Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion

The rank 1 bosonic ghost vertex algebra, also known as the $\beta \gamma$ ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor $C_2$-cofinite. We identify a module category, denoted category $\mathscr{F}$, which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category $\mathscr{F}$ has sufficiently many projective and injective modules, give a classification of all indecomposable modules, show that fusion is rigid and compute all fusion products. The fusion product formulae turn out to perfectly match a previously proposed Verlinde formula, which was computed using a conjectured generalisation of the usual rational Verlinde formula, called the standard module formalism. The bosonic ghosts therefore exhibit essentially all of the rich structure of rational theories despite satisfying none of the standard rationality assumptions such as $C_2$-cofiniteness, the vertex algebra being isomorphic to its restricted dual or having a one-dimensional conformal weight 0 space. In particular, to the best of the authors' knowledge this is the first example of a proof of rigidity for a logarithmic non-$C_2$-cofinite vertex algebra.


I
A vertex algebra is called logarithmic if it admits reducible yet indecomposable modules on which the Virasoro L 0 operator acts non-semisimply, giving rise to logarithmic singularities in the correlation functions of the associated conformal field theory. There is a general consensus within the research community that many of the structures familiar from rational vertex algebras such as modular tensor categories [1] and, in particular, the Verlinde formula should generalise in some form to the logarithmic case, at least for sufficiently nice logarithmic vertex algebras. To this end, considerable work has been done on developing non-semisimple or non-finite generalisations of modular tensor categories [2][3][4].
However, progress has been hindered by a severe lack of examples, making it hard to come up with the right set of assumptions.
Ghost systems have been used extensively in theoretical physics and quantum algebra. Their applications include gauge fixing in string theory [5], Wakimoto free field realisations [6], quantum Hamiltonian reduction [7] and constructing the chiral de Rham complex on smooth manifolds [8]. Fermionic ghosts at central charge c = −2 in the form of symplectic fermions have received a lot of attention in the past [9][10][11], due to their even subalgebra being one of the first known examples of a logarithmic vertex algebra. In particular, they are one of the few known examples of C 2 -cofinite yet logarithmic vertex algebras [12][13][14]. This family also provides the only known examples of logarithmic C 2 -cofinite vertex algebras with a rigid fusion product [12,15].
Here we study the rank 1 bosonic ghosts at central charge c = 2. One of the motivations for studying this algebra is that it is simple enough to allow many quantities to be computed explicitly, while simultaneously being distinguished from better understood algebras in a number of interesting ways. For example, the bosonic ghosts are not C 2 -cofinite and they were shown to be logarithmic by D. Ridout and the second author in [16], in which the module category to be studied here, denoted category F , was introduced. The main goals of [16] were determining the modular properties of characters in category F and computing the Verlinde formula, using the standard module formalism pioneered by D. Ridout and T.
Creutzig [17][18][19], to predict fusion product formulae. Later, D. Adamović and V. Pedić computed the dimensions of spaces of intertwining operators among the simple modules of category F in [20], which turned out to match the predictions made by the Verlinde formula in [16]. Here we show that fusion (in the sense of the P(w)-tensor products of [21]) equips category F with the structure of a braided tensor category. This, in particular, implies that category F is closed under fusion, that is, the fusion product of any two objects in F has no contributions from outside F and is hence again an object 2010 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 super conformal field theory and two-dimensional conformal field theory [22], where the bosonic ghosts appear as one of the smaller examples on the two-dimensional side. Within this context the bosonic ghosts are the first member of a family of vertex algebras called the B p algebras [23,24]. The B p -module categories are conjectured to satisfy interesting tensor categorical equivalences to the module category of the unrolled restricted quantum groups of sl 2 . It will be an interesting future 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 algebra and certain important automorphisms called conjugation and spectral flow; construct category F , the module category to be 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 0 has rank 2 Jordan blocks. We further show that P is both an injective hull and a projective cover of the vacuum module (the bosonic ghost vertex algebra as a module over itself), and classify all projective modules in category F , thereby showing that category F has sufficiently many projectives and injectives. In Section 4 we complete the analysis of category F as an abelian category by classifying all indecomposable modules. In Section 5 we show that fusion equips category F with the structure of a vertex tensor category, the main obstruction being showing that certain conditions, sufficient for the existence of associativity isomorphisms, hold. We further show that the simple projective modules of F are rigid. In Section 6 we show that category F is rigid and determine direct sum decompositions for all fusion products of modules in category F . In Appendix A we review an argument by Yang [25], which provides sufficient conditions for a technical property, called convergence and extension, required for the existence of associativity isomorphisms. We adjust the argument of Yang slightly to remove certain assumptions on module categories. This adjusted argument proves Theorem 5.7, which should also prove useful for the generalisations of category F to other vertex 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 helpful and stimulating discussions regarding the subtleties of vertex tensor categories, and Ehud Meir for the same regarding finitely generated modules. Both authors are very grateful to Thomas Creutzig for drawing their attention to a mistake relating to convergence and extension properties in a previous version of this manuscript. RA's research is supported by the EPSRC Doctoral Training Partnership grant EP/R513003/1. SW's research is supported by the Australian Research Council Discovery Project DP160101520.

B
In this section we introduce the bosonic ghost vertex algebra, along with its gradings and automorphisms. We define the module category which will be the focus of this paper. We also introduce useful tools for the classification of modules and calculation of fusion products, including two free field realisations. Note that we will make specific choices of conformal structure for all vertex algebras considered in this paper and so will not distinguish between vertex algebras, vertex operator algebras 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 to the Weyl algebra. Their defining relations resemble each other and the Zhu algebra of the bosonic ghosts is isomorphic to the Weyl algebra. The bosonic ghosts are therefore also often referred to as the Weyl vertex algebra. Due to these connections, 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 to the relations [p, q] = 1, (2.1) and no additional relations beyond those required by the axioms of an associative algebra. The grading operator is the element 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 cV.
(3) C[x, x −1 ]x λ , λ ∈ C \ Z, where p acts as ∂/∂x and q acts as x. Note that shifting λ by an integer yields an isomorphic module. 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 −1 ], where p acts as ∂/∂x and q acts as x. Denote this module by W + 0 . This module is uniquely characterised by the non-split exact sequence , where p acts as x and q acts as −∂/∂x. Denote this module by W − 0 . This module is uniquely characterised by the non-split exact sequence A module on which N = qp acts semisimply is called a weight module. Note that N acts semisimply on all modules above. [26]). Any simple A-module on which N acts semisimply is isomorphic to one of those listed in Definition 2.2, Parts (1) -(3).

Proposition 2.3 (Block
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 where 1 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 a grading on β and γ called ghost weight (or ghost number), that is, (2.9) Note that for the distinguished elements β, γ, J, and T we suppress the field map symbol Y : G → G z, z −1 . For generic elements A ∈ G we will use both Y(A, z) and A(z) to denote the field corresponding to A, depending on what is easier to read 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 1 = 1. (2.10) The second is conjugation which is given by These automorphisms satisfy the relation At the level of fields, these automorphisms act as The primary utility of the conjugation and spectral flow automorphisms lies in constructing new modules from known ones through twisting.

Definition 2.5.
Let M be a G-module and α an automorphism. The α-twisted module αM is defined to be M as a vector space, but with the action of G redefined to be 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 under conjugation and spectral flow.
Since c 2 β n = −β n and c 2 γ n = −γ n , we have c 2 M M, for any G-module M. We shall later see that spectral flow has infinite order and thus the relations (2.12) imply that at the level of the module category spectral flow and conjugation generate the infinite dihedral group.
Theorem 2.6. For any G-modules M and N, conjugation and spectral flow are compatible with fusion products in the following sense. conjugation was denoted by σ and spectral flow by ρ ℓ . There the automorphism g corresponds to σ −1 c = cσ here. These formulae mean that the fusion of modules twisted by spectral flow is determined by the fusion of untwisted modules, a simplification we shall make frequent use of.

