Fusion and positivity in chiral conformal field theory

Conformal nets and vertex operator algebras are axiomatizations of two-dimensional chiral conformal field theories, both of which have had a significant impact on related mathematical areas of study. The two settings are expected to be equivalent under suitable hypotheses, but in practice such an equivalence has proven elusive, especially with regard to fusion products. This article develops a framework for the systematic comparison of fusion products between the contexts of VOAs and conformal nets. This framework is based on the geometric technique of 'bounded localized vertex operators,' which realizes algebras of observables via insertion operators localized in partially thin Riemann surfaces. We apply our framework to solve old problems about conformal nets and subfactors (e.g. rationality of all WZW conformal nets), obtain new results about VOAs (e.g. unitarity of representation categories), and give short proofs of many old results. We also consider a general class of positivity phenomena for VOAs, and use this to outline a program for identifying unitary tensor product theories of VOAs and conformal nets even for badly-behaved models.


Introduction
Two-dimensional chiral conformal field theories (CFTs) have had significant impact on several areas of mathematics. At present there is no single definition of a CFT which captures all of the key models (as examples) and phenomena (as theorems). Instead, there are three approaches to the study of CFTs, each of which has enjoyed partial success in this endeavor. The most developed definitions are conformal nets, which axiomatize algebras of observables following the Haag-Kastler approach [HK64,FRS92], and vertex operator algebras, which axiomatize the fields in the spirit of the Wightman axioms [Bor86,FLM88]. The third framework is Segal CFTs, which are functorial field theories (similar to the better known Atiyah-Segal axioms of topological quantum field theory) [Seg04,MS89]. It is an important undertaking to develop the mathematical relationship between these frameworks to the point where important theorems, when proven in one context, can thereby be deduced in the others, with the goal of eventually obtaining a single unified framework for the mathematical study of CFTs.
One of the most important (and most difficult) areas of study in CFT is the theory of tensor products ('fusion products') of representations. Tensor products in CFT are connected to topics like modular tensor categories, fusion and braiding of primary fields, and subfactors. The study of tensor products has led to landmark results such as Wassermann's analysis of loop group subfactors [Was98] and Huang's proof of modularity via the Verlinde conjecture [Hua08b,Hua08a]. Many important questions regarding tensor products have been solved in one of the axiomatizations but are challenging open problems in the other.
In this article we develop a framework for the systematic comparison of tensor product theories in VOAs and conformal nets, using the idea of Segal CFT (as extended by Henriques [Hen14]) as a bridge. We also provide many applications of this framework. For example, we use VOAs to give a short proof of the rationality of conformal nets associated to WZW models, which implies that all of the subfactors associated to these models have finite index. The problem of showing that these subfactors have finite index has been open for several decades (see e.g. [GF93]). We also prove rationality of conformal nets assigned to W -algebras of type ADE. As another application, we use conformal nets to demonstrate the unitarity of representation categories for all WZW VOAs (building upon [Gui19b,Gui19c,Gui18b]) as well as many W -algebra examples. We are also able to recover existing results related to tensor product theory (e.g. many results from [Was98,TL97,Lok94,Gui18b]) in a uniform and straightforward manner. The framework directly links existing lines of research in conformal nets and VOAs, allowing future progress in one area of study to benefit the other. Our approach is sufficiently general that the methods used should apply to any model (without any additional assumption, such as rationality), and we describe how it could pave the way for a uniform understanding of the relationship between VOAs and conformal nets.

Bounded localized vertex operators
We now give a brief overview of the geometric method for relating VOAs and conformal nets developed in [Ten19,Ten18b]. For a more detailed overview, see Section 2.3 or [Ten18b,§1.2]. Given a unitary VOA and numbers R > 1 > r, the operator R −L 0 Y (v, z)r L 0 inserts the state v at the point z inside the annulus {R > |w| > r}.
For these values of z, we expect R −L 0 Y (v, z)r L 0 to define a continuous operator on the Hilbert space completion of V . While this construction encodes the insertion operators of a VOA in an analytic language which is closer to that of conformal nets, the resulting operators are not local in the sense of algebraic conformal field theory. A breakthrough idea of Henriques was to consider insertion operators localized in 'degenerate' or 'partially thin' annuli (see [Hen14]). Specifically, we replace the operators r L 0 and R −L 0 with operators A and B corresponding to partially thin annuli supported in an interval I of the unit circle. We say that a unitary VOA V has bounded localized vertex operators if the von Neumann algebra A V (I) generated by insertions BY (v, z)A supported in I are bounded and local in the sense of ACFT, in which case A V (I) is a conformal net.
In [Ten19,Ten18b] it was shown that many VOAs have bounded localized vertex operators, including all WZW models, their tensor products, subtheories, and so on. A key step was to interpret the degenerate insertion operators as arising from Segal CFT, thereby simultaneously linking the three frameworks. The geometric part of this argument was developed in [Ten17], and the analytic part is explored further in [PT18].
If M is a unitary V -module, then the A V -representation π M associated to M (if it exists) is the one which takes insertion operators to insertion operators. That is, π M (BY (v, z)A) = BY M (a, z)A, where we have allowed the degenerate annuli B and A to act on the Hilbert space completion of M via the Virasoro action (see Section 2.3). The correspondence M ↔ π M was explored in [Ten18b].

Translating fusion between VOAs and conformal nets
In this third article on geometric realization of CFTs, we will use this correspondence to relate the tensor product of representations for VOAs V and conformal nets A V . Our approach is informed by the geometry of Segal CFT, and unlike other approaches (e.g. [CKLW18,Gui19b,Gui19c]) deals exclusively with continuous operators The first challenge that one encounters is that the definitions of tensor product for conformal nets and VOAs are very different in nature. For conformal nets, the tensor product of representations π and λ is given by composition of DHR endomorphisms [DHR71,FRS92], or equivalently the Connes-Sauvageot relative tensor product, known as 'Connes fusion' in this context [Was98] (see also Section 2.2). The construction proceeds by putting a new inner product on a subspace of H π ⊗ C H λ , and building a representation π λ on its Hilbert space completion. The tensor product of two representations always exists, and is given explicitly, although in practice it is quite difficult to identify the resulting representation (i.e. as a direct sum of known representations).
The VOA tensor product of modules, on the other hand, is generally defined by a universal property as opposed to an explicit construction. The universal property makes explicit reference to a choice of category of modules, and the correct choices of modules (and universal property) are not clear for poorly behaved unitary VOAs (such as Virasoro VOAs with central charge c ≥ 1).
If one accepts the existence of a correspondence between conformal net representations and VOA modules then the Connes fusion representation π M π N of A V should be of the form π K for some V -module K. Moreover, this VOA module K should be given by an explicit construction, since the fusion product of conformal net representations π M π N is.
The translation of Connes' fusion into the language of VOAs appears in Section 3, where we introduce the notion of a transport module M t N for a pair of unitary modules M and N . We can give two versions of the definition of M t N : the first via a characterizing property, and the second as an explicit construction. In order to describe the property which characterizes M t N , we introduce a sesquilinear form on M ⊗N called the transport form: [ M is a certain canonical intertwining operator. A key feature is that the transport form is given explicitly in terms of VOA data, and the specific formula can be obtained by translating the Connes' fusion inner product formula for conformal nets into the language of VOAs. A transport module is then a unitary module M t N with an intertwining operator Y t such that Y t (a 1 , z)b 1 , Y t (a 2 , w)b 2 M tN = [a 1 ⊗ b 1 , a 2 ⊗ b 2 ] z,w .
A transport module, if it exists, is given by an explicit construction: the transport form [a 1 ⊗ b 1 , a 2 ⊗ b 2 ] z,z is positive semidefinite, and there is a canonical way to put a VOA module structure on a dense subspace of the Hilbert space completion (see Section 3.3). Conversely, if this construction produces a VOA module, then it is a transport module.
case of WZW models. The positivity of transport forms also feature in Wassermann's work, and was studied by Gui [Gui19b,Gui19c] for strongly rational VOAs. In Section 3 we develop basic theory of transport modules in a way which does not depend on any special properties of the VOA, and in Section 4.1 we show that the framework of bounded localized vertex operators is a powerful tool for studying these conjectures (see Proposition 4.2, Theorem 4.4, and Corollary 4.5 for precise statements).
Theorem C. Let V be a simple unitary VOA with bounded localized vertex operators, let M and N be unitary V -modules, and assume that π N exists. Then the transport form for M and N is positive semidefinite.
We can apply Theorem C to show that the transport form is positive (verifying Conjecture B) for many modules for badly non-rational VOAs, such as irreducible Virasoro modules L(c, h) with c ≥ 1. We describe in Section 3.3 the role that positivity of the transport form plays in a potential unitary tensor product theory for such VOAs.
The strongest consequences of Theorem C, however, are for strongly rational VOAs, like the WZW models. Theorem C implies that for these VOAs we can construct a unitary tensor product M N inside the Hilbert space completion of M ⊗ N with respect to the transport form, analogous to the Connes' fusion construction. We can then combine Theorem C with [Gui19c,Thm. 7.9] to see that the mere existence of the modules π N implies the unitarity of categories of VOA representations (Corollary 4.6). As a consequence, we can prove unitarity of representation categories for the remaining open cases of WZW models, namely type E and F, to obtain the following (Theorem 5.5).
Theorem D. Let V be a VOA which is either • a WZW model V (g, k) for g a finite-dimensional simple complex Lie algebra and k a positive integer • a W -algebra W (g, ) in the discrete series, for g of type A or E Then for every pair of simple unitary V -modules M and N , there is a unitary structure on (M N, Y ) making it into a transport module. Hence Gui's construction [Gui19b,Gui19c] makes the category of unitary V -modules into a unitary modular tensor category.
A key feature of our approach is that we are able to obtain positivity and unitarity results for VOAs simply from the existence of conformal net representations π M , without requiring any hard analysis on intertwining operators. In previous approaches, a modelby-model analysis of intertwining operators was often necessary, and this analysis often relied on model specific features of these operators. In contrast, the treatment of examples in Section 5 does not require any technical details, and the methods easily apply to other models as well.

Geometric analysis of Connes' fusion
Transport modules are a translation of Connes' fusion from the language of conformal nets to VOAs. We should therefore expect to be able to relate the two constructions, and the framework of bounded localized vertex operators is very well suited to that task. We show in Section 4.2 that if π M and π N exist, and there is a transport module M t N , then there is a natural and explicit unitary U : H M H N → H M tN . Here, H M H N is the Hilbert space of the conformal net fusion product, and H M tN is the Hilbert space of the VOA transport module.
In the present article, we do not show in full generality that U is an isomorphism between π M π N and π M tN (although such a result seems within reach of current techniques; see Section 5.4). Instead, we identify a maximal submodule M loc N ⊂ M t N such that π M loc N is embedded in π M π N via U , as well as hypotheses which guarantee that this embedding is a unitary equivalence. The following is Theorem 4.16.
Theorem E. Let V be a simple unitary VOA with bounded localized vertex operators, let M and N be unitary V -modules such that π M and π N exist, and suppose that the creation operator Y + M ∈ I M M V is localized. If M loc N is the maximal local submodule of a transport module, then the natural unitary U : H M H N → H M tN realizes π M loc N as a subrepresentation of π M π N . In particular, if M t N = M loc N , then π M π N ∼ = π M tN .
Recalling that when V is strongly rational a transport module is isomorphic to the tensor product, we see that Theorem E provides a means for identifying fusion rules between VOAs and conformal nets. The somewhat technical hypothesis regarding Y + M and M loc N could appear somewhat ominous, as they involve analytic properties of intertwining operators. However, in [Ten18b, §5.4] we provided relatively robust hypotheses for establishing locality of intertwining operators in the necessary sense. In particular, most of the explicit fusion rule calculations done for e.g. type A WZW in [Was98] or type G in [Gui18b] can be obtained systematically from Theorem E. The theorem also provides information about fusion rules for many other conformal nets coming from VOAs, such as type E WZW, W -algebras, and others. At present, it would require a case-by-case analysis to see if the relevant inclusions (e.g. the diagonal inclusion E 8,k+1 ⊂ E 8,k ⊗ E 8,1 ) produce enough intertwining operators to compute the full fusion rules. We should point out that only VOA branching information is required from these inclusions, as the analytic technicalities are resolved by the present article. We do not attempt to do such calculations here, as we believe it is in reach of current techniques to develop a more model independent approach (see Section 5.4).
The techniques used to prove Theorem E should be thought of as an adaptation of Wassermann's approach to a general VOA setting. One difference with Wassermann's work is that in [Was98] the fusion of primary fields was computed explicitly via the KZ equation. In the present setting, the fusion of fields is a part of the tensor product theory of VOAs, which our construction takes as input. We should point out that Connes fusion and transport modules, as presented here, have the potential to provide an alternate approach to unitary tensor products of VOA modules via conformal nets (again, we refer to Section 5.4 for a brief discussion).
As an application and demonstration of our framework for comparing VOA and conformal net tensor products, we use it to show finiteness of the index of subfactors and rationality of conformal nets. A representation π of a conformal net is said to have finite index if the associated Jones-Wassermann subfactor π I (A(I)) ⊂ π I (A(I )) has finite index. A conformal net is said to be rational if it has finitely many irreducible representations, each with finite index. Over the past several decades, the problem of showing specific examples (e.g. WZW models) are rational has proven to be an extremely difficult one, and arguments have relied on specific features of the model under consideration. However, Theorem E is perfectly suited to studying finiteness of the index and rationality of nets, as this information can be obtained from incomplete information about fusion rules (by work of Longo [Lon90]). In Section 4.4 we apply this idea to develop powerful tools for establishing these properties.
Theorem F. Let V be simple unitary VOA with bounded localized vertex operators, let W be a unitary (not necessarily conformal) subalgebra which is regular, and let M be a W -submodule of V . Then π M has finite index.
Corollary G. Let V be a simple unitary VOA with bounded localized vertex operators and let W be a unitary conformal subalgebra which is regular. Then the inclusion A W ⊂ A V has finite index, so that A W is rational if and only if A V is.
The equivalence of rationality of A W and A V is due to Longo [Lon03]. In Section 5, we apply these results to give a short proof of the rationality of WZW conformal nets as well as conformal nets corresponding to W -algebras of type ADE (Theorem 5.1). This was previously known for WZW models of type A, C, and G at all levels [Was98,Xu00,Gui18a], type D at odd levels [TL97], and W -algebras of type A 1 [KL04], as well as a few other isolated cases. The proof we give covers these cases as well.
Theorem H. Let g be a finite-dimensional complex simple Lie algebra, let k be a positive integer, and let V = V (g, k) be the corresponding affine VOA. Then V has bounded localized vertex operators and A V is rational. The same holds if V = W (g, ) is a W -algebra in the discrete series and g is of type ADE.
The proof of Theorem H is quite flexible, and can be adapted to other models if desired. To obtain rationality of WZW models and W -algebras when g is simply laced, we use important results of Arakawa, Creutzig, and Linshaw [Ara15a, Ara15b, ACL19]. Looking forward, future VOA research will immediately provide additional information about the corresponding conformal nets via Theorems E and F. The framework presented here allows for passing information back and forth between conformal nets and VOAs with a minimum of technical overhead, and we hope that it will provide new opportunities and inspiration for collaboration between VOA and conformal net specialists.

