Fusion categories for affine vertex algebras at admissible levels
aa r X i v : . [ m a t h . QA ] J u l Fusion categories for affine vertex algebrasat admissible levels
Thomas Creutzig ∗ Abstract
The main result is that the category of ordinary modules of an affine vertexoperator algebra of a simply laced Lie algebra at admissible level is rigid andthus a braided fusion category. If the level satisfies a certain coprime propertythen it is even a modular tensor category. In all cases open Hopf links coincidewith the corresponding normalized S-matrix entries of torus one-point func-tions. This is interpreted as a Verlinde formula beyond rational vertex operatoralgebras.A preparatory Theorem is a convenient formula for the fusion rules of ra-tional principal W-algebras of any type. ∗ Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AlbertaT6G 2G1, Canada andResearch Institute for Mathematical Sciences, Kyoto University, Kyoto Japan 606-8502.email: [email protected] Introduction
Vertex algebras are a rigorous formulation of chiral algebras of two dimensional con-formal field theories of physics. They appear in many interesting problems of bothmathematics and physics. In our context the relation to braided tensor categoriesand modular forms is of interest. By now one understands this well in the instanceof strongly rational vertex operator algebras [1, 2]. In this case one has a modulartensor category and Verlinde’s formula holds [2] (conjectured by Verlinde [3]), i.e.normalized Hopf links coincide with the corresponding normalized S-matrix entriesof torus one-point functions. The category of a nice class of modules of a vertexoperator algebra is in general expected to form a rigid vertex tensor category. Butalready proving the existence of a vertex tensor category is rather difficult as onehas to verify that quite a few assumptions hold. This theory has been developed byHuang, Lepowsky and Zhang in a series of many papers [4]-[11] and in joint workwith Huang and Yang all these assumptions have been verified to hold for the cate-gory O ℓ ( g ) of ordinary modules of affine vertex operator algebras L ℓ ( g ) at admissiblelevel ℓ [12]. Tomoyuki Arakawa has proven that this category is semi-simple [14]and so it is natural to wonder if this category is even a ribbon or modular tensorcategory. In the case of g = sl we proved that O ℓ ( sl ) is always ribbon and modu-lar if and only if the denominator of ℓ is odd [12]. The natural conjecture is then asimilar outcome for any O ℓ ( g ) and indeed: Main Theorem 1.
Let g be simply-laced and let ℓ be an admissible level for g , thenthe category O ℓ ( g ) is ribbon. Note that by a ribbon category we mean a rigid, braided, semi-simple tensorcategory with twist and with only finitely many inequivalent simple objects. Thetwist in a vertex tensor category is given by the action of e πiL with L the zero-mode of the Virasoro field. The statement to prove is thus rigidity. This Theorem isCorollary 7.3 of the file and it proves Conjecture 1.1 of [12] for simply-laced g . Thefusion rules are also found as a Corollary of proof, see Corollary 7.4. We remark thatCorollary 4.2.3 of [13] is a rigidity statement for O ℓ ( g ), but it is not explained whythe map in their equation (11) is invertible. This invertibility statement is highlynon-trivial as already in the rational case the proof of invertibility of this map wasvery involved and Yi-Zhi Huang had to first prove and then use Verlinde’s formulafor it [1].The proof requires to combine insights of three different directions. First ofall, we of course need the existence of vertex tensor category structure proven in[12]. Secondly, we need a relation of L ℓ ( g ) to a better understood family of vertexoperator algebras and this relation is given by the coset realization of principal W-algebras of simply-laced Lie algebras recently proven in joint work with Arakawa andLinshaw [15]. Thirdly we need the theory of vertex algebra extensions developed incollaboration with Kanade and McRae [16]. This theory uses that vertex algebraextensions are in one-to-one correspondence to commutative, associative, haploidalgebras in the vertex tensor category [17, 18, 19]. Combining all these insights wefind a fully faithful braided tensor functor from a subcategory of modules of the2ational principal W-algebras (rationality is proven in [20]) onto a category that wedenote by e O ℓ ( g ), this is Theorem 7.1. This category inherits the ribbon structurefrom the subcategory of modules of the rational principal W-algebra and since e O ℓ ( g )and O ℓ ( g ) differ by the action of certain simple currents the latter is ribbon as well.The main Theorem and its proof have quite a few interesting consequences.One of them is a proof of an admissible level version of a conjecture of Aganagic,Frenkel and Okounkov [21] made in the context of the quantum geometric Langlandsprogram, see Remark 7.2. Combining various coset statements with vertex tensorcategories one can prove further statements related to their conjecture and that iswork in progress. A second consequence is that one can now study certain vertexoperator superalgebras that are extensions of affine vertex operator algebras at ad-missible levels and rational W-algebras. This is done in the instance of L ℓ ( osp (1 | L ℓ ( sl ) times a rational Virasoro vertex operator algebra in [22, 23]and more complicated examples as e.g. L ℓ ( osp (2 | L ( d (2 , α )) are feasiblefuture aims. The other consequences will now be explained in detail. In order to explain the next result I will explain some well-known background thatis quite useful for this work. I assume familiarity with braided tensor categoriesand refer to the textbook [24] as reference. Let C be a braided tensor category withtwists and to simplify exposition we assume that it is strict. We will denote thetensor bifunctor by ⊠ and the tensor unit by . Our field is End( ) = C . Thenatural families of twists and braidings are denoted by θ • and c • , • The category is called rigid if for each object M in the category, there is adual M ∗ and morphisms b M ∈ Hom( , M ⊠ M ∗ ) (the co-evaluation) and d M ∈ Hom( M ∗ ⊠ M, ) (the evaluation) such that(Id M ⊠ d M ) ◦ ( b M ⊠ Id M ) = Id M , ( d M ⊠ Id M ∗ ) ◦ (Id M ∗ ⊠ b M ) = Id M ∗ . Rigidity implies that there is a trace. Let f in End( M ), thentr( f ) = d M ◦ c M,M ∗ ◦ ( θ M ⊠ Id M ∗ ) ◦ ( f ⊠ Id M ∗ ) ◦ b M ∈ End( ) = C . The partial trace is defined similarly. Let f in End( M ⊠ N ), thenptr L ( f ) = d M ◦ c M,M ∗ ◦ ( θ M ⊠ Id M ∗ ) ◦ ( f ⊠ Id M ∗ ) ◦ b M ∈ End( N ) . For us the most important players are the monodromy M M,N := c M,N ◦ c M,N ∈ End( M ⊠ N )and its partial trace and trace the open and closed Hopf linksΦ M,N = ptr L ( M M,N ) ∈ End( N ) , S m M,N = tr( M M,N ) ∈ C . The use of these lie in the fact [25, 26] that for any object N of C , the mapΦ · ,N : Obj( C ) → End ( N ) , M Φ M,N
3s a representation of the tensor ring. Let us assume that our category C is semi-simple so that X ⊠ Y ∼ = M Z ∈ Sim( C ) N ZX,Y Z where Sim( C ) denotes the set of inequivalent simple objects of C . The collection ofnumbers N ZX,Y for
X, Y, Z in Sim( C ) are called the fusion rules of C . We have forany W Φ X,W ◦ Φ Y,W = X Z ∈ Sim( C ) N ZX,Y Φ Z,W . Let W be simple so that Φ X,W = S m X,W S m ,W Id W . This identity follows from taking the trace of both sides. It follows that S m X,W S m ,W S m Y,W S m ,W Id W = X Z ∈ Sim( C ) N ZX,Y S m Z,W S m ,W Id W for all W ∈ Sim( C ) . Theorem 8.1 says
Main Theorem 2.
