecomposition pairing C ( , X ) ⊗ k C ( X, ) → C ( , ) ∼ = k ). Lemma 3.16.
Let A be a simple orbifold datum in C . For M ∈ C A and f ∈ End C A ( M ) one has: tr C A f · tr C ψ = tr C ( ω M ◦ f ) . (3.26)In particular, | M | C A · tr C ψ = tr C ω M . Proof.
From expressions (3.7) one gets: ( ∗ ) = tr C A f · ⇒ = tr C A f , where in ( ∗ ) we used that A is simple in C A . Theorem 3.17.
Let A be a simple orbifold datum in C . Theni) C A is a modular fusion category;ii) tr C ψ (cid:54) = 0 and Dim C A = Dim C φ · (tr C ψ ) .28 roof. Let i ∈ C , µ ∈ A C A , ∆ ∈ C A be simple objects. One has the followingdecompositions (the label over the isomorphism sign indicates the category it holdsin) a) µ C ∼ = (cid:77) k ∈ Irr C k ⊗ C ( k, µ ) , b) ∆ A C A ∼ = (cid:77) ν ∈ Irr A C A ν ⊗ A C A ( ν, ∆) , c) AiA A C A ∼ = (cid:77) ν ∈ Irr A C A ν ⊗ C ( i, ν ) , d) P ( µ ) C A ∼ = (cid:77) Λ ∈ Irr C A Λ ⊗ A C A ( µ, Λ) , (3.27)Here, the forgetful functors U : C A → A C A and U AA : A C A → C are not written out.The isomorphisms in the second row follow from the biadjunctions Ind A (cid:97) U AA and P (cid:97) U , respectively. For a simple µ ∈ A C A and f ∈ End AA ( µ ), let (cid:104) f (cid:105) ∈ k be such that f = (cid:104) f (cid:105) · id µ . For a fixed simple ∆ ∈ C A , let L ∆ ∈ End C A (∆) be themorphism as on the left hand side of (3.24) (now understood as a string diagramin C A ). Use Lemma 3.16 and the decompositions above to obtain the equalities (inthis computation, all string diagrams are written in C A ) L ∆ · tr C ψ = (cid:88) Λ ∈ Irr C A | Λ | C A · tr C ψ = (cid:88) Λ ∈ Irr C A tr C ω = (cid:88) Λ ,ν tr C ω ν · dim A C A ( ν, Λ) (3.27 d) = (cid:88) ν ∈ Irr A C A tr C ω ν = (cid:88) ν ∈ Irr A C A | ν | C · (cid:104) ω ν (cid:105) = (cid:88) ν ∈ Irr A C A | ν | C (3.27 a) = (cid:88) ν,k | k | C · dim C ( k, ν ) (3.27 c) = (cid:88) k ∈ Irr C | k | C . (3.28)Next, use the expressions (3.7) for (co-)evaluation maps and (3.22) for braidings29o compute (in the following the string diagrams are again in C ):tr C (cid:0) ω ◦ L ∆ · tr C ψ (cid:1) (3.28) = (cid:88) k ∈ Irr C | k | C deform = (3.25) = Dim C (cid:88) α ( ∗ ) = Dim C φ (cid:88) α ( ∗∗ ) = Dim C φ (cid:88) α = Dim C φ (cid:88) α = Dim C φ (cid:88) α φ . (3.29)Step ( ∗∗ ) is best checked in reverse: the A -strings can be removed using the inter-twining properties of τ i and ∆-separability. Step ( ∗ ) consists of two computations,each of which combines two of the four T -loops into one. We will only show thefirst: (O5) = (T16 (cid:48) ) = (T6) deform = (T2) = (T5) (O7) = (O8) = · φ − . The last term in (3.29) contains the average of a morphism as defined in (3.23)which projects onto C A ( A, ∆). The sum over α therefore computes the trace of thisprojection and one hastr C (cid:0) ω ◦ L ∆ · tr C ψ (cid:1) = Dim C φ · dim C A ( A, ∆) = Dim C φ · δ A, ∆ . (3.30)It follows that tr C ψ (cid:54) = 0, as the right hand side is non-zero for A = ∆. Thisproves the first claim in part (ii) of the theorem.Recall from Section 2.1 that since C A is fusion, | ∆ | C A (cid:54) = 0 for all simple ∆. Usingthis, we finally get L ∆ = tr C A L ∆ · tr C ψ | ∆ | C A · tr C ψ · id ∆ (3.26) = tr C ( ω ◦ L ∆ ) | ∆ | C A · tr C ψ · id ∆ (3.31)31 = Dim C| ∆ | C A · φ · (tr C ψ ) · δ A, ∆ · id ∆ . (3.32)Lemma 3.15 now implies part (i) and the remaining claim in part (ii).
4. Examples
In this section we look into examples 1 and 2 in the introduction, that is, the casesof an orbifold datum obtained from a commutative simple ∆-separable Frobeniusalgebra, and from a spherical fusion category.
Let A ∈ C be a commutative ∆-separable Frobenius algebra. We call an A -module M local (or dyslectic ) if = . (4.1)The category of local modules will be denoted by C loc A (see [FFRS] for more detailsand further references on local modules).Note that since A is commutative, for any A -module M the morphisms on bothsides of (4.1) define right A -actions on M , which yields two bimodules M + and M − .Local modules are precisely those for which one has M + = M − . One uses the tensorproduct of bimodules to equip C loc A with tensor product and duals. Furthermore, C induces the braiding and the twists on C loc A . It was proven in [KO] that if A is haploid (i.e. dim C ( , A ) = 1, cf. [FS]), then C loc A is in fact a modular fusioncategory.For the remainder of the section, let A be a haploid ∆-separable commutativeFrobenius algebra in a modular fusion category C . Then it is automatically sym-metric (see [FRS, Cor. 3.10]), and as shown in [CRS3, Sec. 3.4], it gives an orbifolddatum A = (cid:0) A , T = A , α = α = ∆ ◦ µ , ψ = id A , φ = 1 (cid:1) , (4.2)where one uses commutativity of A = T to treat it as A - AA -bimodule. The restof the section is dedicated to proving the following Theorem 4.1.