Structure of the article
In Sections 2.1 and 2.2 we give standard background information on unitary VOAs and conformal nets, respectively. Section 2.3 gives an overview of bounded localized vertex operators, adapted from [Ten18b]. Section 3 introduces transport modules from a purely VOA-theoretic perspective and establishes basic results relating transport, positivity, and tensor products. Section 4 connects transport modules to Connes fusion of conformal net representations. Our main theorems are proven in this section, exploiting this connection between VOAs and conformal nets. Section 5 applies our main theorems to families of examples, primarily WZW models and W -algebras, and also describes opportunities for future work to unify the tensor product theory of VOAs and conformal nets.

Unitary VOAs and their representations
We assume that the reader has some familiarity with unitary vertex operator algebras (VOAs) and their representation theory. For more detailed background on these subjects, we suggest [Ten18b,Gui19b,CKLW18,DL14]. For an overview of tensor product theory for VOAs, see e.g. [CKM17] for a concise treatment (or alternatively [HL13] and related articles). For completeness and convenience of the reader, we will now provide the relevant definitions and more specific references.

Unitary VOAs
Definition 2.1. A vertex operator algebra is given by: • a vector space V • vectors Ω, ν ∈ V called the vacuum vector and the conformal vector, respectively.
is the vector space of formal series with coefficients in End(V ).
This data must satisfy: • For every a ∈ V , we have a (n) Ω = 0 for n ≥ 0 and a (−1) Ω = a.
• For every a, b ∈ V and every m, k, n ∈ Z, we have the Borcherds (or Jacobi) identity: • If we write Y (ν, x) = n∈Z L n x −n−2 , then the L n give a representation of the Virasoro algebra. That is, for a number c ∈ C, called the central charge.
• If we write V α = ker(L 0 − α1 V ), then we have a decomposition of V as an algebraic Note that the eigenvalues of L 0 are non-negative by assumption. If a ∈ V α , then we say that a is homogeneous, with conformal dimension α.
The special cases of the Borcherds identity with n = 0 is called the Borcherds commutator formula, and it says There are natural notions of subalgebra, ideal, and homomorphism for VOAs, and a VOA with no ideals besides {0} or V is called simple (see [FHL93] or [CKLW18]).
Definition 2.2. A unitary vertex operator algebra is a VOA V equipped with an inner product and an antilinear automorphism θ satisfying:

Ω, Ω = 1
We write H V for the Hilbert space completion of V , consisting of vectors n∈Z ≥0 v n with v n ∈ V n and v n 2 < ∞.
A unitary VOA is simple if and only if dim V 0 = 1, which means that simple unitary VOAs are of "CFT type" [CKLW18,Prop. 5.3]. All VOAs considered in this article will be simple and unitary. Some ways to construct new simple unitary VOAs from old ones include tensor products, fixed points under unitary automorphisms, or more generally unitary subalgebras and their cosets (see [CKLW18,§5] or [Ten19, §2.2]).

Unitary modules
Definition 2.3. Let V be a vertex operator algebra. A V -module is given by a vector space M along with a state-field correspondence Y M : which is required to be linear, and satisfy the following additional properties.
• For every a ∈ V and b ∈ M , we have a M (n) b = 0 for n sufficiently large. • For every homogeneous a, b ∈ V and m, k, n ∈ Z, the Borcherds(/Jacobi) identity holds: Our definition of module is sometimes called a 'strong' or 'ordinary' module, in light of the requirement that M be graded by finite-dimensional L 0 -eigenspaces. We restrict to this class of modules for convenience, although many of the ideas of this article apply in a broader context.
There are obvious notions of submodules and direct sums of V -modules, as well as V -module homomorphisms. If M has no proper, non-trivial submodules then it is called a simple module.
We will be interested in unitary modules over unitary VOAs, which first appeared in [DL14].
for all a ∈ V and b, d ∈ M . We call M a unitary module if it is equipped with an invariant inner product and only countably many of the weight spaces M α are non-zero.
A unitary module M has a Hilbert space completion H M , and the requirement that only countably many weight spaces are non-zero is equivalent to requiring that H M be separable. We impose this restriction to avoid considering non-separable Hilbert spaces.
Unitary modules provide positive energy representations of the Virasoro algebra, so M α is non-zero only when α ≥ 0 (see [DL14,Lem. 2.5] where we have denoted by b → b the natural conjugate linear map M → M (as well as its extension to the algebraic completion of M ). The reader is cautioned that M does not refer to the algebraic completion, as that notion will make only a brief appearance in this article. An invariant inner product on M can be understood as an isomorphism between M and the contragredient module M (see [Ten18b,Lem. 2.18]). Definition 2.5. Let V be a VOA, and let M, N and K be V -modules. An intertwining

Intertwining operators
which satisfies the following properties.
1. For every a ∈ M , b ∈ N and k ∈ C, we have a Y (k+n) b = 0 for all sufficiently large n ∈ Z.
2. For every a ∈ M , Y satisfies the L −1 -derivative property: 3. For every homogeneous a ∈ V , b ∈ M , every m, n ∈ Z, and every k ∈ C, the Borcherds(/Jacobi) identity holds: We denote by I K M N the vector space of all intertwining operators of the indicated type.
The module operators Y M are intertwining operators of type M V M , and dim I M V M = 1 precisely when M is simple.
Specializing the Borcherds identity to n = 0 produces the Borcherds commutator formula Thus if the conformal dimensions of all vectors in M , N , and K are real (in particular, if they are unitary modules), then a Y (n) = 0 unless n ∈ R. In the following we assume that this is the case.
Of course any intertwining operator can be made dominant by replacing K with a submodule.
If M , N , and K are unitary modules and Y ∈ I K M N , then there is a unique intertwining operator Y † ∈ I N M K such that for all a ∈ N , b ∈ M , and d ∈ K The creation operator is obtained from Y M by the braiding [FHL93, §5.4], and does not require unitarity. The creation and annihilation operators first appeared in the context of unitary VOAs in [Gui19b,§1]. Observe that creation and annihilation operators are compatible with direct sums of modules.
In this article we will be interested in analytic properties for intertwining operators, and the most fundamental thing we can ask is that an intertwining operator take values in the Hilbert space completion.
To interpret the condition Y(a, z)b ∈ H K , recall that in general Y(a, z)b lies in the algebraic completion n∈R ≥0 K n . Since the K n are pairwise orthogonal, we may realize H K as the subspace of the algebraic completion consisting of vectors whose components have square-summable norms. It follows from basic properties of VOAs that Y M ∈ I Hilb for any unitary V -module M (as in [Ten19, Prop. 2.15]).
It is almost, but not quite, obvious that if Y ∈ I Hilb K M N then for a ∈ M and b ∈ N the expression Y(a, z)b defines a multi-valued holomorphic function from the punctured unit disk into H K . For completeness we include the details. Proof. By standard results [TL86, Ch. 5] it suffices to prove that Y(a, z)b, ξ is holomorphic for every ξ ∈ H K . 1 It also suffices to establish the result when a and b are homogeneous. By definition, since Y ∈ I Hilb we have and thus n∈R a Y (n) bz −n−1 converges in H K to Y(a, z)b. It follows that for every z in the punctured unit disk we have While this series is not a power series, one can apply the usual arguments to see that since it converges for every z in the punctured unit disk, it must in fact converge absolutely and locally uniformly. Thus Y(a, z)b, ξ is locally given by a single-valued holomorphic function.

Modes of intertwining operators as unbounded operators
We briefly present the basic notions of unbounded operators on Hilbert spaces. More detail can be found in any standard functional analysis textbook (e.g. [Ped89]). An unbounded operator on a Hilbert space H is a pair (x, D) where D ⊆ H is a subspace of H (generally required to be dense), and x : D → H is a linear map. The space D is called the domain of x. An unbounded operator is called closed if its graph Γ := {(ξ, xξ) : ξ ∈ D} is a closed subspace of H ⊕ H, and it is called closable if the closure Γ is the graph of a (necessarily closed) operator, which is called the closure of x. A dense subspace D ⊂ D is called a core for a closed operator x if the closure of x| D is equal to x.
We then use the Riesz representation theorem to characterize x * ξ by x * ξ, η = ξ, xη for all ξ ∈ D * and η ∈ D. An operator (D, x) is closable if and only if D * is dense. It was observed in [CKLW18,§6] that if V is a unitary VOA, then if we regard v (n) as an unbounded operator with domain V , then V is also contained in the domain of v (n) * . Thus the modes v (n) are closable operators. The same argument shows that the modes of a unitary module v M (n) are closable. The reader is cautioned that subtleties regarding domains of unbounded operators are somewhat notorious. It will be an extremely important fact for us (Lemma 3.6) that if Y ∈ I Hilb K M N is dominant then span{Y(a, z)b : a ∈ M, b ∈ N } is a core for the closure of v K (n) whenever 0 < |z| < 1.   The data is required to satisfy:

If
4. There is a unique (up to scalar) unit vector Ω ∈ H, called the vacuum vector, which satisfies U (γ)Ω = Ω for all γ ∈ Möb(D). This vacuum vector is required to be cyclic for the von Neumann algebra A(S 1 ) := I∈I A(I). • A(I) is a type III factor for every interval I ∈ I, unless H = C.
Definition 2.11. A representation of a conformal net A is a Hilbert space H π and a family of representations (i.e. normal * -homomorphisms) π I : A(I) → B(H π ), indexed by I ∈ I, which satisfy π I | J = π J when J ⊂ I.
The defining representation H of A is called the vacuum sector. We point out that when H π is separable (as it always will be in this article) the normality of π I is automatic. By Haag duality, U (γ) ∈ A(I) when γ ∈ Diff c (I), and so given a representation of A we obtain strongly continuous representations π I • U of every Diff c (I) which are compatible with inclusions I ⊆ J. By [Hen19, Thm. 12], Diff (∞) c (S 1 ) is the colimit of the Diff c (I) along the inclusions Diff c (I) → Diff c (S 1 ) in the category of topological groups. Thus the representations π I • U assemble to a strongly continuous representation U π of Diff (∞) c (S 1 ) on H π (this was originally proven for irreducible representations in [DFK04]). The representation π is covariant with respect to U π (as in [KL04,Lem. 3 c (S 1 ). Using [Wei06,Thm. 3.8] one can show that the generator of rotation U π (r θ ) has positive energy (see [Ten18b,§2.2]).
If π and λ are representations of A, and I ∈ I, then the local intertwiners are given by Haag duality says that in the vacuum sector we have End A(I) (H) = A(I ).
We now discuss 'fusion' products of representations of a conformal net. Often this is done in the context of localized endomorphisms and DHR theory [DHR71,FRS92,BKLR15]. For our purposes it will be more convenient to use the "Connes' fusion" formulation of the fusion product first used by Wassermann [Was98] and put on solid categorical footing by Gui [Gui18a].
Given a choice of interval I ∈ I, the corresponding fusion product H π I H λ of two representations is the Hilbert space completion/quotient of the algebraic tensor product Hom A(I ) (H, H π ) ⊗ C Hom A(I) (H, H λ ) with respect to the inner product We denote by x I y the image of x ⊗ y in H π I H λ . Observe that A(I) acts and A(I ) act on H π H λ by (π I λ) I (r)(x I y) = π I (r)x I y and (π I λ) I (r)(x I y) = x I λ I (r)y.
We embed Hom A(I) (H, H λ ) → H λ by y → yΩ. The map x ⊗ y → x I y extends continuously to a map Hom A(I) (H, H π ) ⊗ C H λ → H π I H λ , which we again denote by I . If r ∈ A(I), we have xπ I (r) I ξ = x I λ I (r)ξ for all ξ ∈ H λ .
We want to define actions of A(J) on H π H λ for all intervals J in such a way as to construct a representation of A. When the Hilbert spaces are separable, the usual DHR argument shows the existence of such a representation. If one has the equivalence of local and global intertwiners for localized endomorphisms (e.g. for strongly additive nets or when the sectors have finite index [GL96, Thm. 2.3]) then such an extension is unique, but it is not clear that this is always the case.
Instead of the DHR argument, we follow the approach in [Gui18a, §2] which gives a natural A-representation structure on H π I H λ . To describe this representation, we observe that whenĨ ⊂ I is a subinterval, the inclusion Hom A(Ĩ ) (H, H π ) ⊂ Hom A(I ) (H, H π ) extends to a unitary map Gui showed that there is a (necessarily unique) way to endow each H π I H λ with the structure of a sector, compatible with the natural actions of A(I) and A(I ), in such a way that the U I←Ĩ are isomorphisms of sectors. We denote this representation π I λ, or π λ when the particular interval I is not important. In the same article, Gui also shows that this 'Connes' fusion' tensor product makes the category of A-representations into a braided tensor category.