Let g be simply laced and let ℓ = − h ∨ + uv be an admissiblenumber for g . Then open Hopf links Φ L ℓ ( λ ) , L ℓ ( µ ) coincide with character S χ in O ℓ ( g ) ,i.e. Φ L ℓ ( λ ) , L ℓ ( µ ) = S m L ℓ ( λ ) , L ℓ ( µ ) S m L ℓ (0) , L ℓ ( µ ) Id L ℓ ( µ ) = S χ L ℓ ( λ ) , L ℓ ( µ ) S χ L ℓ (0) , L ℓ ( µ ) Id L ℓ ( µ ) . for all simple modules L ℓ ( λ ) , L ℓ ( µ ) in O ℓ ( g ) . The character S χ -matrix is introduced in the main text. I call this Theorem aVerlinde formula for ordinary modules. A speculation is that such a formula mighthold for ordinary modules of a big class of quasi-lisse vertex operator algebras asthey have certain modularity properties [27]. We remark that also for C -cofinitevertex operator algebras the open Hopf links seem to be related to the charac-ters S χ but there one needs to introduce modified traces [28, 29]. We even expectsimilar behaviour if we go to larger representation categories of L ℓ ( g ) then just or-dinary modules. So far, David Ridout and I, we conjectured a Verlinde formula forrelaxed-highest weight modules of L ℓ ( sl ) at admissible level [30, 31] and indeed theconjecture restricted to ordinary modules is true [12, Cor. 7.7].As mentioned before, Yi-Zhi Huang first proved Verlinde’s formula for rationalvertex operator algebras and then used this result to deduce rigidity [1, 2]. It mightbe possible to repeat this analysis for ordinary modules, i.e. first relate a modular S -matrix to Hopf links and then use the result to deduce rigidity.If the Hopf link matrix is invertible then we can immediately express the fusionrules in terms of the Hopf links, that is N ZX,Y = X W ∈ Sim( C ) S m X,W S m Y,W ( S m ) − Z,W S m ,W . C is a modular tensor category. An equivalent formulationof having a modular tensor category is the following. Let W := M X ∈ Sim( C ) X (1)then C is a modular tensor category if and only if the map from the Grothendieckring K ( C ) to End( W ),Φ · ,W : K ( C ) → End( W ) , X Φ X,W (2)is a ring isomorphism. This statement is used to prove
Main Theorem 3.
Let g be simply laced and let ℓ = − h ∨ + uv be an admissiblenumber for g with u, v positive coprime integers. Let N be the level of the weightlattice P of g and let ( N, v ) = 1 . Then O ℓ ( g ) is a modular tensor category. Fusion rules of W-algebras are needed for the proof of the first main Theorem. Theyhave been proven in the simply-laced case if a certain coprime property holds forthe denominator of the level [32, 33]. The proof was based on Verlinde’s formulaand I take the slightly different point of view that the map (2) is a ring isomorphismfrom the Grothendieck ring of the regular W-algebra to the endomorphism ring ofthe direct sum of all inequivalent simples (1).It turns out that then fusion rules can be determined in full generality in the senseof the following Theorem. For this let P m + ( P ∨ ; m + ) be the set of integrable highest-(co)weight modules of g at level m ∈ Z > , let h ( h ∨ ) be the (dual) Coxeter numberof g . Then simple modules of the simple and rational principal W-algebra W k ( g )are labelled by pairs ( λ, λ ′ ) with λ ∈ P u − h ∨ + and λ ′ ∈ P ∨ ; v − h + where k = − h ∨ + uv isadmissible for g . They are denoted by L k ( λ, λ ′ ). This notation is taken from [33]. Main Theorem 4.