Module category.
Every G-module is a G-module, however, the converse is not true (consider for example the adjoint 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 the category 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 conformal field 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.
Condition (1) can be reduced to the non-degeneracy of two and three-point conformal blocks. The non-degeneracy of two-point conformal blocks requires the module category to be closed under taking restricted duals, while non-degeneracy of three-point conformal blocks requires the module category to be closed under fusion (as, for example, constructed by the P(w)-tensor product of Huang-Lepowsky-Zhang). Conformal blocks at higher genera can be constructed from those on the sphere provided there is a well-defined action of the modular group on characters. Thus Condition (2) requires characters to be well-defined, that is, for all modules to decompose into direct sums of finite dimensional simultaneous generalised J 0 and L 0 eigenspaces. On any simple such module both L 0 and J 0 will act semisimply. Further, the action of J 0 is semisimple on a fusion product if J 0 acts semisimply on both factors of the product. We can therefore restrict ourselves to a category of J 0 -semisimple modules without endangering closure under fusion. We cannot, however, assume that L 0 will act semisimply in general.
The main tool for identifying and classifying vertex operator algebra modules is Zhu's algebra. Sadly Zhu's algebra is only sensitive to modules containing vectors annihilated by all positive modes. Any simple such module is a simple module in the category called R below. We will see that R is closed under taking restricted duals, however, as can be seen later in Section 6, category R is not closed under fusion. Further, it was shown in [16] that the action of the modular group does not close on its characters. Thus a larger category is needed, which will be denoted F below. It was shown in [16] that the action of the modular group closes on the characters of F and strong evidence was presented that fusion does as well. We will see in Section 6 that category F is indeed closed under fusion and that it satisfies numerous other nice properties.
The definition of the module categories mentioned above requires the following choice of parabolic triangular decomposition of G. G ± = span{β ±n , γ ±n : n ≥ 1}, This decomposition is parabolic, because G 0 is not abelian and thus not a choice of Cartan subalgebra. The role of the Cartan subalgebra will instead be played by span{1, J 0 }, which is technically a subalgebra of the completion of U(G) rather than 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 (or weak) G-modules M (we follow the conventions of [28] regarding smooth modules) which in addition satisfy that J 0 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 G + m is finite dimensional.
(3) Let F be the full subcategory of G-WMod consisting all finite length extensions of arbitrary spectral flows of modules in R with real J 0 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 0 and γ 0 act as −p and q, respectively, and G − act freely. We denote (1) V Ind V, the vacuum module or bosonic ghost vertex algebra as a module over itself.
(2) cV σ −1 V Ind cV, the conjugation twist of the vacuum module.
Note that due to the simple nature of the G commutation relations (2.7) Ind M is simple whenever M is, that is, the modules 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.
The conjugation twists of simple modules in F satisfy The indecomposable modules W ± 0 satisfy the non-split exact sequences This proposition was originally given in [16,Proposition 1], however, Part (1) is an immediate consequence of Block's classification of simple Weyl modules [26]. We shall show in Proposition 3.2 that, up to spectral flow twists, the indecomposable modules W ± 0 are the only indecomposable length 2 extensions of spectral flows of the vacuum module. In Section 4 we extend the indecomposable modules W ± 0 to infinite families of indecomposable modules.
2.3. Restricted duals. As mentioned above, conformal field theories require their representation categories to be closed under 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 the necessary definitions here.
Definition 2.10. Let M be a weight G-module. The restricted dual (or contragredient) module is defined to be where the action of G is characterised by and where Y(A, z) opp is given by the formula 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 (2) The restricted duals of spectral flows of the indecomposable modules in Definition 2.8 can be identified as (3) Denote by * the composition of twisting by c and taking the restricted dual, then (2.26) (4) Let A, B ∈ F and ℓ ∈ Z, then the homomorphism and first extension groups satisfy Proof. Part (1) follows immediately from Definition 2.10.
Part (2): Since σ ℓ V is simple, σ ℓ V ′ 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 + 1 and γ m , m ≥ −ℓ act locally nilpotently and therefore The modes β n , n ≥ ℓ + 1 and γ m , m ≥ 1 − ℓ act locally nilpotently and therefore σ ℓ W λ ′ is an object of σ −ℓ R. Further, for J 0 homogeneous m ∈ σ ℓ W λ and ψ ∈ σ ℓ W λ ′ , consider Finally, the duals of σ ℓ W ± 0 follow from that fact that the duality functor is exact and contravariant, and by applying it to 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 2 Heisenberg 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 the defining operator product expansions 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 symmetric bilinear lattice form corresponding to the above operator product expansions, that is (ψ, ψ) = −(θ, θ) = 1 and (ψ, θ) = 0. 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 mode of a Heisenberg vertex algebra field a(z), a ∈ L acts as scalar multiplication by (a, λ). We assign to the highest weight vector |λ of F λ the vertex operators V λ (z) : F µ → F µ z, z −1 z (λ,µ) given by the expansion 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,0 (b, λ)e λ , e λ |µ = |λ + µ . (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 simple modules of V K are given by Note that the pairing (Λ, ψ + θ) is well-defined, since it does not depend in the choice of representative λ ∈ Λ. It will occasionally 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 induces an embedding φ 1 : G → V K . Restricting to the image of this embedding, V K -modules can be identified with G-modules as where (Λ, ψ) is the coset in R/Z formed by pairing all representatives of Λ with ψ.
induces an embedding φ 2 : G → V K . Restricting to the image of this embedding, V K -modules can be identified with G-modules as 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 characters and was shown in [30,Proposition 4.7] and [20, (1) Let S 1 = Res V ψ (z), then ker S 1 : that is, S 1 is a screening operator for the free field realisation φ 1 of G. Further the sequence is exact and is therefore a Felder complex.
, that is, S 2 is a screening operator for the free field realisation φ 2 of G. Further the sequence 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 images of β and γ in V K are which are total derivatives in z implying that S 1 = Res V ψ (z) is a screening operator and that im φ 1 ⊂ ker S 1 . Therefore, S 1 commutes with G and hence defines a G-module map F 0 → F ψ . The identification (2.34) implies F 0 σW − 0 and F ψ σ 2 W − 0 . By comparing composition factors we see that the kernel must be either im φ 1 V or all of F 0 , so it is sufficient to show that the map S 1 : F 0 → F ψ is non-trivial. A quick calculation reveals that and thus S 1 is not trivial. By comparing the composition factors of the sequence (2.37) we also see that the sequence is an exact complex if each arrow is non-zero. Finally, the arrows are non-zero because Remark. The existence of Felder complexes will not specifically be needed for any of the results that follow, however, it is interesting to note that the bosonic ghosts admit such complexes. These complexes were crucial in [16] for computing the character formulae needed for the standard module formalism via resolutions of simple modules.

