What structure does Rep(vertex algebra) have?

Question

Let $V$ be a vertex algebra. If $V$ is particularly nice, it is known that its category $\text{Rep} V$ of modules is a modular tensor category, see e.g. [1] [2].

However, this has always seemed to me like a bit of a special case: it ignores e.g. affine VOAs at all levels except positive integers, I don't know if it includes W algebras, etc.

Question. What is the expected (maximal) structure shared by $\text{Rep} V$ for all $V$? Or at least for a large class of $V$ including all examples one usually cares about, or the analogue for some closely related notion like chiral algebra. Are there any physics explanations for what kind of structures you should expect?

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[1] Rigidity and modularity of vertex tensor categories Yi-Zhi Huang

[2] Vertex operator algebras, the Verlinde conjecture and modular tensor categories Yi-Zhi Huang

Answer 1

I suspect the expected structure of $Rep(V)$ common to all vertex algebras $V$ is something like "abelian pseudomonoidal category" and I don't think you can say much else. The abelian structure follows from treating modules as discrete modules for the topological enveloping algebra. Intertwining operators yield tensor product "presheaves" on the module category that are in general not representable. I'm not sure to what extent we can even say that associativity holds for these tensor product presheaves.

You can't expect too much from physical intuition, since "all vertex algebras" encompasses unphysical examples, like "all commutative rings", or strings propagating on infinite dimensional manifolds with bad signature.