Classifying lisse conformal vertex algebras using singularities of associated varieties

Question

For the sake of keeping terminologies consistent, let me say that a conformal vertex algebra is a vertex algebra (VA) with a specified conformal vector, and a vertex operator algebra (VOA) is a conformal VA that is also $\mathbb{Z}$-graded. These are in keeping with the terminologies of Arakawa's paper "A remark on the $C_2$-cofiniteness condition on vertex algebras".

In that paper, there is Theorem 3.3.4, which asserts that if $V$ is a conformal VA and $M$ is a strongly finitely generated $V$-module, then $M$ will be lisse (which is to say that its singular support is $0$-dimensional; see Subsection 3.3) if and only if its associated variety $X_M$ (top of p. 13) is $0$-dimensional. One says that $V$ is lisse if it is so as a module over itself.

Given this theorem, I would like to ask the following:

How may one begin to classify lisee (conformal) VAs, particularly by examination of their singular support ?

I have absolutely zero idea how to even begin looking into this, so any suggestion would be very much appreciated, especially for the special case wherein $V$ is a VOA and $X_V$ is discrete, since I would like to have an impression of the relationship between singularities of $X_V$ and the structure of $V$ first of all.

Thanks in advance!

Answer 1

Before I answer your question, let me begin with a brief rant about terminology. The term "lisse" is bad when applied to vertex algebras:

1. It is French for "smooth", but has nothing to do with smoothness of various geometric objects attached to vertex algebras. For example, it is not equivalent to smoothness of the associated variety. People who use it get confused and ask questions like this one.

2. We already have reasonable terms that are commonly used: It was originally Zhu's "finiteness condition C", but "$C_2$-cofinite" has been standard for about 30 years. (I suppose if I had to choose a name, something like "Poisson-finite" might be more informative, and would remove the mysterious letter C.)

The answer to your question is: a zero dimensional variety has no singularities, so the singularities of $X_V$ can't help you with classification of $C_2$-cofinite vertex algebras. In most of the interesting examples, $X_V$ is a single point. More generally, $C_2$-cofinite vertex algebras can't be classified up to isomorphism, because there are too many of them. See my answer to this earlier, very similar question.

However, there is some work on coarser notions of equivalence, similar to classifying lattices up to genus. Recall that two lattices are in the same genus if and only if their direct sums with the hyperbolic even unimodular lattice $I\!I_{1,1}$ are isomorphic. Moriwaki defines genus equivalence by asserting that two vertex algebras are genus-equivalent if and only if their tensor products with the lattice vertex algebra $V_{I\!I_{1,1}}$ are isomorphic. However, Hoehn defines the genus of a regular vertex operator algebra by the pair (modular tensor category, central charge) - this implicitly uses Huang's modularity theorem. In the case of lattice vertex operator algebras, both notions of equivalence coincide with lattice genera, so they are both valid generalizations.