Let g be a simple Lie algebra let k = − h ∨ + vu be an admissiblelevel for g with u, v positive coprime integers. Let ℓ ∈ Z >h ∨ and let N g ℓ , φλ,ν the fusioncoefficients of L ℓ − h ∨ ( g ) , i.e. they satisfy L ℓ − h ∨ ( λ ) ⊠ L ℓ − h ∨ ( ν ) ∼ = M φ ∈ P ℓ − h ∨ + N g ℓ φλ,ν L ℓ − h ∨ ( φ ) . (3) Let λ, ν ∈ P u − h ∨ + and λ ′ , ν ′ ∈ P ∨ ; v − h + , then we have L k ( λ, ⊠ L k (0 , λ ′ ) ∼ = L k ( λ, λ ′ ) L k ( λ, ⊠ L k ( ν, ∼ = M φ ∈ P u − h ∨ + N g u φλ,ν L k ( φ,
0) (4)
Moreover L k (0 , λ ′ ) centralizes L k ( λ, for all λ ∈ Q . ecall that Feigin-Frenkel duality states that W k ( g ) ∼ = W L k ( L g ) for r ∨ ( k + h ∨ )( L k + L h ∨ ) = 1 [34, 35] (see also [15] ) so that we have L k (0 , λ ′ ) ⊠ L k (0 , ν ′ ) ∼ = M φ ′ ∈ L P q − Lh ∨ + N L g v φ ′ λ ′ ,ν ′ L k (0 , φ ′ ) , q = v if ( v, r ∨ ) = 1 q = vr ∨ if ( v, r ∨ ) = r ∨ where one identifies coroots with roots of the dual Lie algebra via L α = α ∨ √ r ∨ andcorrespondingly coweights with weights of the dual Lie algebra. By L P we then meanthe weight lattice of the dual Lie algebra L g and by r ∨ the lacity of g . This is Theorem 6.1 together with the first point of Remark 6.2. The fusion rulesare completely determined by this Theorem due to associativity and commutativityof fusion.That L k (0 , λ ′ ) centralizes L k ( λ,
0) for all λ ∈ Q (the root lattice) means that themonodromy between these two modules is trivial and this is important for the proofof the first main Theorem as well.In [36] rationality of subregular W -algebras of sl at certain admissible levels wasproven. Moreover it was shown that the Heisenberg coset is isomorphic to a regular W -algebra of type A . The proof relied on the fusion rules of regular W-algebras oftype A and so in that work it could only be done if the coprime condition of [32, 33]was satisfied. The exact same proof now works in general and so Corollary 1.
Let n be a positive integer, such that ( n + 4 , n + 1) = 1 . Let k = − n +43 and let ℓ = − n + n +4 n +1 . Then W k ( sl , f subregular ) is rational and C -cofiniteand let H be its Heisenberg vertex subalgebra, then Com ( H, W k ( sl , f subregular )) ∼ = W ℓ ( sl n , f regular ) . These types of relations between regular and subregular W-algebras of type A isexpected to hold in general and the cases of the subregular W-algebras of sl and sl are already known as well [37, 38]. Moreover, Andrew Linshaw has found a differentmethod to prove these statements [39, Thms. 10.4, 10.5 and 10.6]. Acknowledgements
I am very grateful to Terry Gannon for many discussions on related issues andto Jinwei Yang, Yi-Zhi Huang, Andrew Linshaw, Tomoyuki Arakawa, ShashankKanade and Robert McRae for the collaborations on the works that were needed forthis paper. I am supported by NSERC
Vertex algebra extensions
The key player for studying vertex operator algebra extensions are commutativealgebras in the tensor category, see [18, 24]. We denote by A • , • , • the associativityconstraint in C . Definition 2.1. [24, Def. 7.8.1] Let C be a braided tensor category. A algebra in C is an object A in C together with a multiplication map m : A ⊠ A → A and a unit u : → A such that the multiplication is associative and compatible with left and right multi-plication, i.e. the following three diagrams commute:( A ⊠ A ) ⊠ A A A,A,A / / m ⊠ Id A (cid:15) (cid:15) A ⊠ ( A ⊠ A ) Id A ⊠ m (cid:15) (cid:15) A ⊠ A m & & ▲▲▲▲▲▲▲▲▲▲▲ A ⊠ A m x x rrrrrrrrrrr A ⊠ A ℓ A / / u ⊠ Id A (cid:15) (cid:15) A Id A (cid:15) (cid:15) A ⊠ r A / / Id A ⊠ u (cid:15) (cid:15) A Id A (cid:15) (cid:15) A ⊠ A m / / A A ⊠ A m / / A (5)The algebra A is called haploid if the dimension of Hom C ( , A ) is one. A is com-mutative if the diagram A ⊠ A m ●●●●●●●●● c A,A / / A ⊠ A m { { ✇✇✇✇✇✇✇✇✇ A (6)commutes.There are two natural module categories associated to commutative algebras. Definition 2.2.
Let C be a tensor category and A an algebra in C . Then thecategory C A has objects ( X, m X ) with X an object of C and m X ∈ Hom C ( A ⊠ C X, X )a multiplication morphism that is 7. Associative, i.e. the diagram commutes: A ⊠ ( A ⊠ X ) A − A,A,X / / Id A ⊠ m X (cid:15) (cid:15) ( A ⊠ A ) ⊠ W m ⊠ Id X (cid:15) (cid:15) A ⊠ W m X & & ▼▼▼▼▼▼▼▼▼▼▼ A ⊠ W m X x x qqqqqqqqqqqq X (7)2. Unit: The composition X ℓ X −−→ ⊠ X u ⊠ Id X −−−−→ A ⊠ X m X −−→ X is the identity on X .Morphisms of C A are all C morphisms f ∈ Hom C ( X, Y ) such that A ⊠ X Id A ⊠ f / / m X (cid:15) (cid:15) A ⊠ Y m Y (cid:15) (cid:15) X f / / Y (8)commutes.The category C A is tensor but usually not braided. Definition 2.3.