Let A be a haploid ∆-separable commutative Frobenius algebrain C , and let A be the orbifold datum in (4.2). Then A is simple and one has C A ∼ = C loc A as k -linear ribbon categories. 32 roof. Define a functor F : C loc A → C A as follows: Given a local module M , equipit with the canonical bimodule structure and define the T -crossings to be= := . (4.3)All of the axioms then hold and are easy to check, e.g.= = ( ∗ ) = = = . In ( ∗ ) one uses the fact that the right action of M comes from (4.1). A morphismin C A is precisely an A -module morphism, i.e. F is fully faithful. Since A is simpleas a left module over itself (because A is haploid), this shows that the orbifolddatum A is simple.It is easy to check that F preserves tensor products, braidings and twists, henceit only remains to check that it is an equivalence. We show that F is essentiallysurjective.Let ( M, τ , τ , τ , τ ) ∈ C A . Since τ , τ are A - AA -bimodule morphisms, one has:= = , = = . (4.4)For example in the first equality for τ we think of the right A action as the actionof the first tensor factor of A ⊗ A and in the second equality as the action of thesecond tensor factor. Since M ∼ = A ⊗ A M ∼ = M ⊗ A A , the T -crossings τ , τ canbe recovered from the following invertible A -module morphisms (cid:98) τ , (cid:98) τ : M → M : (cid:98) τ i := , i = 1 , . (4.5)33e can then relate the left and right action on M as follows: (4.4) = = = ⇒ = . Similarly, the identities for τ in (4.4) imply= . Hence M is a local module with the canonical bimodule structure. It remains toshow that the T -crossings are as in (4.3). Using the identity (T1) one has= ⇔ = . Examining both sides of the last equality gives:left hand side: = = = (cid:98) τ , right hand side: = = = (cid:98) τ ◦ (cid:98) τ . Hence one has (cid:98) τ = (cid:98) τ ◦ (cid:98) τ and since it is invertible, (cid:98) τ = id M , which in turn impliesthat τ is precisely as in (4.3). The identity (T3) implies the same for τ .34ombining the above result with Theorem 3.17 gives an independent proof that C loc A is modular. For the orbifold datum (4.2) one has tr C ψ = | A | C and φ = 1, sothat the second part of Theorem 3.17 yieldsDim C loc A = Dim C| A | C . (4.6)Both, modularity and the above dimension formula are already known from [KO]. In this section, fix a spherical fusion category S with Dim S (cid:54) = 0 (this condition isonly relevant if char k (cid:54) = 0 see Section 2.1). We will not assume S to be strict; itsassociator and unitors will be denoted by a X,Y,Z : ( XY ) Z → X ( Y Z ), l X : X → X , r X : X → X , for all X, Y, Z ∈ S .Recall that the Drinfeld centre Z ( S ) consists of pairs ( X, γ ), where X ∈ S and γ : X ⊗ − ⇒ − ⊗ X is a natural transformation, satisfying the hexagonidentity, i.e. (id U ⊗ γ V ) ◦ a UXV ◦ ( γ U ⊗ id V ) = a U,V,X ◦ γ UV ◦ a X,U,V for all
U, V ∈ S .A morphism f : ( X, γ ) → ( Y, δ ) is a morphism f : X → Y in S , such that(id U ⊗ f ) ◦ γ U = δ U ◦ ( f ⊗ id U ) for all U ∈ S . Z ( S ) is a ribbon category with monoidal product( X, γ ) ⊗ ( Y, δ ) := ( X ⊗ Y, Γ XY ) . (4.7)where, for all U ∈ S ,Γ XYU := ( XY ) U a X,Y,U −−−−→ X ( Y U ) id X ⊗ δ −−−−→ X ( U Y ) a − X,U,Y −−−−→ ( XU ) Y γ U ⊗ id Y −−−−→ ( U X ) Y a U,X,Y −−−−→ U ( XY ) . (4.8)The braiding and the twist are given by (cid:2) c ( X,γ ) , ( Y,δ ) : ( X ⊗ Y, Γ XY ) → ( Y ⊗ X, Γ Y X ) (cid:3) := (cid:2) X ⊗ Y γ Y −→ Y ⊗ X (cid:3) , (4.9) θ ( X,γ ) := X r − X −−→ X id X ⊗ coev X −−−−−−−→ X ( XX ∗ ) a − X,X,X ∗ −−−−−→ ( XX ) X ∗ γ X ⊗ id X ∗ −−−−−→ ( XX ) X ∗ a X,X,X ∗ −−−−−→ X ( XX ∗ ) id X ⊗ (cid:101) ev X −−−−−→ X r X −→ X . Let us recall from [CRS3, Sec. 4] how one can associate to S an orbifold da-tum A S in the trivial modular fusion category Vect k of finite dimensional k -vectorspaces. For brevity, denote I := Irr S and | X | := | X | S (recall the conventions Our conventions here differ from those in [CRS3]. For example, instead of T = (cid:76) i,j,l S ( l, ij )as in (4.11), in [CRS3] the bimodule (cid:76) i,j,l S ( ij, l ) is used. The convention used here is bettersuited for the equivalence proof.
35n Section 2.1). For each i ∈ I , fix square roots | i | / and define the naturaltransformation ψ : Id S ⇒ Id S by taking for each X ∈ S ψ X := (cid:88) i,π | i | / [ X π −→ i ¯ π −→ X ] . (4.10)Here, i in the sum ranges over I , π over a basis of S ( X, i ), and ¯ π is the correspond-ing element of the dual basis of S ( i, X ) with respect to the composition pairing S ( i, X ) ⊗ k S ( X, i ) → S ( i, i ) ∼ = k . Now define A S = ( A, T, α, α, ψ, φ ) with: A = (cid:77) i ∈I S ( i, i ) ∼ = (cid:77) i ∈I k , T = (cid:77) l,i,j ∈I S ( l, ij ) ,α : (cid:77) l,a,i,j,k S ( l, ia ) ⊗ k S ( a, jk ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ = (cid:76) l,i,j,k S ( l, i ( jk )) −→ (cid:77) l,b,i,j,k S ( l, bk ) ⊗ k S ( b, ij ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ = (cid:76) l,i,j,k S ( l, ( ij ) k ) , (cid:104) l f −→ i ( jk ) (cid:105) (cid:55)−→ (cid:20) l f −→ i ( jk ) a − i,j,k −−−→ ( ij ) k ψ − ij ⊗ id k −−−−−→ ( ij ) k (cid:21) ,α : (cid:77) l,b,i,j,k S ( l, bk ) ⊗ k S ( b, ij ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ = (cid:76) l,i,j,k S ( l, ( ij ) k ) −→ (cid:77) l,a,i,j,k S ( l, ia ) ⊗ k S ( a, jk ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ = (cid:76) l,i,j,k S ( l, i ( jk )) , (cid:104) l g −→ ( ij ) k (cid:105) (cid:55)−→ (cid:20) l g −→ ( ij ) k a i,j,k −−−→ i ( jk ) id i ⊗ ψ − jk −−−−−→ i ( jk ) (cid:21) ,ψ : (cid:104) i f −→ i (cid:105) (cid:55)−→ (cid:104) i f −→ i ψ i −→ i (cid:105) , φ = 1Dim S = (cid:32)(cid:88) i ∈I | i | (cid:33) − . (4.11)Here, we abuse notation by denoting the morphism ψ : A → A in the orbifolddatum and the natural transformation ψ : Id S ⇒ id S from (4.10) with the samesymbol. The left action of [ f : k → k ] ∈ A on [ m : l → ij ] ∈ T , i, j, k, l ∈ I isprecomposition and the first (resp. second) right action is postcomposition with( f ⊗ id j ) (resp. (id i ⊗ f )) (if the composition is undefined, the corresponding actionis by 0). The isomorphisms in the definitions of α , α given by composition. Forexample, in the source object of α , the explicit form of the isomorphism is f ⊗ k g (cid:55)→ (id i ⊗ g ) ◦ f .Our goal is to prove the following Theorem 4.2.