Bounded localized vertex operators
In [Ten19,Ten18b] we introduced the notion of bounded localized vertex operators as a way of formalizing the relationship between unitary VOAs and conformal nets (an alternate approach to the same problem was given in [CKLW18]). In these articles we gave definitions of what it means for a conformal net A to come from a unitary VOA V , and what it means for an A-representation to come from a unitary V -module. We will now describe the aspects of this relationship which are necessary for this article, and refer the reader to [Ten18b] for more details.
The Segal-Neretin semigroup of annuli consists of compact Riemann surfaces which are isomorphic to annuli, equipped with boundary parametrizations [Seg04,Ner90]. Sufficiently nice representations of Diff (∞) c (S 1 ) extend to representations of a central extension of the semigroup of annuli. The annuli {1 > |z| > r} and {R > |z| > 1} act by r L 0 and R −L 0 , respectively. If V is a VOA, the operator R −L 0 Y (v, z)r L 0 corresponds to inserting a state inside the annulus {R > |z| > r}. In general this should be a bounded operator. We depict this situation: While the field Y (v, z) is local in the Wightman sense, the operator R −L 0 Y (v, z)r L 0 encoding the insertion of v in the annulus is not local in the sense of algebraic conformal field theory. That is, if A V is the conformal net which is associated to the same CFT, then the insertion in the annulus {R > |z| > r} should not lie in any A V (I).
A breakthrough idea of Henriques was to extend the semigroup of annuli to also include 'degenerate' (or 'partially thin') annuli, where the incoming boundary and outgoing boundary are allowed to overlap [Hen14]. If one knows that the representation of the semigroup of annuli on H V extends to this larger semigroup, one can consider operators which insert states in 'local' annuli. Here, an annulus is localized in an interval I if the incoming and outgoing boundary parametrizations agree on I .
Specifically, we wish to consider partially thin annuli Σ whose outgoing boundary lies in {|z| ≥ 1} and whose incoming boundary lies in {|z| ≤ 1}. We divide Σ into two annuli, B := Σ ∩ {|z| ≥ 1} and A := Σ ∩ {|z| ≤ 1}. The unit circle is the incoming boundary of B and the outgoing boundary of A, and we parametrize both by the identity. If π is the representation of the semigroup of partially thin annuli on the positive energy representation H V , then the operator which inserts the state v at z ∈ Σ is π(B)Y (v, z)π(A). We depict the situation as follows: We now attempt to construct a conformal net A V whose local algebras A V (I) are generated by operators of the form π(B)Y (v, z)π(A), as v runs over all of the states of V , (B, A) runs over all pairs (B, A) localized in I, and z ∈ Σ = B • A. At present, the theory of the semigroup of degenerate annuli and its representations has not been completed, although we hope to take up this question in future work with Henriques. What we do have is a way of assigning bounded operators to certain degenerate annuli, and direct verification that these operators behave in many of the ways that one would expect if they did come from a semigroup representation. We do not wish to focus too much on the specifics of these operators, and so we abstract out certain key properties that they satisfy. We call the resulting structure a system of generalized annuli. Just below we will introduce systems of generalized annuli, and then later discuss how to carry out the construction of conformal nets from VOAs in this context.

Systems of generalized annuli
A detailed discussion of generalized annuli is given in [Ten18b,§3]. We will only use certain properties of generalized annuli in this article, and so we omit certain aspects of the definition here.
Generalized annuli will be certain operators acting on the Hilbert space completions . For every such c there is a strongly continuous unitary representation U c,0 of Diff c (S 1 ) on H c,0 , and the local diffeomorphisms generate a conformal net A c called the Virasoro net of central charge c. For more details, see [Car04,§2.4] and references therein.
We write Ann I (H c,0 ) for the class of operators which are left localizable in I and which have dense image. We write Ann r I (H c,0 ) for the class of operators which are right localizable in I and are injective. We abbreviate these to Ann /r I when c is understood from context.
Left localizable operators should represent degenerate annuli whose outgoing boundary is the unit circle parametrized by the identity and the parametrization of the incoming boundary maps I to the thin part of the annulus. The auxiliary diffeomorphism γ adjusts the outgoing parametrization so that the incoming and outgoing parametrizations agree on I . Observe that A ∈ Ann I if and only if A * ∈ Ann r I ; we think of the adjoint as reflecting the degenerate annulus through the unit circle.
Our system of generalized annuli will consist of pairs (B, A) of right/left localizable operators which satisfy additional properties. All of these properties are clear for representations of degenerate annuli.
Definition 2.13 (System of incoming/outgoing generalized annuli with central charge c). A system of incoming generalized annuli of central charge c is a family of subsets A in I ⊂ Ann I (H c,0 ) for all intervals I, such that: for all γ ∈ Diff c (S 1 ) and all θ ∈ R. The associated system of generalized outgoing annuli is given by the adjoints A system of generalized incoming annuli must come equipped with an interior function "int" which assigns to every A ∈ A in I the interior of a degenerate annulus not containing 0. That is, int(A) = D \ D where D is the open unit disk and D ⊂ D is a Jordan domain with C ∞ boundary with 0 ∈ D. We further require that if U (γ)A ∈ A c (I), then γ −1 (I ) ⊆ ∂D ∩ S 1 . We do not rule out the possibility that int(A) = ∅. This interior function must be compatible with reparametrization: The generalized annuli B • A in which we will insert states have the property that the parametrization of the outgoing boundary of B agrees with the parametrization of the incoming boundary of A on I . Thus B and A will be simultaneously localizable in the following sense.
Definition 2.14. Let Ann I be the collection of pairs (B, A) with A ∈ Ann I and B ∈ Ann r I for which there exists a common γ ∈ Diff c (S 1 ) such that U (γ)A, BU (γ) * ∈ A c (I).
We now give the definitions of a system of generalized annuli. The pairs (B, A) represent degenerate annuli divided along the unit circle, and the properties listed in the definition reflect basic properties that such annuli should have.
7. if r θ ∈ Rot(S 1 ), I is an interval containing both the intervals J and r θ (J), and In [Ten18b, §3.4], we constructed a system of generalized annuli explicitly in terms of exponentials of smeared Virasoro fields. Definition 2.15 should be a thought of as a list of properties of those operators. The abstract notion of a system of generalized annuli also allows the possibility to replace the systems constructed in [Ten18b] with arbitrary degenerate annuli once the representation theory of this semigroup has been established.

VOAs with bounded localized vertex operators
Given a fixed system of generalized annuli of central charge c and a unitary VOA with the same central charge, we would like to define insertion operators BY (a, z)A for all (B, A) ∈ A I . The generalized annuli A and B are operators on the vacuum Hilbert space H c,0 of the Virasoro net. We use the Virasoro net to transport the action of A and B to H V , noting that H V decomposes as a direct sum of irreducible representations of the Virasoro algebra and therefore carries a natural representation π of A c . Thus for left localizable annuli A ∈ Ann I we define an action π I (A) = U π (γ) * π I (U (γ)A) which is independent of a choice of localizing unitary γ. More generally, if π is any representation of A c , we have an action π I (A) = U π (γ) * π I (U (γ)A) for any liftγ of γ to Diff (∞) c (S 1 ), which is well-defined up to a multiple of e 2πiL 0 . We can similarly define the action of right localizable annuli, and we have π I (A) * = π I (A * ). These actions are described in detail in [Ten18b,§3.2]. For simplicity, we drop the notation π I (A) and simply let A act on representations of A c , and similarly for B.
We will interpret BY (v, z)A, as a sesquilinear form. We introduce the notion of incoming and outgoing annuli to ensure that it is densely defined.
Definition 2.16. Let M be a unitary representation of the Virasoro algebra on which L 0 is diagonalizable, and let H be the Hilbert space completion of M . An incoming generalized annulus on H is an operator A ∈ B(H) such that M ⊂ Ran(A) and A −1 (M ) is dense in H. An operator B ∈ B(H) is an outgoing generalized annulus if A * is an incoming generalized annulus. We write Ann in (H) and Ann out (H) for the set of incoming and outgoing generalized annuli, respectively.
A is a densely defined sesquilinear form, which may or may not extend to a bounded operator.
Definition 2.17 (Bounded insertions). Let V be a simple unitary VOA of central charge c and fix a system A of generalized annuli with the same central charge. We say that V has bounded insertions for A if for every interval I ∈ I and every (B, The map BY (s L 0 ·, z)A corresponds to the picture Definition 2.18 (Bounded localized vertex operators). If V has bounded insertions for A , we define the local algebras corresponding to intervals I ⊂ S 1 by We say that V has bounded localized vertex operators if A V (I) and A V (J) commute whenever I and J are disjoint.

Representation theory and (local) intertwiners
If M is a unitary V -module, then it decomposes as a direct sum of irreducible representations of the Virasoro algebra, and thus H M carries a representation of the Virasoro net A c . If (B, A) ∈ A I , then B and A are simultaneously localizable in I. As described above, the actions of A and B on H M are defined using a liftγ ∈ Diff (∞) c (S 1 ) of a localizing diffeomorphism γ, and they are in general only well-defined up to a a multiple of e 2πiL 0 . The sesquilinear form BY M (a, z)A, however, is canonically defined by requiring that B and A act using the same liftγ. Provided A ∈ Ann in (H M ) and B ∈ Ann out (H M ) (see Definition 2.16), the sesquilinear form BY M (a, z)A is densely defined.
Definition 2.19. Let V be a simple unitary VOA with bounded localized vertex operators, and let M be a unitary V -module. Suppose that for every interval I and every (B, A) ∈ A I we have A ∈ Ann in (H M ), B ∈ Ann out (H M ). Then the A V -representation corresponding to M , if it exists, is the representation π M on H M such that for every I and (B, A) as above, and every z ∈ int(B, A) and a ∈ V we have In particular this implies that the forms BY M (a, z)A are bounded. In fact, since H M is separable as a part of our definition of unitary V -module, if the representation π M exists it can be implemented by some unitary operator u.
Our next step is to consider operators of the form BY(a, z)A for (B, A) ∈ A I and Y ∈ K M N . We need to assume that K and N are unitary so that A and B act on H N and H K , respectively. Moreover, in order for BY(a, z)A to be densely defined we need to assume that B ∈ Ann out (H K ) and Ann in (H N ). In practice, it is easiest to just assume that π N and π K exist. We also generally assume that M is a unitary module, although strictly speaking that is not necessary.
Unlike the case of BY M (a, z)A, there is in general a genuine ambiguity in the definition of BY(a, z)A. While we can simultaneously localize B and A, the e 2πiL 0 ambiguities in their action on H K and H N do not necessarily cancel. However, we will often be interested in the special case where (1, A) ∈ A I , in which case A ∈ A c (I). We then define the action of A on H N in the natural way using id ∈ Diff Definition 2.20. Let V be a simple unitary vertex operator algebra, let M , N , and K be unitary V -modules, and assume that π N and π K exist. An intertwining operator Y ∈ I K M N is said to have bounded insertions if for every interval I, every (B, A) ∈ A I , and every z ∈ int(B, A) there is some s > 0 such that the operator BY(s L 0 −, z)A defines a bounded bilinear map H M × H N → H K . If moreover we always have for all a ∈ M then we say that Y is localized. We write I loc K M N for the space of localized intertwining operators.
Note that even though the actions of A and B are generally only defined up to a multiple of e 2πiL 0 , the boundedness and locality of the operators is independent of those choices.
Remark 2.21. In the definition of bounded localized vertex operators we required that BY (s L 0 −, z)A define a bounded map from the Hilbert space tensor product H V ⊗ H V into H V for some s > 0. In contrast, in Definition 2.20 we have required that BY(s L 0 −, z)A define a bounded bilinear map H M × H N → H K , or equivalently that it define a bounded map on the projective tensor product H M ⊗ π H N , which is a weaker condition. The weaker condition is more appropriate for intertwining operators (as e.g. the proof of Proposition 2.22 would run into problems). In fact, it is possible that the weaker condition would be more appropriate for bounded localized vertex operators as well, and all of the results from [Ten19,Ten18b] would go through verbatim under the weaker hypothesis. However we have not made this change to the definition of bounded localized vertex operators here so as not to muddle the relationship between this article and the preceding ones. We also point out that if BY (s L 0 −, z)A defines a bounded bilinear map and r L 0 is trace class on H V , then BY ((rs) L 0 −, z)A defines a bounded map on the Hilbert space tensor product. Thus so long as r L 0 is trace class for some r the weaker and stronger definitions are actually equivalent.
By the Banach-Steinhaus theorem, BY(s L 0 −, z)A defining a bounded bilinear map is equivalent to saying that for every a ∈ M the sesquilinear form BY(a, z)A defines a bounded map, and the map a → BY( We in saw in [Ten18b,Lem. 4.8] that if BY(s L 0 −, z)A was bounded then for any r < s the maps BY(r L 0 −, w)A are uniformly bounded in norm when |w − z| is sufficiently small. In particular this implies by [Ten18b,Lem. 4.1] that for any a ∈ M the map BY(a, z)A is holomorphic from int(B, A) to B(H N , H K ).
We will establish one last property of localized intertwining operators.
Proposition 2.22. Let V be a simple unitary VOA with bounded localized vertex operators, let M , N , and K be unitary V -modules, and assume that π K and π N exist. Suppose that Hence taking adjoints we have for any r > 0 Hence if we can find r small enough that θ sz r L 0 is a bounded operator then a → BY † ((rs) L 0 a, z)A will be bounded as well.
Recalling that θ w a = e wL 1 e iπL 0 w −2L 0 a, it suffices to show that for any w there is a r such that e wL 1 r L 0 is bounded. Equivalently, we must show that r L 0 e rwL 1 is bounded. We saw in [Ten18b,Lem. 4.6] that when r + |rw| < 1 the operator e wrL −1 r L 0 is bounded, and so taking adjoints we see that r L 0 e rwL 1 is bounded as well. Thus BY † ((rs) L 0 −, z)A defines a bounded bilinear form for r sufficiently small.

Transport modules and positivity for unitary VOAs
In this section we introduce the transport module associated to a pair of unitary modules M and N for some unitary VOA V . We will see in Section 3.2 that if V is sufficiently nice (e.g. regular) VOA and a transport module exists, then it is isomorphic to the tensor product M N and moreover gives a canonical inner product on M N . Closely related ideas have been used by Gui [Gui19b,Gui19c,Gui19a] to construct unitary modular tensor categories of V -modules for many examples. The results of this section and Section 5 will provide many examples of transport modules, and thereby more examples where Gui's machinery can be applied.
Of additional interest is the fact that transport modules can be defined for an arbitrary VOA V , and we conjecture that they exist for any unitary modules M and N which are reasonably nice. This provides a proposal for a unitary tensor product theory for an arbitrary unitary VOA, just as any conformal net has a tensor category of representations. In fact, the transport form (Definition 3.1) is a translation of the Connes' fusion inner product from conformal nets into the language of VOAs, which we will use in Section 4 to compare VOA and conformal net tensor products. A version of this translation was used by Wassermann in his work on WZW models of type A [Was98], and both our approach and the approach used by Gui were inspired by Wassermann's work.