Let C be a braided tensor category and A a commutative algebrain C . Then the category C loc A ⊂ C A of local modules is the full tensor subcategorywhose objects are local with respect to A , i.e. A ⊠ X m X ●●●●●●●●● M A,X / / A ⊠ X m X { { ✇✇✇✇✇✇✇✇✇ X (9)commutes.This is the category of interest as it inherits the structure of a braided tensorcategory of C . The very important result of Huang, Kirillov and Lepowsky makes contact of vertexoperator algebras with a corresponding categorical notion:8 heorem 3.1. [19, Thms 3.2 and 3.4]
Let C be a vertex tensor category of a vertexoperator algebra V . Then a vertex operator algebra extension V ⊂ A in the category C is equivalent to a commutative associative algebra in the braided tensor category C with trivial twist and injective unit. Moreover, the category of local C -algebramodules C loc A is isomorphic to the category of modules in C for the extended vertexoperator algebra A . The first main result of [16] is that C loc A and the vertex tensor category of A -modules that lie in C are equivalent as braided tensor categories: Theorem 3.2. [16, Thm. 3.65]
Let V be a vertex operator algebra and let C be a fullvertex tensor category of V -modules. Let A ∈ C be a vertex operator algebra exten-sion of V . Then the isomorphisms of categories of Theorems 3.1 and the categoryof vertex algebraic A -modules is an equivalence of braided tensor categories. The second main result of [16] is that the induction functor is a vertex tensorfunctor. This is a functor that maps objects of C to algebra objects: F : C → C A , X ( A ⊠ C X, m ⊠ C Id X ) . Theorem 3.3. [16, Thm. 3.68]
Induction is a vertex tensor functor from the fullsubcategory of V -modules to C A with respect to the vertex tensor category structureof these categories constructed by Huang, Lepowsky and Zhang [11] In general the induced object is not local and the criterion that guarantees local-ity is Proposition 2.65 of [16], saying that F ( Y ) ∈ C loc A if and only if M A,Y = Id A ⊠ C Y .One says that Y centralizes A . Remark 3.4.
This centralizing property is sometimes very easy to verify in theinstance that C is a modular tensor category. Let I be a set of inequivalent simpleobjects such that A = M X ∈I X is a commutative algebra object in C . Let Y be a simple object of C such that X ⊠ C Y is simple for all X ∈ I . Then M X,Y = a Id X ⊠ C Y for some number a that is easily computed by taking the trace of both sides, so that a = S m X,Y dim X dim Y and thus in this instance Y centralizes A if and only if S m X,Y = dim X dim Y for all X ∈ I . Theorem 3.5.
Let V ∼ = M i W i ⊗ Z i be a vertex operator algebra extending the vertex operator algebra W ⊗ Z with W i ⊗ Z i simple W ⊗ Z modules and W i = W j , Z i = Z j for i = j . Let C W ⊠ C Z be a vertextensor category of W ⊗ Z modules such that V is an object of C W ⊠ C Z . Let D ⊂ C W be a full rigid braided tensor subcategory of C , such that every simple object of D lifts to a simple local V -module under the induction functor F , i.e. F ( X ⊠ Z ) islocal and simple for all simple objects X in D . Then F ( D ) is rigid braided tensoras well and equivalent to D as a braided tensor category.Proof. The induction functor is a braided tensor functor by Theorem 2.67 of [16].It is fully faithful by Proposition 8.1 of [40]. It thus gives a braided equivalence of D and F ( D ). Moreover the induction functor preserves duals and hence rigidity byLemma 1.16 of [18], see also Proposition 2.77 of [16]. I use notation and also many statements of the work of Arakawa and van Ekeren[33]. Let g be a simple Lie algebra and let k be a complex number then we denote thesimple affine vertex operator algebra of g at level k by L k ( g ). Let k = − h ∨ + uv for u, v co-prime positive integers, such that k is a admissible number for g . Let L k ( λ )be the simple quotient of the highest-weight representation of highest-weight λ atlevel k . For a positive integer m , we denote by P m + be the set of weights such that { L m ( λ ) | λ ∈ P m + } gives the complete set of inequivalent simple modules of L m ( g ).Then the category of ordinary modules O k ( g ) is a semi-simple category [14] whosesimple objects are { L k ( λ ) | λ ∈ P u − h ∨ + } . It is a vertex tensor category: Theorem 4.1. [12, Thm. 6.6]
Let g be a simple Lie algebra and k a admissiblenumber then the category of ordinary modules O k ( g ) has a vertex tensor categorystructure. Ordinary modules are a subcategory of principal admissible modules at level k .These have the property that their trace functions (as Jacobi forms) converge incertain regions and can be meromorphically continued to meromorphic Jacobi forms[41]. The latter then turn out to close under the modular group, i.e. let ch[ M ]( u, τ )be the meromorphic continuation of the principal admissible module M at level k ,then especially ch[ M ] (cid:18) uτ , − τ (cid:19) = γ ( u ) X S χM,N ch[ N ]( u, τ )10here the sum is over all inequivalent simple principal admissible level k modulesand γ is the usual automorphy factor for Jacobi forms. Then for λ, µ ∈ P u − h ∨ + by[41, 33] S χ L k ( λ ) , L k ( µ ) S χ L k (0) , L k ( µ ) = χ µ ( e ( − v/u ); λ ) χ µ ( e ( − v/u ); 0) (10)with χ µ ( e ( − v/u ); λ ) := X w ∈ W ǫ ( w ) e − πi vu ( λ + ρ,w ( µ + ρ )) , (11)and here W denotes the Weyl group of g and ρ the Weyl vector. W -algebras are vertex operator algebras obtained from affine vertex operator alge-bras via quantum Hamiltonian reduction. One associates to a Lie algebra g anda nilpotent element f and a complex number k the W -algebra of g correspondingto f at level k . We are interested in regular nilpotent elements f and we denotethe resulting W -algebra by W k ( g ) and its simple quotient by W k ( g ). This vertexoperator algebra is often called the regular or principal W -algebra of g at level k .For literature, we refer to [34, 42, 43].Tomoyuki Arakawa has proven [20, 44] that W k ( g ) is a rational and C -cofinitevertex operator algebra if the level satisfies the following conditions k + h ∨ = uv ∈ Q > , ( u, v ) = 1 , u, v > ( u ≥ h ∨ , v ≥ h ( v, r ∨ ) = 1 u ≥ h, v ≥ r ∨ h ∨ ( v, r ∨ ) = r ∨ Here h is the Coxeter number, h ∨ the dual Coxeter number and r ∨ is the lacetyof g . We take the notation of [33] for W-algebra modules, i.e. simple modules aredenoted by L k ( λ, µ ) with k + h ∨ = uv and ( λ, µ ) ∈ P u − h ∨ + × P ∨ ,v − h + . Moreover L k ( λ, µ ) ∼ = L k ( λ ′ , µ ′ ) if and only if ( λ ′ , µ ′ ) = ( w ( λ ) , w ( µ )) for some w ∈ f W + . Here f W + is the subgroup of automorphisms that preserve the coroot basis. If g is simply-laced, then it is isomorphic to the discriminant of the root lattice Q , i.e. P/Q , with P the weight lattice.It has been a long standing open conjecture that regular W-algebras of simply-laced Lie algebras have a coset realization inside L k ( g ) ⊗ L ( g ). In joint work withTomoyuki Arakawa and Andrew Linshaw this problem has been solved in full gen-erality [15]. We need the version at admissible level. This is the main Theorem 3part (b) of [15]. Denote by W k ( g ) the simple regular W -algebra of g at level k . Theorem 5.1. [15, Main Theorem 3 (b)]
Let ℓ = − h ∨ + uv be admissible with ( u, v ) = 1 and let g be simply-laced then L ℓ ( µ ) ⊗ L ( ν ) ∼ = M λ ∈ P u + v − h ∨ + λ = µ + ν mod Q L ℓ +1 ( λ ) ⊗ L k ( λ, µ )11 or µ ∈ P u − h ∨ + and k = − h ∨ + u + vu . Recall that by the main Theorem of [12] we have vertex tensor category structureon O ℓ ( g ) where the simple objects of O ℓ ( g ) are the L ℓ ( µ ) for µ in P u − h ∨ + . Wealso have vertex tensor category on W k ( g ) = L k (0 , k is admissible [20]. We will use thisfact soon, but first we need to explicitly determine fusion rules of the W -algebra. W -algebras The aim of this section is to prove
Theorem 6.1.
Let g be a simple Lie algebra let k = − h ∨ + vu be an admissible levelfor g with u, v positive integers that satisfy ( u, v ) = 1 . Let ℓ ∈ Z >h ∨ and let N g ℓ , φλ,ν the fusion coefficients of L ℓ − h ∨ ( g ) , i.e. they satisfy L ℓ − h ∨ ( λ ) ⊠ L ℓ − h ∨ ( ν ) ∼ = M φ ∈ P ℓ − h ∨ + N g ℓ φλ,ν L ℓ − h ∨ ( φ ) . (12) Let λ, ν ∈ P u − h ∨ + and λ ′ , ν ′ ∈ P ∨ ; v − h + , then we have L k ( λ, ⊠ L k (0 , λ ′ ) ∼ = L k ( λ, λ ′ ) L k ( λ, ⊠ L k ( ν, ∼ = M φ ∈ P u − h ∨ + N g u φλ,ν L k ( φ,
0) (13)
Moreover L k (0 , λ ′ ) centralizes L k ( λ, for all λ ∈ Q . Remark 6.2.
Three remarks are in order:1. Recall that Feigin-Frenkel duality states that W k ( g ) ∼ = W L k ( L g ) for r ∨ ( k + h ∨ )( L k + L h ∨ ) = 1 [34, 35, 15] so that we have L k (0 , λ ′ ) ⊠ L k (0 , ν ′ ) ∼ = M φ ′ ∈ L P q − Lh ∨ + N L g q φ ′ λ ′ ,ν ′ L k (0 , φ ′ ) , q = v if ( v, r ∨ ) = 1 q = vr ∨ if ( v, r ∨ ) = r ∨ where one identifies coroots with roots of the dual Lie algebra via L α = α ∨ √ r ∨ and correspondingly coweights with weights of the dual Lie algebra. By L P we then mean the weight lattice of the dual Lie algebra L g .2. Fusion rules have been known for g simply-laced, see [33, Thm. 8.5] and [32,Thm. 4.3] but with the assumption that q is coprime to the order of the12iscriminant P/Q . Together with commutativity and associativity the fusionrules stated by us completely determine all fusion rules: L k ( λ, λ ′ ) ⊠ L k ( ν, ν ′ ) ∼ = L k ( λ, ⊠ L k (0 , λ ′ ) ⊠ L k ( ν, ⊠ L k (0 , ν ′ ) ∼ = L k ( λ, ⊠ L k ( ν, ⊠ L k (0 , λ ′ ) ⊠ L k (0 , ν ′ ) ∼ = M φ ∈ P u − h ∨ + N g u φλ,ν L k ( φ, ⊠ M φ ′ ∈ L P q − Lh ∨ + N L g q φ ′ λ ′ ,ν ′ L k (0 , φ ′ ) ∼ = M φ ∈ P u − h ∨ + φ ′ ∈ L P q − Lh ∨ + N g u φλ,ν N L g q φ ′ λ ′ ,ν ′ L k ( φ, φ ′ ) .
3. The fusion rules of (13) tell us that the map L ℓ − h ∨ ( λ ) L k ( λ,
0) gives a ringhomomorphism from the tensor ring of the affine vertex operator algebra tothe one of the W-algebra. It is not injective if v − h = 0.Recall from the introduction that a modular tensor category has the propertythat the map K ( C ) → End (cid:16) M W ∈ Sim( C ) W (cid:17) , X X W ∈ Sim( C ) Φ X,W = X W ∈ Sim( C ) S m X,W S m ,W Id W is a ring isomorphism. Heret the set of inequivalent simple objects of C is denotedby Sim( C ). We can thus prove fusion rules by computing all open Hopf links. Therest of the section is the proof of the Theorem: Proof.