Let S be a spherical fusion category, A S the orbifold datum asin (4.11) and C = Vect k . Then A S is simple and C A S ∼ = Z ( S ) as k -linear ribboncategories.Together with Theorem 3.17 this gives an alternative proof that Z ( S ) is modular.Furthermore, for the orbifold datum (4.11) one has tr C ψ = Dim S , so the second36art of Theorem 3.17 yields Dim Z ( S ) = (Dim S ) . (4.12)Modularity and the dimension of Z ( S ) are of course already known from [M¨u].The proof of Theorem 4.2 is somewhat lengthy and technical and is organisedas follows: In Section 4.2.1 we define an auxiliary category A ( S ) which is provedto be equivalent to the centre Z ( S ) as a linear category. Then in Sections 4.2.2and 4.2.3 we show that C A S ∼ = A ( S ) as linear categories, and that the orbifolddatum A S is simple. Composing the two equivalences gives a linear equivalence F : Z ( S ) → C A S . In Section 4.2.4 we equip F with a monoidal structure and showthat it preserves braidings and twists. A ( S ) and equivalence to Z ( S )Definition 4.3. Define the category A ( S ) to have • objects : triples ( X, t X , b X ), where X ∈ S and t X : X ⊗ ( − ⊗ − ) ⇒ ( X ⊗ − ) ⊗ − , b X : X ⊗ ( −⊗− ) ⇒ −⊗ ( X ⊗− ) are natural transformations between endofunctorsof S × S , such that the following diagrams commute for all
U, V, W ∈ S : X ( U ( V W ))( XU )( V W )(( XU ) V ) WX (( U V ) W ) ( X ( U V )) W t XU,V W a − XU,V,W id X ⊗ a − U,V,W t XUV,W t XU,V ⊗ id W , (4.13) X ( U ( V W )) U ( X ( V W )) U (( XV ) W )( U ( XV )) WX (( U V ) W ) ( X ( U V )) W b XU,V W id U ⊗ t XV,W a − U,XV,W id X ⊗ a − U,V,W t XUV,W b XU,V ⊗ id W , X ( U ( V W )) U ( X ( V W )) U ( V ( XW ))( U V )( XW ) X (( U V ) W ) b XU,V W id U ⊗ b XV,W a − U,V,XW id X ⊗ a − U,V,W b XUV,W ;(4.14)37 morphisms: ϕ : ( X, t X , b X ) → ( Y, t Y , b Y ) is a natural transformation ϕ : X ⊗ − ⇒ Y ⊗ − , such that the following diagrams commute for all U, V ∈ S X ( U V ) Y ( U V ) (
Y U ) V ( XU ) V ϕ UV t YU,V t XU,V ϕ U ⊗ id V , X ( U V ) Y ( U V ) U ( Y V ) U ( XV ) ϕ UV b YU,V b XU,V id U ⊗ ϕ V . (4.15) Proposition 4.4.
The functor E : Z ( S ) → A ( S ), acting • on objects: E ( X, γ ) := (
X, t X , b X ), where for all U, V ∈ S t XU,V := (cid:20) X ( U V ) a − X,U,V −−−−→ ( XU ) V (cid:21) ,b XU,V := (cid:20) X ( U V ) a − X,U,V −−−−→ ( XU ) V γ U ⊗ id V −−−−→ ( U X ) V a U,X,V −−−−→ U ( XV ) (cid:21) ; (4.16) • on morphisms: E (cid:0) [( X, γ ) f −→ ( Y, δ )] (cid:1) := { X ⊗ U f ⊗ id U −−−→ Y ⊗ U } U ∈S .is a linear equivalence. Proof.
It is easy to see that E ( X, γ ) is indeed an object in A ( S ) and that E ( f ) isa morphism in A ( S ). In the remainder of the proof we show that E is essentiallysurjective and fully faithful.As a preparation, given an object ( X, t, b ) ∈ A ( S ) ∈ A ( S ) we derive some prop-erties of t and b . For V, W ∈ S , consider the following diagram, whose ingredientswe proceed to explain: X ( ( V W )) ( X )( V W ) (( X ) V ) WX (( V ) W ) ( X ( V )) WX ( V W ) X ( V W ) ( XV ) WX ( V W ) ( XV ) W t ,V W a − X ,V,W id X ⊗ a − ,V,W t V,W t ,V ⊗ id W (cid:92) t ,V W a − X,V,W id t V,W (cid:100) t ,V ⊗ id W ∼ ∼ ∼ ∼ ∼ . (4.17)38e abbreviate t X by t , and we use the following notation for all U ∈ S : (cid:100) t ,U := (cid:20) XU id X ⊗ l − U −−−−−→ X ( U ) t ,U −−→ ( X ) U r X ⊗ id U −−−−→ XU (cid:21) , (cid:100) t U, := (cid:20) XU id X ⊗ r − U −−−−−→ X ( U ) t U, −−→ ( XU ) r XU −−→ XU (cid:21) , (4.18) (cid:100) b ,U := (cid:20) XU id X ⊗ l − U −−−−−→ X ( U ) b ,U −−→ ( XU ) l XU −−→ XU (cid:21) , (cid:100) b U, := (cid:20) XU id X ⊗ r − U −−−−−→ X ( U ) b U, −−→ U ( X ) id U ⊗ r X −−−−−→ U X (cid:21) . This notation will be used in the remainder of this section, too.By taking U = in (4.13), the inner pentagon in (4.17) commutes and all squarescommute by definition of (cid:100) t ,U , by naturality or by monoidal coherence, and hencethe outer pentagon commutes as well. Leaving out the identity edge, we get thefollowing commutative diagram for all V, W ∈ S : X ( V W ) X ( V W )( XV ) W ( XV ) W (cid:92) t ,V W a − X,V,W t V,W (cid:100) t ,V ⊗ id W . (4.19)Similarly, by taking V = and W = in (4.13) one in the end gets the followingtwo commuting diagrams: ∀ U, W ∈ S : X ( U W ) ( XU ) W ( XU ) W t U,W t U,W (cid:100) t U, ⊗ id W , ∀ U, V ∈ S : X ( U V ) ( XU ) VX ( U V ) t U,V (cid:92) t UV, t U,V . (4.20)These diagrams imply that for all U ∈ S one has (cid:99) t U, = id U .Repeating the above procedure of setting individual objects to also for twodiagrams in (4.14) yields three more conditions. Namely, for all U, V, W ∈ S onehas (cid:100) b ,U = id U and the following diagrams commute: X ( U W ) U ( XW ) U ( XW )( U X ) W ( XU ) W b U,W id U ⊗ (cid:100) t ,W a − U,X,W t U,W (cid:100) b U, ⊗ id W , X ( U V ) U ( XV ) U ( V X )( U V ) X b U,V id U ⊗ (cid:100) b V, a − U,V,X (cid:92) b UV, . (4.21)39et η U := [ XU (cid:100) t ,U −−→ XU ] and γ U := [ XU η − U −−→ XU (cid:100) b U, −−→ U X ] for all U ∈ S . We claim that γ is a half-braiding for X and that η is an isomorphism ( X, t, b ) → E ( X, γ ) in A ( S ).We start by showing that η is indeed a morphism in A ( S ). First note that (4.19)now reads t U,V = (cid:20) X ( U V ) η UV −−→ X ( U V ) a − X,U,V −−−−→ ( XU ) V η − U ⊗ id V −−−−−→ ( XU ) V (cid:21) . (4.22)Together with the definition of E ( X, γ ) in (4.16) we see that this is precisely thefirst condition in (4.15). Plugging (4.22) into the first diagram in (4.21) results in b U,V = X ( U V ) η UV −−→ X ( U V ) a − X,U,V −−−−→ ( XU ) V γ U ⊗ id V −−−−→ ( U X ) V a U,X,V −−−−→ U ( XV ) id U ⊗ η − V −−−−−→ U ( XV ) . (4.23)This is precisely the second condition in (4.15).Checking that γ satisfies the hexagon condition is now a direct consequence ofplugging (4.23) into the second diagram in (4.21).Altogether, this shows that E is essentially surjective.To get that E is fully faithful, let ϕ : E ( X, γ ) → E ( Y, δ ) be a morphism in A ( S ).Setting U = in the first condition in (4.15) yields that for all V ∈ S one has ϕ V = (cid:99) ϕ ⊗ id V , where (cid:99) ϕ := (cid:20) X r − X −−→ X ϕ −→ Y r Y −→ Y (cid:21) . (4.24)Setting V = in the second condition in (4.15) shows that (cid:99) ϕ commutes with thehalf-braidings γ and δ . Altogether, ϕ is in the image of E . D from A ( S ) to C A S In this subsection we define a functor D : A ( S ) → C A S . We start by defining D onobjects. Let ( X, t, b ) ∈ A ( S ) and denote the components of D ( X, t, b ) by D ( X, t, b ) =: (
M, τ , τ , τ , τ ) . (4.25)We will go through the definition of the constituents step by step, starting withthe A - A -bimodule M .For n ≥
1, an A ⊗ n -module is an I × n -graded vector space and a morphismbetween modules is a grade-preserving linear map. In particular, an A - A -bimodule40 is a vector space with a decomposition M = (cid:76) i,j ∈I M ij , where for M ij only the S ( i, i )- S ( j, j ) action is non-trivial. For the bimodule M in (4.25) we set M = (cid:77) i,j ∈I M ij with M ij = S ( i, Xj ) , (4.26)with action of S ( i, i ) (from the left) and S ( j, j ) (from the right) given by pre- andpost-composition, respectively.Next we turn to defining τ i and τ i . We will need two ingredients. The first arecertain A - AA -bimodule isomorphisms σ x , x = 0 , ,
2, which are defined as M ⊗ T = (cid:77) l,i,j,a ∈I S ( l, Xa ) ⊗ k S ( a, ij ) σ −−→ (cid:77) l,i,j ∈I S ( l, X ( ij ))[ l f −→ Xa ] ⊗ k [ a g −→ ij ] (cid:55)−→ [ l (id X ⊗ g ) ◦ f −−−−−−→ X ( ij )] ,T ⊗ M = (cid:77) l,i,j,a ∈I S ( l, aj ) ⊗ k S ( a, Xi ) σ −−→ (cid:77) l,i,j ∈I S ( l, ( Xi ) j )[ l f −→ aj ] ⊗ k [ a g −→ Xi ] (cid:55)−→ [ l ( g ⊗ id j ) ◦ f −−−−−→ ( Xi ) j )] ,T ⊗ M = (cid:77) l,i,j,a ∈I S ( l, ia ) ⊗ k S ( a, Xj ) σ −−→ (cid:77) l,i,j ∈I S ( l, i ( Xj ))[ l f −→ ia ] ⊗ k [ a g −→ Xj ] (cid:55)−→ [ l (id i ⊗ g ) ◦ f −−−−−→ i ( Xj ))] . (4.27)To describe the second ingredient, it will be useful to relate linear maps betweenmorphism spaces in S to actual morphisms in S as described in the followingremark. Remark 4.5.