P
In this section we construct reducible yet indecomposable modules P on which the L 0 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 are projective covers and injective hulls of σ ℓ V for any ℓ ∈ Z. We refer readers unfamiliar with homological algebra concepts such as injective and projective modules or extension groups to the book [31] and recall the following result for later use.
for modules A, B. then this implies that the following two sequences are exact, for any module M.
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 projective and 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) = 0 for all simple objects M ∈ F . Injectivity in F then follows by applying the * functor and noting that W λ * W λ . Let M ∈ F be simple, then a necessary condition for the short exact sequence 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 = 0, let j ∈ λ and let v be a non-zero vector in the ghost and conformal weight [ j, 0] space of the submodule M = W λ ⊂ N and let w ∈ N be a representative of a non-zero coset in the [ j, 0] weight space of the quotient N/W λ . Without loss of generality, we can assume that w is a J 0 -eigenvector and a generalised L 0 -eigenvector. A necessary condition for the indecomposability of N, is the existence of an element U in the universal enveloping algebra U(G) such that Uv = w. Since v has minimal generalised conformal weight all positive modes annihilate v, thus Uv can be expanded as a sum of products of β 0 and γ 0 with each summand containing as many β 0 as γ 0 factors, that is, Uv = f (J 0 )v can be expanded as a polynomial in J 0 acting on v. Since N ∈ F , 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 0, then by applying the * and σ functors, Thus the sign of ℓ can be chosen at will and we can assume without loss of generality that ℓ ≥ 1. 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 of ghost weight j in σ ℓ W λ has positive least conformal weight. The least conformal weight of the ghost weight j − 1 space is 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 0 eigenvalue 0, and hence a representative of a non-trivial coset of ghost and conformal weight [ j, 0] in W λ N/σ ℓ W λ . Further let w ∈ σ ℓ W λ ⊂ N be a non-zero vector of ghost and conformal weight [ j − 1, −k]. Both v and w lie in one-dimensional weight spaces and hence span them. If N is indecomposable, then there must exist an element U of ghost and conformal weight [−1, −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 those with lesser index and γ n is placed to the right of β n for any n ∈ Z.
is an element of U(G) of ghost and conformal weight [0, i − k]. In W λ , γ 0 acts bijectively on the space of conformal weight 0 vectors, hence there exists aṽ ∈ N such that γ 0ṽ = v. Since at ghost weight j the conformal weights of N are non-negative, we have γ nṽ = 0, , 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 σ k V, σ ℓ V = Ext V, σ k−ℓ V = Ext V, σ ℓ−k V . So without loss of generality, it is sufficient to consider the extension groups Ext V, σ ℓ V or equivalently short exact sequences of the form 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 0 -eigenvector and a choice of representative of the highest weight vector in V M/σ ℓ V. We first show that these sequences necessarily split if ℓ 1. Assume ℓ = 0, then the exact sequence can only be non-split if there exists a ghost and conformal weight Without loss of generality we can replace U bỹ Since the conformal weights of V are bounded below by 0, they satisfy the same bound in M and β n ω = γ n ω = 0, n ≥ 1, soŨω can be expanded as a sum of products of β 0 and γ 0 acting on ω, with each summand containing the same number of β 0 and γ 0 factors. Equivalently,Ũω can be expanded as a polynomial in J 0 acting on ω.
Assume ℓ ≥ 2. The ghost and conformal weights of . Further, from the spectral flow formulae (2.15), one can see that the weight spaces of ghost and conformal weight Since the conformal weight of U is −ℓ, every summand of the expansion of Uω into β and γ modes must contain factors of γ n or β n with n ≥ 1 and we can choose a Poincaré-Birkhoff-Witt ordering where these modes are placed to the right. Thus Uω = 0, contradicting indecomposability and the exact sequence splits.
Assume ℓ = 1, then σW + 0 provides an example for which the exact sequence does not split and the dimension of the corresponding extension group is at least 1. We show that it is also at most 1. Let ω and σΩ be defined as for ℓ ≥ 2. By arguments analogous to those for ℓ ≥ 2, it follows that the [1, h] weight space vanishes for h < 0 and the [−1, h] weight space vanishes for h < −1. Thus β n ω = γ n+1 ω = 0, n ≥ 1. The [−1, −1] weight space of σV is one-dimensional and is hence spanned by σΩ. If M is indecomposable, there must exist a ghost and conformal weight [−1, Hence the isomorphism class of M is determined by the value of γ 1 ω in the one-dimensional [−1, −1] weight space and 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 ≥ 1, ℓ · λ = Z. The nodes represent the (spectral flows of) relaxed highest weight vectors of each module. Weight spaces are filled in grey. Conformal weight increases from top to bottom and 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 will turn out to be projective covers and injective hulls of σ ℓ V.

Proposition 3.3. Recall that by the first free field realisation φ 1 of Proposition 2.12, we can identify
and determining the action of all other fields in G through normal ordering and taking derivatives, where any vertex operator V λ (z) whose Heisenberg weight λ is in the coset [ψ] = [−θ] is defined to act as 0 on F 0 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 action where ⊕ is meant as a direct sum of vector spaces without considering the module structure. Denote the module with this S 1 -twisted action by P.
The zero mode J 0 therefore acts semisimply and L 0 has rank 2 Jordan blocks. The vectors |−ψ , |−ψ − θ , |θ , |0 ∈ P satisfy the relations The module P is indecomposable and satisfies the non-split exact sequences which implies that its composition factors are σ ±1 V and V with multiplicities 1 and 2, respectively. (4) P is an object in F . Figure 2 for an illustration of how the composition factors of P are linked by the action of G.

See
The composition factors of P with the nodes representing the spectral flows of the highest weight vectors of σ ℓ V for −1 ≤ ℓ ≤ 1. The arrows give the action of G modes on the highest-weight vectors of each factor. In this diagram, ghost weight increases to the left and conformal weight increases downwards. 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. The field identifications (3.8) of Part (2) follow by evaluating definitions introduced there, while the relations (3.9) follow by applying the field identifications.
To conclude the first exact sequence of Part (3) note that the action of β and γ closes on F 0 σW − 0 , because V −θ (z) acts trivially and quotienting by F 0 leaves only F −ψ W − 0 . To conclude the second exact sequence, let Ω be the highest weight vector of V and let σ ℓ Ω be the spectral flow images of Ω. Then |0 ∈ F 0 σ −1 W − 0 can be identified with Ω in the V composition factor of σ −1 W − 0 and |−ψ − θ can be identified with σΩ in the σV composition factor. Further, |−ψ ∈ F −ψ W − 0 can be identified with Ω in the V composition factor and |θ can be identified with σ −1 Ω in the σ −1 V composition factor. See Figure 2 for a diagram of the action of β and γ modes on P and how they connect the different composition factors. It therefore follows that |0 generates an indecomposable module whose composition factors are σ −1 V and V, with V as a submodule and σ −1 V as a quotient. The module therefore satisfies the same non-split exact sequence (2.20) as W + 0 does and since the extension groups in (3.4) are one-dimensional, this submodule is isomorphic to W + 0 . After quotienting by the submodule generated by |θ , the formulae above imply that the quotient is isomorphic to σW + 0 and the second exact sequence of Part (3) follows. Part (4) follows because J 0 acts diagonalisably on P and because P has only finitely many composition factors all of which lie in R or σR. Theorem 3.4. For every ℓ ∈ Z the indecomposable module σ ℓ P is projective and injective in F , and hence is a projective cover 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 than all spectral flow twists of P. We first show that P is injective by showing that dim Ext(W, P) = 0 for any simple module W ∈ F . Following that we will show P * = P, which, since * is an exact invertible contravariant functor, implies P is also projective.
A necessary condition for the non-triviality of such an extension is ghost weights differing only by integers. We therefore need not consider extensions by σ ℓ W λ , λ Z, so we restrict our attention to short exact sequences of the form 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 one of the composition factors of P. By Proposition 3.2 the above sequence must split if |ℓ| ≥ 3 and we therefore only consider |ℓ| ≤ 2.
If ℓ = 2, then the composition factor of P non-trivially extending σ 2 V must be σV. If the extension is non-trivial, then this subquotient must be isomorphic to σ 2 W − 0 . Further, if σ 2 Ω is the spectrally flowed highest weight vector of σ 2 V and |−ψ − θ ∈ P (see Figure 2) is the spectrally flowed highest weight vector of the σV composition factor of P, then (3.12) However, β −1 σ 2 Ω has conformal and ghost weight [−1, −2] and this weight space vanishes for both P and σ 2 V. Thus β −1 σ 2 Ω and hence a = 0, which is a contradiction.
If ℓ = 1, then the composition factor of P non-trivially extending σV must be V. There are two such composition factors in P. Any such non-trivial extension must be isomorphic to σW − 0 . If the non-trivial extension involves the composition factor whose spectrally flowed highest weight vector is represented by |−ψ , then β −1 σΩ = a|−ψ , a ∈ C \ {0}. The relations (3.9) thus imply However, β 0 σΩ = 0, so a = 0, which is a contradiction. If the non-trivial extension involves the composition factor whose spectrally flowed highest weight vector is represented by |0 , then there would exist a ∈ C \ {0} such that β −1 σΩ = a|0 .
If ℓ = 0, then the composition factor of P non-trivially extending V must be σV or σ −1 V. If there is a subquotient isomorphic to a non-trivial extension of V by σ −1 V, that is, isomorphic to W − 0 , then there exists a ∈ C \ {0} such that β 0 Ω = a|θ . But then, by the relations (3.9), β 0 (Ω − a)|θ = 0. Hence (Ω − a)|θ generates a direct summand isomorphic to V, making the extension trivial. An analogous argument rules out the existence of subquotient isomorphic a non-trivial extension of V by σ −1 V.
The cases ℓ = −2 and ℓ = −1 follow the same reasoning as ℓ = 2 and ℓ = 1, respectively. Now that we have established that P is injective, we can apply the functors Hom W − 0 , − and Hom σW + 0 , − to the short exact sequences (3.10a) and (3.10b), respectively, to deduce dim Ext The indecomposable 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 P * . This in turn implies Ext(P, −) = 0 and hence that σ ℓ P is projective for all ℓ ∈ Z.

