Examples of simple vertex operator algebras (VOAs)

Question

A vertex operator algebra $V$ is called simple if $V$ is a simple $V$-module. What are some examples of simple VOAs? Are there lots of examples or this is a very strong condition? Is there a classification? In particular I am interested if the following VOAs are simple or not and under what conditions:

1. The rank $d$ Heisenberg (free field) VOA;
2. A lattice VOA of some non-degenerate lattice;
3. Affine Lie algebra at level $k$ for a semisimple Lie algebra $\mathfrak{g}$;
4. $\mathcal{W}_k(\mathfrak{g},f)$, $\mathcal{W}$ algebras associated to a semisimple Lie algebras at level $k$ and nilpotent element $f$. What if $\mathcal{W}_k(\mathfrak{g},f)$ is the principal $\mathcal{W}$-algebra?

Answer 1

I expect there will never be a classification of simple VOAs, unless perhaps one is only sorting according to very rough criteria. This is because there are too many of them - even the rational case is wide open. For your examples, we have the following:

1. The irreducible modules of the rank $d$ free boson are naturally parametrized by points in $d$-dimensional space. The VOA corresponds to the zero vector, hence is simple.

2. The irreducible modules of a lattice VOA are naturally parametrized by cosets of the positive definite even lattice in its dual lattice. The lattice VOA corresponds to the zero coset, hence is simple.

3. You need to specify a vertex operator algebra here. We often see $V^k(\mathfrak{g})$ used for the vacuum module, and $V_k(\mathfrak{g})$ for its unique simple quotient. They differ when $k$ is a positive integer (and perhaps other cases that I don't recall right now).

4. The W-algebra $W^k(\mathfrak{g},f)$ is given by the functor $H^0_f$ applied to $V^k(\mathfrak{g})$, and ideals are taken to ideals. In particular, the quotient by $H^0_f$ applied to the unique maximal submodule is the unique simple quotient $W_k(\mathfrak{g},f)$.