The transport forms and transport modules
Let V be a simple unitary VOA and let M and N be unitary V -modules. If K is a unitary V -module equipped with an intertwining operator Y ∈ I Hilb K M N , then we can define a family of pre-inner products on M ⊗ N by for z in the punctured unit disk (equipped with a choice of log z), where θ z is as in Section 2.1.
Under favorable circumstances (see Lemma 3.6) vectors of the form Y(a, z)b span a dense subspace of H K , and so we can recover the Hilbert space structure of H K as the Hilbert space completion (and quotient) of M ⊗ N . In fact, the V -module structure on K can also be recovered from the fact that the map M ⊗ N → H K is supposed to come from an intertwining operator (more precisely, the map M ⊗ N → H K should be a P (z)intertwining map).
In the context of vertex tensor categories, we may hope to write the product for certain intertwining operators Y 1 and Y 2 . This sort of computation makes sense for any unitary V -module K and any intertwining operator Y, but a unitary tensor product should be a pair (K, Y) such that we have (3.1) is the annihilation operator -see Section 2.1.3). Thus the pre-inner product [ · , · ] which characterizes both the inner product and V -module structure on a unitary tensor product of M and N can be explicitly written in terms of the data of these modules. The main idea of this section is to introduce the notion of transport module, which is a formalization of the notion of unitary tensor product module characterized by an intrinsic pre-inner product. Now given unitary V -modules M and N , we define a formal series-valued sesquilinear pairing · , · x,y on M ⊗ N by where θ y a 2 = e yL 1 e iπL 0 y −2L 0 a 2 . Here a → a is the canonical antilinear map M → M . We wish to evaluate the above pairing at complex numbers z, w ∈ C \ {0}, and indeed one should expect that the corresponding series converges to a multi-valued holomorphic function on |z| > w −1 − z > 0. However, for our purposes it will suffice to consider a more restricted domain. Let erators with appropriate target and source modules). As a convenient bonus, we will see now that a 1 ⊗ b 1 , a 2 ⊗ b 2 z,w is naturally single-valued on R, assuming that the relevant series converges. Let us first consider when M is a simple module. In this case, the conformal weights of M lie in ∆ + Z for some ∆ ∈ R. In this case, the powers of (y −1 − x) which arise in (3.2) lie in −2∆ + Z. The second source of multi-valuedness in (3.2) is the term θ y a 2 , which introduces a factor of the form y −2∆+m for an integer m. Thus if we expand the series (3.2) we obtain where f (x, y) is a formal series involving only integral powers of x and y. When |z| , |w| < 1, we evaluate (1 − wz) −2∆ using the standard branch of log(1 − wz). Thus the series defining a 1 ⊗ b 1 , a 2 ⊗ b 2 z,w is well-defined and single-valued for (z, w) ∈ R.
So far, we have only considered simple modules M . In general a unitary V -module M may be written M = M i and the annihilation operator for M decomposes similarly. Thus the transport form a 1 ⊗ b 1 , a 2 ⊗ b 2 z,w is naturally single-valued in general.
Definition 3.1. Let V be a simple unitary VOA and let M and N be unitary V -modules. We say that the transport form for M and N exists if the series converges absolutely for every where P s is the projection onto vectors of conformal dimension s. In this case we write instead of the sum (3.4). As explained in the previous paragraph, we regard a 1 ⊗ b 1 , a 2 ⊗ b 2 z,w as a singlevalued function on R using the standard branch of log(1−wz). This function is holomorphic in z and antiholomorphic in w.
In addition to the transport form, we will be interested in the sesquilinear form where Y ∈ I Hilb K M N and (z, w) ∈ R. Recall (Section 2.1.3) that the condition Y ∈ I Hilb means that Y(a i , z)b i ∈ H K , so the inner product in (3.5) makes sense. While a priori (3.5) depends on choices of log z and log w, there is a natural single-valued meaning to this expression on R. Indeed, if (z, w) ∈ R then |1 − wz| < 1, which implies that z and w lie in a common half-plane cut out by a line through the origin of C. We evaluate (3.5) using values of log z and log w coming from a common branch of log defined on this half-plane. The value of (3.5) is independent of the choice of branch, and of the choice of half-plane, yielding a single-valued meaning (3.5). Indeed, this is the unique branch of (3.5) on R with the property that Remark 3.2. Alternatively, note that if U ⊂ C × is an open set such that U × U ⊂ R, and z ∈ U, then U cannot contain any positive multiple of −z. Thus there exists a holomorphic branch of log on U, and we define (3.5) on U × U using any fixed branch of log on both factors. The value of (3.5) for other (z, w) ∈ R can be obtained by analytic continuation.
as single-valued functions on Note that the left-hand and right-hand sides of (3.6) are interpreted as single-valued functions on R as described in the paragraphs preceding Definitions 3.1 and (3.3), respectively.
Remark 3.4. Recall that a dominant intertwining operator Y is one for which terms a Y (k) b span the target module. The requirement of Definition 3.3 that Y t be dominant should be thought of as a non-degeneracy requirement, and if Y t satisfies (3.6) but is not dominant then the submodule generated by the modes of Y t is a transport module.
Remark 3.5. Modulo subtleties about multi-valued functions and domains, the transport module condition is equivalent to for appropriate value of w and z. Thus it is a particular example of what is called the equivalence of products and iterates in the VOA literature. The key feature is that we may write the iterate of Y N and Y − M as a product of an intertwining operator and its adjoint.
We will see below (Proposition 3.7) that a transport module (M t N, Y t ), if it exists, is unique up to unique unitary isomorphism. In particular M t N is unique up to unitary isomorphism, and we will speak of the transport module.
It is clear that if K andK are unitary V -modules satisfying Definition 3.3, with corresponding intertwiners Y t andỸ t , then there is an isometry defined on the subspace of H K spanned by vectors of the form Y t (a, z)b by Y t (a, z)b →Ỹ t (a, z)b. We must check that the domain and range of this map are dense, and that it gives a homomorphism of V -modules.
Recall (Section 2.1.4) that the modes v K (k) and vK (k) are closable operators densely defined on K ⊂ H K andK ⊂ HK, respectively, so it suffices to check that vectors of the form Y t (a, z)b are a core for the closure of v K (k) . This fact, which we establish just below in Lemma 3.6, is both philosophically and technically important. It justifies our approach to understand tensor products by studying the image of intertwining operators, and it is crucial for rigorously applying functional analytic tools to VOAs.
Lemma 3.6. Let V be a simple unitary VOA, let M , N , and K be unitary V -modules, and let Y ∈ I Hilb K M N be a dominant intertwining operator. Let z ∈ C with 1 > |z| > 0, and fix a choice of log z. Set Then K z is dense in H K . Moreover K z is an invariant core for the closure of v K (n) , for every v ∈ V and n ∈ Z.
Proof. We begin by pointing out that K z depends on the choice of log z. For v ∈ V and a ∈ M we will write v (n) for v K (n) and a (k) for a Y (k) . Let D be the domain of the closure of v (n) . We show first that K z ⊂ D. It suffices to show that Y(a, z)b ∈ D when a and b are homogeneous. Since Y(a, z)b ∈ H K the partial sums of the series k∈R a (k) b, in which all terms are orthogonal, provide a sequence in K converging to Y(a, z)b. Now by the Borcherds commutator formula, Each sum indexed by R on the right-hand side is of the homogeneous components of a vector in H K , and thus converges. Hence the partial sums of k∈R a (k) bz −k−1 provide a witness for the vector Y(a, z)b lying in D, and moreover as one expects. In particular we see that K z is invariant under (the closure of) v (n) .
As a next step, we will prove that K z is dense in H K . Recall from Lemma 2.9 that functions of the form Y(a, z)b are holomorphic on the punctured disk. It follows from the definition of an intertwining operator that d dz Y (a, z) (3.8) Thus K w ⊂ K z for |z − w| sufficiently small, where the choice of log w is obtained by analytic continuation. It follows that e iθL 0 maps K z into K z when θ is sufficiently small, and therefore K z is invariant under e iθL 0 for all θ. Since L 0 is diagonalizable on H K , the closed subspace K z decomposes as a direct sum of L 0 -eigenspaces, and it follows that a (k) b ∈ K z for all a ∈ M and b ∈ N . Since Y was assumed dominant, this means that K ⊂ K z and thus K z is dense. Since We have D z ⊆ D, and must prove the reverse inclusion.
We begin by showing that D z = D w for any pair z and w in the punctured disk, and any choices of log z and log w. By (3.7), the function v (n) Y(a, z)b is holomorphic, and by standard VOA 1 a, z)b, and so when |z − w| is sufficiently small we may Taylor expand as before Thus the partial sums of ∞ m=0 (w−z) m m! Y(L m −1 a, z)b witness the fact that Y(a, w)b ∈ D z . We have shown that K w ⊂ D z , and it follows that D w ⊂ D z . But w and z were interchangeable (provided |z − w| is sufficiently small), and so locally we have D z = D w , where the choice of log w is obtained by analytically continuing that of z. It follows that D z does not depend on z or the choice of log z, and so we denote this single domain byD.
Let Γ z ⊂ H K ⊕ H K be the graph of v (n) | Kz , and let Γ z be the corresponding graph when log z is replaced by log z + 2πi. We have shown that Γ z = Γ z =:Γ is the graph of v (n) |D.
Let τ = e 2πiL 0 be the twist on H K . Then (τ ⊕ τ )Γ z = Γ z , so (τ ⊕ τ )Γ =Γ. It follows that the projection ontoΓ commutes with τ ⊕ τ , and therefore commutes with its spectral projections. For h ∈ R let p h to be the projection onto the closed subspace of H K spanned by homogeneous vectors with conformal dimension equal to h mod 1. Then p h is the projection onto the e 2πih -eigenspace of τ , and thus p h ⊕ p h leavesΓ invariant. Now for k ∈ R and homogeneous a ∈ M and b ∈ N , let h be the conformal dimension of a (k) b. Then the function is a single-valued holomorphic function on the punctured disk with values inΓ. Thus In particular a (k) b ∈D. Since Y is dominant, we have K ⊂D. Hence D ⊂D, as D is the domain of the closure of v (n) | K . We conclude that D =D and thus K z is a core for v (n) .
We now prove the uniqueness of transport modules up to unitary isomorphism.
Proposition 3.7. Let V be a simple unitary VOA, let M and N be unitary V -modules, and let (K, Y t ) and (K,Ỹ t ) be transport modules for M and N . Then there is a unique unitary u : H K → HK such that uY t =Ỹ t , and u restricts to an isomorphism of V -modules K ∼ =K.
Proof. Fix a point z such that (z, z) ∈ R (i.e. such that 1 > |z| > 2 −1/2 ), along with a choice of log z. The condition that (K, Y t ) and (K,Ỹ t ) are both transport modules for M and N means that Recall that Y t andỸ t are dominant and lie in I Hilb by the definition of transport module. Thus by Lemma 3.6, K z andK z are dense, and so u extends to a unitary u : H K →H K . We now verify that u gives an isomorphism of V -modules. Let v ∈ V and for n ∈ Z let v K (n) denote the closure of the mode of Y K (v, x), and similarly for vK (n) . Note that K z and K z are invariant cores for v K (n) and vK (n) , respectively, by Lemma 3.6. For ξ ∈ K z we have vK (n) uξ = uv K (n) ξ by (3.7). Hence vK (n) u = uv K (n) as closed operators. In particular, u intertwines the actions of L K 0 and LK 0 , and thus u maps K ontoK. Restricting the identity v (n) u = uv (n) to finite energy vectors we see that u is a V -module isomorphism.
Suppose a ∈ M and b ∈ N are homogeneous. Then since u preserves conformal dimensions and uY t (a, z)b =Ỹ t (a, z)b, we must have ua Yt Uniqueness of u is clear, as Y is dominant by the universal property of tensor products and so vectors of the form Y t (a, z)b span a dense subspace of H K by Lemma 3.6.
We now compute transport for pairs of modules one of which is the vacuum. The definition of the transport module for M and N is not symmetric in the inputs, and it turns out that the transport module for V and N is much simpler than the opposite order. Proof. As a consequence of the Jacobi identity for modules ( [FHL93], see also [Gui19b,Prop. 2.13]), whenever |z | > |z| > |z − z | we have with both series converging absolutely. Note that both series involve integer powers of z, so the functions are single-valued. Thus if (z, w) ∈ R we apply the fact that N is a unitary module to obtain Since Y N is trivially dominant and Y N ∈ I Hilb N V N we conclude that (N, Y N ) is a transport module.
Next, we compute a transport module for M and V . This is somewhat more subtle, as it involves intertwining operators as opposed to module operators. The key step is to establish a 'fusion relation' for annihilation and creation operators (defined in Section 2.1.3).
Lemma 3.9. Let V be a simple unitary VOA and let M be a unitary V -module. Let a 1 ∈ M , a 2 ∈ M , and u, v ∈ V . Then the series converges absolutely when |z| > |w − z| > 0, and Proof. Assume without loss of generality that all vectors are homogeneous. We first prove the lemma when v = Ω.
By part 1 of [Gui19b, Lem. 2.16], if |w| > |z| and we choose log w − z close to log w when z is small, we have , u with the sum converging absolutely. Note that the expansion of e zL −1 in the last expression has only finitely many non-zero terms since L k 1 b = 0 for k sufficiently large. On the other with the sum having only finitely many non-zero terms as before. Comparing these two computations, we see that the lemma has been proven when v = Ω. Now if v ∈ V is homogeneous with conformal weight ∆ v , we apply the Borcherds commutator formula twice to obtain Note that the sums in j and are finite, and thus by the case v = Ω completed above we have that Y − M (a 2 , w)Y + M (a 1 , z)v, u converges absolutely when |w| > |z| > 0 and On the other hand, Comparing (3.11) and (3.12), we see that it suffices to show that the double sum in (3.12) converges absolutely when |z| > |w − z| > 0 and s∈R j≥0 when |w| > |z| > |z − w| > 0.
In fact, we establish (3.13) with the order of summation in s and j reversed, but since we will establish absolute convergence the desired result will still follow. So assume |z| > |z − w| > 0, and observe j≥0 s∈R 14) The first equality in (3.14) is the fact that v (j) raises energy by ∆ v − j − 1, the second is reindexing the sum, the third is the Borcherds commutator formula, and the fourth uses the fact that the sum in and j is finite. The first of the two terms remaining at the end of (3.14) converges absolutely to the first term in (3.13), using the case v = Ω already established. We now show absolute convergence of the second term at the end of (3.14) to the second term of (3.13).
In fact, since the sum over is finite, it suffices to verify absolute convergence for fixed . We have j≥0 s∈R Here we have used the fact that when |z| > |w − z| > 0 we have absolute convergence which is easily established for = 0 using geometric series and then the general case may be obtained by differentiating in w. Hence the sum in (3.15) converges absolutely to Y (Y − M (v ( ) a 2 , w − z)a 1 , z)Ω, u (−w) −1− . Plugging this expression into the end of (3.14), we see that (3.13) has been proven, completing the lemma. Proof. Observe that if 1 > |z| > 0, then by Lemma 3.9, we have convergence of for all a ∈ M and b ∈ V , and thus Y + M ∈ I Hilb M M V . By the same lemma, whenever (z, w) ∈ R the series defining converges absolutely and is equal to as multi-valued functions which agree when log w − z is taken close to log w when z is small. Note that Y + M (a 1 , z)b 1 , Y + M (a 2 , w)b 2 is in fact single-valued, and the definition of · , · z,w as a single-valued function R (preceding Definition 3.1) agrees with taking log w close to log w − z when z is small, completing the proof.