C
In this section, we give a classification of all indecomposable modules in category F . We already know any simple module 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 and projective. We now complete the classification by finding all the reducible indecomposables which can be built as finite length 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 of indecomposable 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 there are 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 m n and T m n , m, n ∈ Z, n ≥ 1, are dual to each other with respect to * , that is, B m n * = T m n , and further satisfy the identifications The superscript m is the number of composition factors or length of the module. As a visual aid, we represent these indecomposable modules using Loewy diagrams.
Here the edges indicate the action of G and the vertices represent the composition factors.
The indecomposable modules B m n and T m n can have either an even or an odd number of composition factors which are constructed inductively by different extensions. Each chain is the result of extending either V, B 2 or T 2 repeatedly by the length two indecomposables B 2 n or T 2 n , as either quotients or submodules. For example, the even length module B 2m is constructed by repeatedly extending B 2 = σW − 0 by spectrally flowed copies of itself, as submodules, as outlined in the diagram below.
The dotted boxes separate the component modules and the dashed lines indicate a non-trivial action of the algebra present only in the extended modules. Similarly for the odd length module B 2m+1 , the chain is built by repeated extensions of B 1 = V by spectrally flowed copies of T 2 = σW + 0 as quotients.
When applying the * functor, the composition factors stay the same, however, all of the arrows corresponding to the action of the algebra between the composition factors are reversed. Thus the top and bottom row are switched in the Loewy diagram and B type indecomposables become T type indecomposables. The letters T and B indicate the composition factor isomorphic to V being either in the top or bottom row, respectively.
Recall from Proposition 3.2 that non-split extensions only exist between composition factors V i , V j with |i − j| = 1. This explains the sequential order of the spectral flows of composition factors in the chains. We will show that these Loewy diagrams uniquely characterise the reducible indecomposable modules, that is, no two non-isomorphic indecomposables have the same Loewy diagram. This is essentially due to certain extension groups being one-dimensional. These diagrams therefore provide a convenient way for reading off all submodules and quotients of a given indecomposable module and hence provide a shortcut for computing the dimensions of Hom groups.
along with the non-split short exact sequences below, uniquely characterise the modules B n and T n .
(2) Any reducible indecomposable module in F is isomorphic to one of the following.   This proposition allows us to find the Hom groups of indecomposables by examining their submodule and quotient structure 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 the injective hull and projective cover. Once these are known, we can construct injective and projective presentations which we 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 we prepare these here.

Proposition 4.3. The dimensions of Hom groups for the indecomposable modules
Further, the dimensions of Ext groups are given by the following table.
Proof. These dimensions follow from the exact sequences (2.20), Proposition 3.2 and Proposition 3.3, and judicious application of Proposition 3.1.
The classification of indecomposable Temperley-Lieb algebra modules in [33] parametrises modules by finite sets of 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 short length indecomposable modules in [33, Propositions 2.17 and 2.18] are equal to those for G-modules after making the identifications in the following table.
The dimensions of the remaining Hom and Ext groups, and therefore the classification, follows from the same homological algebra reasoning as in [33], with no need for exceptions at the boundaries of the finite sets in [33].
Further, projective and injective presentations are characterised by the following.
This data now suffices to show that the extension groups corresponding to the exact sequences (4.1) of Theorem 4.1 are one-dimensional and hence uniquely characterise the indecomposable B and T modules. The data can also be used to show that any non-trivial extension of these indecomposable modules by spectral flows of V will be a direct sum of modules in the list (4.2). Hence Theorem 4.1 follows.
For example, consider all possible extensions involving B 3 and V n , starting with Ext V n , B 3 . Using the tables above, we start with the following injective presentation of Applying the functor Hom(V n , −), Proposition 3.1 gives the Hom-Ext exact sequence We can use (4.3) to calculate these Hom groups, and the vanishing Euler characteristic implies dim Ext V n , B 3 = δ n,−1 + δ n,1 + δ n,3 . These extensions are given by B 4 −1 , B 2 ⊕ T 2 1 and B 4 for n = −1, 1 and 3, respectively. Similarly apply the functor Hom(−, V n ) to the projective presentation The vanishing Euler characteristic then implies dim Ext B 3 , V n = δ n,1 with the extension being given by P 1 . Therefore we see that all extensions involving B 3 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 later sections.
Proof. The above sequences being non-split is intuitively clear from the Loewy diagrams of the B and T indecomposables.

