Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion
aa r X i v : . [ m a t h . QA ] S e p BOSONIC GHOSTBUSTING — THE BOSONIC GHOST VERTEX ALGEBRA ADMITS ALOGARITHMIC MODULE CATEGORY WITH RIGID FUSION
ROBERT ALLEN AND SIMON WOODAbstract. The rank 1 bosonic ghost vertex algebra, also known as the βγ ghosts, symplectic bosons or Weyl vertex algebra,is a simple example of a conformal field theory which is neither rational, nor C -cofinite. We identify a module category,denoted category F , which satisfies three necessary conditions coming from conformal field theory considerations: closure underrestricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second ofthese conditions, with the other two already being known. Further, we show that category F has sufficiently many projectiveand injective modules, give a classification of all indecomposable modules, show that fusion is rigid and compute all fusionproducts. The fusion product formulae turn out to perfectly match a previously proposed Verlinde formula, which was computedusing a conjectured generalisation of the usual rational Verlinde formula, called the standard module formalism. The bosonicghosts therefore exhibit essentially all of the rich structure of rational theories despite satisfying none of the standard rationalityassumptions such as C -cofiniteness, the vertex algebra being isomorphic to its restricted dual or having a one-dimensionalconformal weight 0 space. In particular, to the best of the authors’ knowledge this is the first example of a proof of rigidity for alogarithmic non- C -cofinite vertex algebra. Introduction
A vertex algebra is called logarithmic if it admits reducible yet indecomposable modules on which the Virasoro L operator acts non-semisimply, giving rise to logarithmic singularities in the correlation functions of the associatedconformal field theory. There is a general consensus within the research community that many of the structures familiarfrom rational vertex algebras such as modular tensor categories [1] and, in particular, the Verlinde formula should generalisein some form to the logarithmic case, at least for sufficiently nice logarithmic vertex algebras. To this end, considerablework has been done on developing non-semisimple or non-finite generalisations of modular tensor categories [2–4].However, progress has been hindered by a severe lack of examples, making it hard to come up with the right set ofassumptions.Ghost systems have been used extensively in theoretical physics and quantum algebra. Their applications include gaugefixing in string theory [5], Wakimoto free field realisations [6], quantum Hamiltonian reduction [7] and constructing thechiral de Rham complex on smooth manifolds [8]. Fermionic ghosts at central charge c = − in the form of symplecticfermions have received a lot of attention in the past [9–11], due to their even subalgebra being one of the first knownexamples of a logarithmic vertex algebra. In particular, they are one of the few known examples of C -cofinite yetlogarithmic vertex algebras [12–14]. This family also provides the only known examples of logarithmic C -cofinite vertexalgebras with a rigid fusion product [12, 15].Here we study the rank 1 bosonic ghosts at central charge c = . One of the motivations for studying this algebra is thatit is simple enough to allow many quantities to be computed explicitly, while simultaneously being distinguished frombetter understood algebras in a number of interesting ways. For example, the bosonic ghosts are not C -cofinite and theywere shown to be logarithmic by D. Ridout and the second author in [16], in which the module category to be studiedhere, denoted category F , was introduced. The main goals of [16] were determining the modular properties of charactersin category F and computing the Verlinde formula, using the standard module formalism pioneered by D. Ridout and T.Creutzig [17–19], to predict fusion product formulae. Later, D. Adamović and V. Pedić computed the dimensions of spacesof intertwining operators among the simple modules of category F in [20], which turned out to match the predictionsmade by the Verlinde formula in [16]. Here we show that fusion (in the sense of the P ( w ) -tensor products of [21]) equipscategory F with the structure of a braided tensor category. This, in particular, implies that category F is closed underfusion, that is, the fusion product of any two objects in F has no contributions from outside F and is hence again an object Mathematics Subject Classification.
Primary 17B69, 81T40; Secondary 17B10, 17B67, 05E05. in F . We derive explicit formulae for the decomposition of any fusion product into indecomposable direct summands,and we show that fusion is rigid and matches the Verlinde formula of [16].A further source of interest for the bosonic ghosts is an exciting recent correspondence between four-dimensional superconformal field theory and two-dimensional conformal field theory [22], where the bosonic ghosts appear as one of thesmaller examples on the two-dimensional side. Within this context the bosonic ghosts are the first member of a familyof vertex algebras called the B p algebras [23, 24]. The B p -module categories are conjectured to satisfy interesting tensorcategorical equivalences to the module category of the unrolled restricted quantum groups of sl . It will be an interestingfuture problem to explore these categorical relations in more detail using the results of this paper.The paper is organised as follows. In Section 2, we fix notation by giving an introduction to the bosonic ghost algebraand certain important automorphisms called conjugation and spectral flow; construct category F , the module category tobe studied; and give two free field realisations of the bosonic ghost algebra. In Section 3 we begin the analysis of category F as an abelian category by using the free field realisations of the bosonic ghost algebra to construct a logarithmic module,denoted P , on which the operator L has rank 2 Jordan blocks. We further show that P is both an injective hull and aprojective cover of the vacuum module (the bosonic ghost vertex algebra as a module over itself), and classify all projectivemodules in category F , thereby showing that category F has sufficiently many projectives and injectives. In Section 4we complete the analysis of category F as an abelian category by classifying all indecomposable modules. In Section 5we show that fusion equips category F with the structure of a vertex tensor category, the main obstruction being showingthat certain conditions, sufficient for the existence of associativity isomorphisms, hold. We further show that the simpleprojective modules of F are rigid. In Section 6 we show that category F is rigid and determine direct sum decompositionsfor all fusion products of modules in category F . In Appendix A we review an argument by Yang [25], which providessufficient conditions for a technical property, called convergence and extension, required for the existence of associativityisomorphisms. We adjust the argument of Yang slightly to remove certain assumptions on module categories. Thisadjusted argument proves Theorem 5.7, which should also prove useful for the generalisations of category F to othervertex algebras such as those constructed from affine Lie algebras at admissible levels. Acknowledgements.
SW would like to thank Yi-Zhi Huang, Shashank Kanade, Robert McRae and Jinwei Yang for helpfuland stimulating discussions regarding the subtleties of vertex tensor categories, and Ehud Meir for the same regardingfinitely generated modules. Both authors are very grateful to Thomas Creutzig for drawing their attention to a mistakerelating to convergence and extension properties in a previous version of this manuscript. RA’s research is supported bythe EPSRC Doctoral Training Partnership grant EP/R513003/1. SW’s research is supported by the Australian ResearchCouncil Discovery Project DP160101520.
2. Bosonic ghost vertex algebra
In this section we introduce the bosonic ghost vertex algebra, along with its gradings and automorphisms. We define themodule category which will be the focus of this paper. We also introduce useful tools for the classification of modules andcalculation of fusion products, including two free field realisations. Note that we will make specific choices of conformalstructure for all vertex algebras considered in this paper and so will not distinguish between vertex algebras, vertex operatoralgebras and conformal vertex algebras or other similar naming conventions.2.1.
The algebra and its automorphisms.
The bosonic ghost vertex algebra (also called βγ ghosts) is closely related tothe Weyl algebra. Their defining relations resemble each other and the Zhu algebra of the bosonic ghosts is isomorphicto the Weyl algebra. The bosonic ghosts are therefore also often referred to as the Weyl vertex algebra. Due to theseconnections, we first introduce the Weyl algebra and its modules before going on to consider the bosonic ghosts. Definition 2.1.
The (rank 1)
Weyl algebra A is the unique unital associative algebra with two generators p , q , subject tothe relations [ p , q ] = , (2.1)and no additional relations beyond those required by the axioms of an associative algebra. The grading operator is theelement N = qp . Definition 2.2.
We define the following indecomposable A -modules:(1) C [ x ] , where p acts as ∂/∂ x and q acts as x . Denote this module by V .(2) C [ x ] , where p acts as x and q acts as − ∂/∂ x . Denote this module by c V .(3) C [ x , x − ] x λ , λ ∈ C \ Z , where p acts as ∂/∂ x and q acts as x . Note that shifting λ by an integer yields an isomorphicmodule. Denote the mutually inequivalent isomorphism classes of these modules by W µ , where µ ∈ C / Z , µ , Z and λ ∈ µ .(4) C [ x , x − ] , where p acts as ∂/∂ x and q acts as x . Denote this module by W + . This module is uniquely characterised bythe non-split exact sequence −→ V −→ W + −→ c V −→ . (2.2)(5) C [ x , x − ] , where p acts as x and q acts as − ∂/∂ x . Denote this module by W − . This module is uniquely characterisedby the non-split exact sequence −→ c V −→ W − −→ V −→ . (2.3)A module on which N = qp acts semisimply is called a weight module. Note that N acts semisimply on all modules above. Proposition 2.3 (Block [26]) . Any simple A -module on which N acts semisimply is isomorphic to one of those listed inDefinition 2.2, Parts (1) – (3) . Definition 2.4.
The bosonic ghost vertex algebra G is the unique vertex algebra strongly generated by two fields β, γ ,subject to the defining operator product expansions γ ( z ) β ( w ) ∼ z − w , β ( z ) β ( w ) ∼ γ ( z ) γ ( w ) ∼ , (2.4)and no additional relations beyond those required by vertex algebra axioms.The bosonic ghost vertex algebra admits a one-parameter family of conformal structures. Here we choose the Virasorofield (or energy momentum tensor) to be T ( z ) = − : β ( z ) ∂γ ( z ): , (2.5)thus determining the central charge to be c = and the conformal weights of β and γ to be and , respectively. Thebosonic ghost fields can thus be expanded as formal power series with the mode indexing chosen to reflect the conformalweights. β ( z ) = ∑ n ∈ Z β n z − n − , γ ( z ) = ∑ n ∈ Z γ n z − n . (2.6)The operator product expansions of β and γ fields imply that their modes generate the bosonic ghost Lie algebra G satisfying the Lie brackets [ γ m , β n ] = δ m + n , , [ β m , β n ] = [ γ m , γ n ] = , m , n ∈ Z , (2.7)where is central and acts as the identity on any G -module, since it corresponds to the identity (or vacuum) field.Within G there is a rank 1 Heisenberg vertex algebra generated by the field J ( z ) = : β ( z ) γ ( z ): . (2.8)A quick calculation reveals that J is a free boson of Lorentzian signature, not a conformal primary, and that J defines agrading on β and γ called ghost weight (or ghost number), that is, J ( z ) J ( w ) ∼ − z − w ) , T ( z ) J ( w ) ∼ − z − w ) + J ( w )( z − w ) + ∂ J ( w ) z − w , J ( z ) β ( w ) ∼ β ( w ) z − w , J ( z ) γ ( w ) ∼ − γ ( w ) z − w . (2.9) R ALLEN AND S WOOD
Note that for the distinguished elements β, γ, J , and T we suppress the field map symbol Y : G → G J z , z − K . For genericelements A ∈ G we will use both Y ( A , z ) and A ( z ) to denote the field corresponding to A , depending on what is easier toread in the given context.We make frequent use of two automorphisms of G . The first is spectral flow, which acts on the G modes as σ ℓ β n = β n − ℓ , σ ℓ γ n = γ n + ℓ , σ ℓ = . (2.10)The second is conjugation which is given by c β n = γ n , c γ n = − β n , c = . (2.11)These automorphisms satisfy the relation c σ ℓ = σ − ℓ c . (2.12)At the level of fields, these automorphisms act as σ ℓ β ( z ) = β ( z ) z − ℓ , σ ℓ γ ( z ) = γ ( z ) z ℓ , σ ℓ J ( z ) = J ( z ) + ℓ z − , σ ℓ T ( z ) = T ( z ) − ℓ J ( z ) z − − ℓ ( ℓ − z − , c β ( z ) = γ ( z ) , c γ ( z ) = − β ( z ) , c J ( z ) = − J ( z ) + z − , c T ( z ) = T ( z ) + ∂ J ( z ) + J ( z ) z − . (2.13)The primary utility of the conjugation and spectral flow automorphisms lies in constructing new modules from knownones through twisting. Definition 2.5.
Let M be a G -module and α an automorphism. The α -twisted module α M is defined to be M as a vectorspace, but with the action of G redefined to be A ( z ) · α m = α − ( A ( z )) m , A ∈ G , m ∈ M , (2.14)where the action of G on the right-hand side is the original untwisted action of G on M . Remark.
Due to being algebra automorphisms, spectral flow and conjugation twists both define exact covariant functors.Further, the respective ghost and conformal weights [ j , h ] of a vector m in a G -Module M transform as follows underconjugation and spectral flow. σ ℓ : (cid:2) j , h (cid:3) (cid:2) j − ℓ, h + ℓ j − ℓ ( ℓ + (cid:3) , c : (cid:2) j , h (cid:3) (cid:2) − j , h (cid:3) . (2.15)Since c β n = − β n and c γ n = − γ n , we have c M (cid:27) M , for any G -module M . We shall later see that spectral flow hasinfinite order and thus the relations (2.12) imply that at the level of the module category spectral flow and conjugationgenerate the infinite dihedral group. Theorem 2.6.
For any G -modules M and N , conjugation and spectral flow are compatible with fusion products in thefollowing sense. σ ℓ M ⊠ σ m N (cid:27) σ ℓ + m ( M ⊠ N ) , c M ⊠ c N (cid:27) c σ ( M ⊠ N ) . (2.16)The behaviour of spectral flow under fusion was proven for vertex algebras with finite dimensional conformal weightspaces in [27, Proposition 2.4]. However, the proof does not rely on this fact, and so we can apply the result to G -modules,as in [20, Proposition 3.1]. The behaviour of conjugation under fusion was proven in [20, Proposition 2.1], whereconjugation was denoted by σ and spectral flow by ρ ℓ . There the automorphism g corresponds to σ − c = c σ here. Theseformulae mean that the fusion of modules twisted by spectral flow is determined by the fusion of untwisted modules, asimplification we shall make frequent use of.2.2. Module category.
Every G -module is a G -module, however, the converse is not true (consider for example theadjoint G -module). The category of smooth G -modules consists of precisely those modules which are also G -modules.Such modules are also commonly called weak G -modules and we shall use these terms interchangeably. Unfortunately thecategory of all smooth modules is at present too unwieldy to analyse and so we must invariably consider some subcategory. In this section we define the module category, which we believe to be the natural one from the perspective of conformalfield theory, because it is compatible with the following two necessary conformal field theoretic conditions.(1) Non-degeneracy of n -point conformal blocks (chiral correlation functions) on the sphere.(2) Well-definedness of conformal blocks at higher genera.Condition (1) can be reduced to the non-degeneracy of two and three-point conformal blocks. The non-degeneracy oftwo-point conformal blocks requires the module category to be closed under taking restricted duals, while non-degeneracyof three-point conformal blocks requires the module category to be closed under fusion (as, for example, constructed bythe P ( w ) -tensor product of Huang-Lepowsky-Zhang). Conformal blocks at higher genera can be constructed from thoseon the sphere provided there is a well-defined action of the modular group on characters. Thus Condition (2) requirescharacters to be well-defined, that is, for all modules to decompose into direct sums of finite dimensional simultaneousgeneralised J and L eigenspaces. On any simple such module both L and J will act semisimply. Further, the actionof J is semisimple on a fusion product if J acts semisimply on both factors of the product. We can therefore restrictourselves to a category of J -semisimple modules without endangering closure under fusion. We cannot, however, assumethat L will act semisimply in general.The main tool for identifying and classifying vertex operator algebra modules is Zhu’s algebra. Sadly Zhu’s algebrais only sensitive to modules containing vectors annihilated by all positive modes. Any simple such module is a simplemodule in the category called R below. We will see that R is closed under taking restricted duals, however, as can beseen later in Section 6, category R is not closed under fusion. Further, it was shown in [16] that the action of the modulargroup does not close on its characters. Thus a larger category is needed, which will be denoted F below. It was shownin [16] that the action of the modular group closes on the characters of F and strong evidence was presented that fusiondoes as well. We will see in Section 6 that category F is indeed closed under fusion and that it satisfies numerous othernice properties.The definition of the module categories mentioned above requires the following choice of parabolic triangular decom-position of G . G ± = span { β ± n , γ ± n : n ≥ } , G = span { , β , γ } . (2.17)This decomposition is parabolic, because G is not abelian and thus not a choice of Cartan subalgebra. The role of theCartan subalgebra will instead be played by span { , J } , which is technically a subalgebra of the completion of U ( G ) ratherthan G . Definition 2.7. (1) Let G -WMod be the category of smooth weight G -modules, that is the category whose objects are all smooth (orweak) G -modules M (we follow the conventions of [28] regarding smooth modules) which in addition satisfy that J acts semisimply and whose arrows are all G -module homomorphisms.(2) Let R be the full subcategory of G -WMod consisting of those modules M ∈ G -WMod satisfying • M is finitely generated, • G + acts locally nilpotently, that is, for all m ∈ M , U (cid:0) G + (cid:1) m is finite dimensional.(3) Let F be the full subcategory of G -WMod consisting all finite length extensions of arbitrary spectral flows of modulesin R with real J weights.The A -modules of Definition 2.2 induce to modules in category R . Definition 2.8.
Let M be a A -module, then we induce M to a G -module Ind M in R by having G + act trivially on M , β and γ act as − p and q , respectively, and G − act freely. We denote(1) V (cid:27) Ind V , the vacuum module or bosonic ghost vertex algebra as a module over itself.(2) c V (cid:27) σ − V (cid:27) Ind c V , the conjugation twist of the vacuum module.(3) W λ (cid:27) Ind W λ with λ ∈ C / Z , λ , Z .(4) W ± (cid:27) Ind W ± . R ALLEN AND S WOOD
Note that due to the simple nature of the G commutation relations (2.7) Ind M is simple whenever M is, that is, themodules listed in parts (1) – (3) are simple. Proposition 2.9. (1)
Any simple module in R is isomorphic to one of those listed in Parts (1) – (3) of Definition 2.8. (2) Any simple module in F is isomorphic to one of the following mutually inequivalent modules. σ ℓ V , σ ℓ W λ , ℓ ∈ Z , λ ∈ R / Z , λ , Z . (2.18)(3) The conjugation twists of simple modules in F satisfy c σ ℓ V (cid:27) σ − − ℓ V , c σ ℓ W λ (cid:27) σ − ℓ W − λ , ℓ ∈ Z , λ ∈ R / Z , λ , Z . (2.19)(4) The indecomposable modules W ± satisfy the non-split exact sequences −→ V −→ W + −→ σ − V −→ , (2.20a) −→ σ − V −→ W − −→ V −→ . (2.20b)This proposition was originally given in [16, Proposition 1], however, Part (1) is an immediate consequence of Block’sclassification of simple Weyl modules [26]. We shall show in Proposition 3.2 that, up to spectral flow twists, theindecomposable modules W ± are the only indecomposable length 2 extensions of spectral flows of the vacuum module.In Section 4 we extend the indecomposable modules W ± to infinite families of indecomposable modules.2.3. Restricted duals.
As mentioned above, conformal field theories require their representation categories to be closedunder taking restricted duals. They are also an essential tool for the computation of fusion products using the Huang-Lepowsky-Zhang (HLZ) double dual construction [21, Part IV], also called the P ( w ) -tensor product, and so we record thenecessary definitions here. Definition 2.10.