Regular VOAs and positivity of transport matrices
Let V be a simple unitary VOA and let M and N be unitary V -modules. Recall from Section 3.1 that the transport forms for M and N are defined by for (z, w) ∈ R := {(z, w) ∈ C 2 : 1 > |z| > w −1 − z > 0 and 1 > |w| > 0}, provided the relevant series converges absolutely (in which case we say that the transport form exists for M and N ). In this section, we discuss the transport form in more detail in the context of regular VOAs. Regularity is a strong semisimplicity condition on the representation theory of V originally introduced in [DLM97]. Under the mild assumption that a VOA is of CFT type (which is always true for simple unitary VOAs), "regular" is equivalent to "rational and C 2 -cofinite" [ABD04]. We will not discuss the precise definition of regularity, and instead list the relevant consequences of this property. A regular VOA V has finitely many simple modules up to isomorphism, every V -module can be written as a direct sum of simple modules, and the spaces of intertwining operators I K M N are finite-dimensional. Most importantly, the tensor product theory of Huang and Lepowsky applies to regular VOAs with the additional property of being "strong CFT type," which always holds for simple unitary VOAs [HL13, Hua05, Hua08b, Hua08a]. Proof. First we prove (1). Let a ∈ M and b ∈ N be homogeneous vectors, and suppose 0 < |z| < 1. Observe that Y(a, z)b ∈ H K if and only if the series defining Y(a, z)b, Y(a, z)b converges. As series, we have By Huang's work [Hua05], the series on the right-hand side converges since |z| −1 > 1 > |z|, and thus we conclude that Y(a, z)b ∈ H K .
The proof of (2) is similar, as the convergence of the transport form on R is a special case of Huang's convergence of iterated intertwining operators in the same reference.
A similar proof [KS17, §4.2] actually shows that a ⊗ b → Y(s L 0 a, z)r L 0 b extends to a Hilbert-Schmidt map H M ⊗ H N → H K for appropriate s, r, and z, but we will not use this fact.
We now consider a fixed simple unitary regular VOA V , and fix a set Irr(V ) of isomorphism class representatives of simple V -modules. If M and N are V -modules, then the tensor product M N is a V -module which represents the functor K → I K M N . That is, M N is a V -module equipped with an intertwining operator Y ∈ M N M N such that for any Y ∈ K M N there is a unique V -module homomorphism ϕ : M N → K such that Y = ϕY . 2 The tensor product always exists, and is given by  for all a i ∈ M , b i ∈ N , as single-valued functions on R.
Proof. By the work of Huang [Hua05], there exists a V -module K and intertwining operators Y 1 ∈ I K M N and Y 2 ∈ I N M K such that which establishes existence. Uniqueness is clear since Y ∈ I Hilb by Proposition 3.11 and thus the span of vectors of the form Y (a, z)b is dense in H M N by Lemma 3.6. Definition 3.13. Under the hypotheses of Proposition 3.12, the endomorphism Λ ∈ End V (M N ) satisfying (3.17) is called the transport endomorphism 3 associated to M and N .
Note that Λ depends on both the inner product chosen on M N , as well as the intertwining operator Y making (M N, Y ) into a tensor product. However, we will be interested in the positivity of Λ, which will not depend on these choices. Note that Λ extends to a bounded operator on the Hilbert space completion H M N , and as usual we use the same symbol for this extension.
Remark 3.14. In the work of Gui [Gui19b,Gui19c], the operator Λ is studied as a matrix by choosing bases for the spaces I K M N . We can compare the two approaches as follows. Let Irr(V ) = {M 1 , . . . , M n } be a complete list of simple modules, equipped with a unitary structure. Let M N = α K α be an orthogonal decomposition into simple modules, and for each α choose an isometry v α : M i(α) → M N such that Ran(v α ) = K α , for the appropriate i(α) ∈ {1, . . . , n}. Then by the universal property of the tensor product, {v * α Y } is a set of intertwining operators comprising a basis for n i=1 I M i M N . Since the M i are simple, we have v * β Λv α = Λ αβ v * β v α for some scalars Λ αβ . By construction id M N = α v α v * α , and so we can expand the identity (3.17) characterizing Λ to obtain (for appropriate z and w) where the product Y † β (a 2 , w)Y α (a 1 , z) is understood as 0 if i(α) = i(β). Thus the matrix Λ αβ is exactly the transport matrix [Gui19c, Eqn. (6.16)] considered by Gui. By construction, the operator Λ ≥ 0 if and only if the matrix Λ αβ ≥ 0.
The following easy proposition outlines the relationship between positivity, tensor products, and transport modules for regular VOAs. In Section 3.3, we will expand on this relationship for arbitrary unitary VOAs. 1. For every (z, z) ∈ R, the sesquilinear form · , · z,z on M ⊗N is positive semi-definite.
2. For some (z, z) ∈ R, the sesquilinear form · , · z,z on M ⊗N is positive semi-definite. 5. There exists a transport module for M and N .
Proof. We prove (1) =⇒ (2) =⇒ (3) =⇒ (4) =⇒ (5) =⇒ (1). The implication (1) =⇒ (2) is a tautology. Now assume (2). By Proposition 3.11 as single-valued functions on R. Recall (preceding Definition 3.3) that the right-hand side is defined on R by requiring that Y (a 1 , z)b and Y (a 2 , z)b 2 be interpreted with the same value of log z. Since vectors Y (a, z)b span a dense subspace of H M N by Lemma 3.6, the operator Λ is positive semi-definite because the form · , · z,z is. To see that Λ is positive definite (i.e. invertible), we need to invoke the deep work of Huang which showed that the category of V -modules is rigid. The corollary that the transport operator is invertible was shown in step 3 of the proof of [Gui19c, Thm. 6.7] (see also [HK07]), in the equivalent setting of transport matrices (see Remark 3.14). Now assume (3). Then the invariant inner product Λ · , · on M N makes M N into a transport module by Proposition 3.12.
The implication (4) =⇒ (5) is a tautology so we now assume (5). If (M t N, Y t ) is a transport module for M and N , then is clearly positive semi-definite for any (z, z) ∈ R.
Remark 3.16. When the conditions of Proposition 3.15 hold, we obtain a new construction of the tensor product M N as a subspace of the Hilbert space completion of M ⊗ N with respect to the transport form. The intertwining operator Y can be recovered from the fact that the canonical projection of M ⊗ N onto its Hilbert space completion should be a P (z)-intertwining map. Thus a unitary tensor product M N is constructed from just the transport form; this is described in more detail in [Ten18a] (see also Section 3.3). Note that this construction is 'manifestly unitary,' in that M N comes already endowed with an inner product (in fact, it was constructed from the inner product). We will verify the conditions of Proposition 3.15 in many examples in Section 5, and so this new manifestly unitary construction is known to produce unitary tensor products in these examples.
Remark 3.17. The hypothesis throughout this section that M N admits a unitary structure is expected to be vacuous, as it is conjectured that for unitary regular VOAs every module admits a unitary structure. However, it has not even been shown that if M and N are unitary modules, then M N admits a unitary structure. The construction described in Remark 3.16 provides one possibility for proving this second, weaker conjecture. If the transport forms are positive semidefinite, then one can attempt to build M N inside the Hilbert space completion of M ⊗ N in a manifestly unitary way.

Transport, tensor products, and positivity
We conclude Section 3 by describing the role of transport modules in a unitary tensor product theory for unitary VOAs, and by presenting conjectures related to transport modules.

The positivity conjecture
Let V be a simple unitary VOA and let M and N be unitary V -modules. In Definition 3.1, we defined a family of sesquilinear forms a 1 ⊗ b 1 , a 2 ⊗ b 2 z,w on M ⊗ N which are a single-valued function of (z, w) ∈ R, where R = {(z, w) ∈ C 2 : 1 > |z| > w −1 − z > 0 and 1 > |w| > 0}.
In Proposition 3.15 and Remark 3.16 we described how the transport forms provide a 'manifestly unitary' approach to the study of tensor products of modules for regular VOAs, so long as we know that a certain sesquilinear form is positive semidefinite. Moreover, we saw that this gave the same answer as the standard approach to tensor products of modules.
For VOAs which are not regular, the tensor product theory can be quite a bit more complicated. As a first hurdle, our definition of module (with L 0 diagonalizable and finitedimensional weight spaces) is too restrictive to study the representation theory of general unitary VOAs, as even the tensor product of simple unitary modules should not be a direct sum of such modules. In [Ten18a], we sketched a proposal for a class of unitary modules which should be well-behaved from the perspective of tensor product theory. For the rest of Section 3.3 we will use the word 'module' to mean something more general than the strong sense used throughout the rest of the article.
Beyond the problem of selecting the correct category of modules, there is a fundamental issue of defining the tensor product M N , even when M and N are simple. As an example of what can go wrong, we consider the Virasoro VOAs L(c, 0) when c > 1. For such VOAs, the simple modules are parametrized by h ∈ C, and dim I h 3 h 1 h 2 = 1 regardless of the values of h i . In light of this, it seems unwise to attempt to use the same definition of tensor product as in the case of regular VOAs.
Instead, we propose the notion of transport module as a candidate tensor product of unitary V -modules. The motivation for this conjecture comes from the study of conformal nets. Like unitary VOAs, conformal nets are supposed to axiomatize two-dimensional chiral conformal field theory, and so one expects a dictionary between the phenomena which arise in the two contexts. However, the study of tensor products in the context of conformal nets looks quite different than in the context of VOAs. While the tensor product of VOA modules is understood in terms of a universal property, the tensor product of conformal net representations is given by an explicit construction. This construction is independent of any 'niceness' properties of the conformal net, and produces a braided tensor category [BKLR15]. If one believes in the dictionary between conformal nets and VOAs, then there should be a corresponding VOA construction. The notion of 'transport module' is precisely this analogue. Since the tensor product of conformal net representations always exists, we conjecture the same for transport modules. Note that a part of Conjecture 3.18 is finding the correct class of (generalized) modules for which the statement is true. One immediate consequence of Conjecture 3.18 would be the following.
Conjecture 3.19 (Positivity conjecture). Let V be a simple unitary VOA and let M and N be V -modules. Then for every (z, z) ∈ R the transport forms · , · z,z on M ⊗ N are positive semidefinite.
In Section 4.1, we will show that the positivity conjecture holds for many modules over many VOAs, even badly non-rational ones like Vir c with c ≥ 1. We will do so by identifying the transport form with the Connes fusion inner product from conformal nets. By Proposition 3.15, this implies the existence of many transport modules for regular VOAs. We believe these results provide strong evidence for Conjectures 3.18 and 3.19. In fact, the positivity conjecture should be a major component of proving the existence of transport modules in the non-regular regime.

From the positivity conjecture to transport modules
It is clear that the existence of transport modules implies the positivity conjecture. We now discuss how to go in the opposite direction and construct a transport module given positivity. This is a conceptually important step, as the positivity conjecture feels more approachable than the problem of constructing a transport module, as it is entirely intrinsic to the modules M and N . What we will now see is that the existence of transport modules can also be set up as a problem intrinsic to the modules M and N .
If the positivity conjecture holds for M and N , we let H t be the Hilbert space completion/quotient of M ⊗ N with respect to the transport form for some z. This Hilbert space comes equipped with a map M ⊗ N → H t , which we denote t . To construct a transport module, we need to put a V -module structure on a dense subspace K of H t , and show that there is an intertwining In general the image of t will have trivial intersection with our desired subspace K ⊂ H t , but because of Lemma 3.6 we can still hope that the image of t is a core for the action of modes on finite energy vectors. Thus we proceed in the only way we can: we define an action of V on the image of t by We conjecture that the induced densely defined action of L 0 on H t is essentially self-adjoint, and let K = A≥0 p [0,A] H t , where p [0,A] is the spectral projection for L 0 corresponding to the interval [0, A]. The modes v (n) should be closable with K a core and satisfy the Borcherds identity, at which point we will have successfully constructed a transport module. While there are many steps in the above construction which look difficult to prove, we point out that if a transport module exists, then a posteriori this construction indeed recovers that transport module. Since we demonstrate the existence of many transport modules in Section 5, we have many examples where this construction works. Thus Conjecture 3.18 is equivalent to the conjecture that this construction always produces a transport module. In particular, in the case of regular VOAs, we conjecture that this construction always produces a tensor product M N (in light of Proposition 3.15). Our construction is in principal not so different than the 'working backwards' construction of Huang-Lepowsky [HL95]. The primary difference is that Huang-Lepowsky first construct a V -module on a certain complicated subspace of the algebraic dual (M ⊗ N ) * , whereas our proposed construction produces the tensor product module on the nose.

From positivity to transport in vertex tensor categories
In a vertex tensor category of V -modules, for any modules M and N one expects to be able to find a module K and intertwining operators Y 1 ∈ I K M N and in the sense of matrix coefficients and multi-valued functions. Thus if there exists a vertex tensor category of unitary V -modules, we expect to be able to find a unitary K with this property. Crucially, this should hold even when the category is not rigid. It is not hard to show that if a decomposition (3.19) exists and the positivity conjecture holds for M and N , then there exists a transport module for M and N . We now sketch the argument.
Then the identity (3.19) becomes a 1 , z). From the convergence of the transport form, we see that Y ∈ I Hilb . Let M t N ⊂K be the subspace generated by the modes of Y acting on N , and let v : HK → H M tN be the projection. If we set Λ := vϕv * and Y t = vY, then we have a 1 , z), and now Y t is dominant. From here we can argue as in Proposition 3.15 that there is an inner product on M t N making (M t N, Y t ) in to a transport module. Indeed, since the transport from is positive and Λ is manifestly a bounded operator, we can explicitly define the new inner product on M t N via Λ · , · .

