Quasi-lisse vertex algebras were introduced by Arakawa and Kawasetsu in Quasi-lisse vertex algebras and modular linear differential equations . 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.


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, but Arakawa recently showed that the associated varieties coincide with the Moore-Tachikawa varieties (as defined by Braverman-Finkelberg-Nakajima) 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, and the 71 is a conjecture of Schellekens that is "mostly solved". The lower bound for $c=32$ is from Oliver King (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).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.