Let M be a weight G -module. The restricted dual (or contragredient) module is defined to be M ′ = M h , j ∈ C Hom (cid:16) M ( j )[ h ] , C (cid:17) , Hom (cid:16) M ( j )[ h ] , C (cid:17) = { m ∈ M : ( J − j ) m = , ( L − h ) n m = , n ≫ } , (2.21)where the action of G is characterised by h Y ( A , z ) ψ, m i = h ψ, Y ( A , z ) opp m i , A ∈ G , ψ ∈ M ′ , m ∈ M , (2.22)and where Y ( A , z ) opp is given by the formula Y ( A , z ) opp = Y (cid:16) e zL (cid:0) − z − (cid:1) L A , z − (cid:17) . (2.23) Proposition 2.11.
The vertex algebra G and its modules have the following properties. (1) The modes of the generating fields and the Heisenberg field satisfy β opp n = − β − n , γ opp n = γ − n , J opp n = δ n , − J − n . (2.24)(2) The restricted duals of spectral flows of the indecomposable modules in Definition 2.8 can be identified as (cid:0) σ ℓ V (cid:1) ′ (cid:27) σ − − ℓ V , (cid:0) σ ℓ W λ (cid:1) ′ (cid:27) σ − ℓ W − λ , (cid:0) σ ℓ W ± (cid:1) ′ (cid:27) σ − ℓ W ± . (2.25)(3) Denote by ∗ the composition of twisting by c and taking the restricted dual, then (cid:0) σ ℓ V (cid:1) ∗ (cid:27) σ ℓ V , (cid:0) σ ℓ W λ (cid:1) ∗ (cid:27) σ ℓ W λ , (cid:0) σ ℓ W ± (cid:1) ∗ (cid:27) σ ℓ W ∓ . (2.26)(4) Let A , B ∈ F and ℓ ∈ Z , then the homomorphism and first extension groups satisfy Hom( A , B ) = Hom( c A , c B ) = Hom (cid:0) σ ℓ A , σ ℓ B (cid:1) = Hom (cid:0) B ′ , A ′ (cid:1) = Hom (cid:0) B ∗ , A ∗ (cid:1) , Ext( A , B ) = Ext( c A , c B ) = Ext (cid:0) σ ℓ A , σ ℓ B (cid:1) = Ext (cid:0) B ′ , A ′ (cid:1) = Ext (cid:0) B ∗ , A ∗ (cid:1) . (2.27) Proof.
Part (1) follows immediately from Definition 2.10.Part (2): Since σ ℓ V is simple, (cid:0) σ ℓ V (cid:1) ′ is too, due to taking duals being an invertible exact contravariant functor. Further,by the action given in Definition 2.10 it is easy to see that β n , n ≥ ℓ + and γ m , m ≥ − ℓ act locally nilpotently and therefore (cid:0) σ ℓ V (cid:1) ′ is an object of both σ − ℓ R and σ − − ℓ R . Thus, (cid:0) σ ℓ V (cid:1) ′ (cid:27) σ − − ℓ V .Similarly, since σ ℓ W λ is simple, (cid:0) σ ℓ W λ (cid:1) ′ is too. The modes β n , n ≥ ℓ + and γ m , m ≥ − ℓ act locally nilpotently andtherefore (cid:0) σ ℓ W λ (cid:1) ′ is an object of σ − ℓ R . Further, for J homogeneous m ∈ σ ℓ W λ and ψ ∈ (cid:0) σ ℓ W λ (cid:1) ′ , consider h J ψ, m i = h ψ, (1 − J ) m i . (2.28)Thus, (cid:0) σ ℓ W λ (cid:1) ′ (cid:27) σ − ℓ W − λ . Finally, the duals of σ ℓ W ± follow from that fact that the duality functor is exact and contravariant, and by applying itto the exact sequences (2.20).Part (3) follows from composing the formulae of Part (2) with the conjugation twist formulae of Proposition 2.9.Part (4) follows from c , σ and ′ being exact invertible functors, the first two covariant and the last contravariant.2.4. Free field realisation.
We present two embeddings of G into a rank 1 lattice algebra constructed from a rank 2Heisenberg vertex algebra. We refer to [29] for a comprehensive discussion of Heisenberg and lattice vertex algebras.Let H be the rank 2 Heisenberg vertex algebra with choice of generating fields ψ, θ normalised such that they satisfy thedefining operator product expansions ψ ( z ) ψ ( w ) ∼ z − w ) , θ ( z ) θ ( w ) ∼ − z − w ) , ψ ( z ) θ ( w ) ∼ . (2.29)By a slight abuse of notation we also use ψ and θ to denote a basis of a rank 2 lattice L Z = span Z { ψ, θ } with symmetricbilinear lattice form corresponding to the above operator product expansions, that is ( ψ, ψ ) = − ( θ, θ ) = and ( ψ, θ ) = . Let L = span R { ψ, θ } be the extension of scalars of L Z by R . We denote the Fock spaces of H by F λ , λ ∈ L , where the zero modeof a Heisenberg vertex algebra field a ( z ) , a ∈ L acts as scalar multiplication by ( a , λ ) . We assign to the highest weightvector | λ i of F λ the vertex operators V λ ( z ) : F µ → F µ J z , z − K z ( λ,µ ) given by the expansion V λ ( z ) = e λ z λ ∏ m ≥ exp (cid:18) λ − m m z m (cid:19) exp (cid:18) − λ m m z − m (cid:19) , (2.30)where e λ ∈ C [ L ] is the basis element in the group algebra of L corresponding to λ ∈ L and satisfies the relations [ b n , e λ ] = δ n , ( b , λ ) e λ , e λ | µ i = | λ + µ i . (2.31)Finally, let V K be the lattice vertex algebra extension of H along the indefinite rank 1 lattice K = span Z { ψ + θ } . The simplemodules of V K are given by F Λ = M λ ∈ Λ F λ , Λ ∈ L / K and ( Λ , ψ + θ ) ∈ Z . (2.32)Note that the pairing ( Λ , ψ + θ ) is well-defined, since it does not depend in the choice of representative λ ∈ Λ . It willoccasionally be convenient to label the lattice modules by a representative λ ∈ Λ rather than the coset itself, that is F λ = F Λ .Note also that our notation differs from conventions common in theoretical physics literature. There, for a ∈ L , V a ( z ) would be denoted by : e a ( z ) : and a ( z ) by ∂ a ( z ) . Proposition 2.12. (1)
The assignment β ( z ) V θ + ψ ( z ) , γ ( z ) : ψ ( z )V − θ − ψ ( z ): (2.33) induces an embedding φ : G → V K . Restricting to the image of this embedding, V K -modules can be identified with G -modules as F ℓψ (cid:27) σ ℓ + W − , F Λ (cid:27) σ ( Λ ,ψ + θ ) + W ( Λ ,ψ ) , Λ ∈ L / K , ( Λ , ψ + θ ) ∈ Z and ( Λ , ψ ) , Z , (2.34) R ALLEN AND S WOOD where ( Λ , ψ ) is the coset in R / Z formed by pairing all representatives of Λ with ψ . (2) The assignment β ( z ) : ψ ( z )V θ + ψ ( z ): , γ ( z ) V − θ − ψ ( z ) (2.35) induces an embedding φ : G → V K . Restricting to the image of this embedding, V K -modules can be identified with G -modules as F ℓψ (cid:27) σ ℓ W + , F Λ (cid:27) σ ( Λ ,ψ + θ ) W ( Λ ,ψ ) , Λ ∈ L / K , ( Λ , ψ + θ ) ∈ Z and ( Λ , ψ ) , Z , (2.36) where ( Λ , ψ ) is the coset in R / Z formed by pairing all representatives of Λ with ψ . The embeddings are well known and the identifications of V K -modules with G -modules follow by comparing charactersand was shown in [30, Proposition 4.7] and [20, Proposition 4.1]. Theorem 2.13. (1)
Let S = Res V ψ ( z ) , then ker (cid:0) S : V K → F ψ (cid:1) = im φ , where φ : G → V K is the embedding of Proposition 2.12. (1) ,that is, S is a screening operator for the free field realisation φ of G . Further the sequence · · · S −→ F − ψ S −→ F S −→ F ψ S −→ · · · (2.37) is exact and is therefore a Felder complex. (2) Let S = Res V − ψ ( z ) , then ker (cid:0) S : V K → F − ψ (cid:1) = im φ , where φ : G → V K is the embedding of Proposition 2.12. (2) ,that is, S is a screening operator for the free field realisation φ of G . Further the sequence · · · S −→ F ψ S −→ F S −→ F − ψ S −→ · · · (2.38) is exact and is therefore a Felder complex.Proof. We prove part (1) only, as part (2) follows analogously. The operator product expansion of V ψ ( z ) with the imagesof β and γ in V K are V ψ ( z ) β ( w ) ∼ , V ψ ( z ) γ ( w ) ∼ − V − θ ( w )( z − w ) , (2.39)which are total derivatives in z implying that S = Res V ψ ( z ) is a screening operator and that im φ ⊂ ker S . Therefore, S commutes with G and hence defines a G -module map F → F ψ . The identification (2.34) implies F (cid:27) σ W − and F ψ (cid:27) σ W − . By comparing composition factors we see that the kernel must be either im φ (cid:27) V or all of F , so it issufficient to show that the map S : F → F ψ is non-trivial. A quick calculation reveals that S |− ψ − θ i = |− θ i , (2.40)and thus S is not trivial. By comparing the composition factors of the sequence (2.37) we also see that the sequence is anexact complex if each arrow is non-zero. Finally, the arrows are non-zero because S |− ψ + m θ i = | m θ i , ∀ m ∈ Z . (2.41) Remark.
The existence of Felder complexes will not specifically be needed for any of the results that follow, however, itis interesting to note that the bosonic ghosts admit such complexes. These complexes were crucial in [16] for computingthe character formulae needed for the standard module formalism via resolutions of simple modules.
3. Projective modules
In this section we construct reducible yet indecomposable modules P on which the L operator has rank 2 Jordan blocks.We further prove that the modules σ ℓ P and σ ℓ W λ are both projective and injective, and that in particular the σ ℓ P areprojective covers and injective hulls of σ ℓ V for any ℓ ∈ Z . We refer readers unfamiliar with homological algebra conceptssuch as injective and projective modules or extension groups to the book [31] and recall the following result for later use. Proposition 3.1.
For a module R which is both projective and injective, the Hom-Ext sequences terminate. That is, if wehave the short exact sequence −→ A −→ R −→ B −→ , (3.1) for modules A , B . then this implies that the following two sequences are exact, for any module M . −→ Hom( M , A ) −→ Hom( M , R ) −→ Hom( M , B ) −→ Ext( M , A ) −→ , (3.2) −→ Hom( B , M ) −→ Hom( R , M ) −→ Hom( A , M ) −→ Ext( B , M ) −→ . (3.3) Furthermore,
Hom( R , − ) and Hom( − , R ) are exact covariant and exact contravariant functors respectively. This proposition assists with the calculation of Hom and Ext groups, when all but one of the dimensions in the sequenceare known. Using the fact that the Euler characteristic (the alternating sum of the dimensions of the coefficients) of anexact sequence vanishes, there is only one possibility for the remaining group.
Proposition 3.2.
The first extension groups of simple modules in F satisfy dim Ext (cid:0) σ k V , σ ℓ V (cid:1) = , | k − ℓ | = , otherwise , dim Ext (cid:0) σ k W λ , M (cid:1) = dim Ext (cid:0) M , σ k W λ (cid:1) = , (3.4) where λ ∈ R / Z , λ , Z , k , ℓ ∈ Z and M is any module in F . In particular the simple modules σ k W λ are both projectiveand injective in F .Proof. To conclude that σ k W λ is projective in F it is sufficient to show that dim Ext( W λ , M ) = for all simple objects M ∈ F . Injectivity in F then follows by applying the ∗ functor and noting that W λ ∗ (cid:27) W λ . Let M ∈ F be simple, then anecessary condition for the short exact sequence −→ M −→ N −→ W λ −→ , M ∈ F (3.5)being non-split is that the respective ghost and conformal weights of W λ and M differ only by integers. For simple M this rules out M = σ ℓ V or M = σ ℓ W µ , µ , λ . So we consider M = σ ℓ W λ . Assume ℓ = , let j ∈ λ and let v be a non-zerovector in the ghost and conformal weight [ j , space of the submodule M = W λ ⊂ N and let w ∈ N be a representative ofa non-zero coset in the [ j , weight space of the quotient N / W λ . Without loss of generality, we can assume that w is a J -eigenvector and a generalised L -eigenvector. A necessary condition for the indecomposability of N , is the existenceof an element U in the universal enveloping algebra U ( G ) such that Uv = w . Since v has minimal generalised conformalweight all positive modes annihilate v , thus Uv can be expanded as a sum of products of β and γ with each summandcontaining as many β as γ factors, that is, Uv = f ( J ) v can be expanded as a polynomial in J acting on v . Since N ∈ F , J acts semisimply, hence f ( J ) v ∝ v . Since v is not a scalar multiple of w , this contradicts the indecomposability of N .Thus the exact sequence splits or, equivalently, the corresponding extension group vanishes.Assume M = σ ℓ W λ with ℓ , , then by applying the ∗ and σ functors, we have Ext (cid:0) W λ , σ ℓ W λ (cid:1) = Ext (cid:0) σ ℓ W λ , W λ (cid:1) = Ext (cid:0) W λ , σ − ℓ W λ (cid:1) . Thus the sign of ℓ can be chosen at will and we can assume without loss of generality that ℓ ≥ .Further, from the formulae for the conformal weights of spectral flow twisted modules (2.15), the conformal weights of W λ and σ ℓ W λ differ by integers if and only if ℓ · λ = Z . Let j ∈ λ be the minimal representative satisfying that the space ofghost weight j in σ ℓ W λ has positive least conformal weight. The least conformal weight of the ghost weight j − spaceis a negative integer, which we denote by − k . See Figure 1 for an illustration of how the weight spaces are arranged. Let v ∈ N be a non-zero vector of ghost weight j and generalised L eigenvalue 0, and hence a representative of a non-trivialcoset of ghost and conformal weight [ j , in W λ (cid:27) N /σ ℓ W λ . Further let w ∈ σ ℓ W λ ⊂ N be a non-zero vector of ghostand conformal weight [ j − , − k ] . Both v and w lie in one-dimensional weight spaces and hence span them. If N isindecomposable, then there must exist an element U of ghost and conformal weight [ − , − k ] in U ( G ) , such that Uv = w .We pick a Poincaré-Birkhoff-Witt ordering such that generators with larger mode index are placed to the right of thosewith lesser index and γ n is placed to the right of β n for any n ∈ Z . Thus Uv = ∑ ki = U ( i ) γ i v , where U ( i ) is an element of U ( G ) of ghost and conformal weight [0 , i − k ] . In W λ , γ acts bijectively on the space of conformal weight 0 vectors, hence there exists a ˜ v ∈ N such that γ ˜ v = v . Since at ghost weight j the conformal weights of N are non-negative, we have γ n ˜ v = , n ≥ and thus Uv = ∑ ki = U ( i ) γ i γ ˜ v = ∑ ki = U ( i ) γ γ i ˜ v = , contradicting the indecomposability of N .Next we consider the extensions of spectral flows of the vacuum module. By judicious application of the ∗ and σ functors, we can identify Ext (cid:0) σ k V , σ ℓ V (cid:1) = Ext (cid:0) V , σ k − ℓ V (cid:1) = Ext (cid:0) V , σ ℓ − k V (cid:1) . So without loss of generality, it is sufficientto consider the extension groups Ext (cid:0) V , σ ℓ V (cid:1) or equivalently short exact sequences of the form −→ σ ℓ V −→ M −→ V −→ , ℓ ∈ Z ≥ , M ∈ F . (3.6)Let σ ℓ Ω ∈ σ ℓ V ⊂ M denote the the spectral flow image of the highest weight vector of V and let ω ∈ M be a J -eigenvectorand a choice of representative of the highest weight vector in V (cid:27) M /σ ℓ V . We first show that these sequences necessarilysplit if ℓ , . Assume ℓ = , then the exact sequence can only be non-split if there exists a ghost and conformal weight [0 , element U in U ( G ) such that U ω = a σ ℓ Ω − b ω , a , b ∈ C , a , . Without loss of generality we can replace U by ˜ U = U − b to obtain ˜ U ω = a σ ℓ Ω . Since the conformal weights of V are bounded below by 0, they satisfy the same boundin M and β n ω = γ n ω = , n ≥ , so ˜ U ω can be expanded as a sum of products of β and γ acting on ω , with each summandcontaining the same number of β and γ factors. Equivalently, ˜ U ω can be expanded as a polynomial in J acting on ω .Since ω is a J -eigenvector ˜ U ω ∝ ω . Since ω is not a scalar multiple of σ ℓ Ω , ˜ U ω = contradicting indecomposability,and the exact sequence splits.Assume ℓ ≥ . The ghost and conformal weights of σ ℓ Ω are [ − ℓ, − ℓ ( ℓ + ] . Further, from the spectral flow formulae(2.15), one can see that the weight spaces of ghost and conformal weight [ − , h ] of σ ℓ V vanish for h < ( ℓ + ℓ − andsimilarly the [1 , h ] weight spaces of σ ℓ V vanish for h < ( ℓ + ℓ + . Since we are assuming ℓ ≥ , ( ℓ + ℓ ± ≥ . Thus γ n ω = β n ω = , n ≥ . If M is indecomposable, there must exist a ghost and conformal weight [ − ℓ, − ℓ ( ℓ + ] element U in U ( G ) such that U ω = σ ℓ Ω . Since the conformal weight of U is − ℓ , every summand of the expansion of U ω into β and γ modes must contain factors of γ n or β n with n ≥ and we can choose a Poincaré-Birkhoff-Witt ordering where thesemodes are placed to the right. Thus U ω = , contradicting indecomposability and the exact sequence splits.Assume ℓ = , then σ W + provides an example for which the exact sequence does not split and the dimension of thecorresponding extension group is at least 1. We show that it is also at most 1. Let ω and σ Ω be defined as for ℓ ≥ . Byarguments analogous to those for ℓ ≥ , it follows that the [1 , h ] weight space vanishes for h < and the [ − , h ] weightspace vanishes for h < − . Thus β n ω = γ n + ω = , n ≥ . The [ − , − weight space of σ V is one-dimensional andis hence spanned by σ Ω . If M is indecomposable, there must exist a ghost and conformal weight [ − , − element U in U ( G ) such that U ω = σ Ω . Thus, U ω can be expanded as f ( J ) γ ω = f (0) γ ω = a Ω , where f ( J ) is a polynomial.Hence the isomorphism class of M is determined by the value of γ ω in the one-dimensional [ − , − weight space and dim Ext( V , σ V ) = . ˜ v v w γ U σ ℓ W λ W λ Figure 1.
This diagram is a visual aid for the proof of the inextensibility of the simple module W λ ∈ F , λ ∈ R / Z , λ , Z . Here ℓ ≥ , ℓ · λ = Z . The nodes represent the (spectral flows of) relaxed highest weightvectors of each module. Weight spaces are filled in grey. Conformal weight increases from top to bottomand ghost weight increases from right to left. Armed with the above results on extension groups, we can construct indecomposable modules σ ℓ P ∈ F , which willturn out to be projective covers and injective hulls of σ ℓ V . Proposition 3.3.