We continue to take notation and set-up of [33]. Let g be a simple Liealgebra and let k = − h ∨ + vu be an admissible level for g and let λ, µ ∈ P u − h ∨ + and λ ′ , µ ′ ∈ P ∨ ; v − h + . Let I ( p,q )+ = (cid:16) P u − h ∨ + × P ∨ ; v − h + (cid:17) / f W + and recall the definition of χ µ ( e ( − v/u ); λ ) in (11). Let me take the short hand notation for this proof χ µ ( e ( − v/u ); λ ) := χ µ ( λ ) , χ µ ′ ( e ( − u/v ); λ ′ ) := χ ′ µ ′ ( λ ′ ) = X w ∈ W ǫ ( w ) e − πi uv ( λ ′ + ρ,w ( µ ′ + ρ )) so that the modular S -matrix is given by [33, Cor. 8.4]ch[ L k ( λ, λ ′ )] (cid:18) − τ , τ − L [0] u (cid:19) = X µ,µ ′ ∈ I p,q + S χ ( λ,λ ′ ) , ( µ,µ ′ ) ch[ L k ( µ, µ ′ )] ( τ, u ) S χ ( λ,λ ′ ) , ( µ,µ ′ ) = e πi (( λ + ρ,µ ′ + ρ )+( λ ′ + ρ,µ + ρ )) ( uv ) ℓ/ | J | / χ µ ( λ ) χ ′ µ ′ ( λ ′ ) . (14) ℓ is the rank of g and J is the groupoid of simple currents. We note that [33]introduced a second variable u in order to distinguish characters that otherwisewould be linearly dependent. This then confirmed an earlier result [32]. It follows13hat χ µ (0) and χ ′ µ ′ (0) are non-zero as the modular S χ -matrix and the Hopf link S m in a rational and C -cofinite vertex operator algebra are proportional to each other(and non-zero) and the dimension of a simple object in a semi-simple rigid tensorcategory is always non-zero. We thus have S χ ( λ,λ ′ ) , ( µ,µ ′ ) S χ (0 , , ( µ,µ ′ ) = e πi (( λ ′ ,µ + ρ )+( λ,µ ′ + ρ )) χ µ ( λ ) χ ′ µ ′ ( λ ′ ) χ µ (0) χ ′ µ ′ (0) . Hence for any µ, µ ′ we have that S χ ( λ,λ ′ ) , ( µ,µ ′ ) S χ (0 , , ( µ,µ ′ ) = e πi (( λ ′ ,µ + ρ )+( λ,µ ′ + ρ )) χ µ ( λ ) χ ′ µ ′ ( λ ′ ) χ µ (0) χ ′ µ ′ (0)= e πi (( λ,µ ′ + ρ )+(0 ,µ + ρ )) χ µ ( λ ) χ ′ µ ′ (0) χ µ (0) χ ′ µ ′ (0) e πi ((0 ,µ ′ + ρ )+( λ ′ ,µ + ρ )) χ µ (0) χ ′ µ ′ ( λ ′ ) χ µ (0) χ ′ µ ′ (0)= S χ ( λ, , ( µ,µ ′ ) S χ (0 , , ( µ,µ ′ ) S χ (0 ,λ ′ ) , ( µ,µ ′ ) S χ (0 , , ( µ,µ ′ ) . Since the W -algebra is rational [20] so that its module category is modular [1] themap from the Grothendieck ring to the endomorphism ring of the direct sum of allsimples via open Hopf links is an isomorphism and so it follows that L k ( λ, ⊠ L k (0 , λ ′ ) ∼ = L k ( λ, λ ′ ) . We now check the centralizing property. Using the symmetry that χ λ ( µ ) = χ µ ( λ )and that for any two objects X, Y of a strongly rational vertex operator algebra onehas S m X,Y S m ,Y = S χX,Y S χ ,Y we get that S m ( λ, , (0 ,λ ′ ) S m (0 , , (0 ,λ ′ ) S m (0 , , ( λ, = S m ( λ, , (0 ,λ ′ ) S m (0 , , ( λ,λ ′ ) = S m ( λ, , (0 ,λ ′ ) S m (0 , , ( λ,λ ′ ) S m (0 , , (0 ,λ ′ ) S m (0 , , (0 ,λ ′ ) S m (0 , , (0 , S m (0 , , (0 , = S χ ( λ, , (0 ,λ ′ ) S χ (0 , , ( λ,λ ′ ) S χ (0 , , (0 ,λ ′ ) S χ (0 , , (0 ,λ ′ ) S χ (0 , , (0 , S χ (0 , , (0 , = S χ ( λ, , (0 ,λ ′ ) S χ (0 , , ( λ,λ ′ ) = e πi (( λ ′ + ρ,λ + ρ )+( ρ,ρ )) χ ( λ ) χ ′ λ ′ (0) e πi (( λ + ρ,ρ )+( λ ′ + ρ,ρ )) χ λ (0) χ ′ λ ′ (0) = e πi (( λ ′ + ρ,λ + ρ )+( ρ,ρ )) e πi (( λ + ρ,ρ )+( λ ′ + ρ,ρ )) = e πi ( λ,λ ′ ) . It follows that L k (0 , λ ′ ) centralizes L k ( λ,
0) for all λ ∈ Q by Remark 3.4.We will now be more explicit on the fusion ring. For this let r ∈ { , v } andconsider the ring R r = Z ( e ( − r/ ( N u ))) with N the level of the weight lattice P , that14s the smallest positive integer N , such that N P is an integral lattice. For examplefor sl n we have n = N . Let S r = < χ µ ( e ( − r/u ); 0) > be the monoid generated by χ µ ( e ( − r/u ); 0) so that S χ ( λ, , ( µ,µ ′ ) S χ (0 , , ( µ,µ ′ ) = e πi (( λ,µ ′ + ρ )) χ µ ( λ ) χ µ (0) ∈ S − v R v . Let σ v be the surjective ring homomorphism generated by e ( − / ( N u )) e ( − v/ ( N u )), i.e. σ v : S − R → S − v R v , e ( − / ( N u )) e ( − v/ ( N u )) . (15)So that we see that σ v (cid:18) χ µ ( e ( − /u ); λ ) χ µ ( e ( − /u ); 0) (cid:19) = χ µ ( e ( − v/u ); λ ) χ µ ( e ( − v/u ); 0) . The former are the normalized Hopf links of L u − h ∨ ( g ) and satisfy χ µ ( e ( − /u ); λ ) χ µ ( e ( − /u ); 0) χ µ ( e ( − /u ); ν ) χ µ ( e ( − /u ); 0) = X φ N g u φλ,ν χ µ ( e ( − /u ); φ ) χ µ ( e ( − /u ); 0)with N g u φλ,ν the fusion coefficients of L u − h ∨ ( g ): L u − h ∨ ( λ ) ⊠ L u − h ∨ ( ν ) ∼ = M φ ∈ P u − h ∨ + N g u φλ,ν L u − h ∨ ( φ ) . Note that N g u φλ,ν = 0 implies that λ + ν = φ mod Q since the fusion product is ahomomorphic image of the tensor ring of the compact Lie group. It thus followsthat S χ ( λ, , ( µ,µ ′ ) S χ (0 , , ( µ,µ ′ ) S χ ( ν, , ( µ,µ ′ ) S χ (0 , , ( µ,µ ′ ) = e πi (( λ + ν,µ ′ + ρ )) χ µ ( λ ) χ µ (0) χ µ ( ν ) χ µ (0)= e πi (( λ + ν,µ ′ + ρ )) σ v (cid:18) χ µ ( e ( − /u ); λ ) χ µ ( e ( − /u ); 0) (cid:19) σ v (cid:18) χ µ ( e ( − /u ); ν ) χ µ ( e ( − /u ); 0) (cid:19) = e πi (( λ + ν,µ ′ + ρ )) X φ N g u φλ,ν σ v (cid:18) χ µ ( e ( − /u ); φ ) χ µ ( e ( − /u ); 0) (cid:19) = X φ N g u φλ,ν e πi (( φ,µ ′ + ρ )) σ q (cid:18) χ µ ( e ( − /u ); φ ) χ µ ( e ( − /u ); 0) (cid:19) = X φ N g u φλ,ν S χ ( φ, , ( µ,µ ′ ) S χ (0 , , ( µ,µ ′ ) , and hence L k ( λ, ⊠ L k ( ν, ∼ = M φ ∈ P u − h ∨ + N g u φλ,ν L k ( φ, . Fusion categories for ordinary modules
Let g be simply-laced. We can now proceed in analogy to section 7 of [12]. Firstly, let ℓ + h ∨ = uv and let C u be the category of right modules of W k ( g ) with k + h ∨ = u + vu ,i.e. the simple objects are the L k (0 , µ ) with µ ∈ P u − h ∨ + and we have just seen thatthis is a full vertex tensor subcategory for W k ( g ). Moreover, consider O ℓ ( g ) ⊠ O ( g )and let e O ℓ ( g ) be its subcategory whose simple objects are the L ℓ ( λ ) ⊗ L ( ν ) with( λ, ν ) ∈ P u − h ∨ + × P such that λ + ν ∈ Q . Then Theorem 7.1.
The categories e O ℓ ( g ) and C u are equivalent as braided tensor cate-gories with twist. In particular e O ℓ ( g ) is rigid and thus it is a ribbon category. Remark 7.2.
Aganagic, Frenkel and Okounkov [21] conjecture a braided equiva-lence between W -algebra modules and Weyl modules of the affine vertex operatoralgebra at generic levels if they satisfy the relation β = 1 k + h ∨ + r ∨ N where r ∨ is the lacety of g and β is the shifted level of the W -algebra. The originalconjecture (Conjecture 6.3 of [21]) is the case N = 0, but they later say that thisshould generalize to any integral N . Our Theorem here is the simply-laced versionat admissible level of this at N = 1 and we see that the statement has to be slightlytweaked as we get an equivalence between e O ℓ ( g ) and C u . Note that β = u + vu hereand k + h ∨ = uv so this fits. Proof.
Let V = L ℓ +1 ( g ) ⊗ W k ( g ) and let A = L ℓ ( g ) ⊗ L ( g ). Then V has a vertextensor category structure C V given by the Deligne product of the category of ordinarymodules of L ℓ +1 and the modular tensor category of W k ( g ). By Theorem 5.1 thevertex operator algebra A is an object in this category and thus by Theorem 3.1corresponds to a commutative and associative algebra in C V . We can view C u as afull tensor subcategory of C V via the embedding C u → C V , L k (0 , µ ) L ℓ +1 ( g ) ⊠ L k (0 , µ ) . Let F : C V → C A be the induction functor. By Theorem 6.1 every object in C u centralizes A and so F maps C u to C loc A . We are thus in the situation of Theorem3.5 and the image of L ℓ +1 ( g ) ⊠ L k (0 , µ ) under induction is F ( L ℓ +1 ( g ) ⊠ L k (0 , µ )) ∼ = V A ⊠ V L ℓ +1 ( g ) ⊠ L k (0 , µ ) ∼ = V M λ ∈ P u + v − h ∨ + ∩ Q ( L ℓ +1 ( λ ) ⊗ L k ( λ, ⊠ V ( L ℓ +1 ( g ) ⊠ L k (0 , µ )) ∼ = V M λ ∈ P u + v − h ∨ + ∩ Q L ℓ +1 ( λ ) ⊗ L k ( λ, µ ) ∼ = V L ℓ ( µ ) ⊗ L ( ν ) ∈ e O ℓ ( g ) . L ℓ ( µ ) ⊗ L ( ν ) ∈ e O ℓ ( g ). The weight ν is the unique one of L ( g ) such that µ + ν ∈ Q .The claim is now proven due to Theorem 3.5.This Theorem has a series of nice corollaries. Corollary 7.3.