R
In this section we prove that fusion furnishes category F with the structure of a rigid tensor category and define evaluation 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 rigidity to [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 conformal weight h ∈ C, which take values in B and A, respectively. We will use the notations M ( j) and M [h] to denote the homogeneous spaces with respect to the additional grading or generalised conformal weight, respectively, and denote the simultaneous homogeneous space by M ( j) . The action of V on M is required to be compatible with the A and B gradation, that is, and where 1 is the vacuum vector and ω is the conformal vector.
[h+k] = 0, whenever k ∈ Z is sufficiently negative. The vertex algebra V is called strongly graded with respect to A if it is strongly graded as a module 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,  for sufficiently large k ∈ Z.
(2) For any holds. Here ι + means expanding about t = 0 such that the exponents of t are bounded below and the action of for sufficiently large k ∈ Z.
Denote by LGR(M 1 , M 2 ) the vector space of all P(w)-local grading restricted functionals. Define Remark. The variable w in P(w) denotes the insertion point of the tensor product constructed in [21], where it is usually denoted z and hence the tensor product is referred to as the P(z)-tensor product. shown 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 recommend the survey [36], which relates this construction of fusion to others in the literature.  Conditions (1) -(4) of Theorem 5.6 hold by construction for category F , so all that remains is verifying Conditions (5) and (6).  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 subspace of G. The lemma then follows by verifying that all modules in F are discretely strongly graded and graded C 1 -cofinite as modules over V.
All modules in F are discretely strongly graded by ghost weight j ∈ R. To prove this, we need to check that the simultaneous ghost and conformal weight spaces are finite dimensional and that every ghost weight homogeneous space has lower bounded conformal weights. The simultaneous ghost and conformal weight spaces of objects in R and therefore also those of σ ℓ R are finite dimensional by construction. Thus, since the objects of F are finite length extensions of those in σ ℓ R, the objects of F also have finite dimensional simultaneous ghost and conformal weight spaces. Similarly we 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 ≥ 0} and is isomorphic to W 1+∞ W 3,−2 ⊗ H where W 3,−2 is the singlet algebra at c = −2 and H is a rank 1 Heisenberg algebra [38,39]. Note that the conformal vector of W 1+∞ is usually chosen so as to have a central charge of 1. 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 2, we choose conformal vector of our Heisenberg algebra H so that its central charge is 4 (the conformal structure of W 3,−2 is unique). Fortunately, this does not complicate matters, as the simple modules over H are just Fock spaces regardless of the central charge or conformal vector. The tensor factors of W 1+∞ decompose nicely with respect to the free field realisation of Proposition 2.12.(2). The Heisenberg algebra H is generated by θ(z) and the singlet algebra W 3,−2 is a vertex subalgebra 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 0 and θ 0 . All simple V W 1+∞ modules can 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 3,−2 -module, is the simple quotient of the submodule of F (λ,ψ) generated by the highest weight vector. The homogeneous space σ ℓ V ( j) is simple, as a V-module [38, Lemma 4.1], see also [41,42]. Recall from Proposition 2.12.
(3) If M 1 has composition factors only in σ k R and σ k−1 R, and has composition factors only in σ ℓ R and σ ℓ−1 R, W ∈ F and W has composition factors only in σ k+ℓ+i R, −3 ≤ i ≤ 0.
Proof. Due to the compatibility of fusion with spectral flow, see Theorem 2.6, it is sufficient to only consider k = ℓ = 0. We prove Part (1) first. Let M 1 , M 2 , be modules in F . Let v ∈ G be the vector corresponding to the field J(z) and take the residue with respect to x 0 and x 1 in the Jacobi identity (5.8). This yields Hence, since the fusion factors M i are graded by ghost weight, the fusion product will be too. This means that the intertwining operator will be grading compatible and W must be doubly graded. Next we prove Part (3). Assume that M 1 , M 2 have composition factors only in R and σ −1 R. Note that J n , n ≥ 1 acts locally nilpotently on any object in F and that β n−ℓ , γ n+ℓ , n ≥ 1 act locally nilpotently on any object in σ ℓ R (recall that local nilpotence is one of the defining properties of σ ℓ R). We first show that J n , β n+1 , γ n , n ≥ 1 acting locally nilpotently on M 1 , M 2 implies that J n , β n+3 , γ n , n ≥ 1 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 0 x n+h−1 1 , n, k ∈ Z and take residues with respect to x 0 and x 1 . This yields This implies the local nilpotence of γ n , n ≥ 1 on Y(m 1 , x 2 )m 2 from its local nilpotence on m 1 and m 2 . Next consider v = J (and thus h = 1) and k = 1 in (5.16) to obtain Since J k , k ≥ 1 is nilpotent on both m 1 and m 2 , we see that J n − x 2 J n−1 is nilpotent for n ≥ 2. Recall that the series expansion of the intertwining operator satisfies a lower truncation condition, that is, for fixed s, if there exists a u ∈ C satisfying m (u,s) 0, then there exists a minimal representative t ∈ u + Z such that m (t,s) 0 and m (t ′ ,s) = 0 for all t ′ < t. Since J n − x 2 J n−1 is nilpotent on Y(m 1 , x 2 )m 2 it is also nilpotent on the leading term m (t,s) . By comparing coefficients of x 2 and log x 2 it then follows that J n , n ≥ 2 acts nilpotently on m (t,s) and by induction also on all coefficients of higher powers of x 2 . To show that J 1 acts locally nilpotently, assume that m 1 has J 0 -eigenvalue j and set n = 1, k = 0 in (5.16) to obtain Thus J 1 − x 2 j is nilpotent, which by the previous leading term argument implies that J 1 is too. Finally, consider v = β (and thus h = 1) and k = 2 in (5.16) to obtain By leading term arguments analogous to those used for J n , this implies that β n acts locally nilpotently for n ≥ 4. Consider the subspace V ⊂ W annihilated by β n+3 , γ n , n ≥ 1. Then V is a module over four commuting copies of the Weyl algebra respectively generated by the pairs (β 0 , γ 0 ), (β 1 , γ −1 ), (β 2 , γ −2 ), (β 3 , γ −3 ). Further, V is closed under the action of J n , n ≥ 1 and restricted to acting on V, the first few J n modes expand as We show that on any composition factor of V at least three of the four Weyl algebras have a generator acting nilpotently and that thus the induction of such a composition factor is an object in one of the categories σ i R, −3 ≤ i ≤ 0. Let C 0 ⊗ C 1 ⊗ C 2 ⊗ C 3 be isomorphic to a composition factor of V, where C i is a simple module over the Heisenberg algebra generated by the pair (β i , γ −i ). Since J 1 , J 2 , J 3 act locally nilpotently on V they must also do so on C 0 ⊗ C 1 ⊗ C 2 ⊗ C 3 using the expansions (5.22). If we assume that neither β 3 nor γ 0 act locally nilpotently on C 3 and C 0 , respectively, that is there exist c 3 ∈ C 3 and c 0 ∈ C 0 such that U(β 3 )c 3 and U(γ 0 )c 0 are both infinite dimensional, and choose c 1 , c 2 , to be non-zero vectors in C 1 and C 2 , respectively. Then U(J 3 )(c 0 ⊗ c 1 ⊗ c 2 ⊗ c 3 ) will be infinite dimensional contradicting the local nilpotence of J 3 . So assume β 3 acts locally nilpotently but γ 0 does not, and let c 3 ∈ C 3 be annihilated by β 3 and c 0 , c 1 , c 2 be non-zero vectors in C 0 ,C 2 ,C 3 , respectively. On this vector J 2 evaluates to By the same reasoning as before, unless either β 2 or γ 0 act nilpotently, we have a contradiction to the nilpotence of J 2 , so β 2 must act nilpotently on c 2 . Repeating this argument for J 1 and assuming β 2 c 2 = 0 we have a contradiction to the nilpotence of J 1 unless β 1 acts nilpotently. The composition factor isomorphic to C 0 ⊗ C 1 ⊗ C 2 ⊗ C 3 thus induces to an object in R. Repeating the previous arguments, assuming that γ 0 acts locally nilpotently but β 3 does not, implies that γ −1 and γ −2 must act locally nilpotently to avoid contradictions to the local nilpotence of J 1 , J 2 , J 3 . Such a composition factor would induce to a module in σ −3 R. Finally assume both β 3 and γ 0 act locally nilpotently, then analogous arguments to those used above applied to the action of J 1 imply that at least one of β 2 or γ −1 act locally nilpotently. Such a composition factor would induce to an object in σ −2 R or σ −1 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 classification of indecomposable modules in Theorem 4.1, a finite length indecomposable module with composition factors only in σ i R, −3 ≤ i ≤ 0 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 same nilpotence conditions as above, however for β one needs to reconsider (5.16) with k = 1 to conclude that β n , n ≥ 2 is nilpotent. 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 implies that category F is an additive braided tensor category. Additionally, since category F is abelian, it is a braided tensor category.