Let T nB : S × · · · × S → S be the functor which takes the n -foldtensor product with a given bracketing B . Consider the I × ( n +1) -graded vectorspace V B := (cid:76) l,i ,...,i n S ( l, T nB ( i , . . . , i n )). For two bracketings B , B (cid:48) , one has alinear isomorphism (cid:8) natural transformations T nB ⇒ T nB (cid:48) (cid:9) ∼ −−→ (cid:8) graded linear maps V B → V B (cid:48) (cid:9) , given by post-composition. That is, it takes a natural transformation ϕ to thegraded linear map (cid:104) l f −→ T nB ( i , . . . , i n ) (cid:105) (cid:55)−→ (cid:104) l f −→ T nB ( i , . . . , i n ) ϕ i ,...,in −−−−−→ T nB (cid:48) ( i , . . . , i n ) (cid:105) . (4.28)This is easily generalised for functors obtained from T nB by fixing some of thearguments. 41ecall the natural transformations t , b that form part of the object ( X, t, b ) onwhich we are defining the functor D . The second ingredient needed to define τ i , τ i are four families of morphisms ( τ (cid:48) i ) UV , ( τ i (cid:48) ) UV in S which are natural in U, V ∈ S : (cid:2) X ( U V ) ( τ (cid:48) ) UV −−−−→ ( XU ) V (cid:3) := (cid:2) X ( U V ) t UV −−→ ( XU ) V ψ − XU ⊗ id V −−−−−→ ( XU ) V (cid:3) , (cid:2) X ( U V ) ( τ (cid:48) ) UV −−−−→ U ( XV ) (cid:3) := (cid:2) X ( U V ) b UV −−→ U ( XV ) id U ⊗ ψ − XV −−−−−−→ U ( XV ) (cid:3) , (cid:2) ( XU ) V ( τ (cid:48) ) UV −−−−→ X ( U V ) (cid:3) := (cid:2) ( XU ) V t − UV −−→ X ( U V ) id X ⊗ ψ − UV −−−−−−→ X ( U V ) (cid:3) , (cid:2) U ( XV ) ( τ (cid:48) ) UV −−−−→ X ( U V ) (cid:3) := (cid:2) U ( XV ) b − UV −−→ X ( U V ) id X ⊗ ψ − UV −−−−−−→ X ( U V ) (cid:3) . (4.29)Combining these two ingredients, we define τ i , τ i in (4.25) to be: τ := (cid:2) M ⊗ T σ −→ S ( l, X ( ij )) ( τ (cid:48) ) ij ◦ ( − ) −−−−−→ S ( l, ( Xi ) j ) σ − −−→ T ⊗ M (cid:3) ,τ := (cid:2) M ⊗ T σ −→ S ( l, X ( ij )) ( τ (cid:48) ) ij ◦ ( − ) −−−−−→ S ( l, i ( Xj )) σ − −−→ T ⊗ M (cid:3) ,τ := (cid:2) T ⊗ M σ −→ S ( l, ( Xi ) j ) ( τ (cid:48) ) ij ◦ ( − ) −−−−−−→ S ( l, X ( ij )) σ − −−→ M ⊗ T (cid:3) ,τ := (cid:2) T ⊗ M σ −→ S ( l, i ( Xj )) ( τ (cid:48) ) ij ◦ ( − ) −−−−−−→ S ( l, X ( ij )) σ − −−→ M ⊗ T (cid:3) . (4.30)The verification that these morphisms satisfy (T1)-(T7) will be part of the proofof Proposition 4.6 below.The action of D on a morphism ϕ : ( X, t X , b X ) → ( Y, t Y , b Y ) in A ( S ) is D ( ϕ ) := (cid:104) D ( X, t X , b X ) = (cid:77) i,j ∈I S ( i, Xj ) ϕ j ◦ ( − ) −−−−−→ (cid:77) i,j ∈I S ( i, Y j ) = D ( Y, t Y , b Y ) (cid:105) . (4.31) Proposition 4.6.
The functor D : A ( S ) → C A S is well-defined and a linear equiv-alence. Proof.
The proof that D ( X, t, b ) is indeed an object in C A S is a little tedious andwill be given in Subsection 4.2.3 below. For now we assume that this has beendone and continue with the remaining points.To see that D ( ϕ ) : D ( X, t X , b X ) → D ( Y, t Y , b Y ) is a morphism in C A S we haveto verify the identities in (M). We will demonstrate this for τ as an example.Denote the underlying A - A -bimodules of D ( X, t X , b X ) and D ( Y, t Y , b Y ) as M and42 , respectively, and consider the following diagram: M ⊗ T T ⊗ MN ⊗ T T ⊗ N ⊕S ( l, X ( ij )) ⊕S ( l, ( Xi ) j ) ⊕S ( l, ( Xi ) j ) ⊕S ( l, Y ( ij )) ⊕S ( l, ( Y i ) j ) ⊕S ( l, ( Y i ) j ) ⊕ ( t Xij ) ∗ ⊕ ( ψ − Xi ⊗ id) ∗ ⊕ ( t Yij ) ∗ ⊕ ( ψ − Y i ⊗ id) ∗ ⊕ ( ϕ ij ) ∗ ⊕ ( ϕ i ⊗ id) ∗ ⊕ ( ϕ i ⊗ id) ∗ σ σ σ σ τ M τ N D ( ϕ ) ⊗ i d i d ⊗ D ( ϕ ) (4.32)Here, all direct sums run over i, j, l ∈ I . The notation ( − ) ∗ stands for post-composition with the corresponding morphism. The left innermost square com-mutes by (4.15), and the right innermost square commutes by naturality of ψ . Thetop and bottom squares are just the definition of τ in (4.29) and (4.30). That therightmost square commutes is immediate from the definition of σ in (4.27), whilefor the leftmost square one needs to invoke in addition the naturality of ϕ .So far we have shown that the functor D is well-defined. We now check that itis essentially surjective and fully faithful.Let ( M, τ , τ , τ , τ ) be an arbitrary object in C A S . As above, we decompose M = (cid:76) i,j ∈I M ij , where for M ij only the S ( i, i )- S ( j, j ) action is non-trivial. Since τ is an A - AA -bimodule isomorphism M ⊗ T ∼ −→ T ⊗ M , we have a graded linearisomorphism τ : (cid:77) i,j,l,a ∈I M la ⊗ k S ( a, ij ) ∼ −−→ (cid:77) i,j,l,b ∈I S ( l, bj ) ⊗ k M bi , (4.33)Specialising to i = gives linear isomorphisms, for all l, j ∈ I , (cid:77) a ∈I M la ⊗ k S ( a, j ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ = M lj ∼ −−→ (cid:77) b ∈I S ( l, bj ) ⊗ k M b , (4.34)Setting X = (cid:76) b ∈I b ⊗ M b ∈ S , we see that this implies M ∼ = (cid:76) l,j S ( l, Xj )as A - A -bimodules. We may thus assume without loss of generality that in fact M = (cid:76) l,j S ( l, Xj ) for some X ∈ S .Define t, b by inverting the first two defining relations in each of (4.29) and (4.30)(this is possible by Remark 4.5). We need to verify that t, b satisfy the conditionsin (4.13) and (4.14). 43onsider condition (T1) satisfied by τ . Along the same lines as was donein (4.32), one can translate (T1) into an equality of two graded linear maps (cid:76) S ( l, X ( i ( jk ))) → (cid:76) S ( l, (( Xi ) j ) k ). Both of these maps are given by post-composition, resulting in a commuting diagram of morphisms in S , for all i, j, k : X ( i ( jk )) ( Xi )( jk ) (( Xi ) j ) kX (( ij ) k ) ( X ( ij )) k ( Xi )( jk ) (( Xi ) j ) kX (( ij ) k ) (( Xi ) j ) kX (( ij ) k ) ( X ( ij )) k t i,jk ψ − Xi ⊗ id jk a − Xi,j,k ψ − Xi ) j ⊗ id k id X ⊗ a − ijk id X ⊗ ( ψ − ij ⊗ id k ) i d X ⊗ ( ψ i j ⊗ i d k ) t ( i j ) , k ψ − X ( i j ) ⊗ i d k t ij ⊗ id k ( ψ − Xi ⊗ id j ) ⊗ id k (4.35)Since t , a and ψ are natural transformations, one can cancel all arrows with ψ ,which then yields precisely the diagram (4.13). Similarly, (T2), (T3) give the twodiagrams in (4.14).It remains to show that D is fully faithful. Faithfulness is clear from (4.31). Forfullness, let f : D ( X, t X , b X ) → D ( Y, t Y , b Y ) be a morphism in C A S . By Remark 4.5, f is given by post-composition with a natural transformation ϕ : X ⊗ − ⇒ Y ⊗ − .The identities (M) impose that the two diagrams in (4.15) commute. Thus ϕ is amorphism in A ( S ) and f = D ( ϕ ). Corollary 4.7.
