Quasi-lisse vertex algebras were introduced by Arakawa and Kawasetsu in [Quasi-lisse vertex algebras and modular linear differential equations](https://arxiv.org/pdf/1610.05865.pdf) . They satisfy the property that the normalized character of an ordinary representation of a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. I am wondering how much is known about the classification of these algebras?


1. Is a complete classification possible?
2. What are some examples of quasi-lisse vertex algebras apart from the one's discussed in the paper (affine vertex algebras associated with the Deligne exceptional series at non-admissible levels) and traditional ones (such as affine VOAs at admissible levels and their W-algebras through work of Kac and Wakimoto)?
3. A special case is the classification of rational and $C\_2$-cofinite vertex algebras? Can these be classified?
4. What are some examples where the quasi-modular forms of the characters have been explicitly calculated?


My interest is mainly in the modular invariance of vertex algebras and I am looking for lots of different examples where such explicit computations of characters have been made or could be made.



Classification of quasi-lisse vertex algebras
I do not have a complete answer to your questions, but this is what I can say for now:


Question 1: A classification is impossible (see the response to question 3).


Question 2: Additional examples are mentioned in the introduction to the Arakawa-Kawasetsu paper you have linked. In particular, there is a large family of examples coming from $N=2$ superconformal field theory in 4 dimensions. This was mentioned as a conjecture in [Arakawa-Kawasetsu](https://arxiv.org/abs/1610.05865), but [Arakawa recently showed](https://arxiv.org/abs/1811.01577) that the associated varieties coincide with the Moore-Tachikawa varieties (as defined by [Braverman-Finkelberg-Nakajima](https://arxiv.org/abs/1706.02112)) of the 4d theories.


Question 3: Even in this special case, classification is hopeless, because it subsumes the problem of classifying all positive definite even lattices. Even the question of holomorphic $C\_2$-cofinite vertex operator algebras (those whose representation category is semisimple with only one irreducible object) is essentially impossible: The number of isomorphism classes in this special case is given by


* 1 for central charge 0 (the trivial one dimensional VOA)
* 1 for central charge 8 (the $E\_8$ lattice VOA)
* 2 for central charge 16 (from the $E\_8 \times E\_8$ and $D\_{16}^+$ lattices)
* 71 for central charge 24 (conjecturally)
* more than $1.1 \times 10^9$ for central charge 32


The first few figures are a theorem of [Dong and Mason](https://arxiv.org/abs/math/0203005), and the 71 is a conjecture of [Schellekens](https://arxiv.org/abs/hep-th/9205072) that is "[mostly solved](https://arxiv.org/abs/1804.02521)". The lower bound for $c=32$ is from [Oliver King](https://arxiv.org/abs/math/0012231) (top of page 17). We can get large lower bounds in large central charge because of the explosive growth of unimodular lattice isomorphism classes. 


Question 4: Character computations are scattered throughout the literature. For lattice vertex algebras, the characters are given by the modular functions $\Theta\_{L+\lambda}(\tau)/\eta^{\text{rank }L}(\tau)$, where $\lambda \in L^\vee/L$. For some elementary cases like minimal models, I think you can find them in conformal field theory texts. For other cases, I suggest a search (or asking experts for characters of specific objects rather than all of them).