(1) All modules in category F are strongly graded by ghost weight j ∈ R. Further, by Lemma 5. (6) Since the P(w)-tensor product is right exact, by [21,Part IV,Proposition 4.26], and since category F has sufficiently many projectives, that is, every module can be realised as a quotient of a direct sum of indecomposable projectives, we can without loss of generality assume M 1 and M 2 are indecomposable projective modules, as Condition (6) holding for projective modules implies that it also holds for their quotients. Further, due to the compatibility of fusion with spectral flow, we can pick M 1 and M 2 to be isomorphic to W λ or P. Let ν ∈ COMP(M 1 , M 2 ) be doubly homogenous and assume that the module M ν generated by ν is lower bounded. By assumption, the functional ν therefore satisfies all the properties of P(w)-local grading restriction except for the finite dimensionality of the doubly homogeneous spaces of 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 admits an infinite direct sum. Then the partial sums define an ascending filtration whose union is the entire module. Hence after some finite number of steps all generators must appear within this filtration, but if this finite sum contains all generators, it must be equal to the entire module and hence all later direct summands must be zero. Remark. Note that the above proof did not make any use of M ν being lower bounded to conclude that M ν ∈ F and that membership of category F implies lower boundedness.
We will prove the above lemma by showing that W λ W −λ has exactly one submodule isomorphic to P. This requires finding linear functionals which satisfy P(w) compatibility. This is very difficult to do in practice, since (5.11) needs to be checked for every vector v ∈ V. Fortunately there is a result by Zhang which cuts this down to generators. Zhang originally formulated the theorem below for a related type of fusion product called the Q(z)-tensor product, so we have translated his result to the P(w)-tensor product, which we use here. We further prepare some helpful identities.
Proof. These identities follow by evaluating (5.12) for the fields β, γ, J and T . Let ψ ∈ Hom(W λ ⊗ W −λ , C) satisfy βt k+n (t −1 − w) n ψ = γt k+n (t −1 − w) n ψ = 0 for all m ≥ 1. Thus by Theorem 5.11, ψ satisfies the P(w)-compatibility property and β m ψ = γ m+1 ψ = 0, m ≥ 1. If in addition ψ is doubly homogeneous, then ψ lies in W λ W −λ . By assumption the left-hand sides of (5.27) and (5.28) vanish for k ≥ 1. These relations imply that the value of ψ on any vector in W λ ⊗ W −λ is determined by its value on tensor products of relaxed highest weight vectors, because negative modes on one factor can be traded for less negative modes on the other factor. For example, for k = 1, n = 0 in (5.27), we have the relation Let u ± j ∈ W ±λ , j ∈ ±λ be a choice of normalisation of relaxed highest weight vectors satisfying u ± j−1 = γ 0 u ± j . This implies β 0 u ± j = ± ju 1± j . Since the negative β and γ modes act freely on the simple projective modules W λ and W −λ , there are no relations in addition to those coming from βt k+n (t −1 − w) n ψ = γt k+n (t −1 − w) n ψ = 0 for all m ≥ 1 . Thus there is a linear isomorphism Clearly, K(W λ W −λ ) is a subspace of {ψ ∈ W λ W −λ : β n ψ = γ n+1 ψ = 0, n ≥ 1} and so we impose the remaining two relations, the vanishing of J 0 and J 1 , via (5.25). The vanishing of J 0 ψ implies Thus ψ vanishes on u j ⊗ u −i unless i = j − 1. The vanishing of J 1 ψ implies Thus ψ is completely characterised by its value on a two pairs of relaxed highest weight vectors, say u j ⊗ u 1− j and u j+1 ⊗ u − j . Therefore, the subspace K(W λ W −λ ) is two dimensional.
Next we show that that L 0 has a rank two Jordan block on it when acting on this space. Let ψ ∈ K(W λ W −λ ). If ψ 0, then there exist a, b ∈ C, not both zero, such that The evaluation of L 0 ψ on u j ⊗ u 1− j and u j+1 ⊗ u − j is then Therefore if a b (a choice which we can make as K(W λ W −λ ) is two dimensional), the vectors ψ and L 0 ψ are linearly independent and span K(W λ W −λ ), which also shows that L 0 has a rank two Jordan block.
Remark. In [16, Section 7] the above fusion product was computed using the NGK algorithm up to certain conjectured additional conditions. In light of the survey [36] explaining the equivalence of the HLZ double dual construction and the NGK algorithm, the authors thought it appropriate to supplement the NGK calculation of [16] with an HLZ double dual calculation here.
yield the identity maps 1 M and 1 M ∨ , respectively. Here w 1 , w 2 are distinct non-zero complex numbers satisfying |w 2 | > |w 1 | and |w 2 | > |w 2 − w 1 |; ⊠ w indicates the relative positioning of insertion points of fusion factors, that is, the right most factor will be inserted at 0, the middle factor at w 1 and the left most at w 2 ; Technically there exist distinct notions of left and right duals and the above properties are those for left duals. We prove below that M = σ ℓ W λ is left rigid. Right rigidity follows from left rigidity due to category F being braided. For M = σ ℓ W λ we take the tensor dual to be M ∨ = σ 1−ℓ W −λ and we will construct the evaluation and coevaluation morphisms using the first free field realisation (2.33) given in Proposition 2.12.(1). In particular, we have 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 Fock space weights. Thus the fusion product over V K of the modules corresponding to σ ℓ W λ and σ 1−ℓ W −λ is given by Therefore we have the V K -module map Y : F −λ(θ+ψ)−ℓψ ⊠ ff F λ(θ+ψ)+(ℓ−1)ψ → F −ψ given by the intertwining operator that maps the kets in the Fock space F λ(θ+ψ)+(ℓ−1)ψ to vertex operators, that is, operators of the form (2.30). Since V K -module maps are also G-module maps by restriction and since the fusion product of two modules over a vertex subalgebra is a quotient of the fusion product over the larger vertex algebra, Y also defines a G-module map F −λ(θ+ψ)−ℓψ ⊠ F λ(θ+ψ)+(ℓ−1)ψ → F −ψ W − 0 . Furthermore, the screening operator S 1 = V ψ (z)dz defines a G-module map S 1 : F −ψ → F 0 with the image being the bosonic 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 1 .
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. It is this copy of V which the coevaluation shall map to. Since V is the vector space underlying the vertex algebra G and any vertex algebra is generated from its vacuum vector, we characterise the coevaluation map by the image of the vacuum vector.
where the first arrow is the inclusion of V into F 0 W − 0 ⊂ P, S −1 1 denotes picking preimages of S 1 and j the unique representative of the coset λ satisfying 0 < j < 1. Note that the ambiguity of picking preimages of S 1 in the second arrow is undone by reapplying S 1 in the fourth arrow and hence the map is well-defined. This map maps to F 0 , which is a submodule 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 proportionality factors 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 − 1)ψ + ( j − ℓ)θ ∈ F λ(ψ+θ)+(ℓ−1)θ σ ℓ W λ . Following the sequence of maps in (5.36a) we get where 0,w 2 denotes a contour about 0 and w 2 but not w 1 , w 1 ,w 2 denotes a contour about w 1 and w 2 but not 0. The proportionality factor is obtained by pairing the above with the dual of the Fock space highest weight vector, which we denote by an empty bra |, and thus equal to the matrix element 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 radial ordering, 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 matrix elementĨ Since both matrix elements are equal, I(w 1 , w 2 ) =Ĩ(w 1 , w 2 ), rigidity follows by showing that they are non-zero.
We evaluate the four integrals appearing in I(w 1 , w 2 ). We simplify the first integral using the substitution z = w 1 x.
where the second equality follows by deforming the contour about 0 and 1 to a dumbbell or dog bone contour, whose end points vanish because the contributions from the end points are O(ε 1+ j ) and O(ε 1− j ) respectively, and 0 < j < 1; and the third equality is the integral representation of the hypergeometric function and B is the beta function. Similarly, For the integrals with contours about w 1 and w 2 we use the substitution z = w 2 − (w 2 − w 1 )x and then again obtain integral representations of the hypergeometric function.
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 1− j ) for 0 and 1 respectively. Making use of the hypergeometric and beta function identities the proportionality factor I(w 1 , w 2 ) simplifies to Since j Z, I(w 1 , w 2 ) can only vanish, if one of the hypergeometric factors does. We specialise the complex numbers w 1 , w 2 , such that w 2 = 2w 1 . Then, and the relationship between contiguous functions implies F 2 1 − j, j; 1; Thus I(w 1 , w 2 ) 0 and we can rescale the evaluation map by I(w 1 , w 2 ) −1 so that the sequences of maps (5.36) are equal to the identity maps on M and M ∨ . Thus σ ℓ W λ is rigid.

