Some examples of vertex algebra modules

Question

Recently, I've been learning the vertex modules. In the paper, there are many abstract theories about the module theory, for instance, the $C_{2}$-cofinite conditions and associated variety. I hope to find fundamental examples to get some intuition for these theories. As we all know, a vertex algebra is a module of itself. But this is too trivial to reveal some ideas behind it. Are there some nontrivial basic examples of vertex algebra modules, for instance, over Heisenberg vertex algebra, Affine vertex algebra, or Virasora vertex algebra? I would appreciate it if you could provide some details or references.

Answer 1

I don't know what paper you are reading, but you can find examples in most textbooks. For example, Frenkel and Ben-Zvi's book "Vertex algebras and algebraic curves" has a treatment of modules in chapter 5 that goes into some detail for the Heisenberg case. One thing they don't mention is that there are non-trivial self-extensions of irreducible modules, given by a rather easy Jordan block method. There is a brief note in 5.5.5 about various rational quotients of affine and Virasoro vertex algebras, with references to the literature.

Any conformal vertex algebra is a module for the Virasoro vertex algebra of that central charge, so this particular case has an overabundance of examples.