Recall that by the first free field realisation φ of Proposition 2.12, we can identify F ℓψ (cid:27) σ ℓ + W − . Definethe S -twisted action of G on F − ψ ⊕ F by assigning β ( z ) φ ( β ( z )) = V ψ + θ ( z ) , γ ( z ) φ ( γ ( z )) − V − θ ( z ) z = : ψ ( z )V − ψ − θ ( z ): − V − θ ( z ) z , (3.7) and determining the action of all other fields in G through normal ordering and taking derivatives, where any vertexoperator V λ ( z ) whose Heisenberg weight λ is in the coset [ ψ ] = [ − θ ] is defined to act as on F and as usual on F − ψ . (1) The assignment is well-defined, that is, it represents the operator product expansions of G , and hence defines an actionof G on F − ψ ⊕ F , where ⊕ is meant as a direct sum of vector spaces without considering the module structure. Denotethe module with this S -twisted action by P . (2) The composite fields J ( z ) = : β ( z ) γ ( z ): , T ( z ) = − : β ( z ) ∂γ ( z ): act as J ( z ) φ ( J ( z )) = − θ ( z ) , T ( z ) φ ( T ( z )) + V ψ ( z ) z = : ψ ( z ) : − : θ ( z ) :2 − ∂ ψ ( z ) − θ ( z )2 + V ψ ( z ) z . (3.8) The zero mode J therefore acts semisimply and L has rank 2 Jordan blocks. The vectors |− ψ i , |− ψ − θ i , | θ i , | i ∈ P satisfy the relations β |− ψ i = | θ i , γ |− ψ i = −|− ψ − θ i , γ | θ i = −| i , β − |− ψ − θ i = | i , L |− ψ i = | i . (3.9)(3) The module P is indecomposable and satisfies the non-split exact sequences −→ σ W − −→ P −→ W − −→ , (3.10a) −→ W + −→ P −→ σ W + −→ , (3.10b) which implies that its composition factors are σ ± V and V with multiplicities 1 and 2, respectively. (4) P is an object in F . See Figure 2 for an illustration of how the composition factors of P are linked by the action of G . |− ψ i | i | θ i |− ψ − θ i σ − V σ V V γ β − β γ Figure 2.
The composition factors of P with the nodes representing the spectral flows of the highestweight vectors of σ ℓ V for − ≤ ℓ ≤ . The arrows give the action of G modes on the highest-weightvectors of each factor. In this diagram, ghost weight increases to the left and conformal weight increasesdownwards. Note that there are two copies of V , illustrated by a small vertical shift in their weights. Proof.
Part (1) follows from [32], where a general procedure was given for twisting actions by screening operators. Thefield identifications (3.8) of Part (2) follow by evaluating definitions introduced there, while the relations (3.9) follow byapplying the field identifications.
To conclude the first exact sequence of Part (3) note that the action of β and γ closes on F (cid:27) σ W − , because V − θ ( z ) actstrivially and quotienting by F leaves only F − ψ (cid:27) W − .To conclude the second exact sequence, let Ω be the highest weight vector of V and let σ ℓ Ω be the spectral flow imagesof Ω . Then | i ∈ F (cid:27) σ − W − can be identified with Ω in the V composition factor of σ − W − and |− ψ − θ i can be identifiedwith σ Ω in the σ V composition factor. Further, |− ψ i ∈ F − ψ (cid:27) W − can be identified with Ω in the V composition factor and | θ i can be identified with σ − Ω in the σ − V composition factor. See Figure 2 for a diagram of the action of β and γ modeson P and how they connect the different composition factors. It therefore follows that | i generates an indecomposablemodule whose composition factors are σ − V and V , with V as a submodule and σ − V as a quotient. The module thereforesatisfies the same non-split exact sequence (2.20) as W + does and since the extension groups in (3.4) are one-dimensional,this submodule is isomorphic to W + . After quotienting by the submodule generated by | θ i , the formulae above imply thatthe quotient is isomorphic to σ W + and the second exact sequence of Part (3) follows.Part (4) follows because J acts diagonalisably on P and because P has only finitely many composition factors all ofwhich lie in R or σ R . Theorem 3.4.
For every ℓ ∈ Z the indecomposable module σ ℓ P is projective and injective in F , and hence is a projectivecover and an injective hull of the simple module σ ℓ V .Proof. Since spectral flow is an exact invertible functor, it is sufficient to prove projectivity and injectivity of P , rather thanall spectral flow twists of P . We first show that P is injective by showing that dim Ext( W , P ) = for any simple module W ∈ F . Following that we will show P ∗ = P , which, since ∗ is an exact invertible contravariant functor, implies P is alsoprojective.A necessary condition for the non-triviality of such an extension is ghost weights differing only by integers. Wetherefore need not consider extensions by σ ℓ W λ , λ , Z , so we restrict our attention to short exact sequences of the form −→ P −→ M −→ σ ℓ V −→ . (3.11)If the above extension is non-split, then there must exist a subquotient of M which is a non-trivial extension of σ ℓ V by oneof the composition factors of P . By Proposition 3.2 the above sequence must split if | ℓ | ≥ and we therefore only consider | ℓ | ≤ .If ℓ = , then the composition factor of P non-trivially extending σ V must be σ V . If the extension is non-trivial,then this subquotient must be isomorphic to σ W − . Further, if σ Ω is the spectrally flowed highest weight vector of σ V and |− ψ − θ i ∈ P (see Figure 2) is the spectrally flowed highest weight vector of the σ V composition factor of P , then β − σ Ω = a |− ψ − θ i , a ∈ C \ { } . The relations (3.9) thus imply a | i = a β − |− ψ − θ i = a β − β − σ Ω = a β − β − σ Ω . (3.12)However, β − σ Ω has conformal and ghost weight [ − , − and this weight space vanishes for both P and σ V . Thus β − σ Ω and hence a = , which is a contradiction.If ℓ = , then the composition factor of P non-trivially extending σ V must be V . There are two such composition factorsin P . Any such non-trivial extension must be isomorphic to σ W − . If the non-trivial extension involves the compositionfactor whose spectrally flowed highest weight vector is represented by |− ψ i , then β − σ Ω = a |− ψ i , a ∈ C \ { } . The relations(3.9) thus imply a | θ i = a β |− ψ −i = a β β − σ Ω = a β − β σ Ω . (3.13)However, β σ Ω = , so a = , which is a contradiction. If the non-trivial extension involves the composition factor whosespectrally flowed highest weight vector is represented by | i , then there would exist a ∈ C \ { } such that β − σ Ω = a | i .But then, by the relations (3.9), β − ( σ Ω − a ) |− ψ − θ i = . Hence ( σ Ω − a ) |− ψ − θ i generates a direct summand isomorphicto σ V , making the extension trivial.If ℓ = , then the composition factor of P non-trivially extending V must be σ V or σ − V . If there is a subquotientisomorphic to a non-trivial extension of V by σ − V , that is, isomorphic to W − , then there exists a ∈ C \ { } such that β Ω = a | θ i . But then, by the relations (3.9), β ( Ω − a ) | θ i = . Hence ( Ω − a ) | θ i generates a direct summand isomorphic to V , making the extension trivial. An analogous argument rules out the existence of subquotient isomorphic a non-trivialextension of V by σ − V .The cases ℓ = − and ℓ = − follow the same reasoning as ℓ = and ℓ = , respectively.Now that we have established that P is injective, we can apply the functors Hom (cid:0) W − , − (cid:1) and Hom (cid:0) σ W + , − (cid:1) to theshort exact sequences (3.10a) and (3.10b), respectively, to deduce dim Ext (cid:0) W − , σ W − (cid:1) = = dim Ext (cid:0) σ W + , W + (cid:1) . Theindecomposable module P is therefore the unique module making the short exact sequences (3.10a) and (3.10b) non-split.By applying the functor ∗ to these exact sequences, we see that P ∗ also satisfies these same sequences and hence P (cid:27) P ∗ .This in turn implies Ext( P , − ) = and hence that σ ℓ P is projective for all ℓ ∈ Z .
4. Classification of indecomposables
In this section, we give a classification of all indecomposable modules in category F . We already know any simplemodule is isomorphic to either σ m W λ or σ m V , and we also know that the σ m W λ are inextensible due to being injective andprojective. We now complete the classification by finding all the reducible indecomposables which can be built as finitelength extensions with composition factors isomorphic to spectral flows of V . To unclutter formulae, we use the notation M n = σ n M for any module M . The classification of indecomposable modules in F closely resembles the classification ofindecomposable modules over the Temperley-Lieb algebra with parameter at roots of unity given in [33]. Conveniently,the majority of the reasoning in [33] also applies to the G -modules, with only minor modifications — primarily, that thereare no exceptional cases to consider for the bosonic ghost modules.The reducible yet indecomposable modules constituting the classification are the spectral flows of the projective module P , and two infinite families. These two families, denoted B mn and T mn , m , n ∈ Z , n ≥ , are dual to each other with respect to ∗ , that is, (cid:0) B mn (cid:1) ∗ = T mn , and further satisfy the identifications B = T = V , B = σ W − and T = σ W + . The superscript m is the number of composition factors or length of the module. As a visual aid, we represent these indecomposable modulesusing Loewy diagrams. VV − V VP V V T V V B Here the edges indicate the action of G and the vertices represent the composition factors.The indecomposable modules B mn and T mn can have either an even or an odd number of composition factors which areconstructed inductively by different extensions. Each chain is the result of extending either V , B or T repeatedly by thelength two indecomposables B n or T n , as either quotients or submodules. For example, the even length module B m isconstructed by repeatedly extending B = σ W − by spectrally flowed copies of itself, as submodules, as outlined in thediagram below. V V V V V V B B B B m The dotted boxes separate the component modules and the dashed lines indicate a non-trivial action of the algebra presentonly in the extended modules. Similarly for the odd length module B m + , the chain is built by repeated extensions of B = V by spectrally flowed copies of T = σ W + as quotients. V V V V V V V T T T B m + When applying the ∗ functor, the composition factors stay the same, however, all of the arrows corresponding to theaction of the algebra between the composition factors are reversed. Thus the top and bottom row are switched in the Loewydiagram and B type indecomposables become T type indecomposables. The letters T and B indicate the compositionfactor isomorphic to V being either in the top or bottom row, respectively. V V V V V T V V V V V B Recall from Proposition 3.2 that non-split extensions only exist between composition factors V i , V j with | i − j | = . Thisexplains the sequential order of the spectral flows of composition factors in the chains. We will show that these Loewydiagrams uniquely characterise the reducible indecomposable modules, that is, no two non-isomorphic indecomposableshave the same Loewy diagram. This is essentially due to certain extension groups being one-dimensional. These diagramstherefore provide a convenient way for reading off all submodules and quotients of a given indecomposable module andhence provide a shortcut for computing the dimensions of Hom groups.
Theorem 4.1. (1)
The initial identifications B = T = V , B = σ W − and T = σ W + , along with the non-split shortexact sequences below, uniquely characterise the modules B n and T n . −→ B n − −→ B n + −→ T n − −→ , (4.1a) −→ B n − −→ B n −→ B n − −→ , (4.1b) −→ B n − −→ T n + −→ T n − −→ , (4.1c) −→ T n − −→ T n −→ T n − −→ . (4.1d)(2) Any reducible indecomposable module in F is isomorphic to one of the following. P m = σ m P , B nm = σ m B n , T nm = σ m T n , m , n ∈ Z , n ≥ . (4.2)Theorem 4.1 follows by first computing dimensions of Hom and Ext groups to prove the existence of all indecomposableslisted above, and then showing that the list is closed under extensions by simple modules.For a module M , we recall the following two well known substructures. The first is the maximal semisimple submoduleof M , called the socle and which we denote soc M . The second, called the head, is the maximal semisimple quotient of M , defined to be the quotient of M by its radical (the intersection of its maximal proper submodules), which we denote hd M . We also let J [ M ] and P [ M ] denote the injective hull and the projective cover of M respectively. Proposition 4.2.
For any module M ∈ F , we have Hom( V n , M ) (cid:27) Hom( V n , soc M ) , Hom( M , V n ) (cid:27) Hom(hd M , V n ) , (4.3) and J [ M ] (cid:27) J [soc M ] , P [ M ] (cid:27) P [hd M ] . (4.4)This proposition allows us to find the Hom groups of indecomposables by examining their submodule and quotientstructure and applying (4.3), and we can use Hom-Ext exact sequences to fill in the gaps. We also know that P [ V n ] = J [ V n ] = P n , and therefore knowledge of the submodules and quotients of the indecomposable modules immediately determines theinjective hull and projective cover. Once these are known, we can construct injective and projective presentations whichwe use with the Hom-Ext exact sequence to determine the remaining Ext groups. The dimensions of Hom and Ext groups involving indecomposables of large length can be computed inductively from short length indecomposables and so weprepare these here. Proposition 4.3.
The dimensions of Hom groups for the indecomposable modules V m , B m , T m , P m are given by thefollowing table. MN dim Hom( N , M ) V m T m B m P m V n δ n , m δ n , m + δ n , m δ n , m T n δ n , m δ n , m + δ n , m + δ n , m δ n , m − + δ n , m B n δ n , m − δ n , m δ n , m − + δ n , m δ n , m − + δ n , m P n δ n , m δ n , m + δ n , m + δ n , m + δ n , m + δ n , m − + δ n , m + δ n , m + Further, the dimensions of Ext groups are given by the following table. MN dim Ext( N , M ) V m T m B m V n δ n , m − + δ n , m + δ n , m + δ n , m − T n δ n , m + δ n , m + + δ n , m + B n δ n , m − δ n , m − + δ n , m − Proof.
These dimensions follow from the exact sequences (2.20), Proposition 3.2 and Proposition 3.3, and judiciousapplication of Proposition 3.1.The classification of indecomposable Temperley-Lieb algebra modules in [33] parametrises modules by finite setsof integers. The analogue here is the subscript m in (4.2) parametrising spectral flow, which is an infinite index set.However, away from the end points of these finite sets of integers the dimensions of Hom and Ext groups for shortlength indecomposable modules in [33, Propositions 2.17 and 2.18] are equal to those for G -modules after making theidentifications in the following table. G V m B nm T nm B m T m P m TL I m B n − m T n − m C m S m P m , J m The dimensions of the remaining Hom and Ext groups, and therefore the classification, follows from the same homologicalalgebra reasoning as in [33], with no need for exceptions at the boundaries of the finite sets in [33].
Proposition 4.4. [33, Corollary 3.3, Propositions 3.6 and 3.7] The indecomposable modules B nm , T nm have the followingprojective covers and hulls. M B k + m B km T k + m T km P [ M ] L k − i = P m + i + L k − i = P m + i + L ki = P m + i L k − i = P m + i J [ M ] L ki = P m + i L k − i = P m + i L k − i = P m + i + L k − i = P m + i + Further, projective and injective presentations are characterised by the following.
M B k + m B km T k + m T km ker ( P [ M ] → M ) B k − m B km + T k + m T km − coker ( M → J [ M ]) B k + m B km − T km T km + This data now suffices to show that the extension groups corresponding to the exact sequences (4.1) of Theorem 4.1are one-dimensional and hence uniquely characterise the indecomposable B and T modules. The data can also be usedto show that any non-trivial extension of these indecomposable modules by spectral flows of V will be a direct sum ofmodules in the list (4.2). Hence Theorem 4.1 follows.For example, consider all possible extensions involving B and V n , starting with Ext (cid:0) V n , B (cid:1) . Using the tables above,we start with the following injective presentation of B −→ B −→ P ⊕ P −→ B − −→ . (4.5)Applying the functor Hom( V n , − ) , Proposition 3.1 gives the Hom-Ext exact sequence −→ Hom (cid:0) V n , B (cid:1) −→ Hom( V n , P ⊕ P ) −→ Hom (cid:0) V n , B − (cid:1) −→ Ext (cid:0) V n , B (cid:1) −→ . (4.6)We can use (4.3) to calculate these Hom groups, and the vanishing Euler characteristic implies dim Ext (cid:0) V n , B (cid:1) = δ n , − + δ n , + δ n , . These extensions are given by B − , B ⊕ T and B for n = − , 1 and 3, respectively. Similarly applythe functor Hom( − , V n ) to the projective presentation −→ V −→ P −→ B −→ . (4.7)The vanishing Euler characteristic then implies dim Ext (cid:0) B , V n (cid:1) = δ n , with the extension being given by P . Thereforewe see that all extensions involving B and V n return direct sums of classified indecomposable modules.We end this section with some properties of the classified indecomposable modules which will prove helpful in latersections. Proposition 4.5.
The evaluation of the ∗ functor of Proposition 2.11 on reducible indecomposable modules is given by ( P n ) ∗ (cid:27) P n , (cid:0) B mn (cid:1) ∗ (cid:27) T mn , (cid:0) T mn (cid:1) ∗ (cid:27) B mn . (4.8) Proof.
The action of the ∗ functor on the B and T modules follows from their defining sequences (4.1a) – (4.1d) beingdual to each other, these sequences uniquely characterising the B and T modules, and proceeding by induction, startingwith (cid:0) W ± (cid:1) ∗ = W ∓ from Proposition 2.11.(3). The self duality of P is a consequence of Proposition 3.3.(3). Corollary 4.6.
The B and T indecomposable modules satisfy the non-split exact sequences. −→ V −→ B n −→ T n − −→ , (4.9a) −→ V n −→ B n + −→ B n −→ , (4.9b) −→ B n − −→ B n −→ V n − −→ , (4.9c) −→ B n − −→ B n −→ B −→ , (4.9d) −→ B n − −→ T n −→ V −→ , (4.9e) −→ T n −→ T n + −→ V n −→ , (4.9f) −→ V n − −→ T n −→ T n − −→ . (4.9g) Proof.
The above sequences being non-split is intuitively clear from the Loewy diagrams of the B and T indecomposables.
5. Rigid tensor category
In this section we prove that fusion furnishes category F with the structure of a rigid tensor category and defineevaluation and coevaluation maps for the simple projective modules to verify that these modules and maps satisfy the conditions required for rigidity. We refer readers unfamiliar with tensor categories or related notions such as rigidityto [34]. Theorem 5.1.
Category F with the tensor structures defined by fusion is a braided tensor category. This theorem follows by verifying certain conditions which were proved to be sufficient in [21], and [35]. To this end,we recall some necessary definitions and results.
Definition 5.2.
Let V be a vertex algebra and let M be a module over V . Let A ≤ B be abelian groups.(1) The module M is called doubly-graded if both M and V are equipped with second gradations, in addition to conformalweight h ∈ C , which take values in B and A , respectively. We will use the notations M ( j ) and M [ h ] to denote thehomogeneous spaces with respect to the additional grading or generalised conformal weight, respectively, and denotethe simultaneous homogeneous space by M ( j )[ h ] = M ( j ) ∩ M [ h ] . The action of V on M is required to be compatible withthe A and B gradation, that is, v n V ( i ) ⊂ V ( i + j ) , v n M ( k ) ⊂ M ( j + k ) , v ∈ V ( j ) , n ∈ Z , i , j ∈ A , k ∈ B , (5.1)and ∈ V (0)[0] , ω ∈ V (0)[2] , (5.2)where is the vacuum vector and ω is the conformal vector.(2) The module M is called lower bounded if it is doubly graded and if for each j ∈ B , M ( j )[ h ] = for Re h sufficientlynegative.(3) The module M is called strongly graded with respect to B if it is doubly graded; it is the direct sum of its homogeneousspaces, that is, M = M h ∈ C j ∈ B M ( j )[ h ] , (5.3)where the homogeneous spaces M ( j )[ h ] are all finite dimensional; and for fixed h and j , M ( j )[ h + k ] = , whenever k ∈ Z issufficiently negative. The vertex algebra V is called strongly graded with respect to A if it is strongly graded as amodule over itself.(4) The module M is called discretely strongly graded with respect to B if all conformal weights are real and for any j ∈ B , h ∈ R the space M ˜ h ∈ R ˜ h ≤ h M ( j )[˜ h ] (5.4)is finite dimensional.(5) For j ∈ B , let C ( M ) ( j ) = span C (cid:8) u − h w ∈ M ( j ) : u ∈ V [ h ] , h > , w ∈ M (cid:9) . A strongly graded module M is called graded C -cofinite if ( M / C ( M )) ( j ) is finite dimensional for all j ∈ B . Definition 5.3.