F
In this section we determine the decomposition of all fusion products in category F . A complete list of fusion products among representatives of each spectral flow orbit is collected in Theorem 6.1, while the proofs of these decomposition formulae have been split into the dedicated Subsections 6.1 and 6.2. To simplify some of the decomposition formulae we introduce dedicated notation for certain sums of spectral flows of the projective module P. Consider the polynomial of spectral flows and let 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), in category F among representatives for each spectral flow orbit. All other fusion products are determined from these through 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 For all λ, µ ∈ R/Z, λ, µ, λ + µ Z, For m, n ∈ Z, m ≥ n, such that the lengths of indecomposables below are positive, we have the following fusion product formulae.

Remark.
The fusion product formulae of Theorem 6.1 projected onto the Grothendieck group match the conjectured Verlinde formula of [16, Corollaries 7 and 10], thereby proving that category F satisfies the standard module formalism version of the Verlinde formula. It will be an interesting future problem to find a more conceptual and direct proof for the validity 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 simple projective modules.
Proposition 6.2. For λ, µ ∈ R/Z, λ, µ, λ + µ Z, we have Proof. Since W λ and W µ both lie in category R, we know, by Lemma 5.9, that the composition factors of the fusion product lie in categories R or σ −1 R. Further, since J(z) is a conformal weight 1 field, its corresponding weight, the ghost weight, adds under fusion. Therefore the only possible composition factors are W λ+µ and σ −1 W λ+µ . Since these composition factors are both projective and injective, they can only appear as direct summands and all that remains is to determine their multiplicity. In [20] Adamović and Pedić computed dimensions of spaces of intertwining operators for fusion products of the simple projective modules. In particular, [20, Corollary 6.1] states that if M is isomorphic to σ ℓ W λ+µ , ℓ = 0, −1. Thus the proposition follows.
Remark. To prove the above proposition directly without citing the literature, we could have used the two free field realisations in Section 2.4 to construct intertwining operators of the type appearing in equation (6.8), thereby showing that the dimension of the corresponding space of intertwining operators is at least 1. This was also done in [20]. An upper bound of 1 can then easily be determined by calculations involving either the HLZ double dual construction (similar to the calculations done in Lemma 5.10) or the NGK algorithm. Proposition 6.3. For λ ∈ R/Z, λ Z, we have 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 = −1, 0. Finally, since ghost weights add under fusion, the ghost weights of the fusion product must lie in Z. Thus the fusion product must be isomorphic to a direct sum of some number of copies of σ −1 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 projective covers 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 is therefore 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 . 6.2. Fusion products of reducible indecomposable modules. In this section we calculate the remaining fusion product formulae 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 calculating certain basic fusion products from which the remainder can be determined inductively.
Proof. Taking the short exact sequence (2.20a) for W + 0 = T 2 −1 and fusing it with W − 0 = B 2 −1 yields the short exact sequence Similarly, fusing the short exact sequence (2.20b) for If either of the above exact sequences splits there is a contradiction, because if σ −1 W + 0 and W + 0 are direct summands of W + 0 ⊠ W − 0 , (6.14) is not exact, and if W − 0 and σ −1 W − 0 are direct summands, (6.15) is not exact. Hence both sequences must be non-split. As can be read off from the tables in Proposition 4.
There is only one candidate for the middle coefficient of these exact sequences, namely σ −1 P. Thus the first fusion rule follows. The other two fusion products by are determined by fusing W ± 0 with the short exact sequences for W ± 0 . The extension groups corresponding to these fused exact sequences are zero-dimensional and hence the sequences split and the lemma follows.
We further prepare the following Ext group dimensions for later use. The corresponding extensions are given by T 2n+m+1 and B 2n+m respectively.
Proof. We start with the following presentation of T 2n+1 Applying the functor Hom −, B m 2n+1 yields The first coefficient vanishes due to T 2n+1 and B m 2n+1 having no common composition factors. The second coefficient can be shown to vanish using the projective cover formulae in Proposition 4.4 and reading off Hom group dimensions from the Loewy diagrams. For the third coefficient, the only composition factor common to both T 2n+2 and B m 2n+1 is V 2n+1 , which occurs as a quotient for T 2n+2 and a submodule for B m 2n+1 , so this gives rise to a one dimensional Hom group. The vanishing Euler characteristic then implies that dim Ext T 2n+1 , B m 2n+1 = 1 as expected. Furthermore, we can examine T 2n+m+1 to see that it has a B m 2n submodule which yields T 2n+1 when quotiented out, therefore this is the unique extension characterised by Ext T 2n+1 , B m 2n+1 .
We can follow the same procedure starting with the projective presentation of B 2n to obtain the following exact sequence By the same argument as above we can calculate the Hom groups,and vanishing Euler characteristic implies dimExt B 2n , B m 2n = 1. Similarly we see that B 2n+m provides an extension of B 2n by B m 2n 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 the following decomposition formulae.
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 2n+1 , We then take the fusion product with B 2 , Because P 2n is projective, the sequence splits and we have the recurrence relation Then, the first fusion product formula of the lemma follows by induction with B 1 = V as the base case.
We next consider the short exact sequence (4.9c) and fuse it with B 2 to obtain Since Ext B 2 2n+1 , B 2 = 0, by the tables in Proposition 4.3, this sequence splits and we obtain the second fusion product of the lemma.
For the final two fusion products, we perform the same exercises with different exact sequences. For the third and fourth fusion products we use (4.9d), with odd and even length respectively. Fusing with T 2 gives the short exact sequences In both cases, the sequences split because P 1 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 decomposition formulae below, for m ≥ n.
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 2n+1 and fuse with B 2m+1 to find Now comparing these exact sequences, and using the fact that P is projective, we find that the sequences cannot both split, as they would give different direct sums. For the first short exact sequence, we use Lemma 6.7, to find dim Ext B 2 , B 2m+2n−1 2 = 1, with the extension being given by B 2m+2n+1 so we can determine the fusion product formulae inductively to get We can deduce the remaining rules from short exact sequences that relate even and odd Bs. Firstly, we take the two short exact sequences (4.9d) and (4.1b), and fuse them with B 2m+1 to get Either of these exact sequences splitting would lead to a contradiction, hence both must be non-split. Further, by Lemma 6.7 we find dim Ext B 2 , B 2n 2 = dim Ext B 2n , B 2 2n = 1, with the corresponding non-split extension given by B 2n+2 . Therefore Finally we fuse (4.9c) with B 2n to find For m ≥ n, dim Ext B 2n 2m+1 , B 2n = 0, which follows because the composition factors are separated by at least two units of spectral flow and Ext(V n , V m ) = 0 for |n − m| > 1, the above sequence splits. In the case when m = n − 1, we have that Ext B 2n 2n−1 , V k = 0 for all the composition factors of B 2n , that is, (0 ≤ k ≤ 2n − 1). Hence dim Ext B 2n 2n−1 , B 2n = 0 and the above sequence again splits. Thus, Proof. We prove the left column of identities. The right column then follows from Corollary 6.5 and applying the * functor to each module. We start with sequences (4.1a) and (4.9d) for odd length B, and fuse them with T 2m+1 to find Specialising to n=1 we have Since P is projective, its spectral flows must appear as direct summands in the middle coefficient of the above exact sequences. Thus, Therefore the module A satisfies the exact sequences Because either of these sequences splitting would lead to a contradiction and the corresponding extension groups are one-dimensional, the sequences uniquely characterise the fusion product. Proceeding by induction, we obtain Next we take two short exact sequences (4.9g) and (4.9e), for T 2m and fuse them with B 2n+1 to get Again either of these sequences splitting would lead to a contradiction, and by Lemma 6.7, dim Ext T 2m−2n−1 2n , B 2n+1 2m−1 = 1, with the extension being given by T 2m 2n , so the second fusion rule follows. Finally, fusing (4.1d) with B 2n , we have In this section we give a proof of Theorem 5.7 by reviewing reasoning presented by Yang in [25] and showing that certain assumptions on the category of strongly graded modules (see [25,  Throughout this section let A ≤ B be abelian groups. Further, let V be an A-graded vertex algebra with a vertex subalgebra 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−1 regardless of the conformal weight of v ∈ V, that is, v n refers to the coefficient of z −n−1 rather than the one which shifts conformal weight by −n.
(1) We say that two B-graded logarithmic intertwining operators Y 1 , Y 2 of respective types W 2 , W 3 satisfy the convergence and extension property for products if for any a 1 , a 2 , ∈ B and any doubly homogeneous elements where N ∈ Z depends only on the intertwining operators Y 1 , Y 2 and a 1 + a 2 , such that as a formal power series the matrix element converges absolutely in the region |z 1 | > |z 2 | > 0 and may be analytically continued to the multivalued analytic function in the region |z 2 | > |z 1 − z 2 | > 0.
(2) We say that two B-graded logarithmic intertwining operators Y 1 , Y 2 of respective types W 0 W 4 , W 3 , W 4 W 1 , W 2 satisfy the convergence and extension property for iterates if for any a 2 , a 3 , ∈ B and any doubly homogeneous elements where N ∈ Z depends only on the intertwining operators Y 1 , Y 2 and a 2 + a 3 , such that as a formal power series the matrix element converges absolutely in the region |z 2 | > |z 1 − z 2 | > 0 and may be analytically continued to the multivalued analytic in the region |z 1 | > |z 2 | > 0.