The orbifold datum A S is simple. Proof.
By Proposition 4.6, the functor D : A ( S ) → C A S is a linear equivalence.Since C A S is semisimple (Proposition 3.13), so is A ( S ). Any object of the form( S , t, b ) is simple in A ( S ), as S is simple in S . For an appropriate choice of t, b we have D ( S , t, b ) ∼ = C A S , the tensor unit of C A S . Using once more that D is anequivalence, we conclude that C A S is simple in C A S . T -crossings Here we complete the proof of Proposition 4.6 by showing that D ( X, t, b ) from(4.25) satisfies conditions (T1)–(T7). 44or condition (T1), the computation is the same as in the proof of essentialsurjectivity of D , just in the opposite direction, i.e. one starts by writing (4.13) as(4.35). Analogously, (4.14) produces (T2), (T3).Conditions (T4) and (T5) are straightforward to check from the definitions (4.29)and (4.30).Since (T6) and (T7) involve duals, it is helpful to express the (vector space) dualbimodule M ∗ of M = (cid:76) l,a S ( l, Xa ) in terms of the bimodule M ∨ := (cid:76) l,a S ( l, X ∗ a ).Given a basis { µ } of S ( l, Xa ), we get the basis { µ ∗ } of the dual vector space S ( l, Xa ) ∗ and the basis { ¯ µ } of S ( Xa, l ), which is dual to { µ } with respect to thecomposition pairing. Let us fix an isomorphism M ∗ → M ∨ as follows: ζ : M ∗ −→ M ∨ , µ ∗ (cid:55)−→ | a || l | (cid:2) a ∼ −→ a coev X ⊗ id −−−−−−→ ( XX ∗ ) a ∼ −→ X ( X ∗ a ) ¯ µ −→ Xl (cid:3) (4.36)Using ζ , one can translate the evaluation and coevaluation maps from vector spaceduals to the new duals M ∨ . For example,coev M := (cid:2) A −→ M ⊗ A M ∗ id ⊗ ζ −−−→ M ⊗ A M ∨ (cid:3) , (cid:103) coev M := (cid:2) A −→ M ∗ ⊗ A M ζ ⊗ id −−→ M ∨ ⊗ A M (cid:3) , (4.37)where the unlabelled arrow is the canonical coevaluation in vector spaces. Explic-itly, this gives the A - A -bimodule mapsev M : M ∨ ⊗ A M → A , ⊗ k (cid:55)→ δ l,k , coev M : A → M ⊗ A M ∨ , id l (cid:55)→ (cid:88) a,µ ⊗ k | a || l | (cid:101) ev M : M ⊗ A M ∨ → A , ⊗ k (cid:55)→ δ l,k | l || a | (cid:103) coev M : A → M ∨ ⊗ A M , id l (cid:55)→ (cid:88) a,µ | l || a | ⊗ k (4.38)The choice (4.36) makes the expression for ev M simpler but the other three dualitymaps still contain the dimension factors. Using isomorphisms given by composition45imilar to those in (4.27), one can also write these maps as (by abuse of notationwe keep the same names for the maps)ev M : S ( l, X ∗ ( Xl )) → S ( l, l ) , λ (cid:55)→ l λ −→ X ∗ ( Xl ) a − X ∗ ,X,l −−−−→ ( X ∗ X ) l ev X ⊗ id l −−−−−→ l l l −→ l , coev M : S ( l, l ) → S ( l, X ( X ∗ l )) , µ (cid:55)→ (cid:34) l µ −→ l l − l −−→ l coev X −−−→ ( XX ∗ ) l a X,X ∗ ,l −−−−→ X ( X ∗ l ) (cid:35) , (cid:101) ev M : S ( l, X ( X ∗ l )) → S ( l, l ) , ν (cid:55)→ l ν −→ X ( X ∗ l ) id X ⊗ ψ − X ∗ l −−−−−−→ X ( X ∗ l ) a − X,X ∗ ,l −−−−→ ( XX ∗ ) l (cid:101) ev X ⊗ id l −−−−→ l l l −→ l ψ l −→ l , (cid:103) coev M : S ( l, l ) → S ( l, X ∗ ( Xl )) , ξ (cid:55)→ l ξ −→ l l − l −−→ l (cid:103) coev X −−−→ ( X ∗ X ) l a X ∗ ,X,l −−−−→ X ∗ ( Xl ) id X ∗ ψ − Xl −−−−−→ X ∗ ( Xl ) id X ∗ ⊗ (id X ⊗ ψ l ) −−−−−−−−−→ X ∗ ( Xl ) . (4.39)For example to get the expression for coev M one uses the identity:= (cid:88) a,µ | a || l | (4.40)Note that these are dualities in A C A . To obtain the dualities in C A S some extra ψ -insertions are needed, see (3.7).Given this reformulation of the duality morphisms, the verification of (T6), (T7)works along the same lines as (T1)–(T3). Denoting the composed functor by F := D ◦ E , we obtain the following corollaryto Propositions 4.4 and 4.6. Corollary 4.8.