Let A ≤ B be abelian groups. Let V be a vertex algebra graded by A and let M , M and M be modulesover V , graded by B . Denote by M { x } (cid:2) log x (cid:3) the space of formal power series in x and log x with coefficient in M , wherethe exponents of x can be arbitrary complex numbers and with only finitely many log x terms. A grading compatiblelogarithmic intertwining operator of type (cid:0) M M , M (cid:1) is a linear map Y : M → Hom( M , M ) { x } (cid:2) log x (cid:3) m Y ( m , x ) = ∑ s ≥ t ∈ C ( m ) t , s x − t − (cid:0) log x (cid:1) s (5.5)satisfying the following properties.(1) Truncation: For any m i ∈ M i , i = , , and s ≥ m ) t + k , s m = (5.6)for sufficiently large k ∈ Z . (2) L − -derivation: For any m ∈ M , Y ( L − m , x ) = dd x Y ( m , x ) . (5.7)(3) Jacobi identity: x − δ (cid:18) x − x x (cid:19) Y ( v , x ) Y ( m , x ) m = x − δ (cid:18) − x + x x (cid:19) Y ( m , x ) Y ( v , x ) m + x − δ (cid:18) x − x x (cid:19) Y ( Y ( v , x ) m , x ) m , (5.8)where Y denotes field map encoding the action of V on either M , M or M and δ denotes the algebraic deltadistribution, that is the formal power series δ (cid:18) y − xz (cid:19) = ∑ r ∈ Z s ≥ (cid:18) rs (cid:19) ( − s x s y r − s z − r . (5.9)(4) Grading compatibility: For any m i ∈ M ( j i ) i , j i ∈ B , i = , , t ∈ C and s ≥ m ) t , s m ∈ M ( j + j )3 . (5.10) Definition 5.4.
Let A ≤ B be abelian groups. Let V be a vertex algebra graded by A and let M and M be modules over V , graded by B . We define the following properties for functionals ψ ∈ Hom( M ⊗ M , C ) .(1) P ( w ) -compatibility:(1) Lower truncation : For any v ∈ V , v n ψ = , for any sufficiently large n ∈ Z .(2) For any v ∈ V and f ∈ C [ t , t − , ( t − − w ) − ] the identity v f ( t ) ψ = v ι + ( f ( t )) ψ (5.11)holds. Here ι + means expanding about t = such that the exponents of t are bounded below and the action of V ⊗ C [ t , t − , ( t − − w ) − ] or V ⊗ C (( t )) on ψ is characterised by h vg ( t ) ψ, m ⊗ m i = h ψ, ι + ◦ T w (cid:0) v opp g ( t − ) (cid:1) m ⊗ m i + h ψ, m ⊗ ι + (cid:0) v opp g ( t − ) (cid:1) m i , (5.12)where m i ∈ M i , v ∈ V , g ∈ C [ t , t − , ( t − − w ) − ] , T w replaces t by t + w , v opp = e t − L ( − t ) L vt − , and (assuming v has conformal weight h ) vt n m i = v n − h + m i .Denote by COMP( M , M ) the vector space of all P ( w ) -compatible functionals.(2) P ( w ) -local grading restriction:(1) The functional ψ is a finite sum of vectors that are both B -homogeneous and L generalised eigenvectors.(2) Denote the smallest subspace of Hom( M ⊗ M , C ) containing ψ and stable under V ⊗ C [ t , t − ] by M ψ . Then M ψ must satisfy for any r ∈ C , b ∈ B dim (cid:16) M ψ ( b )[ r ] (cid:17) < ∞ , and dim (cid:16) M ψ ( b )[ r + k ] (cid:17) = , (5.13)for sufficiently large k ∈ Z .Denote by LGR( M , M ) the vector space of all P ( w ) -local grading restricted functionals.Define M (cid:27) M = COMP( M , M ) ∩ LGR( M , M ) . Remark.
The variable w in P ( w ) denotes the insertion point of the tensor product constructed in [21], where it is usuallydenoted z and hence the tensor product is referred to as the P ( z ) -tensor product. Theorem 5.5 (Huang-Lepowsky-Zhang [21, Part IV, Theorem 5.44, 5.45, 5.50]) . Let A ≤ B be abelian groups. Let V be a vertex algebra graded by A with a choice of module category C which is closed under restricted duals and let M , M ∈ C be graded by B . Then COMP( M , M ) and M (cid:27) M are modules over V . Further, if M (cid:27) M ∈ C , then M ⊠ M (cid:27) ( M (cid:27) M ) ′ . In [21] M (cid:27) M is originally defined as the image of all intertwining operators with M and M as factors, but it is thenshown that this is equivalent to the definition given above. The construction of fusion products through Definition 5.4 is sometimes called the HLZ double dual construction. In addition to the primary reference [21], the authors also recommendthe survey [36], which relates this construction of fusion to others in the literature. Theorem 5.6 (Huang-Lepowsky-Zhang [21, Part VIII, Theorem 12.15], Huang [35, Theorem 3.1]) . For any vertex algebraand module category C satisfying the conditions below, fusion equips category C with the structures of an additive braidedtensor category. (1) The vertex algebra and all its modules in C are strongly graded and all logarithmic intertwining operators are gradingcompatible. [21, Part III, Assumption 4.1]. (2) C is a full subcategory of the category of strongly graded modules and is closed under the contragredient functor andunder taking finite direct sums [21, Part IV, Assumption 5.30]. (3) All objects in C have real weights and the non-semisimple part of L acts on them nilpotently [21, Part V, Assumption7.11]. (4) C is closed under images of module homomorphisms [21, Part VI, Assumption 10.1.7]. (5) The convergence and extension properties for either products or iterates holds [21, Part VII, Theorem 11.4]. (6)
For any objects M , M ∈ C , let M v be the doubly graded V -module generated by a B -homogeneous generalised L eigenvector v ∈ COMP( M , M ) . If M v is lower bounded then M v is strongly graded and an object in C [35, Theorem3.1]. Here the action of L is defined by (5.12) , given in Definition 5.4. Conditions (1) – (4) of Theorem 5.6 hold by construction for category F , so all that remains is verifying Conditions(5) and (6). Theorem 5.7.
Let A ≤ B be abelian groups, let V be a doubly A -graded vertex operator algebra and let V be a vertexsubalgebra of V (0) . Further, let W i , i = , , , , be doubly B -graded V -modules. Finally let Y , Y , Y and Y belogarithmic grading compatible intertwining operators of types (cid:0) W W , W (cid:1) , (cid:0) W W , W (cid:1) , (cid:0) W W , W (cid:1) and (cid:0) W W , W (cid:1) respectively.If the modules W i , i = , , , (note i = is excluded) are discretely strongly graded, and graded C -cofinite as V -modules,then Y , Y satisfy the convergence and extension property for products and Y , Y satisfy the convergence and extensionproperty for iterates. The above theorem follows from the proof of [25, Theorem 7.2], however, in [25] some assumptions are made on thecategory of strongly graded modules (see [25, Assumption 7.1, Part 3]) which do not hold for G . Fortunately, the proof ofTheorem 5.7 does not depend at all on any categorical considerations or even on the details of the intertwining operators Y i beyond their types. It merely depends on certain finiteness properties of the modules W i . We reproduce the proof ofYang in Appendix A, with some minor tweaks to the arguments, to show that the conclusion of Theorem 5.7 holds, withoutmaking any assumptions on the category of all strongly graded modules. Lemma 5.8.
The convergence and extension properties for products and iterates holds for F .Proof. If, in the assumptions of Theorem 5.7, we set V = G and grade by ghost weight, so that A = Z , then the modules of F are graded by B = R . We further choose V = G (0) , that is, the vertex subalgebra given by the ghost weight 0 subspaceof G . The lemma then follows by verifying that all modules in F are discretely strongly graded and graded C -cofinite asmodules over V .All modules in F are discretely strongly graded by ghost weight j ∈ R . To prove this, we need to check that thesimultaneous ghost and conformal weight spaces are finite dimensional and that every ghost weight homogeneous spacehas lower bounded conformal weights. The simultaneous ghost and conformal weight spaces of objects in R and thereforealso those of σ ℓ R are finite dimensional by construction. Thus, since the objects of F are finite length extensions ofthose in σ ℓ R , the objects of F also have finite dimensional simultaneous ghost and conformal weight spaces. Similarlywe have that the objects in F are graded lower bounded and therefore discretely strongly graded.Next we need to decompose objects of F as V -modules,. It is known that V is generated by { : β ( z )( ∂ n γ ( z )): , n ≥ } and is isomorphic to W + ∞ (cid:27) W , − ⊗ H where W , − is the singlet algebra at c = − and H is a rank 1 Heisenbergalgebra [38, 39]. Note that the conformal vector of W + ∞ is usually chosen so as to have a central charge of . Since we require V to embed conformally into G , that is, to have the same conformal vector as G and the central charge of G is ,we choose conformal vector of our Heisenberg algebra H so that its central charge is 4 (the conformal structure of W , − is unique). Fortunately, this does not complicate matters, as the simple modules over H are just Fock spaces regardlessof the central charge or conformal vector. The tensor factors of W + ∞ decompose nicely with respect to the free fieldrealisation of Proposition 2.12.(2). The Heisenberg algebra H is generated by θ ( z ) and the singlet algebra W , − is a vertexsubalgebra of the Heisenberg algebra generated by ψ ( z ) .We denote Fock spaces over the rank 1 Heisenberg algebras generated by ψ and θ , respectively, by the same symbol F µ ,where, the index µ ∈ C indicates the respective eigenvalues of the zero modes ψ and θ . All simple V (cid:27) W + ∞ modulescan be constructed via its free field realisation as V ( λ,ψ ) ⊗ F ( λ,θ ) [40, Corollary 6.1], where V ( λ,ψ ) , as a W , − -module, is thesimple quotient of the submodule of F ( λ,ψ ) generated by the highest weight vector. The homogeneous space (cid:0) σ ℓ V (cid:1) ( j ) issimple, as a V -module [38, Lemma 4.1], see also [41, 42]. Recall from Proposition 2.12.(2) that with K = span Z { ψ, θ } and Λ ∈ L / K , we can construct the simple projective G -modules as σ ( Λ ,ψ + θ ) W ( Λ ,ψ ) (cid:27) F Λ . To identify the homogeneous space (cid:0) σ ( Λ ,ψ + θ ) W ( Λ ,ψ ) (cid:1) ( j ) as a V -module, we use the fact that J ( z ) = − θ ( z ) , thus the R -grading on F Λ is given by the eigenvalueof − θ . Therefore, for j ∈ R , (cid:0) σ ( Λ ,ψ + θ ) W ( Λ ,ψ ) (cid:1) ( j ) (cid:27) F ( j ) Λ (cid:27) M λ ∈ Λ F ( λ,ψ ) ⊗ F ( λ,θ ) ! ( j ) = ( F ( Λ ,ψ + θ ) + j ⊗ F − j , j ∈ ( Λ , ψ ) , , j < ( Λ , ψ ) . (5.14)For ( Λ , ψ + θ ) + j < Z , F ( Λ ,ψ + θ ) + j is irreducible as a W , − module, by [43, Section 3.2], see also [44, Section 5]. Thus, (cid:0) σ ( Λ ,ψ + θ ) W ( Λ ,ψ ) (cid:1) ( j ) (cid:27) F ( Λ ,ψ + θ ) + j ⊗ F − j = V ( Λ ,ψ + θ ) + j ⊗ F − j . The finite length modules of W , − are all C -cofinite [45,Corollary 14], as are H -modules, since Fock spaces have C -codimension 1. Therefore all V -modules appearing as thehomogeneous spaces of modules in F are C -cofinite and the lemma follows. Lemma 5.9.
Let M , M be modules in F , let W be an indecomposable smooth (or weak) G module and let Y be asurjective logarithmic intertwining operator of type (cid:0) WM , M (cid:1) . (1) The logarithmic intertwining operator Y is grading compatible and the module W is doubly graded. (2) If M ∈ σ k R , M ∈ σ ℓ R , then W ∈ F and W has composition factors only in σ k + ℓ R and σ k + ℓ − R . (3) If M has composition factors only in σ k R and σ k − R , and has composition factors only in σ ℓ R and σ ℓ − R , W ∈ F and W has composition factors only in σ k + ℓ + i R , − ≤ i ≤ .Proof. Due to the compatibility of fusion with spectral flow, see Theorem 2.6, it is sufficient to only consider k = ℓ = .We prove Part (1) first. Let M , M , be modules in F . Let v ∈ G be the vector corresponding to the field J ( z ) and take theresidue with respect to x and x in the Jacobi identity (5.8). This yields J Y ( m , x ) m = Y ( m , x ) J m + Y ( J m , x ) m . (5.15)Hence, since the fusion factors M i are graded by ghost weight, the fusion product will be too. This means that theintertwining operator will be grading compatible and W must be doubly graded.Next we prove Part (3). Assume that M , M have composition factors only in R and σ − R . Note that J n , n ≥ actslocally nilpotently on any object in F and that β n − ℓ , γ n + ℓ , n ≥ act locally nilpotently on any object in σ ℓ R (recall thatlocal nilpotence is one of the defining properties of σ ℓ R ). We first show that J n , β n + , γ n , n ≥ acting locally nilpotentlyon M , M implies that J n , β n + , γ n , n ≥ act locally nilpotently on W . Let h be the conformal weight of v = β, γ or J ,multiply both sides of the Jacobi identity (5.8) by x k x n + h − , n , k ∈ Z and take residues with respect to x and x . This yields ∑ s ≥ (cid:18) ks (cid:19) ( − s x s v n − s Y ( m , x ) m = ∑ s ≥ (cid:18) ks (cid:19) ( − s x k − s Y ( m , x ) v n − k + s m + ∑ s ≥ (cid:18) s − n + k − hs (cid:19) ( − s x n − k + h − s − Y ( v s − h + k + m , x ) m . (5.16)Set v = γ (and thus h = ) and k = in (5.16) to obtain γ n Y ( m , x ) m = Y ( m , x ) γ n m + n ∑ s = (cid:18) s − ns (cid:19) ( − s x n − s − Y ( γ s + m , x ) m . (5.17) This implies the local nilpotence of γ n , n ≥ on Y ( m , x ) m from its local nilpotence on m and m . Next consider v = J (and thus h = ) and k = in (5.16) to obtain ( J n − x J n − ) Y ( m , x ) m = Y ( m , x )( J n − x J n − ) m + n ∑ s = (cid:18) s − ns (cid:19) ( − s x n − s − Y ( J s + m , x ) m . (5.18)Since J k , k ≥ is nilpotent on both m and m , we see that J n − x J n − is nilpotent for n ≥ . Recall that the series expansionof the intertwining operator Y ( m , x ) m = ∑ t ∈ C s ≥ ( m ) ( t , s ) m x − t − (log x ) s (5.19)satisfies a lower truncation condition, that is, for fixed s , if there exists a u ∈ C satisfying m ( u , s ) , , then there exists aminimal representative t ∈ u + Z such that m ( t , s ) , and m ( t ′ , s ) = for all t ′ < t . Since J n − x J n − is nilpotent on Y ( m , x ) m it is also nilpotent on the leading term m ( t , s ) . By comparing coefficients of x and log x it then follows that J n , n ≥ acts nilpotently on m ( t , s ) and by induction also on all coefficients of higher powers of x . To show that J acts locallynilpotently, assume that m has J -eigenvalue j and set n = , k = in (5.16) to obtain J Y ( m , x ) m = Y ( m , x ) J m + x j Y ( m , x ) m + ∑ s ≥ ( − s (cid:18) s − s (cid:19) x − s Y ( J s m , x ) m . (5.20)Thus J − x j is nilpotent, which by the previous leading term argument implies that J is too. Finally, consider v = β (andthus h = ) and k = in (5.16) to obtain ( β n − x β n − + x β n − ) Y ( m , x ) m = Y ( m , x )( β n − x β n − + x β n − ) m + ∑ s ≥ (cid:18) s − n + s (cid:19) ( − s x n − s − Y ( β s + m , x ) m . (5.21)By leading term arguments analogous to those used for J n , this implies that β n acts locally nilpotently for n ≥ .Consider the subspace V ⊂ W annihilated by β n + , γ n , n ≥ . Then V is a module over four commuting copies of theWeyl algebra respectively generated by the pairs ( β , γ ) , ( β , γ − ) , ( β , γ − ) , ( β , γ − ) . Further, V is closed under theaction of J n , n ≥ and restricted to acting on V , the first few J n modes expand as J = β γ , J = β γ + β γ − , J = β γ + β γ − + β γ − . (5.22)We show that on any composition factor of V at least three of the four Weyl algebras have a generator acting nilpotentlyand that thus the induction of such a composition factor is an object in one of the categories σ i R , − ≤ i ≤ . Let C ⊗ C ⊗ C ⊗ C be isomorphic to a composition factor of V , where C i is a simple module over the Heisenberg algebragenerated by the pair ( β i , γ − i ) . Since J , J , J act locally nilpotently on V they must also do so on C ⊗ C ⊗ C ⊗ C using the expansions (5.22). If we assume that neither β nor γ act locally nilpotently on C and C , respectively, thatis there exist c ∈ C and c ∈ C such that U ( β ) c and U ( γ ) c are both infinite dimensional, and choose c , c , to benon-zero vectors in C and C , respectively. Then U ( J )( c ⊗ c ⊗ c ⊗ c ) will be infinite dimensional contradicting thelocal nilpotence of J . So assume β acts locally nilpotently but γ does not, and let c ∈ C be annihilated by β and c , c , c be non-zero vectors in C , C , C , respectively. On this vector J evaluates to J ( c ⊗ c ⊗ c ⊗ c ) = γ c ⊗ c ⊗ β c ⊗ c . (5.23)By the same reasoning as before, unless either β or γ act nilpotently, we have a contradiction to the nilpotence of J ,so β must act nilpotently on c . Repeating this argument for J and assuming β c = we have a contradiction to thenilpotence of J unless β acts nilpotently. The composition factor isomorphic to C ⊗ C ⊗ C ⊗ C thus induces to anobject in R . Repeating the previous arguments, assuming that γ acts locally nilpotently but β does not, implies that γ − and γ − must act locally nilpotently to avoid contradictions to the local nilpotence of J , J , J . Such a composition factorwould induce to a module in σ − R . Finally assume both β and γ act locally nilpotently, then analogous arguments tothose used above applied to the action of J imply that at least one of β or γ − act locally nilpotently. Such a compositionfactor would induce to an object in σ − R or σ − R , respectively. The final potential obstruction to W lying in F is that such a submodule might not be finite length. However, if W had infinite length, it would have to admit indecomposable subquotients of arbitrary finite length, yet by the classificationof indecomposable modules in Theorem 4.1, a finite length indecomposable module with composition factors only in σ i R , − ≤ i ≤ has length at most 5. Therefore W ∈ F .Part (2) follows by a similar but simplified version of the above arguments. J n and γ n continue to satisfy the samenilpotence conditions as above, however for β one needs to reconsider (5.16) with k = to conclude that β n , n ≥ isnilpotent. The remainder of the argument follows analogously. Proof of Theorem 5.1.