Consider the Noetherian ring
. Then for any quadruple of B-graded V-modules W 0 , W 1 , W 2 , W 3 , and any triple (a 1 , a 2 , a 3 ) ∈ B 3 , we define the R-module where all the tensor product symbols denote complex tensor products. We will generally omit the tensor product symbol separating R from the V-modules. The motivation for considering this module is that for any B-graded module W 4 and any pair of grading compatible logarithmic intertwining operators Y 1 , Y 2 of respective types , where h is the combined conformal weight of w ′ 0 , w 1 , w 2 , w 3 and C({x}) is the space of all power series in x with bounded below real exponents (the modules W i , i = 0, 1, 2, 3 will always have real conformal weights below), defined by is the map expanding elements of R such that the powers of z 2 are bounded below. This in turn justifies considering the submodule where the generators are preimages of the relations coming from residues of the Jacobi identity for intertwining operators and where v * k : a 2 ,a 3 ) lies in the kernel of φ Y 1 ,Y 2 for any choice of intertwining operators Y 1 , Y 2 of the correct types.
Proposition A.2. Let the V-modules W i , i = 0, 1, 2, 3 be discretely strongly B-graded and B-graded C 1 -cofinite as Vmodules, then for any a 1 , a 2 , a 3 ∈ B there exists M ∈ Z such that for any r ∈ R F r (T (a 1 ,a 2 ,a 3 ) ) ⊂ F r (J (a 1 ,a 2 ,a 3 ) ) + F M (T (a 1 ,a 2 ,a 3 ) ) and T (a 1 ,a 2 ,a 3 ) ⊂ J (a 1 ,a 2 ,a 3 ) + F M (T (a 1 ,a 2 ,a 3 ) ). (A.13) Proof. By assumption the modules W i , i = 0, 1, 2, 3 are B-graded C 1 -cofinite as V-modules, that is, the spaces 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 1 ,a 2 ,a 3 ) ) defining a filtration. Next assume that F r (T (a 1 ,a 2 ,a 3 ) ) ⊂ F r (J (a 1 ,a 2 ,a 3 ) ) + F M (T (a 1 ,a 2 ,a 3 ) ) is true for all r < s ∈ R for some s > M. We will show that any element of the homogeneous space T can be written as a sum of elements in F s (J (a 1 ,a 2 ,a 3 ) ) and F M (T (a 1 ,a 2 ,a 3 ) ). 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 ′ 0 ⊗ v −1 w 1 ⊗ w 2 ⊗ w 3 ∈ T , where w ′ 0 ∈ W ′ 0 (a 1 +a 2 +a 3 ) , w i ∈ W (a i ) i , i = 1, 2, 3, v ∈ V [h] , h > 0. By computing the degrees of the summands making up A(v, w ′ 0 , w 1 , w 2 , w 3 ) in (A.10) we see that the three sums over k all lie in F s−1 (T (a 1 ,a 2 ,a 3 ) ) ⊂ F s−1 (J (a 1 ,a 2 ,a 3 ) ) + F M (T (a 1 ,a 2 ,a 3 ) ) and that A(v, w ′ 0 , w 1 , w 2 , w 3 ) ∈ F s (J (a 1 ,a 2 ,a 3 ) ). Further, Thus w ′ 0 ⊗ v −1 w 1 ⊗ w 2 ⊗ w 3 lies in the sum F s (J (a 1 ,a 2 ,a 3 ) ) + F M (T (a 1 ,a 2 ,a 3 ) ) and the first inclusion of the proposition follows. The second inclusion follows from F r (T (a 1 ,a 2 ,a 3 ) ) and F r (J (a 1 ,a 2 ,a 3 ) ) defining filtrations.
Proof. Let a 1 , a 2 , a 3 be the respective B-grades of w 1 , w 2 , w 3 , then we can assume that the B-grade of w ′ 0 is a 1 + a 2 + a 3 , because otherwise the matrix element vanishes and the theorem follows trivially. Recall the map φ Y 1 ,Y 2 : T (a 1 ,a 2 ,a 3 ) → z h 1 C({z 2 /z 1 }) z ±1 1 , z ±2 2 , defined by the formula (A.8). Since J (a 1 ,a 2 ,a 3 ) lies in the kernel of φ Y 1 ,Y 2 , we have an induced map φ Y 1 ,Y 2 : T (a 1 ,a 2 ,a 3 ) /J (a 1 ,a 2 ,a 3 ) → z h 1 C({z 2 /z 1 }) z ±1 1 , z ±2 2 . (A.23) The theorem then follows by applying φ Y 1 ,Y 2 to the relations (A.19) of Corollary A.3.(2), using the L −1 derivative property of intertwining operators and expanding in the region |z 1 | > |z 2 | > 0. We need to consider new filtrations in addition to those considered previously. Let R = C[z ±1 1 , z ±2 2 ], then R n = (z 1 − z 2 ) −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 = n∈Z R n and R n · R m ⊂ R m+n . The R-module T (a 1 ,a 2 ,a 3 ) can then also be equipped with a compatible filtration R r (T (a 1 ,a 2 ,a 3 ) ) = ∏ n+h 0 +h 1 +h 2 +h 3 ≤r h i ∈R R n ⊗ W ′ in the sense that R r (T (a 1 ,a 2 ,a 3 ) ) ⊂ R s (T (a 1 ,a 2 ,a 3 ) ), if r ≤ s, T (a 1 ,a 2 ,a 3 ) = r∈R R r (T (a 1 ,a 2 ,a 3 ) ) and R n ·R r (T (a 1 ,a 2 ,a 3 ) ) ⊂ R n+r (T (a 1 ,a 2 ,a 3 ) ).
For any a 1 , a 2 , a 3 ∈ B and any doubly homogeneous vectors w ′ 0 ∈ W ′ 0 (a 1 +a 2 +a 3 ) , w i ∈ (W i ) (a i ) [h i ] , let h = ∑ i h i , let h be the smallest non-negative representative of the coset h + Z and let m J ∈ R h (J (a 1 ,a 2 ,a 3 ) ), m T ∈ F M (T (a 1 ,a 2 ,a 3 ) ) be vectors satisfying Then there exists S ∈ R such that h + S ∈ Z ≥0 and (z 1 − z 2 ) h+S m T ∈ U (a 1 ,a 2 ,a 3 ) .
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 ≥0 and such that for any r ≤ −S , T = 0. Such an S must exist, since the conformal weights of T (a 1 ,a 2 ,a 3 ) are bounded below by assumption. By definition, R r (T (a 1 ,a 2 ,a 3 ) ) is spanned by elements of the form (z 1 − z 2 ) −n f (z 1 , z 2 ) w 0 ⊗ w 1 ⊗ w 2 ⊗ w 3 , where f ∈ R and n + ∑ i wt w i ≤ r. The number S was therefore chosen such that (z 1 − z 2 ) r+S R r (T (a 1 ,a 2 ,a 3 ) ) ⊂ U (a 1 ,a 2 ,a 3 ) whenever r + S ∈ Z. Now, by assumption, The right-hand side of this equality lies in R h (T (a 1 ,a 2 ,a 3 ) ) by construction and therefore so does the left-hand side. Hence (z 1 − z 2 ) h+S m T ∈ U (a 1 ,a 2 ,a 3 ) .