The functor F : Z ( S ) → C A S , acting • on objects: F ( X, γ ) := ( (cid:76) k,l ∈I S ( l, Xk ) , τ , τ , τ , τ ), where for all i, j, l ∈ I the T -crossings and their pseudo-inverses are (we omit writing out the isomorphisms46 i from (4.27) explicitly) τ : S ( l, X ( ij )) → S ( l, ( Xi ) j ) , λ (cid:55)→ (cid:20) l λ −→ X ( ij ) a − X,i,j −−−→ ( Xi ) j ψ − Xi ⊗ id j −−−−−→ ( Xi ) j (cid:21) τ : S ( l, X ( ij )) → S ( l, i ( Xj )) , µ (cid:55)→ l µ −→ X ( ij ) a − X,i,j −−−→ ( Xi ) j γ i ⊗ id j −−−→ ( iX ) j a i,X,j −−−→ i ( Xj ) id i ⊗ ψ − Xj −−−−−→ i ( Xj ) τ : S ( l, ( Xi ) j ) → S ( l, X ( ij )) , ν (cid:55)→ (cid:20) l ν −→ ( Xi ) j a X,i,j −−−→ X ( ij ) id X ⊗ ψ − ij −−−−−→ X ( ij ) (cid:21) τ : S ( l, X ( ij )) → S ( l, i ( Xj )) , ξ (cid:55)→ l ξ −→ i ( Xj ) a − i,X,j −−−→ ( iX ) j γ − i ⊗ id j −−−−−→ ( Xi ) j a X,i,j −−−→ X ( ij ) id X ⊗ ψ − ij −−−−−→ X ( ij ) ; • on morphisms: F ([( X, γ ) f −→ ( Y, δ )]) := (cid:34) S ( l, Xk ) → S ( l, Y k ) ∀ k, l ∈ I [ l g −→ Xk ] (cid:55)→ [ l g −→ Xk f ⊗ id k −−−→ Y k ] (cid:35) is a linear equivalence.Recall that a monoidal structure consists of an isomorphism F : C A S ∼ −→ F ( Z ( S ) ) , (4.41)in C A S as well as a collection of isomorphisms F (( X, γ ) , ( Y, δ )) : F ( X, γ ) ⊗ C A S F ( Y, δ ) → F ( X ⊗ S Y, Γ XY ) , (4.42)in C A S , natural in ( X, γ ) , ( Y, δ ) ∈ Z ( S ), satisfying the usual compatibility condi-tions (see e.g. [TV], Sec.1.4). We set: F : (cid:77) i ∈I S ( i, i ) → (cid:77) i ∈I S ( i, i ) , (cid:104) i f −→ i (cid:105) (cid:55)→ (cid:20) i f −→ i ψ − i −−→ i l − i −−→ i (cid:21) . (4.43)As in Section 4.2.2 we get the isomorphisms F ( X, γ ) ⊗ C A S F ( Y, δ ) ∼ = (cid:77) l,r ∈I S ( l, X ( Y r )) , F ( X ⊗ Y, Γ XY ) ∼ = (cid:77) l,r ∈I S ( l, ( XY ) r ) . (4.44)For all l, r ∈ I , set F (( X, γ ) , ( Y, δ )) : (cid:104) l f −→ X ( Y r ) (cid:105) (cid:55)→ (cid:20) l f −→ X ( Y r ) id X ⊗ ψ Y r −−−−−→ X ( Y r ) a − X,Y,r −−−→ ( XY ) r (cid:21) . (4.45)47ne can check that they are indeed morphisms in C A S and satisfy the compatibil-ities. F = ( F, F , F ) is therefore a monoidal equivalence.Recall, that F is a braided functor if F (( Y, δ ) , ( X, γ )) ◦ c F ( X,γ ) ,F ( Y,δ ) = F ( c ( X,γ ) , ( Y,δ ) ) ◦ F (( X, γ ) , ( Y, δ )) . (4.46)For M = (cid:76) l,r ∈I S ( l, Xr ), N = (cid:76) l,r ∈I S ( l, Y r ) with T -crossings τ Mi , τ Ni , i = 1 , c M,N ∈ C A S explicitly.Recall that the braiding in C A S is obtained by taking the partial trace of the mor-phism Ω defined in (3.10). It amounts to a family of linear maps S ( l, X ( Y ( ij ))) →S ( l, Y ( X ( ij ))), i, j, l ∈ I , which are post-compositions with B ij := 1Dim S · (cid:104) X ( Y ( ij )) id X ⊗ ψ Y ( ij ) −−−−−−−→ X ( Y ( ij )) id X ⊗ ( τ (cid:48) Y ) ij −−−−−−−→ X (( Y i ) j ) ( τ (cid:48) X ) Y i,j −−−−−→ ( Y i )( Xj ) ψ Y i ⊗ ψ Xj −−−−−→ ( Y i )( Xj ) (id Y ⊗ ψ i ) ⊗ (id X ⊗ ψ j ) −−−−−−−−−−−−→ ( Y i )( Xj ) ( τ (cid:48) Y ) i,Xj −−−−−→ Y ( i ( Xj )) id Y ⊗ ( τ (cid:48) X ) ij −−−−−−−→ Y ( X ( ij )) id Y ⊗ ψ X ( ij ) −−−−−−−→ Y ( X ( ij )) (cid:105) . We now need to trace the above morphism over T , for which we need the dual T ∗ .Similar to Section 4.2.3 it is useful to work with T ∨ := (cid:76) i,j,r ∈I S ( ij, r ) instead.Given a basis { α } of S ( r, ij ), the basis { α ∗ } of the dual vector space S ( r, ij ) ∗ and the composition-dual basis { ¯ α } of S ( ij, r ), we fix the isomorphism T ∗ → T ∨ , α ∗ (cid:55)→ ¯ α . Using this isomorphism, the relevant evaluation and coevaluation mapsare[ (cid:101) ev T : T ⊗ , T ∨ → A ] = (cid:20) (cid:76) i,j ∈I S ( l, ij ) ⊗ k S ( ij, r ) → S ( l, l ) f ⊗ k g (cid:55)→ δ k,r g ◦ f (cid:21) , [coev T : A → T ⊗ , T ∨ ] = (cid:20) S ( l, l ) → (cid:76) i,j ∈I S ( l, ij ) ⊗ k S ( ij, l )id l (cid:55)→ (cid:80) i,j,α α ⊗ k ¯ α (cid:21) . All in all, we get the braiding to be the map S ( l, X ( Y r )) → S ( l, Y ( Xr )) , (cid:55)→ (cid:88) i,j,α . (4.47)For M = F ( X, γ ), N = F ( Y, δ ), the T -crossings are as given in Corollary 4.8.Using these expressions, the braiding (4.47) and the monoidal structure (4.45), one48oncludes that the left hand side of (4.46) is a family of linear maps S ( l, X ( Y r )) →S ( l, ( Y X ) r ), i, j, l ∈ I , obtained from post-composition with morphisms X ( Y r ) → ( Y X ) r , which in graphical calculus are as shown in Figure 4.1.In the last equality there we used (cid:88) i,j | i | · | j | · N rij = (cid:88) i,j | i | · | j ∗ | · N j ∗ ir ∗ = (cid:88) i | i | · | i | · | r | = dim S · | r | . (4.48)Substituting the braiding of Z ( S ) as defined in (4.9), one immediately finds theright hand side of (4.46) to be given by post-composition with the morphism inthe last diagram of Figure 4.1. The condition (4.46) then holds and hence F is abraided equivalence.Finally, recall that the twist of M ∈ C A S is given by the morphism in (3.11).Using the calculation in Figure 4.1 and the expressions (4.39) for (co-)evaluationmaps one computes that the twist θ F ( X,γ ) is a family of maps S ( l, Xk ) → S ( l, Xk ),obtained from post-composition with= (4.9) = , (4.49)which is the same morphism as F ( θ ( X,g ) ). F is therefore an equivalence of ribbonfusion categories and with that the proof of Theorem 4.2 is complete.49dim S (cid:88) i,j,α = 1dim S (cid:88) i,j,α · | i | · | j || r | = 1dim S (cid:88) i,j,α · | i | · | j || r | = 1dim S (cid:88) i,j · | i | · | j | · N rij | r | = . Figure 4.1: Left hand side of (4.46). Here, in the first equality one uses the naturaltransformation property of ψ and the half-braidings, in the third weabbreviate N rij = dim S ( r, ij ).. 50 . Appendix A.1. Useful identities for orbifold data and Wilson lines