Positive vectors in VOAs
Positivity is a crucial aspect of the approach to quantum field theory using von Neumann algebras, and a key fact is that positivity is preserved by representations of von Neumann algebras. The study of unitary VOAs would be greatly aided by a corresponding tool, which we now present a candidate for.
Definition 3.20. Let V be a simple unitary VOA, let ξ ∈ H V , let N be a unitary Vmodule, and suppose 0 < |z| < 1. We say that ξ is z-positive acting on M if for every b 1 , b 2 ∈ N the series defining Y N (ξ, z)b 1 , b 2 converges absolutely and In general, one should not expect to be able to find z-positive vectors lying in V itself, only in some completion. Observe that if (z, z) ∈ R, then by Proposition 3.10 for any

Universal property of transport modules?
We believe that the conjectures discussed so far in this section are strongly supported by the available evidence. We close this section with a much more speculative conjecture. A version of this conjecture was first described to the author by A. Henriques in another context.

Identification of VOA and conformal net fusion products 4.1 VOA positivity via conformal nets
In Section 3, we gave several conjectures regarding positivity of certain matrices and sesquilinear forms and described their role in the tensor product theory of unitary VOAs. In general it is quite difficult to establish positivity of sesquilinear forms unless the positivity is manifest from the construction. In contrast to the situation with VOAs, when chiral CFTs are studied using von Neumann algebras (i.e. as conformal nets), then the sesquilinear forms which arise are manifestly positive in light of basic results in the theory of operators on Hilbert spaces.
All of the existing positivity results for specific models require detailed case by case analysis to obtain certain analytic properties of intertwining operators (which are often not conjectured to hold for all VOAs). In this section, we present a method for proving positivity results about VOAs via conformal nets which in contrast uses essentially no information about intertwining operators, and only uses analytic properties which are conjectured to hold for all unitary VOAs. These analytic The main idea in linking conformal net representations and the transport form is that if a representation π N exists, we can "apply π N " to the transport form for M and V and obtain the transport form for M and N . This is extremely useful, as we completely understand the transport form for M and V since we have shown that (M, Y + M ) is a transport module (Proposition 3.10). Since π N preserves positivity, we will be able to deduce positivity of the transport form for M and N from the one for M and V .
The following lemma uses geometric ideas to show that the transport form [ · , · ] z 1 ,z 2 for M and N can be computed via π N when the z i lie in the interior of generalized annuli and are sufficiently close to each other.  (1, A 1 ), (1, A 2 ) ∈ A I , and suppose there is some Then there exists a non-empty open subset U ⊂ int(A 1 ) ∩ int(A 2 ) with the following properties. There is an interval J ⊂ S 1 containing w 0 such that J ⊂ ∂U . We have U × U ⊂ R, and whenever z 1 , z 2 ∈ U , a 1 , a 2 ∈ M , and b 1 , b 2 ∈ N : 1. The series defining the transport form for M and N converges absolutely.
2. The sesquilinear forms x a 1 (z 1 ) and x a 2 (z 2 ) define bounded operators and we have x a 2 (z 2 ) * x a 1 (z 1 ) ∈ A V (I).
Proof. First we choose the set U. Since V has bounded localized vertex operators, for every z ∈ int(A * 2 , A 1 ) there exists some s > 0 such that A * 2 Y (s L 0 ·, z)A 1 defines a bounded map H V ⊗ H V → H V . Note that we used the fact that if (1, A i ) ∈ A I then (A * 2 , A 1 ) ∈ A I by the definition of system of generalized annuli. By [Ten18b,Cor. 4.5], we can find a neighborhood w 0 ∈Ũ ⊂ int(A * 2 , A 1 ) and some s > 0 such that A * 2 Y (s L 0 ·, z)A 1 is bounded for every z ∈Ũ . Since π N exists and is unitarily implemented, A * 2 Y N (s L 0 ·, z)A 1 is bounded for every z ∈Ũ as well. Without loss of generality we may take s < 1. By shrinkingŨ about w 0 ∈ S 1 , we may ensure that |z 1 | > s > z 2 −1 − z 1 for all z 1 , z 2 ∈Ũ . We now as in (3.3)) and there is some interval J ⊂ ∂U containing z 0 .
We now prove (1). Fix a holomorphic branch of log on U (which is possible since U ×U ⊂ R, see Remark 3.2). Without loss of generality assume that the a i and b i are homogeneous. Fix z 1 , z 2 ∈ U, and set Note that the expression defining ξ is multi-valued, but since (z 1 , z 2 ) ∈ R we may make the standard choice, as in the definition of the transport form (Definition 3.1). Since Y − M ∈ I Hilb V M M (by Proposition 3.10) we have ξ ∈ r L 0 H V whenever r > z 2 −1 − z 1 . Since s > z 2 −1 − z 1 , we may fix some r < s such that ξ ∈ r L 0 H V . Let η = r −L 0 ξ, and let t = r/s. Then we have ξ = s L 0 t L 0 η and t < 1.
Since π N exists, there are ξ i ∈ H N such that A i ξ i = b i . Observe that since the b i are homogeneous, there is some ∆ ∈ R such that Y N (P α ξ, z)b 1 , b 2 = 0 unless α ∈ ∆ + Z ≥0 , where P α is the projection onto vectors of conformal weight α. We now compute Now, since A * 2 Y N (s L 0 ·, z 1 )A 1 is a bounded operator and P α η ≤ η , we see that A * 2 Y N (s L 0 P α η, z 1 )A 1 ξ 1 , ξ 2 is bounded independent of α. Thus (4.1) converges absolutely by comparison to a geometric series. This completes the proof of (1).
We now proceed to (2). Let u i ∈ V be homogeneous and pick ξ i such that A i ξ i = u i . Since (Y M , Y + M ) is a transport module for M and V (Proposition 3.10), x a 1 (z 1 )ξ 1 , x a 2 (z 2 )ξ 2 = Y + M (a 1 , z 1 )u 1 , Y + M (a 2 , z 2 )u 2 = Y (ξ, z 1 )u 1 , u 2 = A * 2 Y (ξ, z 1 )A 1 ξ 1 , ξ 2 . (4.2) Note that the series defining Y (ξ, z 1 )u 1 , u 2 converges absolutely by part (1) (applied to N = V ). In using the definition of the transport module, it was important that log z i were chosen from a common branch of log on U.
Since ξ ∈ s L 0 H V and z 1 ∈ U the operator A * 2 Y (ξ, z 1 )A 1 is bounded by the definition of bounded localized vertex operators. Thus examining (4.2) in the case a 1 = a 2 and z 1 = z 2 we see that x a 1 (z 1 ) and x a 2 (z 1 ) are bounded operators. Moreover we have x a 2 (z 2 ) * x a 1 (z 1 ) = A * 2 Y (ξ, z 1 )A 1 ∈ A V (I), again by the definition of bounded localized vertex operators. This completes the proof of (2).
Finally we prove (3). Since ξ ∈ s L 0 H V , there exists a sequence of finite energy vectors Thus from the definition of π N we have As an easy consequence of Lemma 4.1 we can verify the positivity conjecture (Conjecture 3.19) under mild hypotheses, which we will be able to verify for many models (including non-rational ones).
Proposition 4.2. Let V be a simple unitary VOA with bounded localized vertex operators, and let M and N be unitary V -modules. Assume that π N exists and that Y − M ∈ I Hilb V M M . Then there is some r < 1 such that whenever (z, z) ∈ R and |z| > r the transport form Proof. Observe that e iθL 0 a 1 ⊗ e iθL 0 b 1 , e iθL 0 a 2 ⊗ e iθL 0 b 2 z,z = a 1 ⊗ b 1 , a 2 ⊗ b 2 e iθ z,e iθ z and so the existence and positivity of the transport form only depends on |z|.
So choose some interval I and some (1, A) ∈ A I with non-empty interior (which exists by the definition of system of generalized annuli). The boundary of int(1, A) contains an interval of S 1 , and so we may choose some w 0 ∈ int(A * , A). Let U be the open set provided by Lemma 4.1, which necessarily contains points z = (1 − ε)w 0 for ε sufficiently small. Thus it suffices to prove the positivity of the transport form for z ∈ U.
Since H N is a separable Hilbert space (as a part of our definition of V -module), there is a unitary V : H V → H N such that π N I = Ad V . LetÃ = (A| ker(A) ⊥ ) −1 as a map N → H N . Now with notation as in Lemma 4.1 part (3), we have Note that if T is any linear map from a vector space to a Hilbert space, the sesquilinear form T (v 1 ), T (v 2 ) H on V is clearly positive semidefinite. Applying this to T : M ⊗ N → H M given by completes the proof.
Remark 4.3. The positivity conjecture states that · , · z,z is positive semidefinite whenever (z, z) ∈ R; that is, when 1 > |z| > 1/ √ 2. Thus to prove the conjecture for a pair of modules M and N , one has to get the number r from Proposition 4.2 down to 1/ √ 2. In practice, we do not have much control over the value of r provided by Proposition 4.2 as it is stated. This is mainly due to the fact that we are using an abstract system of generalized annuli. In future work, we expect to give a complete treatment of the representation theory of the semigroup of partially thin annuli, which will provide a system of generalized annuli for which the value of r provided by Proposition 4.2 is the optimal 1/ √ 2.
Even in its present form we regard Proposition 4.2 as very strong evidence for the positivity conjecture. Indeed, in Section 5 we will use it to demonstrate that the transport form is positive semidefinite for a wide range of modules for rational and non-rational VOAs, including many representations L(c, h) of the Virasoro algebra with c ≥ 1. Applying the argument of Section 3.3.3 we can construct transport modules for VOAs which are not necessarily rational, so long as they have a tensor category of modules.
We can now provide even stronger evidence of the positivity conjecture for regular VOAs.
Theorem 4.4. Let V be a simple regular unitary VOA, let M and N be unitary V -modules. Assume that M N admits a unitary structure and that π N exists. Then the transport form · , · z,z exists and is positive semidefinite for all (z, z) ∈ R.
Proof. Note that Y − M ∈ I Hilb and the transport forms exist since V is regular (Proposition 3.11). By Proposition 4.2 the transport form is positive semidefinite for some z, and thus by Proposition 3.15 it is positive semidefinite for all (z, z) ∈ R.  Proof. Theorem 4.4 guarantees that for all (z, z) ∈ R the transport form · , · z,z is positive semidefinite. By Proposition 3.15, this is equivalent to the desired conclusion.
In Section 5 we describe many models to which we can apply Theorem 4.4 and Corollary 4.5.
We call the inner product on M N from Corollary 4.5 the standard inner product. Since the inner product depends on the choice of Y , by Proposition 3.7 a transport module pair (M N, Y ) is unique up to unique unitary isomorphism. Thus the standard unitary structure on M N is unique up to canonical unitary isomorphism.
If V is a simple regular unitary VOA, then the work of Huang and Lepowsky demonstrates that the category Rep(V ) of V -modules is a modular tensor category in a natural way [Hua08a] (see also [CKM17] or [Gui19b] for a description of this structure). In order to study the representation theory of regular unitary VOAs from a categorical perspective, however, one needs to consider the category Rep u (V ) of unitary V -modules, and one would like to know that it has the structure of a unitary modular tensor category in a natural way. In order to define a tensor product on Rep u (V ), it is necessary to choose an invariant inner product on M N given such a choice for M and N . One would like to do so in such a way that the associators, braidings, and twists which Huang and Lepowsky construct for Rep(V ) are unitary.
The problem of defining such an inner product is quite different than the analogous problem for unitary representations of groups, for which there is a canonical, easy-toconstruct invariant inner product on the tensor product of unitary representations. On the other hand, it is not obvious that the tensor product of unitary VOA modules even admits an invariant inner product (although the transport form provides a candidate), but this is widely believed to be true. In fact, it is widely believed that every module for a regular unitary VOA admits a unitary structure, although this fails quite badly for VOAs which are not regular (e.g. the Heisenberg VOA).
Gui has recently made significant progress on the problem of making Rep u (V ) into a unitary modular tensor category [Gui19b,Gui19c] in a natural way, but his construction assumes the positivity of certain sesquilinear forms. In light of Corollary 4.5, we are now able to verify the necessary positivity hypothesis in a much wider class of models.
Corollary 4.6. Let V be a simple regular unitary VOA with bounded localized vertex operators. Suppose that every V -module M admits a unitary structure, and that π M exists. Then the standard inner product of Corollary 4.5 makes Rep u (V ) into a unitary modular tensor category in a natural way.
Proof. We will apply [Gui19c, Thm. 7.9]. In order to apply this theorem, we must verify that the transport matrices Λ αβ of (3.18) are positive (semi)definite. By Proposition 3.15, this is equivalent to the positivity of the transport form proven in Theorem 4.4.
We apply the above result to examples such as WZW models and W -algebras in Section 5.