We verify that the assumptions of Theorem 5.6 hold, in numerical order. Theorem 5.6 thus impliesthat category F is an additive braided tensor category. Additionally, since category F is abelian, it is a braided tensorcategory.(1) All modules in category F are strongly graded by ghost weight j ∈ R . Further, by Lemma 5.9.(1), all logarithmicintertwining operators are grading compatible.(2) By Proposition 2.11, category F is closed under taking restricted duals. Closure under finite direct sums holds byconstruction, since category F is abelian.(3) The modules in F have real conformal weights by definition. The only modules on which the non semi-simple partof L acts non-trivially are σ m P n , for which it squares to zero.(4) Closure under images of module homomorphisms holds by construction, since category F is abelian.(5) The convergence and extension properties hold by Lemma 5.8.(6) Since the P ( w ) -tensor product is right exact, by [21, Part IV, Proposition 4.26], and since category F has sufficientlymany projectives, that is, every module can be realised as a quotient of a direct sum of indecomposable projectives, wecan without loss of generality assume M and M are indecomposable projective modules, as Condition (6) holdingfor projective modules implies that it also holds for their quotients. Further, due to the compatibility of fusion withspectral flow, we can pick M and M to be isomorphic to W λ or P . Let ν ∈ COMP( M , M ) be doubly homogenousand assume that the module M ν generated by ν is lower bounded. By assumption, the functional ν therefore satisfies allthe properties of P ( w ) -local grading restriction except for the finite dimensionality of the doubly homogeneous spacesof M ν . We need to show the finite dimensionality of these doubly homogeneous spaces and that M ν is an object in F . Since M ν is finitely generated (cyclic even) it is at most a finite direct sum. To see this, assume the module admitsan infinite direct sum. Then the partial sums define an ascending filtration whose union is the entire module. Henceafter some finite number of steps all generators must appear within this filtration, but if this finite sum contains allgenerators, it must be equal to the entire module and hence all later direct summands must be zero. Denote the directsummands by M ν, i , i ∈ I , where I is some finite index set. By [21, Part IV, Proposition 5.24] there exists a smooth G module W ν, i such that W ′ ν, i (cid:27) M ν, i and a surjective intertwiner of type (cid:0) W ν, i M , M (cid:1) . Hence, by Lemma 5.9, W ν, i ∈ F . Inparticular, since category F is closed under taking restricted duals and all its objects have finite dimensional doublyhomogeneous spaces, we have M ν, i ∈ F and M ν ∈ F . Remark.
Note that the above proof did not make any use of M ν being lower bounded to conclude that M ν ∈ F and thatmembership of category F implies lower boundedness. Lemma 5.10.
For λ ∈ R / Z , λ , Z , the fusion product σ ℓ W λ ⊠ σ k W − λ has exactly one direct summand isomorphic to σ ℓ + k − P . We will prove the above lemma by showing that W λ (cid:27) W − λ has exactly one submodule isomorphic to P . This requiresfinding linear functionals which satisfy P ( w ) compatibility. This is very difficult to do in practice, since (5.11) needs to bechecked for every vector v ∈ V . Fortunately there is a result by Zhang which cuts this down to generators. Zhang originallyformulated the theorem below for a related type of fusion product called the Q ( z ) -tensor product, so we have translated hisresult to the P ( w ) -tensor product, which we use here. Theorem 5.11 (Zhang [37, Theorem 4.7]) . Let A ≤ B be abelian groups. Let V be a vertex algebra graded by A with a setof strong generators S and let M and M be modules over V , graded by B . A functional ψ ∈ Hom( M ⊗ M , C ) is saidto satisfy the strong lower truncation condition for a vector v ∈ V , if there exists an N ∈ N such that for all n , m ∈ Z , with m ≥ N , we have vt m + n ( t − − w ) n ψ = . (5.24) Then ψ ∈ Hom( M ⊗ M , C ) satisfies the P ( w ) -compatibility condition if and only if it satisfies the strong lower truncationcondition for all elements of S . We further prepare some helpful identities.
Lemma 5.12.
Let M , M ∈ F , m i ∈ M i , i = , , and ψ ∈ COMP( M , M ) , then we have the identities. h J n ψ, m ⊗ m i = δ n , h ψ, m ⊗ m i − ∑ i ≥ (cid:18) − ni (cid:19) w − n − i h ψ, J i m ⊗ m i − h ψ, m ⊗ J − n m i , n ∈ Z , (5.25) h L ψ, m ⊗ m i = h ψ, L m ⊗ m i + w h ψ, L − m ⊗ m i + h ψ, m ⊗ L m i , (5.26) (cid:10) β t k + n ( t − − w ) n ψ, m ⊗ m (cid:11) = − ∑ i ≥ (cid:18) − k − ni (cid:19) w − k − n − i h ψ, β n + i m ⊗ m i− ∑ i ≥ (cid:18) ni (cid:19) ( − w ) n − i h ψ, m ⊗ β i − k − n m i , k , n ∈ Z , (5.27) (cid:10) γ t k + n ( t − − w ) n ψ, m ⊗ m (cid:11) = ∑ i ≥ (cid:18) − k − n − i (cid:19) w − k − n − − i h ψ, γ n + i + m ⊗ m i + ∑ i ≥ (cid:18) ni (cid:19) ( − w ) n − i h ψ, m ⊗ γ i − k − n − m i , k , n ∈ Z . (5.28) Proof.
These identities follow by evaluating (5.12) for the fields β, γ, J and T . Proof of Lemma 5.10.
We shall use the HLZ double dual construction of Definition 5.4. By the compatibility of fusionwith spectral flow, Theorem 2.6, it is sufficient to consider the case ℓ = k = . Note since σ − P is both projective andinjective, it must be a direct summand if it appears as either a quotient or a subspace. Further, by Lemma 5.9, allcomposition factors must lie in categories σ i R , i = − , . This implies that the composition factors of W λ (cid:27) W − λ must alllie in σ i R , i = , . Note further, that ( σ − P ) ′ (cid:27) P and so we seek to find a copy of P within W λ (cid:27) W − λ . We do so byconsidering a certain characterising two dimensional subspace of P . For a G -module M consider the subspace K ( M ) = { m ∈ M : β n m = γ n + m = J m = J m = , n ≥ } . (5.29)From the expansions of T ( z ) and J ( z ) in terms of the fields β and γ , it follows that for any m ∈ K ( M ) , L m = L n m = J n m = , n ≥ . In particular, in the notation of Figure 2, K ( P ) = span C {| i , |− ψ i} and thus K ( P ) is two dimensional and L has a rank 2 Jordan block of generalised eigenvalue 0 on this space. Further, P is the only indecomposable module withcomposition factors in categories σ i R , i = , admitting L Jordan blocks. The remaining indecomposable modules withcomposition factors in categories σ i R , i = , all have K ( M ) subspaces of dimension zero or one.Let ψ ∈ Hom( W λ ⊗ W − λ , C ) satisfy β t k + n ( t − − w ) n ψ = γ t k + n ( t − − w ) n ψ = for all m ≥ . Thus by Theorem 5.11, ψ satisfies the P ( w ) -compatibility property and β m ψ = γ m + ψ = , m ≥ . If in addition ψ is doubly homogeneous, then ψ liesin W λ (cid:27) W − λ . By assumption the left-hand sides of (5.27) and (5.28) vanish for k ≥ . These relations imply that the valueof ψ on any vector in W λ ⊗ W − λ is determined by its value on tensor products of relaxed highest weight vectors, becausenegative modes on one factor can be traded for less negative modes on the other factor. For example, for k = , n = in(5.27), we have the relation h ψ, m ⊗ β − m i = − ∑ i ≥ (cid:18) − i (cid:19) w − − i h ψ, β i m ⊗ m i . (5.30)Let u ± j ∈ W ± λ , j ∈ ± λ be a choice of normalisation of relaxed highest weight vectors satisfying u ± j − = γ u ± j . Thisimplies β u ± j = ± ju ± j . Since the negative β and γ modes act freely on the simple projective modules W λ and W − λ , thereare no relations in addition to those coming from β t k + n ( t − − w ) n ψ = γ t k + n ( t − − w ) n ψ = for all m ≥ . Thus there is a linear isomorphism { ψ ∈ W λ (cid:27) W − λ : β n ψ = γ n + ψ = , n ≥ } (cid:27) −→ Hom (cid:0) span C (cid:8) u j ⊗ u − i (cid:9) , C (cid:1) . (5.31)Clearly, K ( W λ (cid:27) W − λ ) is a subspace of { ψ ∈ W λ (cid:27) W − λ : β n ψ = γ n + ψ = , n ≥ } and so we impose the remaining tworelations, the vanishing of J and J , via (5.25). The vanishing of J ψ implies = (cid:10) J ψ, u j ⊗ u − i (cid:11) = (cid:10) ψ, u j ⊗ u − i (cid:11) − (cid:10) ψ, J u j ⊗ u − i (cid:11) − (cid:10) ψ, u j ⊗ J u − i (cid:11) = (1 − j + i ) (cid:10) ψ, u j ⊗ u − i (cid:11) . (5.32)Thus ψ vanishes on u j ⊗ u − i unless i = j − . The vanishing of J ψ implies (2 j − (cid:10) ψ, u j ⊗ u − j (cid:11) − j (cid:10) ψ, u j + ⊗ u − j (cid:11) + (1 − j ) (cid:10) ψ, u j − ⊗ u − j (cid:11) = , (5.33)where we have used J − u − j = ( γ − β + β − γ ) u − j . Thus ψ is completely characterised by its value on a two pairs ofrelaxed highest weight vectors, say u j ⊗ u − j and u j + ⊗ u − j . Therefore, the subspace K ( W λ (cid:27) W − λ ) is two dimensional.Next we show that that L has a rank two Jordan block on it when acting on this space. Let ψ ∈ K ( W λ (cid:27) W − λ ) . If ψ , ,then there exist a , b ∈ C , not both zero, such that (cid:10) ψ, u j ⊗ u − j (cid:11) = a , (cid:10) ψ, u j + ⊗ u − j (cid:11) = b . (5.34)The evaluation of L ψ on u j ⊗ u − j and u j + ⊗ u − j is then (cid:10) L ψ, u j ⊗ u − j (cid:11) = j ( a − b ) , (cid:10) L ψ, u j + ⊗ u − j (cid:11) = − j ( a − b ) . (5.35)Therefore if a , b (a choice which we can make as K ( W λ (cid:27) W − λ ) is two dimensional), the vectors ψ and L ψ are linearlyindependent and span K ( W λ (cid:27) W − λ ) , which also shows that L has a rank two Jordan block. Remark.
In [16, Section 7] the above fusion product was computed using the NGK algorithm up to certain conjecturedadditional conditions. In light of the survey [36] explaining the equivalence of the HLZ double dual construction and theNGK algorithm, the authors thought it appropriate to supplement the NGK calculation of [16] with an HLZ double dualcalculation here.
Proposition 5.13.
For all ℓ ∈ Z and λ ∈ R / Z , λ , Z , the simple module σ ℓ W λ is rigid in category F , with tensor dualgiven by (cid:0) σ ℓ W λ (cid:1) ∨ = σ − ℓ W − λ .Proof. Recall that an object M in a tensor category is rigid if there exists an object M ∨ (called a tensor dual of M ) andtwo morphisms e M : M ∨ ⊠ M → V and i M : V → M ⊠ M ∨ , respectively, called evaluation and coevaluation, such that thecompositions M (cid:27) V ⊠ M i M ⊗ −→ (cid:0) M ⊠ w M ∨ (cid:1) ⊠ w M A − −→ M ⊠ w (cid:0) M ∨ ⊠ w M (cid:1) ⊗ e M −→ M ⊠ V (cid:27) M , (5.36a) M ∨ (cid:27) M ∨ ⊠ V ⊗ i M −→ M ∨ ⊠ w (cid:0) M ⊠ w M ∨ (cid:1) A −→ (cid:0) M ∨ ⊠ w M (cid:1) ⊠ w M ∨ e M ⊗ −→ V ⊠ M ∨ (cid:27) M ∨ , (5.36b)yield the identity maps M and M ∨ , respectively. Here w , w are distinct non-zero complex numbers satisfying | w | > | w | and | w | > | w − w | ; ⊠ w indicates the relative positioning of insertion points of fusion factors, that is, the right most factorwill be inserted at 0, the middle factor at w and the left most at w ; Technically there exist distinct notions of left andright duals and the above properties are those for left duals. We prove below that M = σ ℓ W λ is left rigid. Right rigidityfollows from left rigidity due to category F being braided.For M = σ ℓ W λ we take the tensor dual to be M ∨ = σ − ℓ W − λ and we will construct the evaluation and coevaluationmorphisms using the first free field realisation (2.33) given in Proposition 2.12.(1). In particular, we have σ ℓ W λ (cid:27) F λ ( θ + ψ ) + ( ℓ − ψ , σ − ℓ W − λ (cid:27) F − λ ( θ + ψ ) − ℓψ , ℓ ∈ Z , λ ∈ R / Z , λ , Z . (5.37)We denote fusion over the lattice vertex algebra V K of the free field realisation by ⊠ ff to distinguish it from fusion over G .Recall that the fusion product of Fock spaces over the lattice vertex algebra V K of the free field realisation just adds Fockspace weights. Thus the fusion product over V K of the modules corresponding to σ ℓ W λ and σ − ℓ W − λ is given by F − λ ( θ + ψ ) − ℓψ ⊠ ff F λ ( θ + ψ ) + ( ℓ − ψ (cid:27) F − ψ (cid:27) W − . (5.38) Therefore we have the V K -module map Y : F − λ ( θ + ψ ) − ℓψ ⊠ ff F λ ( θ + ψ ) + ( ℓ − ψ → F − ψ given by the intertwining operator that mapsthe kets in the Fock space F λ ( θ + ψ ) + ( ℓ − ψ to vertex operators, that is, operators of the form (2.30). Since V K -module mapsare also G -module maps by restriction and since the fusion product of two modules over a vertex subalgebra is a quotient ofthe fusion product over the larger vertex algebra, Y also defines a G -module map F − λ ( θ + ψ ) − ℓψ ⊠ F λ ( θ + ψ ) + ( ℓ − ψ → F − ψ (cid:27) W − .Furthermore, the screening operator S = H V ψ ( z )d z defines a G -module map S : F − ψ → F with the image being thebosonic ghost vertex algebra G . Up to a normalisation factor, to be determined later, we define the evaluation map for M = σ ℓ W λ to be the composition of Y and the screening operator S . e M = S ◦ Y : M ∨ ⊠ M → V . (5.39)To define the coevaluation we need to identify a submodule of M ⊠ M ∨ isomorphic to V . By Lemma 5.10, we know that M ⊠ M ∨ has a direct summand isomorphic to P , which by Proposition 3.3 we know has a submodule isomorphic to V . Itis this copy of V which the coevaluation shall map to. Since V is the vector space underlying the vertex algebra G andany vertex algebra is generated from its vacuum vector, we characterise the coevaluation map by the image of the vacuumvector. i M : Ω −→ | i S − −→ |− ψ i −→ V ( j − ψ + ( j − ℓ ) θ ( w ) |− j ψ − ( j − ℓ ) θ i S −→ I w S ( z )V ( j − ψ + ( j − ℓ ) θ ( w ) |− j ψ − ( j − ℓ ) θ i d z , (5.40)where the first arrow is the inclusion of V into F (cid:27) W − ⊂ P , S − denotes picking preimages of S and j the uniquerepresentative of the coset λ satisfying < j < . Note that the ambiguity of picking preimages of S in the second arrowis undone by reapplying S in the fourth arrow and hence the map is well-defined. This map maps to F , which is asubmodule of P as shown in Proposition 3.3.Note that since the modules M and M ∨ considered here are simple, the compositions of coevaluations and evaluations(5.36) are proportional to the identity by Schur’s lemma. Rigidity therefore follows, if we can show that the proportionalityfactors for (5.36a) and (5.36b) are equal and non-zero.We determine the proportionality factor for (5.36a) by applying the map to the ket | ( j − ψ + ( j − ℓ ) θ i ∈ F λ ( ψ + θ ) + ( ℓ − θ (cid:27) σ ℓ W λ . Following the sequence of maps in (5.36a) we get | ( j − ψ + ( j − ℓ ) θ i → | i ⊠ | ( j − ψ + ( j − ℓ ) θ i → I w , w S ( z )V ( j − ψ − ( j − ℓ ) θ ( w )V − j ψ − ( j − ℓ ) θ ( w ) | ( j − ψ − ( j − ℓ ) θ i d z → I , w I w , w S ( z ) S ( z )V ( j − ψ + ( j − ℓ ) θ ( w )V − j ψ − ( j − ℓ ) θ ( w ) | ( j − ψ + ( j − ℓ ) θ i d z d z , (5.41)where H , w denotes a contour about and w but not w , H w , w denotes a contour about w and w but not . Theproportionality factor is obtained by pairing the above with the dual of the Fock space highest weight vector, which wedenote by an empty bra h| , and thus equal to the matrix element I ( w , w ) = I , w I w , w h| S ( z ) S ( z )V ( j − ψ + ( j − ℓ ) θ ( w )V − j ψ − ( j − ℓ ) θ ( w ) | ( j − ψ + ( j − ℓ ) θ i d z d z = f ( w , w ) I , w I w , w ( z − z ) z j − ( z − w ) j − ( z − w ) − j z j − ( z − w ) j − ( z − w ) − j d z d z = f ( w , w ) (cid:18) I , w z j ( z − w ) j − ( z − w ) − j d z I w , w z j − ( z − w ) j − ( z − w ) − j d z − I , w z j − ( z − w ) j − ( z − w ) − j d z I w , w z j ( z − w ) j − ( z − w ) − j d z (cid:19) , (5.42)where f ( w , w ) = ( w − w ) ℓ + j (1 − ℓ ) w ( j − j − ℓ − w ℓ + j (1 − ℓ )1 . (5.43)Note that the second equality of (5.42) is where the associativity isomorphisms are used to pass from compositions(or products) of vertex operators to their operator product expansions (also called iterates). For intertwining operators,associativity amounts to the analytic continuation of their series expansions and then reexpanding in a different domain.On the left-hand side of the second equality the intertwining operators (or here specifically vertex operators) are in radialordering, while on the right-hand side they have been analytically continued and then reexpanded as operator product expansions. By an analogous argument the proportionality factor produced by the sequence of maps (5.36b) is the matrixelement ˜ I ( w , w ) = I , w I w , w h| S ( z ) S ( z )V − j ψ − ( j − ℓ ) θ ( w )V ( j − ψ + ( j − ℓ ) θ ( w ) |− j ψ − ( j − ℓ ) θ i d z d z = f ( w , w ) (cid:18) I , w z j ( z − w ) j − ( z − w ) − j d z I w , w z j − ( z − w ) j − ( z − w ) − j d z − I , w z j − ( z − w ) j − ( z − w ) − j d z I w , w z j ( z − w ) j − ( z − w ) − j d z (cid:19) . (5.44)Since both matrix elements are equal, I ( w , w ) = ˜ I ( w , w ) , rigidity follows by showing that they are non-zero.We evaluate the four integrals appearing in I ( w , w ) . We simplify the first integral using the substitution z = w x . I , w z j ( z − w ) j − ( z − w ) − j d z = − w j − w I , x j (1 − x ) − j (cid:18) − w w x (cid:19) j − d x = − (cid:0) e π i j − (cid:1) w j − w Z x j (1 − x ) − j (cid:18) − w w x (cid:19) j − d x = − (cid:0) e π i j − (cid:1) w j − w B (1 + j , − j ) F (cid:18) − j , + j ; 2; w w (cid:19) , (5.45)where the second equality follows by deforming the contour about 0 and 1 to a dumbbell or dog bone contour, whose endpoints vanish because the contributions from the end points are O ( ε + j ) and O ( ε − j ) respectively, and < j < ; and thethird equality is the integral representation of the hypergeometric function and B is the beta function. Similarly, I , w z j − ( z − w ) j − ( z − w ) − j d z = − (cid:0) e π i j − (cid:1) w j − B( j , − j ) F (cid:18) − j , j ; 1; w w (cid:19) . (5.46)For the integrals with contours about w and w we use the substitution z = w − ( w − w ) x and then again obtain integralrepresentations of the hypergeometric function. I w , w z j − ( z − w ) j − ( z − w ) − j d z = ( − j (cid:0) e π i j − (cid:1) w j − B ( j , − j ) F (cid:18) − j , j ; 1; w − w w (cid:19) , I w , w z j ( z − w ) j − ( z − w ) − j d z = ( − j (cid:0) e π i j − (cid:1) w j B ( j , − j ) F (cid:18) − j , j ; 1; w − w w (cid:19) . (5.47)Note that for the three integrals above, the end point contributions of the contour also vanish due to being O ( ε j ) and O ( ε − j ) for 0 and 1 respectively.Making use of the hypergeometric and beta function identities F (cid:18) − µ, + µ ; 2; w w (cid:19) = w w F (cid:18) − µ, µ ; 1; 1 − w w (cid:19) , F (cid:18) − µ, µ ; 1; 1 − w w (cid:19) = F (cid:18) − µ, µ ; 1; w w (cid:19) , B(1 + µ, − µ ) = µ B( µ, − µ ) = πµ sin( πµ ) , (5.48)the proportionality factor I ( w , w ) simplifies to I ( w , w ) = ( − j f ( w , w ) (cid:0) e π i j − (cid:1) w j − π ( j − π j ) F (cid:18) − j , j ; 1; w − w w (cid:19) F (cid:18) − j , j ; 1; w w (cid:19) . (5.49)Since j < Z , I ( w , w ) can only vanish, if one of the hypergeometric factors does. We specialise the complex numbers w , w , such that w = w . Then, F (cid:18) − j , j ; 1; w w (cid:19) = F (cid:0) − j , j ; 1; (cid:1) = Γ (cid:0) (cid:1) Γ (1) Γ (cid:16) − j (cid:17) Γ (cid:16) + j (cid:17) , , (5.50)and the relationship between contiguous functions implies F (cid:18) − j , j ; 1; w − w w (cid:19) = (cid:0) F (cid:0) − j , j ; 1; (cid:1) + F (cid:0) − j , + µ ; 1; (cid:1)(cid:1) (5.51) = Γ (cid:0) (cid:1) Γ (1) Γ (cid:16) − j (cid:17) Γ (cid:16) + j (cid:17) , . (5.52)Thus I ( w , w ) , and we can rescale the evaluation map by I ( w , w ) − so that the sequences of maps (5.36) are equal tothe identity maps on M and M ∨ . Thus σ ℓ W λ is rigid.