Since α : T ⊗ T → T ⊗ T , α : T ⊗ T → T ⊗ T are A - AAA -bimodule morphisms,one has = = , = = , (A.1)= , = . (A.2)In particular, one can commute the ψ -insertions with α , α as follows:= = = (A.3)= = = (A.4)= , = , = , = , (A.5)= , = , = , = , (A.6)51imilarly, T -crossings, being A - AA -bimodule morphisms τ i : M ⊗ T → T ⊗ i M , τ i : T ⊗ i M → M ⊗ T , i = 1 ,
2, commute with ψ -insertions in the following way:= , = , = , = , (A.7)= , = , = , = , (A.8)The following identities are dual versions of (O6) and (O7):= (O9 (cid:48) ) = (O10 (cid:48) )Finally, one can show that (T1)-(T3) imply:= (T8 (cid:48) ) = (T9 (cid:48) ) = (T10 (cid:48) )= (T11 (cid:48) ) = (T12 (cid:48) ) = (T13 (cid:48) )52 (T14 (cid:48) ) = (T15 (cid:48) ) = (T16 (cid:48) ) A.2. Monadicity for separable biadjunctions
We will use string diagrams for 2-categories, as reviewed in [FSV, Sec. 6.1].Let A , B be categories, and let X : A → B , Y : B → A be biadjoint functorswith units and counits denoted by , , , . (A.9)Furthermore we will assume that this biadjunction is separable, i.e. the naturaltransformation := . (A.10)is invertible. The endofunctor T := [ Y X : A → A ] becomes a ∆-separable Frobe-nius algebra in the strict monoidal category End A via the structure morphisms:= , := , := , := . (A.11)53et A T be the category of T -modules in A . Its objects are pairs (cid:0) U ∈ A , [ ρ : T ( U ) → U ] (cid:1) and a morphism ( U, ρ ) → ( U (cid:48) , ρ (cid:48) ) is a morphism [ f : U → U (cid:48) ] ∈ A , such that thefollowing diagrams commute: T T ( U ) T ( U ) T ( U ) T µ U T ( ρ ) ρρ , U T ( U ) U id η U ρ , T ( U ) T ( U (cid:48) ) U U (cid:48) T ( f ) ρ ρ (cid:48) f . (A.12)Let (cid:63) be the category with only one object and only the identity morphism. Inwhat follows, it is going to be useful to identify any category A with the categoryof functors (cid:63) → A and natural transformations in the obvious way. The conditions(A.12) can then be written graphically as= , = , = . (A.13)Define the functor (cid:98) Y : B → A T to be the same as Y , except that the image isequipped with the following T -action::= . (A.14) Definition A.1.
Let A be a category. • An idempotent [ p : U → U ] ∈ A is called split, if it has a retract, i.e. a triplet( S, e, r ) where S ∈ A , e : S → U , r : U → S , such that e is mono, r ◦ e = id S , e ◦ r = p . 54 A is called idempotent complete if every idempotent is split. Proposition A.2. If B is idempotent complete, then (cid:98) Y is an equivalence. Proof.
We will give an inverse (cid:98) X : A T → B . Let M ∈ A T . Define the followingmorphism [ p M : X ( M ) → X ( M )] ∈ B : p M := . (A.15)One quickly checks that it is an idempotent. Set (cid:98) X ( M ) = im p . To prove that itis indeed an inverse, one computes (cid:98) X (cid:98) Y ( R ) = im , (cid:98) Y (cid:98) X ( M ) = im . (A.16)The morphisms in pairs , and , are then inverses of each other.Now let A , X , Y (and hence also (cid:98) X , (cid:98) Y ) in addition be k -linear additive categoriesand functors. Proposition A.3.
Suppose A is idempotent complete and finitely semisimple. Then so is A T . 55 roof. We first show idempotent completeness of A T . Given an idempotent p : M → M in A T and a retract e : S → M , r : M → S in A with p = e ◦ r , one canequip S with a T -action as follows, ρ S = (cid:2) T ( S ) T ( e ) −−→ T ( M ) ρ M −−→ M r −→ S (cid:3) . (A.17)With respect to this action, e and r are morphisms in A T , so that ( S, e, r ) becomesa retract in A T .Next we show semisimplicity of A T . Let M, N ∈ A T and let ı : M → N be monoin A T . Since A is semisimple, there is (cid:101) π : N → M in A , such that (cid:101) π ◦ ι = id M .Define π := . (A.18)One checks that π : N → M is a morphism in A T and π ◦ ı = = = = id M . It follows that N (cid:39) M ⊕ X for X = ker( ı ◦ π ), and so all subobjects are directsummands. The kernel exists as it is the image of the idempotent id N − ı ◦ π .For finiteness we show that every T -module M ∈ A T is a submodule of an induced T -module, i.e. a one of the form Ind( U ) := [ (cid:63) U −→ A T −→ A ] for some object U ∈ A . Indeed, pick U = M and the following morphisms: Υ : M → Ind( M ) andΠ : Ind( M ) → M Υ := , Π := (A.19)56ne can check that Υ and Π are module morphisms and that Π ◦ Υ = id M , hence M is indeed a submodule of Ind( M ). Every simple T -module is then a submoduleof Ind( V ) where V ∈ A is simple and since there are finitely many of those, A T must have finitely many simple objects. References [CGPW] S.X. Cui, C. Galindo, J.Y. Plavnik, Z. Wang,
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