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][1]
Yi-Zhi Huang

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


  [1]: https://arxiv.org/pdf/math/0502533.pdf
  [2]: https://arxiv.org/pdf/math/0412261.pdf