6. Fusion product formulae
In this section we determine the decomposition of all fusion products in category F . A complete list of fusion productsamong representatives of each spectral flow orbit is collected in Theorem 6.1, while the proofs of these decompositionformulae have been split into the dedicated Subsections 6.1 and 6.2. To simplify some of the decomposition formulae weintroduce dedicated notation for certain sums of spectral flows of the projective module P . Consider the polynomial ofspectral flows f n ( σ ) = n ∑ k = σ k − , n ∈ N , (6.1)and let Q n = f n ( σ ) P = n M k = P k − , n ∈ N . (6.2)Further, let Q nk = σ k Q n , Q m , nk = σ k − f m ( σ ) Q n = m + n − M r = N r P k + r − , N r = min { r , m , n , m + n − r } , m , n ∈ N , k ∈ Z . (6.3) Theorem 6.1. (1)
Category F under fusion is a rigid braided tensor category. (2) The following is a list of all non-trivial fusion products, those not involving the fusion unit (the vacuum module V ), incategory F among representatives for each spectral flow orbit. All other fusion products are determined from thesethrough spectral flow and the compatibility of spectral flow with fusion as given in Theorem 2.6.Since F is rigid, the fusion product of a projective module R with any indecomposable module M is given by R ⊠ M (cid:27) M S [ M : S ] R ⊠ S , (6.4) where the summation index runs over all isomorphism classes of composition factors of M and [ M : S ] is the multiplicityof the composition factor S in M .For all λ, µ ∈ R / Z , λ, µ, λ + µ , Z , W λ ⊠ W µ (cid:27) W λ + µ ⊕ σ − W λ + µ , W λ ⊠ W − λ (cid:27) σ − P . (6.5) For m , n ∈ Z , m ≥ n , such that the lengths of indecomposables below are positive, we have the following fusion productformulae. B m + ⊠ B n + (cid:27) B m + n + ⊕ Q m , n T m + ⊠ T n + (cid:27) T m + n + ⊕ Q m , n B m + ⊠ B n (cid:27) B n ⊕ Q m , n T m + ⊠ T n (cid:27) T n ⊕ Q m , n B m ⊠ B n (cid:27) B n m − ⊕ B n ⊕ Q m − , n T m ⊠ T n (cid:27) T n m − ⊕ T n ⊕ Q m − , n (6.6a) T m + ⊠ B n + (cid:27) T m − n + n ⊕ Q m + , n B m + ⊠ T n + (cid:27) B m − n + n ⊕ Q m + , n T m ⊠ B n + (cid:27) T m n ⊕ Q m , n B m ⊠ T n + (cid:27) B m n ⊕ Q m , n T m ⊠ B n (cid:27) Q m , n B m ⊠ T n (cid:27) Q m , n (6.6b)We split the proof of Theorem 6.1 into multiple parts. Theorem 6.1.(1) is shown in Proposition 6.4. The fusion formulae(6.5),(6.6a),(6.6b) are determined in Propositions 6.2, 6.9 and 6.10 and Lemma 5.10, respectively. Remark.
The fusion product formulae of Theorem 6.1 projected onto the Grothendieck group match the conjecturedVerlinde formula of [16, Corollaries 7 and 10], thereby proving that category F satisfies the standard module formalismversion of the Verlinde formula. It will be an interesting future problem to find a more conceptual and direct proof for thevalidity of the Verlinde formula, rather than a proof by inspection.6.1. Fusion products of simple projective modules.
In this section we determine the fusion products of the simpleprojective modules.
Proposition 6.2.
For λ, µ ∈ R / Z , λ, µ, λ + µ < Z , we have W λ ⊠ W µ (cid:27) W λ + µ ⊕ σ − W λ + µ . (6.7) Proof.
Since W λ and W µ both lie in category R , we know, by Lemma 5.9, that the composition factors of the fusionproduct lie in categories R or σ − R . Further, since J ( z ) is a conformal weight 1 field, its corresponding weight, theghost weight, adds under fusion. Therefore the only possible composition factors are W λ + µ and σ − W λ + µ . Since thesecomposition factors are both projective and injective, they can only appear as direct summands and all that remains is todetermine their multiplicity. In [20] Adamović and Pedić computed dimensions of spaces of intertwining operators forfusion products of the simple projective modules. In particular, [20, Corollary 6.1] states that dim (cid:18) MW λ , W µ (cid:19) = , (6.8)if M is isomorphic to σ ℓ W λ + µ , ℓ = , − . Thus the proposition follows. Remark.
To prove the above proposition directly without citing the literature, we could have used the two free fieldrealisations in Section 2.4 to construct intertwining operators of the type appearing in equation (6.8), thereby showing thatthe dimension of the corresponding space of intertwining operators is at least 1. This was also done in [20]. An upperbound of 1 can then easily be determined by calculations involving either the HLZ double dual construction (similar to thecalculations done in Lemma 5.10) or the NGK algorithm.
Proposition 6.3.
For λ ∈ R / Z , λ , Z , we have W λ ⊠ W − λ (cid:27) σ − P . (6.9) Proof.
By Proposition 5.13, W λ is rigid and hence its fusion product with a projective module must again be projective.Further, by Lemma 5.9, all composition factors must lie in categories σ ℓ R , ℓ = − , . Finally, since ghost weights addunder fusion, the ghost weights of the fusion product must lie in Z . Thus the fusion product must be isomorphic to a directsum of some number of copies of σ − P . By Lemma 5.10, we know there is exactly one such summand. Proposition 6.4.
Category F is rigid.Proof. Category F has sufficiently many injective and projective modules, that is, all simple modules have projectivecovers and injective hulls, and all projectives are injective and vice-versa. Further, the simple projective modules σ ℓ W λ are rigid and generate the non-simple projective modules under fusion, so all projective modules are rigid. Catefory F istherefore a Frobenius category and hence any for short exact sequence with two rigid terms (whose duals are also rigid)the third term is also rigid. This implies that all modules are rigid and hence so is category F . Corollary 6.5.
Let M , N ∈ F , then M ∗ ⊠ N ∗ (cid:27) ( M ⊠ N ) ∗ . (6.10) Proof.
Due to rigidity, the tensor duality functor ∨ defines an equivalence of categories and is therefore exact. Further,the tensor duality functor satisfies M ∨ ⊠ N ∨ (cid:27) ( M ⊠ N ) ∨ . (6.11)This also implies that V ∨ k = V − k . We see that the tensor dual M ∨ agrees with σ ( M ′ ) on all simple modules in F . Asboth ( − ) ∨ and σ ( − ) ′ are exact contravariant invertible functors, we have M ∨ (cid:27) σ ( M ′ ) for any module in F . Recalling ( − ) ∗ = c ( − ) ′ , we further have M ∗ (cid:27) σ c M ∨ . Theorem 2.6 then implies M ∗ ⊠ N ∗ (cid:27) (cid:0) σ c M ∨ (cid:1) ⊠ (cid:0) σ c N ∨ (cid:1) (cid:27) σ c ( M ⊠ N ) ∨ (cid:27) ( M ⊠ N ) ∗ . (6.12)6.2. Fusion products of reducible indecomposable modules.
In this section we calculate the remaining fusion productformulae involving indecomposable modules in F . The main tool for determining these fusion products is that category F is rigid by Proposition 6.4. Hence fusion is biexact and projective modules form a tensor ideal. We begin by calculatingcertain basic fusion products from which the remainder can be determined inductively. Lemma 6.6. T ⊠ B (cid:27) P , B ⊠ B (cid:27) B ⊕ B , T ⊠ T (cid:27) T ⊕ T . (6.13) Proof.
Taking the short exact sequence (2.20a) for W + = T − and fusing it with W − = B − yields the short exact sequence −→ W − −→ W + ⊠ W − −→ σ − W − −→ . (6.14)Similarly, fusing the short exact sequence (2.20b) for W − with W + yields −→ σ − W + −→ W − ⊠ W + −→ W + −→ . (6.15)If either of the above exact sequences splits there is a contradiction, because if σ − W + and W + are direct summands of W + ⊠ W − , (6.14) is not exact, and if W − and σ − W − are direct summands, (6.15) is not exact. Hence both sequences mustbe non-split. As can be read off from the tables in Proposition 4.3, dim Ext (cid:0) σ − W − , W − (cid:1) = dim Ext (cid:0) W + , σ − W + (cid:1) = .There is only one candidate for the middle coefficient of these exact sequences, namely σ − P . Thus the first fusion rulefollows. The other two fusion products by are determined by fusing W ± with the short exact sequences for W ± . Theextension groups corresponding to these fused exact sequences are zero-dimensional and hence the sequences split andthe lemma follows.We further prepare the following Ext group dimensions for later use. Lemma 6.7.
The indecomposable modules T n + , B m n + , B n and B m n satisfy dim Ext (cid:0) T n + , B m n + (cid:1) = dim Ext (cid:0) B n , B m n (cid:1) = . (6.16) The corresponding extensions are given by T n + m + and B n + m respectively.Proof. We start with the following presentation of T n + −→ T n + −→ P [ T n + ] −→ T n + −→ . (6.17)Applying the functor Hom (cid:0) − , B m n + (cid:1) yields −→ Hom (cid:0) T n + , B m n + (cid:1) −→ Hom (cid:0) P [ T n + ] , B m n + (cid:1) −→ Hom (cid:0) T n + , B m n + (cid:1) −→ Ext (cid:0) T n + , B m n + (cid:1) −→ . (6.18)The first coefficient vanishes due to T n + and B m n + having no common composition factors. The second coefficient canbe shown to vanish using the projective cover formulae in Proposition 4.4 and reading off Hom group dimensions fromthe Loewy diagrams. For the third coefficient, the only composition factor common to both T n + and B m n + is V n + ,which occurs as a quotient for T n + and a submodule for B m n + , so this gives rise to a one dimensional Hom group. Thevanishing Euler characteristic then implies that dim Ext (cid:0) T n + , B m n + (cid:1) = as expected. Furthermore, we can examine T n + m + to see that it has a B m n submodule which yields T n + when quotiented out, therefore this is the unique extensioncharacterised by Ext (cid:0) T n + , B m n + (cid:1) . We can follow the same procedure starting with the projective presentation of B n to obtain the following exact sequence −→ Hom (cid:0) B n , B m n (cid:1) −→ Hom (cid:0) P [ B n ] , B m n (cid:1) −→ Hom (cid:0) B n , B m n (cid:1) −→ Ext (cid:0) B n + , B m n (cid:1) −→ . (6.19)By the same argument as above we can calculate the Hom groups,and vanishing Euler characteristic implies dimExt (cid:0) B n , B m n (cid:1) = . Similarly we see that B n + m provides an extension of B n by B m n and must therefore be the unique one.We can now determine fusion products when one factor has length 2 and the other has arbitrary length. Lemma 6.8.
The fusion products of length 2 indecomposables with any indecomposable of types B or T satisfy thefollowing decomposition formulae. B n + ⊠ B (cid:27) B ⊕ Q n T n + ⊠ T (cid:27) T ⊕ Q n B n + ⊠ B (cid:27) B n + ⊕ B ⊕ Q n T n + ⊠ T (cid:27) T n + ⊕ T ⊕ Q n B n + ⊠ T (cid:27) T n ⊕ Q n T n + ⊠ B (cid:27) B n ⊕ Q n B n ⊠ T (cid:27) Q n T n ⊠ B (cid:27) Q n (6.20) Proof.
We prove the left column of identities. The right column then follows from Corollary 6.5 and applying the ∗ functor. We start with the short exact sequence (4.1a) satisfied by B n + , −→ B n − −→ B n + −→ T n − −→ . (6.21)We then take the fusion product with B , −→ B n − ⊠ B −→ B n + ⊠ B −→ P n −→ . (6.22)Because P n is projective, the sequence splits and we have the recurrence relation B n + ⊠ B (cid:27) ( B n − ⊠ B ) ⊕ P n . (6.23)Then, the first fusion product formula of the lemma follows by induction with B = V as the base case.We next consider the short exact sequence (4.9c) and fuse it with B to obtain −→ B n + ⊠ B −→ B n + ⊠ B −→ B n + −→ . (6.24)Since Ext (cid:0) B n + , B (cid:1) = , by the tables in Proposition 4.3, this sequence splits and we obtain the second fusion productof the lemma.For the final two fusion products, we perform the same exercises with different exact sequences. For the third and fourthfusion products we use (4.9d), with odd and even length respectively. Fusing with T gives the short exact sequences −→ B n − ⊠ T −→ B n + ⊠ T −→ P −→ , −→ B n ⊠ T −→ B n + ⊠ T −→ P −→ . (6.25)In both cases, the sequences split because P is projective.We now use Lemma 6.8 to prove the fusion product formulae (6.6a) of Theorem 6.1. Proposition 6.9.
The fusion products of indecomposable modules of types B and T with themselves satisfy the decompo-sition formulae below, for m ≥ n . B m + ⊠ B n + (cid:27) B m + n + ⊕ Q m , n T m + ⊠ T n + (cid:27) T m + n + ⊕ Q m , n B m + ⊠ B n (cid:27) B n ⊕ Q m , n T m + ⊠ T n (cid:27) T n ⊕ Q m , n B m ⊠ B n (cid:27) B n m − ⊕ B n ⊕ Q m − , n T m ⊠ T n (cid:27) T n m − ⊕ T n ⊕ Q m − , n (6.26) Proof.
We prove the left column of identities. The right column then follows from Corollary 6.5 and applying the ∗ functor.First, for both superscripts odd, we take two short exact sequences (4.9d) and (4.1a) for B n + and fuse with B m + to find −→ B n − ⊠ B m + −→ B n + ⊠ B m + −→ B ⊕ Q m −→ , −→ B n − ⊠ B m + −→ B n + ⊠ B m + −→ T n + m − ⊕ Q m n − −→ . (6.27)Now comparing these exact sequences, and using the fact that P is projective, we find that the sequences cannotboth split, as they would give different direct sums. For the first short exact sequence, we use Lemma 6.7, to find dim Ext (cid:0) B , B m + n − (cid:1) = , with the extension being given by B m + n + so we can determine the fusion product formulaeinductively to get B m + ⊠ B (cid:27) B m + ⊕ Q m , B m + ⊠ B (cid:27) B m + ⊕ (cid:0) + σ (cid:1) Q m , B m + ⊠ B n + (cid:27) B m + n + ⊕ m M k = Q n k − = B m + n + ⊕ Q m , n . (6.28)We can deduce the remaining rules from short exact sequences that relate even and odd B s. Firstly, we take the two shortexact sequences (4.9d) and (4.1b), and fuse them with B m + to get −→ B m + ⊠ B n −→ B m + ⊠ B n + −→ B ⊕ Q m −→ , −→ B n ⊕ Q m n + −→ B m + ⊠ B n + −→ B m + ⊠ B n −→ . (6.29)Either of these exact sequences splitting would lead to a contradiction, hence both must be non-split. Further, by Lemma 6.7we find dim Ext (cid:0) B , B n (cid:1) = dim Ext (cid:0) B n , B n (cid:1) = , with the corresponding non-split extension given by B n + . Therefore B m + ⊠ B n (cid:27) B n ⊕ Q m , n . (6.30)Finally we fuse (4.9c) with B n to find −→ B m + ⊠ B n −→ B m + ⊠ B n −→ B n m + −→ . (6.31)For m ≥ n , dim Ext (cid:0) B n m + , B n (cid:1) = , which follows because the composition factors are separated by at least two unitsof spectral flow and Ext( V n , V m ) = for | n − m | > , the above sequence splits. In the case when m = n − , we have that Ext (cid:0) B n n − , V k (cid:1) = for all the composition factors of B n , that is, (0 ≤ k ≤ n − . Hence dim Ext (cid:0) B n n − , B n (cid:1) = andthe above sequence again splits. Thus, B m + ⊠ B n (cid:27) B n m + ⊕ B n ⊕ Q m , n , m ≥ n − . (6.32) Proposition 6.10.
The fusion products of indecomposable modules of types B and T with each other satisfy the decom-position formulae below, for m ≥ n . T m + ⊠ B n + (cid:27) T m − n + n ⊕ Q m + , n B m + ⊠ T n + (cid:27) B m − n + n ⊕ Q m + , n T m ⊠ B n + (cid:27) T m n ⊕ Q m , n B m ⊠ T n + (cid:27) B m n ⊕ Q m , n T m ⊠ B n (cid:27) Q m , n B m ⊠ T n (cid:27) Q m , n (6.33) Proof.
