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



Some examples of vertex algebra modules
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.