Transport modules and Connes fusion
Our original motivation for studying the standard inner product on the tensor product of unitary V -modules is that it is closely connected with the inner product on the Connes fusion of conformal net representations. In this section, we will systematically develop that connection using bounded localized vertex operators. We will work under the hypothesis that V has bounded localized vertex operators, M and N are unitary V -modules such that π M and π N exists, and that a transport module M t N exists. The results of Section 4.1 provide large families of VOA and modules satisfying these hypotheses, and when V is regular we have that M t N is just the usual tensor product M N equipped with the standard inner product.
The goal of this section is to construct unitary maps U I : H M I H N → H M tN relating the Connes fusion to the transport module in a natural way.
The key step in constructing a unitary is to define an appropriate map Thus if we had a unitary equivalence (4.3), then we could obtain from that a map of the form (4.4). One of Wassermann's ideas in [Was98] was to work backwards and first construct (4.4), and then use that map to define an isomorphism (4.3).
The following fundamental lemma combines Lemma 4.1, which shows that the transport form for M and N can be calculated using π N , with the definition of transport module to exhibit a correspondence of the form (4.4).  (1, A 1 ), (1, A 2 ) ∈ A I and suppose that int(A * 2 , A 1 ) ∩ S 1 = ∅. Let U ⊂ int(A 1 )∩int(A 2 ) be the non-empty open subset provided by Lemma 4.1 for some choice of w 0 ∈ int(A * 2 , A 1 ) ∩ S 1 . Since U × U ⊂ R, we may fix a holomorphic branch of log on U (Remark 3.2).
Let (M t N, Y t ) be a transport module for M and N . For a i ∈ M , let using our fixed branch of log on U. Then for all z 1 , z 2 ∈ U, y i are bounded operators and π I (x * 2 x 1 ) = y * 2 y 1 .
Proof. Let b 1 , b 2 ∈ N , and since π N exists we may choose ξ i ∈ H N such that A i ξ i = b i . By Lemma 4.1 and the definition of a transport module we have (4.5) In the second equality, we use the fact that U × U ⊂ R, and the definition of transport module gives us an equality of single-valued functions on R. This step requires that we chose log z i from a single branch of log on U to obtain the correct value of Y t (a 1 , z 1 )b 1 , Y t (a 2 , z 2 )b 2 (see Remark 3.2). Considering the case A 1 = A 2 , z 1 = z 2 and a 1 = a 2 , we have π N (x * x)ξ 1 , ξ 2 = yξ 1 , yξ 2 where x = x 1 = x 2 and y = y 1 = y 2 . Since vectors ξ i ∈ A −1 N are dense (since π N exists), and π N (x * x) is bounded, we see that y is bounded. Returning to (4.5) we see that π N (x * 2 x 1 ) = y * 2 y 1 as desired.
The map Y + M (a, z)A → Y t (a, z)A which appears in Lemma 4.7 will play an important role in our calculation of π M I π N . In order to relate this correspondence to Connes' fusion, we will need to know that Y + M (a, z)A actually lies in Hom A(I) (H V , H M ). Thus our remaining results will require the hypothesis Y + M ∈ I loc M M V . In practice, we can verify this hypothesis in any example for which we know that π M exists, such as all modules for WZW models of type EFG and W-algebras of type E, among others..
As an intermediate step to constructing the natural unitary U I of (4.3), we will construct a family of unitaries depending on additional choices (Lemma 4.8), and then show that the resulting map does not depend on those choices (Lemmas 4.9-4.10). The proofs of these lemmas are motivated by [Was98]. We put assemble the lemmas and construct the unitary U I in Theorem 4.11.
Proof. Note that Y + M ∈ I loc implies that Y − M = (Y + M ) † ∈ I loc and in particular Y − M ∈ I Hilb . Since int(A) is a degenerate annulus by definition, there is necessarily some w 0 ∈ int(A * , A) ∩ S 1 . Let U ⊂ int(A) be the non-empty subset obtained from Lemma 4.1, and observe that U ∩ X = ∅. By replacing U with a smaller set, we may assume U ⊂ X and we restrict our chosen branch of log z to U.
Let S ⊂ Hom A V (I ) (H V , H M ) be given by Note that Y + M (a, z)A is bounded since Y + M ∈ I loc and z ∈ U ⊂ int(A). We first claim that S ⊗ H N is dense in H M I H N . It suffices to prove that SH V is dense in H M . By the definition of bounded localized vertex operators, V ⊂ AH V , and thus Y + M (a, z)b ∈ SH V for all a ∈ M and b ∈ V . By Lemma 3.6 it follows that SH V = H M and thus that S ⊗ H N is dense in H M I H N . Let x 1 , x 2 ∈ S and ξ 1 , ξ 2 ∈ H N . Write x i = Y + M (a i , z i )A, and set Observe that by Lemma 4.7 we have π N (x * 2 x 1 ) = π N ((Y + M (a 2 , z 2 )A) * (Y + M (a 1 , z 1 )A)) = y * 2 y 1 .
That is, the map S ⊗ H N → H M tN given by x ⊗ ξ → yξ is an isometry when S ⊗ H N is given the fusion inner product. Since S ⊗ H N is dense in the fusion product, we obtain an isometry U = U I,A,X : on int(A), and in particular on X . By (4.6), we have for z ∈ U. Both sides of this equality define holomorphic functions on X , and thus we conclude that the equality holds for all z ∈ X . Since b 2 was arbitrary and A −1 H N is dense, we conclude that (4.6) holds for all z ∈ X as well.
Note that the existence of a branch of log on X was required in Lemma 4.8, but such a branch always exists. Indeed, by definition, int(A) is a degenerate annulus not containing 0 whose thick part is supported in the interval I, and thus the connected components of int(A) are simply connected. Now given (1, A) ∈ A I and a connected component X of int(A) equipped with a branch of log, we have constructed a unitary U I,A,X . Our next step is to compare the maps U I,A,X as A and X vary, but in order to do so we need a reasonable way of comparing branches of log on X ⊂ int(A) and X ⊂ int(A ) for some other (1, A ) ∈ A I .
In order to systematically choose a branch of log on every connected component of every A with (1, A) ∈ A I , we will choose a continuous branch of log on the interval I itself.

I ∪ int(A)
Since int(A) is a degenerate annulus with ∂ int(A) ∩ S 1 ⊂ I, a continuous branch of log on I extends canonically to a holomorphic branch on int(A) thereby yields a unitary U I,A,X for every connected component X of int(A). In the following we fix a branch of log on I and show that the resulting unitaries U I,A,X do not depend on A or X .
We build up to that result by considering a special case.
Lemma 4.9. Let V be a simple unitary VOA with bounded localized vertex operators, and let M and N be unitary V -modules. Assume that π M and π N exist, and that Y + M ∈ I loc M M V . Let (M t N, Y t ) be a transport module for M and N . Let I be an interval equipped with a branch of log, and let (1, A 1 ), (1, A 2 ) ∈ A I . Let X i ⊆ int(A i ) be connected components and suppose that S 1 ∩∂X 1 ∩∂X 2 contains an interval. Then U I,A 1 ,X 1 = U I,A 2 ,X 2 .
Proof. Since S 1 ∩ ∂X 1 ∩ ∂X 2 contains an interval, we may choose a w 0 ∈ S 1 with the property that be the open sets obtained by applying Lemma 4.7. Since the boundary of each U ij contains an interval about w 0 , the same is true for the boundary of U := U ij . Hence U ∩ X 1 ∩ X 2 = ∅, and by taking U smaller if necessary we assume U ⊂ X 1 ∩ X 2 . Now for i, j ∈ {1, 2}, a 1 , a 2 ∈ M , z 1 , z 2 ∈ U, if we set where log z i is obtained by holomorphically extending the branch on I to U, then π I (x * 2 x 1 ) = y * 2 y 1 (4.7) by Lemma 4.7. We now construct a unitary U : H M I H N → H M tN in the same way as in the proof of Lemma 4.8. Using the transport relation (4.7), we can argue just as in Lemma 4.8 that the map for i ∈ {1, 2}. Analytically continuing, we see that (4.8) holds for all z ∈ X i . Thus U = U I,A 1 ,X 1 = U I,A 2 ,X 2 , as desired.
SupposeĨ ⊂ I, and θ is a sufficiently small angle that r θ (Ĩ) ⊂ I. For (1, A) ∈ AĨ , set A θ = e iθL 0 Ae −iθL 0 and note that (1, A θ ) ∈ A I by the definition of system of generalized annuli. Moreover int(A θ ) = r θ (int(A)). The next lemma identifies the maps U I,A,X for an annulus with the map for the rotated annulus. Proof. Let J be an interval compactly contained in ∂X ∩ S 1 . Observe that for θ < |J|, X and X θ share a boundary interval and thus the result follows from Lemma 4.9. For the general case, we may iterate the above observation.
We are now ready to prove the main result of this section: such that for all a ∈ M , (1, A) ∈ A I , z ∈ int(A), and ξ ∈ H N we have where the value of log z is obtained by continuing the branch of log from I to int(A).
for which the projection of Y t is localized, and show that the isometry U * I | H M loc N is an isomorphism of sectors. Indeed, M loc N is the largest submodule of M t N for which we could have this intertwining property, so this is the best possible result which does not address the general problem of existence and locality of π M tN and Y t . We now carefully define the module M loc N .
If K is a unitary V -module, then it can be decomposed where the K i are pairwise non-isomorphic simple unitary V -modules and the X i are finitedimensional Hilbert spaces. Let S ⊂ S be the subset consisting of i such that π K i exists. Then by [Ten18b,Prop. 5.5], the maximal submodule of K such that π K exists is Now suppose that M and N are unitary V -modules and that π N exists, and fix Y ∈ I K M N . Let p i be the projection of K onto the isotypical component Each I i is a left-ideal and therefore of the form I i = B(X i )p i,loc for a projection p i,loc . We define K loc , the local part of K with respect to Y, to be and let p loc = 1 ⊗ p i,loc .
Remark 4.13. For any (B, A) ∈ A I and z ∈ int(B, A) we have (1 ⊗ p i,loc )BY i (a, z)A ∈ Hom A(I ) (H N , H K ). In general we do not know that p loc BY(a, z)A is a bounded operator when S is infinite, as (1 ⊗ p i,loc )BY i (a, z)A might not be bounded independent of i. However if we know a priori that p loc BY(a, z)A is bounded then it lies in Hom A(I ) (H N , H K ), as each component p i,loc BY(a, z)A is a local intertwiner. In particular, this is the case if V is regular, as there are only finitely many simple V -modules, and we have p loc Y ∈ I loc K loc M N . Definition 4.14. Let V be a simple unitary VOA with bounded localized vertex operators, let M and N be unitary V -modules, and suppose that a transport module (M t N, Y t ) exists. Then we denote by M loc N the maximal local submodule of M t N with respect to Y t , as described above. We set Y loc := p loc Y.
Observe that if (K, Y t ) and (K,Ỹ t ) are a pair of transport modules, and u : K →K is the canonical unitary equivalence of Proposition 3.7, then uK loc =K loc and uY loc =Ỹ loc . Thus if a transport module for M and N exists then the pair (M loc N, Y loc ) is unique up to canonical unitary equivalence.

Hence
U loc I = U loc I U I←Ĩ .
Since K ⊂Ĩ , and both U I←Ĩ and U loc I commute with the actions of A V (Ĩ ), so does U I loc . In particular, U I loc commutes with the actions of A V (K). Thus we conclude that U I loc is map of A V -representations.
The intertwiner U loc I = p loc U I constructed in Theorem 4.16 is the adjoint of an isometry. The embedding (U loc I ) * is unitary precisely when p loc = 1, which is to say M loc N = M t N . Thus we have: In the special case when V is regular, we have an explicit formula for M loc N , provided a transport module exists (see Remark 4.15). We can restate Corollary 4.17 more explicitly in terms of fusion rules in this case:

Finite index subfactors and rationality of conformal nets
If π is a representation of a conformal net, then we obtain a subfactor π I (A(I)) ⊆ π I (A (I )) for every interval I, called a Jones-Wassermann subfactor. By diffeomorphism covariance (in fact, by Möbius covariance), the subfactors corresponding to the intervals I and J are unitarily conjugate, so we can talk about "the" subfactor corresponding to π. The problem of showing that these subfactors have finite index for key examples of conformal nets is widely accepted to be a challenging, but crucial, open problem. The first major achievement in this direction was the work of Wassermann [Was98] for WZW models of type A, with further progress made by Loke [Lok94], Toledano Laredo [TL97], and Gui [Gui18b].
One purpose of this article is to adapt Wassermann's approach to provide a systematic approach to the relationship between VOA and conformal net fusion, which can be used to show the finiteness of the index of subfactors. The approach taken here is more flexible than previous work, in two important ways. First, we do not rely on any special analytic properties of certain VOAs which are not conjectured to hold for all unitary VOAs. Second, earlier approaches required large amounts of information about intertwining operators to obtain any information about the fusion product of conformal net representations. For example, one might need analytic information about every intertwining operator of type K M N to obtain information about the fusion π M π N . In contrast, our methods provide extremely useful information even with limited information about intertwining operators. We demonstrate that now by applying the results of Section 4 to the problem of finiteness of the index of representations, and more generally rationality of conformal nets.
A conformal net is called rational if it has finitely many irreducible representations all of which have finite index. By the work of Longo-Xu [LX04] and Morinelli-Tanimoto-Weiner [MTW18], rationality is equivalent to "complete rationality," which was introduced by Kawahigashi-Longo-Müger [KLM01], who showed that the representation category of a (completely) rational conformal net is modular. We will use two results of Longo to show that conformal nets are completely rational. The first is [Lon03,Thm. 24], which says that if B ⊂ A is a conformal subnet and [A : B] < ∞, then A is rational if and only if B is. Here, [A : B] is the index of the subfactor B(I) ⊆ A(I) for any(/every) interval I. The second result we will use is [Lon90, Thm. 4.1], which says if π and ρ are irreducible and π ρ contains the vacuum representation as a subsector, then π and ρ have finite index and are mutually contragredient. 4 We use here the fact that the category of representations is braided. .
We thus obtain the following.
If θ is the PCT automorphism of V , then θM is a W -submodule of V which is isomorphic to M , and thus we conclude that π M exists. Thus we may apply Theorem 4.20 to obtain the desired result.
Remark 4.22. The argument for Corollary 4.21 works just as well if there is a V -module N containing M as a W -submodule such that π N and π N exist and Y + N , Y + N ∈ I loc . There are several ways to apply the above results to obtain rationality of conformal nets. Applying Theorem 4.20 we immediately obtain the following. Thus we can obtain rationality with only minimal information about intertwining operators. In future work, we hope to show that every representation of A V is of the form π M , which would streamline Corollary 4.23 further; see Section 5.4.
Another way to apply Theorem 4.20 is via conformal inclusions. We obtain a method for establishing rationality without any discussion of intertwining operators.
Corollary 4.24. Let V be a simple unitary VOA with bounded localized vertex operators, let W be a unitary subalgebra which is regular, and assume that the coset W c is regular.
In In Section 5 we apply these results to many important examples of unitary VOAs.

Applications and outlook
In this section we consider bounded localized vertex operators with respect to the system of generalized annuli constructed in [Ten18b, §3.4].