We prove the left column of identities. The right column then follows from Corollary 6.5 and applying the ∗ functorto each module. We start with sequences (4.1a) and (4.9d) for odd length B , and fuse them with T m + to find −→ T m + ⊠ B n − −→ T m + ⊠ B n + −→ T n − ⊕ Q m n −→ , (6.34) −→ T m + ⊠ B n − −→ T m + ⊠ B n + −→ B m ⊕ Q m −→ . (6.35)Specialising to n=1 we have −→ T m + −→ T m + ⊠ B −→ T ⊕ m M k = σ k + P −→ , (6.36) −→ T m + −→ T m + ⊠ B −→ B m ⊕ m M k = σ k − P −→ . (6.37) Since P is projective, its spectral flows must appear as direct summands in the middle coefficient of the above exactsequences. Thus, T m + ⊠ B (cid:27) A ⊕ m + M k = σ k − P = A ⊕ Q m + . (6.38)Therefore the module A satisfies the exact sequences −→ T m + −→ A ⊕ P −→ T −→ , −→ T m + −→ A ⊕ P m + −→ B m −→ . (6.39)Because either of these sequences splitting would lead to a contradiction and the corresponding extension groups areone-dimensional, the sequences uniquely characterise the fusion product. Proceeding by induction, we obtain T m + ⊠ B (cid:27) T m − ⊕ Q m + , T m + ⊠ B (cid:27) T m − ⊕ (cid:0) + σ (cid:1) Q m + , T m + ⊠ B n + (cid:27) T m − n + n ⊕ Q m + , n . (6.40)Next we take two short exact sequences (4.9g) and (4.9e), for T m and fuse them with B n + to get −→ B n + m − −→ T m ⊠ B n + −→ T m − n − n ⊕ Q m , n −→ , −→ B m + n − ⊕ Q m − , n −→ T m ⊠ B n + −→ B n + −→ . (6.41)Again either of these sequences splitting would lead to a contradiction, and by Lemma 6.7, dim Ext (cid:0) T m − n − n , B n + m − (cid:1) = ,with the extension being given by T m n , so the second fusion rule follows. Finally, fusing (4.1d) with B n , we have −→ T m − ⊠ B n −→ T m ⊠ B n −→ Q n m − −→ , (6.42) T m ⊠ B n (cid:27) m M k = Q n k − = Q m , n . (6.43) Appendix A. Sufficient conditions for convergence and extension – Proof of Theorem 5.7
In this section we give a proof of Theorem 5.7 by reviewing reasoning presented by Yang in [25] and showing thatcertain assumptions on the category of strongly graded modules (see [25, Assumption 7.1, Part 3]) are not required, if oneonly wishes to conclude that convergence and extension properties hold. Instead all that is required is that the modulesconsidered satisfy suitable finiteness conditions. This appendix closely follows the logic of [25, Sections 5 & 6] andalso [46, Section 2].Throughout this section let A ≤ B be abelian groups. Further, let V be an A -graded vertex algebra with a vertexsubalgebra V ⊂ V (0) . In this section only, all mode expansions of fields from a vertex operator algebra V will be of the form Y ( v , z ) = ∑ n ∈ Z v n z − n − regardless of the conformal weight of v ∈ V , that is, v n refers to the coefficient of z − n − rather thanthe one which shifts conformal weight by − n . Definition A.1.
Let W , W , W , W , W be B -graded V -modules.(1) We say that two B -graded logarithmic intertwining operators Y , Y of respective types (cid:0) W W , W (cid:1) , (cid:0) W W , W (cid:1) satisfythe convergence and extension property for products if for any a , a , ∈ B and any doubly homogeneous elements w ′ ∈ W ′ , w ∈ W , w i ∈ W ( a i ) i , i = , , there exist M ∈ Z ≥ , r , . . . , r M , s , . . . s M ∈ R , u , . . . u M , v , . . . v M ∈ Z ≥ andanalytic functions f ( z ) , . . . , f M ( z ) on the disc | z | < satisfying wt w + wt w + s k > N , for each k = , . . ., M , (A.1)where N ∈ Z depends only on the intertwining operators Y , Y and a + a , such that as a formal power series thematrix element h w ′ , Y ( w , z ) Y ( w , z ) w i (A.2) converges absolutely in the region | z | > | z | > and may be analytically continued to the multivalued analytic function M ∑ k = z r k ( z − z ) s k (log z ) u k (log( z − z )) v k f k (cid:18) z − z z (cid:19) (A.3)in the region | z | > | z − z | > .(2) We say that two B -graded logarithmic intertwining operators Y , Y of respective types (cid:0) W W , W (cid:1) , (cid:0) W W , W (cid:1) satisfy the convergence and extension property for iterates if for any a , a , ∈ B and any doubly homogeneous elements w ′ ∈ W ′ , w ∈ W , w i ∈ W ( a i ) i , i = , , there exist M ∈ Z ≥ , r , . . ., r M , s , . . . s M ∈ R , u , . . . u M , v , . . . v M ∈ Z ≥ and analyticfunctions f ( z ) , . . . f M ( z ) on the disc | z | < satisfying wt w + wt w + s k > N , for each k = , . . ., M , (A.4)where N ∈ Z depends only on the intertwining operators Y , Y and a + a , such that as a formal power series thematrix element h w ′ , Y ( Y ( w , z − z ) w , z ) w i (A.5)converges absolutely in the region | z | > | z − z | > and may be analytically continued to the multivalued analyticfunction M ∑ k = z r k z s k (log z ) u k (log z ) v k f k (cid:18) z z (cid:19) (A.6)in the region | z | > | z | > .Consider the Noetherian ring R = C [ z ± , z ± , ( z − z ) − ] . Then for any quadruple of B -graded V -modules W , W , W , W , and any triple ( a , a , a ) ∈ B , we define the R -module T ( a , a , a ) = R ⊗ (cid:0) W ′ (cid:1) ( a + a + a ) ⊗ W ( a )1 ⊗ W ( a )2 ⊗ W ( a )3 , (A.7)where all the tensor product symbols denote complex tensor products. We will generally omit the tensor product symbolseparating R from the V -modules. The motivation for considering this module is that for any B -graded module W andany pair of grading compatible logarithmic intertwining operators Y , Y of respective types (cid:0) W W , W (cid:1) and (cid:0) W W , W (cid:1) itproduces matrix elements via the map φ Y , Y : T ( a , a , a ) → z h C ( { z / z } )[ z ± , z ± ] , where h is the combined conformalweight of w ′ , w , w , w and C ( { x } ) is the space of all power series in x with bounded below real exponents (the modules W i , i = , , , will always have real conformal weights below), defined by φ Y , Y ( f ( z , z ) w ′ ⊗ w ⊗ w ⊗ w ) = ι ( f ( z , z )) h w ′ , Y ( w , z ) Y ( w , z ) w i , (A.8)where ι : R → C J z / z K [ z ± , z ± ] is the map expanding elements of R such that the powers of z are bounded below. Thisin turn justifies considering the submodule J ( a , a , a ) = span R (cid:8) A (cid:0) v , w ′ , w , w , w (cid:1) , B (cid:0) v , w ′ , w , w , w (cid:1) , C (cid:0) v , w ′ , w , w , w (cid:1) , D (cid:0) v , w ′ , w , w , w (cid:1) ∈ T ( a , a , a ) : v ∈ V , w ′ ∈ ( W ′ ) ( a + a + a ) , w i ∈ W ( a i ) , i = , , (cid:9) , (A.9)where the generators A (cid:0) v , w ′ , w , w , w (cid:1) = − w ′ ⊗ v − w ⊗ w ⊗ w + ∑ k ≥ (cid:18) − k (cid:19) ( − z ) k v ∗− − k w ′ ⊗ w ⊗ w ⊗ w − ∑ k ≥ (cid:18) − k (cid:19) ( − ( z − z )) − − k w ′ ⊗ w ⊗ v k w ⊗ w − ∑ k ≥ (cid:18) − k (cid:19) ( − z ) − − k w ′ ⊗ w ⊗ w ⊗ v k w , B (cid:0) v , w ′ , w , w , w (cid:1) = − w ′ ⊗ w ⊗ v − w ⊗ w + ∑ k ≥ (cid:18) − k (cid:19) ( − z ) k v ∗− − k w ′ ⊗ w ⊗ w ⊗ w − ∑ k ≥ (cid:18) − k (cid:19) ( − ( z − z )) − − k w ′ ⊗ v k w ⊗ w ⊗ w − ∑ k ≥ (cid:18) − k (cid:19) ( − z ) − − k w ′ ⊗ w ⊗ w ⊗ v k w , C (cid:0) v , w ′ , w , w , w (cid:1) = v ∗− w ′ ⊗ w ⊗ v − w ⊗ w − ∑ k ≥ (cid:18) − k (cid:19) z − − k w ′ ⊗ v k w ⊗ w ⊗ w − ∑ k ≥ (cid:18) − k (cid:19) z − − k w ′ ⊗ w ⊗ v k w ⊗ w − w ′ ⊗ w ⊗ w ⊗ v − w , D (cid:0) v , w ′ , w , w , w (cid:1) = v − w ′ ⊗ w ⊗ v − w ⊗ w − ∑ k ≥ (cid:18) − k (cid:19) z k + w ′ ⊗ e z − L (cid:0) − z (cid:1) L v k (cid:0) − z − (cid:1) L e − z − L w ⊗ w ⊗ w − ∑ k ≥ (cid:18) − k (cid:19) z − − k w ′ ⊗ w ⊗ e z − L (cid:0) − z (cid:1) L v k (cid:0) − z − (cid:1) L e − z − L w ⊗ w − w ′ ⊗ w ⊗ w ⊗ v ∗− w , (A.10)are preimages of the relations coming from residues of the Jacobi identity for intertwining operators and where v ∗ k : W ′ i → W ′ i denotes the adjoint of v k : W i → W i . Hence J ( a , a , a ) lies in the kernel of φ Y , Y for any choice of intertwining operators Y , Y of the correct types.Next consider the doubly homogeneous space T ( a , a , a )[ r ] = ∏ r , r , r , r ∈ R r + r + r + r = r R ⊗ (cid:0) W ′ (cid:1) ( a + a + a )[ r ] ⊗ ( W ) ( a )[ r ] ⊗ ( W ) ( a )[ r ] ⊗ ( W ) ( a )[ r ] (A.11)to construct the subspaces F r ( T ( a , a , a ) ) = ∏ s ≤ r T ( a , a , a )[ s ] , F r ( J ( a , a , a ) ) = J ( a , a , a ) ∩ F r ( T ( a , a , a ) ) . (A.12)These define filtrations on T ( a , a , a ) and J ( a , a , a ) , respectively, since F s ( T ( a , a , a ) ) ⊂ F r ( T ( a , a , a ) ) and F s ( J ( a , a , a ) ) ⊂ F r ( J ( a , a , a ) ) , if s ≤ r , and S r ∈ R F r ( T ( a , a , a ) ) = T ( a , a , a ) and S r ∈ R F r ( J ( a , a , a ) ) = J ( a , a , a ) . Note that if the W i , i = , , , are discretely strongly graded, then T ( a , a , a )[ r ] is a finite sum of finite dimensional doubly homogeneous spacestensored with R . Hence T ( a , a , a )[ r ] is a finitely generated free R -module. Further, F r ( T ( a , a , a ) ) is also a finite sum andhence also a finitely generated free R -module. Finally, the ring R is Noetherian and so the submodule F r ( J ( a , a , a ) ) is alsofinitely generated. Proposition A.2.
Let the V -modules W i , i = , , , be discretely strongly B -graded and B -graded C -cofinite as V -modules, then for any a , a , a ∈ B there exists M ∈ Z such that for any r ∈ R F r ( T ( a , a , a ) ) ⊂ F r ( J ( a , a , a ) ) + F M ( T ( a , a , a ) ) and T ( a , a , a ) ⊂ J ( a , a , a ) + F M ( T ( a , a , a ) ) . (A.13) Proof.
By assumption the modules W i , i = , , , are B -graded C -cofinite as V -modules, that is, the spaces C ( M ) ( a ) = span C (cid:8) v − h w ∈ M ( a ) : v ∈ V [ h ] h > , w ∈ M (cid:9) (A.14)have finite codimension in M ( a ) for M = W i , i = , , , . Thus M ( a )[ h ] ⊂ C ( M ) ( a ) for sufficiently large conformal weight h ∈ R and hence there exists M ∈ Z such that M n > M T ( a , a , a )[ n ] ⊂ C (cid:0) W ′ (cid:1) ( a + a + a ) ⊗ W ( a )1 ⊗ W ( a )2 ⊗ W ( a )3 + (cid:0) W ′ (cid:1) ( a + a + a ) ⊗ C ( W ) ( a ) ⊗ W ( a )2 ⊗ W ( a )3 + (cid:0) W ′ (cid:1) ( a + a + a ) ⊗ W ( a )1 ⊗ C ( W ) ( a ) ⊗ W ( a )3 + (cid:0) W ′ (cid:1) ( a + a + a ) ⊗ W ( a )1 ⊗ W ( a )2 ⊗ C ( W ) ( a ) . (A.15)We prove the first inclusion of the proposition by induction on r ∈ R . If r ≤ M , then the inclusion is true by F r ( T ( a , a , a ) ) defining a filtration. Next assume that F r ( T ( a , a , a ) ) ⊂ F r ( J ( a , a , a ) ) + F M ( T ( a , a , a ) ) is true for all r < s ∈ R for some s > M . We will show that any element of the homogeneous space T ( a , a , a )[ s ] can be written as a sum of elements in F s ( J ( a , a , a ) ) and F M ( T ( a , a , a ) ) . Since s > M , this homogeneous element is an element of the right-hand side of (A.15).We shall only consider the case of this element lying in the second summand of the right-hand side, as the other cases follow analogously. Without loss of generality we can assume the element has the form w ′ ⊗ v − w ⊗ w ⊗ w ∈ T ( a , a , a )[ s ] ,where w ′ ∈ (cid:0) W ′ (cid:1) ( a + a + a ) , w i ∈ W ( a i ) i , i = , , , v ∈ V [ h ] , h > . By computing the degrees of the summands making up A ( v , w ′ , w , w , w ) in (A.10) we see that the three sums over k all lie in F s − ( T ( a , a , a ) ) ⊂ F s − ( J ( a , a , a ) ) + F M ( T ( a , a , a ) ) and that A ( v , w ′ , w , w , w ) ∈ F s ( J ( a , a , a ) ) . Further, w ′ ⊗ v − w ⊗ w ⊗ w = − A ( v , w ′ , w , w , w ) + ∑ k ≥ (cid:18) − k (cid:19) ( − z ) k v ∗ k w ′ ⊗ w ⊗ w ⊗ w − ∑ k ≥ (cid:18) − k (cid:19) ( − ( z − z )) − − k w ′ ⊗ w ⊗ v k w ⊗ w − ∑ k ≥ (cid:18) − k (cid:19) ( − z ) − − k w ′ ⊗ w ⊗ w ⊗ v k w . (A.16)Thus w ′ ⊗ v − w ⊗ w ⊗ w lies in the sum F s ( J ( a , a , a ) ) + F M ( T ( a , a , a ) ) and the first inclusion of the proposition follows.The second inclusion follows from F r ( T ( a , a , a ) ) and F r ( J ( a , a , a ) ) defining filtrations. T ( a , a , a ) = [ r ∈ R F r ( T ( a , a , a ) ) ⊂ [ r ∈ R (cid:0) F r ( J ( a , a , a ) ) + F M ( T ( a , a , a ) ) (cid:1) = [ r ∈ R F r ( J ( a , a , a ) ) ! + F M ( T ( a , a , a ) ) = J ( a , a , a ) + F M ( T ( a , a , a ) ) . (A.17) Corollary A.3.
Let the V -modules W i , i = , , , be discretely strongly B -graded and B -graded C -cofinite as V -modules. (1) The quotient R -module T ( a , a , a ) / J ( a , a , a ) is finitely generated. (2) For any representative w ∈ T ( a , a , a ) , we denote its coset in T ( a , a , a ) / J ( a , a , a ) by [ w ] . Let w ′ ∈ (cid:0) W ′ (cid:1) ( a + a + a ) and w i ∈ W ( a i ) i i = , , , and consider the submodules of T ( a , a , a ) / J ( a , a , a ) given by M = span R nh w ⊗ L j − w ⊗ w ⊗ w i : j ∈ Z ≥ o , M = span R nh w ⊗ w ⊗ L j − w ⊗ w i : j ∈ Z ≥ o . (A.18) Then M and M are finitely generated, in particular, there exist m , n ∈ Z ≥ and a k ( z , z ) , b ℓ ( z , z ) ∈ R , ≤ k ≤ m , ≤ ℓ ≤ n such that (cid:2) w ⊗ L m − w ⊗ w ⊗ w (cid:3) + a ( z , z ) (cid:2) w ⊗ L m − − w ⊗ w ⊗ w (cid:3) + · · · + a m ( z , z )[ w ⊗ w ⊗ w ⊗ w ] = , (cid:2) w ⊗ w ⊗ L n − w ⊗ w (cid:3) + b ( z , z ) (cid:2) w ⊗ w ⊗ L n − − w ⊗ w (cid:3) + · · · + b n ( z , z )[ w ⊗ w ⊗ w ⊗ w ] = . (A.19) Proof.
Since R is a Noetherian ring, Part (1) holds if T ( a , a , a ) / J ( a , a , a ) is isomorphic to a subquotient of a finitelygenerated module over R . By Proposition A.2 we have the inclusion and identification T ( a , a , a ) / J ( a , a , a ) ⊂ (cid:0) J ( a , a , a ) + F M ( T ( a , a , a ) ) (cid:1) / J ( a , a , a ) (cid:27) F M ( T ( a , a , a ) ) / (cid:0) F M ( T ( a , a , a ) ) ∩ J ( a , a , a ) (cid:1) . (A.20)Thus T ( a , a , a ) / J ( a , a , a ) is isomorphic to a subquotient of the finitely generated module F M ( T ( a , a , a ) ) and Part (1)follows. Part (2) is an immediate consequence of Part (1) and the fact that a submodule of a finitely generated module overa Noetherian ring is again finitely generated. Theorem A.4.
Let the V -modules W i , i = , , , be discretely strongly B -graded and B -graded C -cofinite as V -modules, let W be a B -graded V -module and let Y , Y be logarithmic grading compatible intertwining operators of types (cid:0) W W , W (cid:1) , (cid:0) W W , W (cid:1) , respectively. Then for any homogeneous elements w ′ ∈ W ′ , w i ∈ W i , i = , , , there exist m , n ∈ Z ≥ and a k ( z , z ) , b ℓ ( z , z ) ∈ R , ≤ k ≤ m , ≤ ℓ ≤ n such that the power series expansion of the matrix element h w ′ , Y ( w , z ) Y ( w , z ) w i (A.21) is a solution to the power series expansion of the system of differential equations ∂ m φ∂ z m + a ( z , z ) ∂ m − φ∂ z m − + · · · + a m ( z , z ) φ = , ∂ n φ∂ z n + b ( z , z ) ∂ n − φ∂ z n − + · · · + b n ( z , z ) φ = , (A.22) in the region | z | > | z | > . Proof.