The category O ℓ ( g ) is ribbon.Proof. The category O ℓ ( g ) is a subcategory of O ℓ ( g ) ⊠ O ( g ) under the identificationof L ℓ ( µ ) with L ℓ ( µ ) ⊗ L (0). The L ( ν ) are all rigid simple currents with fusionrules L ( ν ) ⊠ L ( ν ′ ) ∼ = L ( ν + ν ′ ) and ν, ν ′ ∈ P/Q and via the embedding O ( g ) →O ℓ ( g ) ⊠ O ( g ) the objects L ℓ (0) ⊗ L ( µ ) are rigid simple currents as well. The inverseobject of L ℓ (0) ⊗ L ( µ ) is the dual module denoted by L ℓ (0) ⊗ L ( µ ∗ ). It followsthat L ℓ ( µ ) ⊗ L (0) ∼ = ( L ℓ ( µ ) ⊗ L ( µ ∗ )) ⊠ ( L ℓ (0) ⊗ L ( µ ))and hence L ℓ ( µ ) ⊗ L (0) is rigid as tensor product of two rigid modules is rigid. Corollary 7.4.
The fusion rules in O ℓ ( g ) are L ℓ ( λ ) ⊠ L ℓ ( ν ) ∼ = M φ ∈ P u − h ∨ + N g u φλ,ν L ℓ ( φ ) with N g u φλ,ν the fusion rules of L u − h ∨ ( g ) , see (12) .Proof. This is clear, since the induction functor is a tensor functor and since thefusion rules of C u have been determined in Theorem 6.1 and since L ( ν ) ⊠ L ( ν ′ ) ∼ = L ( ν + ν ′ ) for ν, ν ′ ∈ P/Q together with N g u φλ,ν = 0 unless λ + ν = φ mod Q . S m coincide with character S χ We now compare the open Hopf links of admissible affine vertex operator algebra tomodular S χ . For this let g be simply laced and let ℓ = − h ∨ + uv be an admissiblenumber. Theorem 8.1.
Open Hopf links coincide with normalized character S χ in O ℓ ( g ) ,i.e. S m L ℓ ( λ ) , L ℓ ( µ ) S m L ℓ (0) , L ℓ ( µ ) = S χ L ℓ ( λ ) , L ℓ ( µ ) S χ L ℓ (0) , L ℓ ( µ ) = χ µ ( e ( − v/u ); λ ) χ µ ( e ( − v/u ); 0) for all λ, µ ∈ P u − h ∨ + . roof. Theorem 2.89 of [16] tells us that induction preserves Hopf links, so that theHopf links of W k ( g ) with k = − h ∨ + u + vu satisfy S m (0 ,λ ) , (0 ,µ ) S m (0 , , (0 ,µ ) = S m L ( λ ∗ ) , L ( µ ∗ ) S m L (0) , L ( µ ∗ ) S m L ℓ ( λ ) , L ℓ ( µ ) S m L ℓ (0) , L ℓ ( µ ) (16)The Hopf links of interest of W k ( g ) are given by Verlinde’s formula [1] via character’s S χ by S m (0 ,λ ) , (0 ,µ ) S m (0 , , (0 , = S m (0 ,λ ) , (0 ,µ ) S m (0 , , (0 ,µ ) S m (0 , , (0 ,µ ) S m (0 , , (0 , = S χ (0 ,λ ) , (0 ,µ ) S χ (0 , , (0 ,µ ) S χ (0 , , (0 ,µ ) S χ (0 , , (0 , = S χ (0 ,λ ) , (0 ,µ ) S χ (0 , , (0 , = e πi ( λ + µ,ρ ) χ µ ( e ( − ( u + v ) /u ); λ ) χ ( e ( − ( u + v ) /u ); 0) = e − πi (( µ,λ )+ ρ ) χ µ ( e ( − v/u ); λ ) χ ( e ( − v/u ); 0)where for the last equality we used that Weyl reflections change weights by anelement in the root lattice and since e − πi ( µ + ρ,λ + ρ ) = e − πi ( w ( µ + ρ ) ,λ + ρ ) for any w ∈ W we can take this factor out of the sum. It follows that S m (0 ,λ ) , (0 ,µ ) S m (0 , , (0 ,µ ) = S m (0 ,λ ) , (0 ,µ ) S m (0 , , (0 , S m (0 , , (0 , S m (0 , , (0 ,µ ) = e − πi ( µ,λ ) χ µ ( e ( − v/u ); λ ) χ µ ( e ( − v/u ); 0)but this coincides with the normalized character S -matrix of the affine vertex opera-tor algebra at level ℓ times the corresponding lattice vertex operator algebra module,i.e. S m (0 ,λ ) , (0 ,µ ) S m (0 , , (0 ,µ ) = S χ L ( λ ∗ ) , L ( µ ∗ ) S χ L (0) , L ( µ ∗ ) S χ L ℓ ( λ ) , L ℓ ( µ ) S χ L ℓ (0) , L ℓ ( µ ) . (17)By Verlinde’s formula [1] we have S χ L ( λ ∗ ) , L ( µ ∗ ) S χ L (0) , L ( µ ∗ ) = S m L ( λ ∗ ) , L ( µ ∗ ) S m L (0) , L ( µ ∗ ) and since all L ( µ ) are simple currents in a ribbon category the normalized Hopflinks must be roots of unity of order dividing the order of the simple current. Theyare especially all non-zero so that the claim follows by comparing (16) and (17).Recall that a ribbon category C is a modular tensor category if and only if themap K ( C ) → End (cid:16) M W ∈ Sim( C ) W (cid:17) , X X W ∈ Sim( C ) Φ X,W = X W ∈ Sim( C ) S m X,W S m ,W Id W (18)is a ring isomorphism. Recall also that the set of inequivalent simple objects of C isdenoted by Sim( C ). 18 orollary 8.2. With the same notation as in the previous Theorem. Let N be thelevel of the weight lattice P of g and let ( N, v ) = 1 . Then O ℓ ( g ) is a modular tensorcategory.Proof. If (
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