Applications to rationality of conformal nets
The machinery developed in Section 4 allows one to study representations of conformal nets using corresponding information about VOAs. The goal of Section 5.1 is to use this machinery to give a straightforward proof of the following theorem.
Theorem 5.1. Let V be a VOA which is either: • a WZW model V (g, k) for g a finite-dimensional simple complex Lie algebra and k a positive integer • a W -algebra W (g, ) in the discrete series, for g of type ADE Then V has bounded localized vertex operators and A V is rational.
It was shown in [Ten18b,Thm. 7.4] that WZW models have bounded localized vertex operators and the resulting conformal nets are equivalent to the standard loop group conformal nets. Thus Theorem 5.1 resolves the problem of rationality of loop group conformal nets, which first appeared in [GF93]. Theorem 5.1 had previously been established for V (g, k) when g was of type ACG (all levels) and D (odd levels) [Was98,Xu00,TL97,Gui18a]. It had also been established for the W -algebras W (sl 2 , ), the Virasoro minimal models [KL04]. Our proof covers these cases as well.
The proof of Theorem 5.1 will be done in two parts. In Lemma 5.2, we will prove the theorem for WZW models of type ABCDE and W -algebras of type ADE using conformal inclusions (applying Corollary 4.24). In Proposition 5.4, we will prove the theorem for WZW models of type F and G by showing directly that every irreducible representation has finite index (via Corollary 4.21). The union of these two results gives Theorem 5.1. Many of the models considered could be addressed by either method.
First consider when g is of type ADE. For a positive integer k, we have the diagonal embedding V (g, k + 1) ⊂ V (g, k) ⊗ V (g, 1), where V (g, k) is the WZW model of type g at level k. A theorem of Arakawa-Creutzig-Linshaw [ACL19] says that the coset V (g, k + 1) c of this inclusion is the W -algebra W (g, ), where is given by +ȟ = k+ȟ k+ȟ+1 . Moreover, the theorem asserts that V (g, k+1) cc = V (g, k+1). The W -algebras which arise in this context are precisely the discrete series (see the introduction of [ACL19] for more detail), which are known to be regular by work of Arakawa [Ara15a,Ara15b]. In the special case when g = sl 2 , these W -algebras give the discrete series of Virasoro minimal models, recovering the well-known result of Goddard-Kent-Olive [GKO86].
Using these inclusions, we may apply Corollary 4.24 to obtain a short proof of rationality of the WZW and W -algebras of type ADE. For the remaining classical types, we apply a similar argument using conformal inclusions arising from inclusions of groups.
Lemma 5.2. Let V be a VOA which is either: k) for positive integer k and g is of type ABCDE • V = W (g, ) is in the discrete series for g of type ADE Then A V is rational.
Proof. We first consider g of type ADE, and begin with the case V = V (g, 1). The Frenkel-Kac-Segal construction asserts that V is a lattice VOA. Bischoff gave a version of the Frenkel-Kac-Segal construction for conformal nets, and verified that the loop group conformal nets corresponding to g at level 1 were rational [Bis12, Prop. 4.1.17] (see also [DX06]). By [Ten18b, §7.1], A V is isomorphic to the loop group conformal net, and so we have rationality of A V in these cases.
We can now give an inductive argument to obtain rationality at higher levels. By [ACL19] we have a conformal inclusion for the appropriate value of . Since V (g, k +1) is regular by [DLM97] (see also [FZ92]) and W (g, ) is regular by [Ara15a,Ara15b], and tensor products of regular VOAs are regular [DLM97], we can invoke Corollary 4.24 and conclude that We have used here that Since A 1 ⊗ A 2 is rational if and only if both A i are (e.g. by µ-index considerations; see [KLM01]), we can conclude that if A V (g,k) is rational then so are A V (g,k+1) and A W (g, ) .
Repeating this procedure completes the proof in the case where g is of type ADE.
When g is classical, we can appeal to inclusions of groups. First recall that V (so(n), 1) is the even part of the (real) free fermion VOA F ⊗n . Thus A V (so(n),1) is the even part of A ⊗n F , which has µ-index 1 (as we explicitly identified A ⊗2 F with the complex free fermion net in [Ten19], which has µ-index 1). Thus A V (so(n),1) has µ-index 4 by [KLM01], and is in particular rational. We now apply Corollary 4.24 to the conformal inclusions V (so(n), m) ⊗ V (so(m), n) ⊂ V (so(nm), 1) and V (C n , m) ⊗ V (C m , n) ⊂ V (so(4nm), 1) to obtain rationality of WZW models of type B and C (and D, again).
We now turn our attention to V = V (g, k) when g is of type F or G. We make use of the well known inclusion of groups F 4 × G 2 ⊂ E 8 [Dyn52]. This gives rise to a conformal inclusion of VOAs V (F 4 , 1) ⊗ V (G 2 , 1) ⊂ V (E 8 , 1) (note that F 4 and G 2 must both occur at level 1, as otherwise the central charge of V (F 4 , 1) ⊗ V (G 2 , 1) would exceed that of V (E 8 , 1)). Both V (F 4 , 1) and V (G 2 , 1) have only two simple modules. On the other hand, V (E 8 , 1) is holomorphic and therefore cannot be equal to V (F 4 , 1)⊗V (G 2 , 1), and it follows that every V (g, 1) module occurs inside V (E 8 , 1) for g = F 4 and G 2 . We were unable to find a proof of the following higher level analog of this fact in the literature, and so we include one here.
Lemma 5.3. Let g be G 2 or F 4 . Then every simple V (g, k)-module arises as a submodule of V (E 8 , 1) ⊗k .
Proof. We will only do the case g = F 4 , as the case of G 2 is very similar but slightly simpler (and also appears implicitly in [Gui18b, §5.2]). We assume that the reader has some familiarity with the representation theory of affine VOAs (see [FZ92] for background).  = [1, 1, 0, 0]. The simple V (F 4 , k)-modules are given by L k (M λ ) for λ a dominant weight with λ · θ ≤ k, where M λ is the corresponding irreducible g-module.
Alternatively, if some other n i = 0 we can apply the same argument, using the inductive hypothesis for k − 3, k − 2, or k − 1.
We point out that Lemma 5.3 gives a proof of local equivalence for loop group conformal nets of type F and G (see [Ten18b, §7.2]). We can in fact deduce much more from this branching information.
Proposition 5.4. Let V = V (g, k) for g of type E, F , or G. Then A V has bounded localized vertex operators and every irreducible A V -representation is of the form π M for a simple unitary V -module M . Moreover π M exists and has finite index for all unitary V -modules M . Hence A V is rational.
Proof. Bounded localized vertex operators was established in [Ten18b,Thm. 7.4]. We have that every V (g, k)-module occurs inside V (E 8 , 1) ⊗k for type F and G by Lemma 5.3. The same fact for g of type E is more standard (a short proof can be given using the regularity of cosets considered above, see [Ten18b,Thm. 7.11]). It follows that π M exists for every unitary V (g, k)-module M [Ten18b, Thm. 5.6].
Henriques showed that every irreducible representation of the loop group nets can be obtained from a representation of the loop group [Hen19, Thm. 26], which gives an upper bound on the number of irreducible representations. Using this fact, one can use a counting argument [Ten18b,Lem. 7.8] to show that if π M exists for every unitary V -module, then every irreducible representation of A V is of the form π M . Finally, since every such M is a V (g, k)-submodule of V (E 8 , 1) ⊗k , we have that π M has finite index by Corollary 4.21.

Unitary and positivity for VOA tensor product modules
A consequence of the work in Section 4.1 is that positivity of transport matrices and unitarity of tensor categories for VOAs can be obtained from the existence of representations π M . Proposition 5.4 demonstrates the existence of such representations for V = V (g, k) when g is of type EFG. The classical cases (along with G 2 ) were addressed by Gui [Gui18b, Thm. 6.1], and putting these together we obtain a complete solution for WZW models.
Theorem 5.5. Let V be a VOA which is either • a WZW model V (g, k) for g a finite-dimensional simple complex Lie algebra and k a positive integer • a W -algebra W (g, ) in the discrete series, for g of type A or E Then for every pair of simple unitary V -modules M and N , the corresponding transport matrix is positive and there is a unitary structure on (M N, Y ) making it into a transport module. Hence Gui's construction [Gui19b,Gui19c] makes the category of unitary Vmodules into a unitary modular tensor category.
Proof. By [Gui19c, Thm. 7.8], it suffices to verify positivity of transport matrices, which is equivalent to existence of transport modules by Proposition 3.15. The WZW models of type ABCDG were addressed by Gui in [Gui19c,Gui18b] (and implicitly by Wassermann [Was98] in type A). For the remaining WZW models, we have the existence of π M for every simple V -module M by Proposition 5.4, and for the W -algebras we have the same conclusion by [Ten18b,Prop. 7.12]. Thus we have the desired positivity by Corollary 4.5.
The only obstruction to extending the argument of Theorem 5.5 to give a reproof of positivity for WZW models of type BD or a proof for W -algebras of type D is the need to control representations of V (so(n), 1). We cannot show the existence of all such modules from embeddings into fermions without discussing the Ramond sector, which we have avoided so far for the sake of simplicity. In practice, such an argument should be possible. However we instead prefer to direct future research towards obtaining model-independent results, as described in Section 5.4.
We are also interested in verifying the positivity conjecture (Conjecture 3.19) for (highly) non-rational VOAs, with an eye towards studying their unitary tensor product theory via transport modules. A typical application of our results here is the following.
Proposition 5.6. Let V be a simple unitary regular VOA of central charge c ≥ 1, let M and N be unitary V -modules, and suppose that π N exists. Suppose that M has a Virasoro primary state of conformal dimension h and N has a Virasoro primary state of conformal dimension h . Then there is some ε > 0 such that the transport form on the Vir c -modules L(c, h) and L(c, h ) exists and is positive semi-definite for 1 > |z| > 1 − ε.
Proof. By Proposition 3.11, Y − M ∈ I Hilb V M M , and thus the restriction to a L(c, 0)intertwining operator lands in I Hilb L(c,0) L(c,h)L(c,h) . By [Ten18b,Thm. 5.6], π L(c,h ) exists. Thus by Proposition 4.2 we have the desired conclusion.
We have numerous examples of regular VOAs such that π M exists for every V -module, and so we can apply Proposition 5.6 to all of their Virasoro primary states, as well as the primary states in the tensor products of modules, and so on. This provides a wide class of Vir c modules for which positivity has been verified, and we regard this as very strong support for Conjecture 3.19. We can apply the same argument to obtain positivity for many other models as well, and if those models have vertex tensor categories of representations, we may use positivity to construct a transport module (see Section 3.3.3). Thus we regard this result as strong suppose for Conjecture 3.18 as well.

On fusion rule calculations
By Corollary 4.17, if V is regular, π M and π N exist, and Y + M ∈ I loc , then π M π N contains a submodule isomorphic to the local submodule M loc N ⊂ M N . Moreover, if M loc N = M N , then we have π M π N ∼ = π M N . It is not an easy problem to directly show that M loc N = M N , although it should be true in general for regular unitary VOAs. In practice, this can be done in many examples by embedding V in a larger unitary VOA W , and inferring locality properties of V -intertwining operators from those of W -intertwining operators.
This strategy is quite similar to the one employed in [Was98, TL97, Gui18b] to compute fusion rules for WZW models of type ABCDG and Virasoro minimal models. In fact, the results in Section 4 can be understood as a way of systematizing these kinds of calculations, and going directly from information about VOA branching rules to theorems about conformal nets without needing to prove any analytic results along the way.
For example, Wassermann uses the embedding of V (sl n , k) into free fermions to deduce analytic properties of sl n -primary fields whose charge space is the vector representation. The same inclusions show that I N M K = I loc N M K where M is the V (sl n , k) representation corresponding to the vector representation. Thus we can show π M π N ∼ = π M N . Similarly, the inclusions used to study WZW models of type BCDG in [Gui18b] produce localized intertwining operators, and many of the same conformal net fusion rule computations can be obtained directly from our results. Moreover, these results can be extended to WZW models of type EF as well, as we do not encounter any of the difficulties described in [Gui18b, §6] following our approach. It is possible that when g is of type E or F , enough information about can be extracted from the inclusions g ⊂ E 8 and g k+1 ⊂ g k × g 1 to fully compute the fusion rules for A V (g,k) . We do not attempt such a calculation here. Instead, we will explore in future work the general framework described in Section 5.4 for obtaining fusion rule calculations without a case-by-case analysis.

Outlook
The long-term goal is to understand the relationship between VOAs and conformal nets well enough to be able to move seamlessly between representations of V and representations of A V . We have seen that the close relationship between transport modules and Connes fusion provides a potential bridge between the two notions, and in this article we have used this connection to establish new results for both conformal nets and VOAs. In order to invoke our theorems, however, various analytic properties of the VOAs in question must be verified. The assumptions which appear include V having bounded localized vertex operators, π M existing, Y + M ∈ I loc , and more generally locality of intertwining operators. We will use the rest of this section to describe a program to establish these properties broadly.
Given a unitary VOA V , there are two approaches to constructing a conformal net: the 'standard' approach which was pursued in [CKLW18] and the 'bounded localized vertex operator' approach used here [Ten19]. At present there does not appear to be a method which could prove that an arbitrary unitary VOA leads to a conformal net. Instead, one begins with a collection of examples for which there is such a correspondence, and shows that the correspondence is preserved under suitable constructions (subtheories, tensor products, extensions, and so on). Work to this effect has been completed in [CKLW18,Ten18b,Gui19a].
The next step is to develop the correspondence between modules and representations. The following conjecture appears to be in reach of current methods.
Conjecture 5.7. Let V be a simple unitary VOA with bounded localized vertex operators, and let π be a representation of A V . Then there exists a (generalized) unitary V -module M such that π ∼ = π M .
Indeed, one can attempt to define Y M on H π by π M (v, z)b = π(Y (v, z)A)A −1 b, although it would take some work to verify that Y M defines a V -module. Note that the converse problem of integrating an arbitrary (generalized) V -module to a representation of A V is likely to be quite a bit more difficult, and would require some hypotheses on the module when V is not e.g. regular.
If one can establish Conjecture 5.7, then one knows a priori that given a pair of representations π M and π N , there must be a module K such that π M π N = π K . In fact, K should be a transport module; that is precisely the motivation for the definition of transport module.
Conjecture 5.8. Let V be a simple unitary VOA with bounded localized vertex operators, let M , N and K be unitary V -modules such that Y + M ∈ I loc and π M π N = π K . Then Y(a, z)b := Y + M (a, z)A A −1 b defines an intertwining operator Y ∈ I K M N . The power of this approach is that the definition of Connes fusion implies that (K, Y) is a transport module, and moreover that Y ∈ I loc . Thus we could conclude that π M π N ∼ = π M tN . In particular if V is regular we would have π M π N ∼ = π M N for all modules M and N such that π M and π N exist and Y + M ∈ I loc . Thus a corollary of Conjecture 5.8 would be that transport modules exist and t is associative, even for poorly behaved VOAs like non-rational Vir c ; we see this as evidence that one could build a tensor category of appropriate V -modules around the notion of transport module, On the other hand, for regular V we would see that integrable modules form a tensor subcategory with the same fusion rules as Rep(V ), and the tools of [Gui18a, Gui19a] provide a pathway for identifying the categories and not just the fusion rules.
We see both Conjecture 5.7 and Conjecture 5.8 as approachable with current methods. Positive results of these conjectures would dramatically reduce the amount of case-by-case analysis necessary to identify the tensor product theory of a conformal net with that of a regular VOA, and provide new insight into unitary VOAs with wild representation theory.