Let a , a , a be the respective B -grades of w , w , w , then we can assume that the B -grade of w ′ is a + a + a ,because otherwise the matrix element vanishes and the theorem follows trivially. Recall the map φ Y , Y : T ( a , a , a ) → z h C ( { z / z } ) (cid:2) z ± , z ± (cid:3) , defined by the formula (A.8). Since J ( a , a , a ) lies in the kernel of φ Y , Y , we have an induced map φ Y , Y : T ( a , a , a ) / J ( a , a , a ) → z h C ( { z / z } ) (cid:2) z ± , z ± (cid:3) . (A.23)The theorem then follows by applying φ Y , Y to the relations (A.19) of Corollary A.3.(2), using the L − derivative propertyof intertwining operators and expanding in the region | z | > | z | > .Systems of differential equations of the form (A.22) have solutions very close to the expansion required if their singularpoints are regular, see for example [47, Appendix B]. A sufficient condition, whose validity we shall verify shortly, forregularity at a given singular point is that the coefficients a i , b j in the system (A.22) have poles of degree at most m − i and n − j respectively. Such singular points are called simple (see [47, Appendix B] for the general definition). The singularpoints relevant for the convergence and extension property for products are z = z and ( z − z ) / z = .We need to consider new filtrations in addition to those considered previously. Let R = C [ z ± , z ± ] , then R n = ( z − z ) − n R , n ∈ Z equips R with the structure of a filtered ring in the sense that R n ⊂ R m , if n ≤ m , R = S n ∈ Z R n and R n · R m ⊂ R m + n . The R -module T ( a , a , a ) can then also be equipped with a compatible filtration R r ( T ( a , a , a ) ) = ∏ n + h + h + h + h ≤ rh i ∈ R R n ⊗ (cid:0) W ′ (cid:1) ( a + a + a )[ h ] ⊗ ( W ) ( a )[ h ] ⊗ ( W ) ( a )[ h ] ⊗ ( W ) ( a )[ h ] , r ∈ R , (A.24)in the sense that R r ( T ( a , a , a ) ) ⊂ R s ( T ( a , a , a ) ) , if r ≤ s , T ( a , a , a ) = S r ∈ R R r ( T ( a , a , a ) ) and R n · R r ( T ( a , a , a ) ) ⊂ R n + r ( T ( a , a , a ) ) .Further, let R r ( J ( a , a , a ) ) = R r ( T ( a , a , a ) ) ∩ J ( a , a , a ) . Proposition A.5.
Let the V -modules W i , i = , , , be discretely strongly B -graded and B -graded C -cofinite as V -modules. Then for any a , a , a ∈ B there exists M ∈ Z such that for any r ∈ R R r ( T ( a , a , a ) ) ⊂ R r ( J ( a , a , a ) ) + F M ( T ( a , a , a ) ) and T ( a , a , a ) = J ( a , a , a ) + F M ( T ( a , a , a ) ) . (A.25) Further, T ( a , a , a ) / J ( a , a , a ) is finitely generated.Proof. The proof of this proposition mimics the proof of Proposition A.2 once one has verified that the elements A ( u , w ′ , w , w , w ) , B ( u , w ′ , w , w , w ) , C ( u , w ′ , w , w , w ) and D ( u , w ′ , w , w , w ) lie in R h ( J ) , where h is the sum ofthe conformal weights of u , w ′ , w , w , w .We also need to consider the R -module U ( a , a , a ) = R ⊗ (cid:0) W ′ (cid:1) ( a + a + a ) ⊗ W ( a )1 ⊗ W ( a )2 ⊗ W ( a )3 and denote by U ( a , a , a )[ r ] the subspace of conformal weight r ∈ R . Thus U ( a , a , a ) = ∏ r ∈ R U ( a , a , a )[ r ] . Lemma A.6.
Let the V -modules W i , i = , , , be discretely strongly B -graded and B -graded C -cofinite as V -modules.For any a , a , a ∈ B and any doubly homogeneous vectors w ′ ∈ (cid:0) W ′ (cid:1) ( a + a + a )[ h ] , w i ∈ ( W i ) ( a i )[ h i ] , let h = ∑ i h i , let h be thesmallest non-negative representative of the coset h + Z and let m J ∈ R h ( J ( a , a , a ) ) , m T ∈ F M ( T ( a , a , a ) ) be vectors satisfying w ′ ⊗ w ⊗ w ⊗ w = m J + m T . (A.26) Then there exists S ∈ R such that h + S ∈ Z ≥ and ( z − z ) h + S m T ∈ U ( a , a , a ) .Proof. Note that the existence of the vectors m J , m T is guaranteed by Proposition A.5. Choose S ∈ R such that h + S ∈ Z ≥ and such that for any r ≤ − S , T ( a , a , a )[ r ] = . Such an S must exist, since the conformal weights of T ( a , a , a ) are boundedbelow by assumption. By definition, R r ( T ( a , a , a ) ) is spanned by elements of the form ( z − z ) − n f ( z , z ) e w ⊗ e w ⊗ e w ⊗ e w ,where f ∈ R and n + ∑ i wt e w i ≤ r . The number S was therefore chosen such that ( z − z ) r + S R r ( T ( a , a , a ) ) ⊂ U ( a , a , a ) whenever r + S ∈ Z . Now, by assumption, m T = w ′ ⊗ w ⊗ w ⊗ w − m J . (A.27)The right-hand side of this equality lies in R h ( T ( a , a , a ) ) by construction and therefore so does the left-hand side. Hence ( z − z ) h + S m T ∈ U ( a , a , a ) . Theorem A.7.
Let the V -modules W i , i = , , , be discretely strongly B -graded and B -graded C -cofinite as V -modules,let W be a B -graded V -module and let Y , Y be logarithmic grading compatible intertwining operators of types (cid:0) W W , W (cid:1) , (cid:0) W W , W (cid:1) , respectively and consider the system of differential equations of Theorem A.4. For the singular points z = z and ( z − z ) / z = there exist coefficients a k ( z , z ) , b l ( z , z ) ∈ R such that these singular points of the system of differentialequations (A.22) satisfied by the matrix elements (A.21) are regular.Proof. We consider first the singular point z = z . By Proposition A.5 and Lemma A.6, for any k ∈ Z ≥ together with avector w ′ ⊗ L k − w ⊗ w ⊗ w ∈ T ( a , a , a ) , where the w i are doubly homogeneous vectors of total conformal weight h ∈ R ,there exist m ( k ) J ∈ R h + k ( J ( a , a , a ) ) and m ( k ) T ∈ F M ( T ( a , a , a ) ) such that w ′ ⊗ L k − w ⊗ w ⊗ w = m ( k ) J + m ( k ) T . (A.28)Let h be the smallest non-negative representative of the coset h + Z . Then, by Lemma A.6, there exists S ∈ R such that h + S ∈ Z ≥ and ( z − z ) h + k + S m ( k ) T ∈ U ( a , a , a ) and thus ( z − z ) h + k + S m ( k ) T ∈ S r ≤ M U ( a , a , a )[ r ] . Since the V -modules W i arediscretely strongly B -graded and B -graded C -cofinite, ∏ r ≤ M U ( a , a , a )[ r ] is a finite sum of finitely generated R -modules andhence also finitely generated. Thus, since R is Noetherian, the submodule generated by the ( z − z ) h + k + S m ( k ) T , k ∈ Z ≥ isalso finitely generated. Hence there exists an m ∈ Z ≥ such that { ( z − z ) h + k + S m ( k ) T : 0 ≤ k ≤ m − } is a finite generatingset for this submodule and subsequently there exist c k ( z , z ) ∈ R such that ( z − z ) h + m + S m ( m ) T + m − ∑ k = c k ( z , z )( z − z ) h + k + S m ( k ) T = . (A.29)Therefore, w ′ ⊗ L m − w ⊗ w ⊗ w + m − ∑ k = c k ( z , z )( z − z ) k − m w ′ ⊗ L k − w ⊗ w ⊗ w = m ( m ) J + m − ∑ k = c k ( z , z ) m ( k ) J . (A.30)Thus in the quotient module T ( a , a , a ) / J ( a , a , a ) , we obtain (where we again use square brackets to denote cosets) (cid:2) w ′ ⊗ L m − w ⊗ w ⊗ w (cid:3) + m − ∑ k = c k ( z , z )( z − z ) k − m (cid:2) w ′ ⊗ L k − w ⊗ w ⊗ w (cid:3) = , (A.31)since m ( k ) J ∈ J ( a , a , a ) . By a similar line of reasoning there exists an n ∈ Z ≥ and d ℓ ( z , z ) ∈ R such that (cid:2) w ′ ⊗ w ⊗ L n − w ⊗ w (cid:3) + m − ∑ ℓ = d ℓ ( z , z )( z − z ) ℓ − n (cid:2) w ′ ⊗ w ⊗ L k − w ⊗ w (cid:3) = . (A.32)Applying the map φ Y , Y defined by (A.8) and using the L − property for intertwining operators will then result in a systemof differential equations for which z = z is a simple, and hence regular, singular point.To show the regularity of the singular point ( z − z ) / z = , we introduce new gradings on R and T ( a , a , a ) . Weassign degree − to the variables z , z , thus giving R a Z grading and then grade T ( a , a , a ) by adding R -degreesand conformal weights. This implies that the elements A ( v , w ′ , w , w , w ) , B ( v , w ′ , w , w , w ) , C ( v , w ′ , w , w , w ) and D ( v , w ′ , w , w , w ) are homogeneous with respect to this new grading if their arguments are doubly homogeneous. Thenew grading therefore descends to T ( a , a , a ) / J ( a , a , a ) . Further, for doubly homogeneous elements w ′ , w , w , w , theelements (cid:2) w ′ ⊗ L k − w ⊗ w ⊗ w (cid:3) , (cid:2) w ′ ⊗ w ⊗ L ℓ − w ⊗ w (cid:3) ∈ T ( a , a , a ) / J ( a , a , a ) , (A.33)are also homogeneous. Thus the coefficients c k ( z , z ) , d ℓ ( z , z ) of equations (A.31) and (A.32) are elements of degree 0 in R and can therefore be written as Laurent polynomials in ( z − z ) / z . It then follows that the singular point ( z − z ) / z = is regular.The fact that the matrix element (A.2) satisfies an expansion of the form (A.3) now follows by the reasoning of [46,Theorem 3.5]. A little care is needed when following the reasoning of [46], since there only modules with a diagonalisableaction of L are considered. However, as noted in [21, Part VII, Proof of Theorem 11.8 and Remark 11.9] the argumentextends easily to modules where L has Jordan blocks. The basic idea is that one can use the L conjugation propertyof intertwining operators (recall that L is the generator of dilations) to rescale the variables in the matrix element (A.2) by z so that it becomes a function in z = ( z − z ) / z only and the system of differential equations (A.31) and (A.32)then becomes an ordinary differential equation for z with a regular singularity at z = . Similar reasoning for the matrixelement (A.5) leads one to conclude that it satisfies the expansion (A.6). Hence Theorem 5.7 follows. References [1] Y-Z Huang. Vertex operator algebras and the Verlinde conjecture.
Commun. Contemp. Math. , 10:103–1054, 2008. arXiv:math/0406291 .[2] J Fuchs, C Schweigert, and C Stigner. From non-semisimple Hopf algebras to correlation functions for logarithmic CFT.
J. Phys. , A46:494008,13. arXiv:1302.4683 [hep-th] .[3] J Fuchs and C Schweigert. Consistent systems of correlators in non-semisimple conformal field theory.
Adv. Math. , 307:598–639, 2017. arXiv:1604.01143 [math.QA] .[4] T Gannon T Creutzig. Logarithmic conformal field theory, log-modular tensor categories and modular forms.
J. Phys. , A50:404004, 2017. arXiv:1605.04630 [math.QA] .[5] D Friedan, E Martinec, and S Shenker. Conformal invariance, supersymmetry and string theory.
Nucl. Phys. , B271:93–165, 1986.[6] M Wakimoto. Fock representation of the algebra A (1)1 . Comm. Math. Phys. , 104:605–609, 1986.[7] B Feigin and E Frenkel. Quantization of the Drinfeld-Sokolov reduction.
Phys. Lett. , 246:75–81, 1990.[8] F Malikov, V Schechtman, and A Vaintrob. Chiral de Rham complex.
Comm. Math. Phys. , 204:439–473, 1999. arXiv:math/9803041 .[9] H Kausch. Curiosities at c = − . DAMTP , 95–52:26, 1995. arXiv:hep-th/9510149 .[10] M Gaberdiel and H Kausch. A rational logarithmic conformal field theory.
Phys. Lett. , B386:131–137, 1996. arXiv:hep-th/9606050 .[11] M Gaberdiel and I Runkel. From boundary to bulk in logarithmic CFT.
J. Phys. , A41:075402, 2008. arXiv:0707.0388 [hep-th] .[12] I Runkel. A braided monoidal category for free super-bosons.
J. Math. Phys. , 55:59, 2014. arXiv:1209.5554 [math.QA] .[13] D Adamović and A Milas. On the triplet vertex algebra W ( p ) . Adv. Math. , 217:2664––2699, 2008. arXiv:0707.1857 [math.QA] .[14] A Tsuchiya and S Wood. On the extended W -algebra of type sl at positive rational level. Int. Math. Res. Not. , 2015:5357––5435, 2015. arXiv:1302.6435 [math.QA] .[15] A Tsuchiya and S Wood. The tensor structure on the representation category of the W ( p ) triplet algebra. J. Phys. , A46:445203, 2013. arXiv:1201.0419 [hep-th] .[16] D Ridout and S Wood. Bosonic ghosts at c = as a logarithmic CFT. Lett. Math. Phys. , 105:279–307, 2015. arXiv:1408.4185 [hep-th] .[17] D Ridout. b sl (2) − / : A case study. Nucl. Phys. , B814:485–521, 2009. arXiv:0810.3532 [hep-th] .[18] T Creutzig and D Ridout. Modular data and Verlinde formulae for fractional level WZW models I.
Nucl. Phys. , B865:83–114, 2012. arXiv:1205.6513 [hep-th] .[19] T Creutzig and D Ridout. Modular data and Verlinde formulae for fractional level WZW models II.
Nucl. Phys. , B875:423–458, 2013. arXiv:1306.4388 [hep-th] .[20] D Adamović and V Pedić. On fusion rules and intertwining operators for the Weyl vertex algebra.
J. Math. Phys. , 60:081701, 2019.[21] Y-Z Huang, J Lepowsky James, and L Zhang. Logarithmic tensor product theory I–VIII. arXiv:1012.4193 [math.QA], arXiv:1012.4196 [math.QA],arXiv:1012.4197 [math.QA], arXiv:1012.4198 [math.QA], arXiv:1012.4199 [math.QA] , arXiv:1012.4202 [math.QA], arXiv:1110.1929 [math.QA],arXiv:1110.1931 [math.QA] .[22] C Beem, M Lemos, P Liendo, W Peelaers, L Rastelli, and B van Rees. Infinite chiral symmetry in four dimensions.
Comm. Math. Phys. ,336:1369–1433, 2015. arXiv:1312.5344 [hep-th] .[23] T Creutzig, D Ridout, and S Wood. Coset constructions of logarithmic (1,p)-models.
Lett. Math. Phys. , 104:553–583, 2014. arXiv:1305.2665[math.QA] .[24] J Auger, T Creutzig, S Kanade, and M Rupert. Braided tensor categories related to B p vertex algebras. Comm. Math. Phys. , 2020.[25] J Yang. A sufficient condition for convergence and extension property for strongly graded vertex algebras.
Contemporary Mathematics ,711:119–141, 2018.[26] R Block. The irreducible representations of the Weyl algebra A . Lecture Notes in Mathematics , 740:69–79, 1979.[27] H Li. The physics superselection principle in vertex operator algebra theory.
J. Algebra , 196(2):436–457, 1997.[28] E Frenkel and D Ben-Zvi.
Vertex Algebras and Algebraic Curves , volume 88 of
Mathematical Surveys and Monographs . American MathematicalSociety, 2001.[29] C Dong and J Lepowsky.
Generalized Vertex Algebras and Relative Vertex Operators . Progress in Mathematics. Birkhäuser, Boston, 1993.[30] S Wood. Admissible level osp (1 | minimal models and their relaxed highest weight modules. Transf. Groups , 2020:57, 2020. arXiv:1804.01200[math.QA] .[31] P Hilton and U Stammbach.
A Course in Homological Algebra . Graduate Texts in Mathematics. Springer-Verlag, 2 edition, 1997.[32] J Fjelstad, J Fuchs, S Hwang, A M Semikhatov, and I Yu Tipunin. Logarithmic conformal field theories via logarithmic deformations.
Nucl. Phys. ,633(3):379–413, 2002. arXiv:hep-th/0201091 .[33] J Belletête, D Ridout, and Y Saint-Aubin. Restriction and induction of indecomposable modules over the Temperley–Lieb algebras.
J. Phys. ,51(4):045201, 2017. arXiv:1605.05159 [math-ph] .[34] P Etingof, G Shlomo, D Nikshych, and V Ostrik.
Tensor Categories . Number volume 205 in Mathematical Surveys and Monographs. AmericanMathematical Society, 2015. [35] Y-Z Huang. On the applicability of logarithmic tensor category theory, 2017. arXiv:1702.00133 [math.QA] .[36] S Kanade and D Ridout. NGK and HLZ: Fusion for physicists and mathematicians. In D Adamovic and P Papi, editors, Affine, Vertex and W -algebras , volume 37 of Springer INdAM , pages 135–181, Cham, 2019. Springer. arXiv:1812.10713 [math-ph] .[37] L Zhang. Vertex tensor category structure on a category of Kazhdan-Lusztig.
New York J. Math. , 14:261–284, 2008. arXiv:math/0701260 .[38] W Wang. W + ∞ algebra, W algebra, and Friedan–Martinec– Shenker bosonization. Comm. Math. Phys. , 195:95 – 111, 1998.[39] A Linshaw. Invariant chiral differential operators and the W algebra. J. Pure Appl. Algebra , 213:632 – 648, 2009.[40] D Adamović. Representations of the vertex algebra W + ∞ with a negative integer central charge. Comm. Algebra , 29(7):3153–3166, 2001. arXiv:math/9904057 .[41] V Kac and A Radul. Representation theory of the vertex algebra W + ∞ , 1996. arXiv:hep-th/9512150 .[42] Y Matsuo. Free fields and quasi-finite representation of W + ∞ algebra. Phys. Lett. , 326(1–2):95—-100, 1994. arXiv:hep-th/9312192 .[43] T Creutzig and A Milas. False theta functions and the verlinde formula.
Adv. Math. , 262:520 – 545, 2014. arXiv:1309.6037 [math.QA] .[44] D Adamović and A Milas. Logarithmic intertwining operators and W (2 , p − algebras. J. Math. Phys. , 48(7):073503, 2007. arXiv:math/0702081[math.QA] .[45] T Creutzig, A Milas, and M Rupert. Logarithmic link invariants of U hq ( sl )) and asymptotic dimensions of singlet vertex algebras. J. Pure Appl.Algebra , pages 3224 – 3247, 2017. arXiv:1605.05634 [math.QA] .[46] Y-Z Huang. Differential equations and intertwining operators.
Commun. Contemp. Math. , 7:375–400, 2005. arXiv:math/0206206 .[47] A Knapp.
Representation theory of semisimple groups: an overview based on examples . Princeton Landmarks in Mathematics. PrincetonUniversity Press, 1986.(Robert Allen)
School of Mathematics, Cardiff University, Cardiff, United Kingdom, CF24 4AG.
E-mail address : [email protected] (Simon Wood) School of Mathematics, Cardiff University, Cardiff, United Kingdom, CF24 4AG.
E-mail address ::