Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 085, 49 pages      arXiv:2111.04616      https://doi.org/10.3842/SIGMA.2022.085

Character Vectors of Strongly Regular Vertex Operator Algebras

Cameron Franc a and Geoffrey Mason b
a) McMaster University, Canada
b) UCSC, USA

Received December 11, 2021, in final form October 13, 2022; Published online October 29, 2022

Abstract
We summarize interactions between vertex operator algebras and number theory through the lens of Zhu theory. The paper begins by recalling basic facts on vertex operator algebras (VOAs) and modular forms, and then explains Zhu's theorem on characters of VOAs in a slightly new form. We then axiomatize the desirable properties of modular forms that have played a role in Zhu's theorem and related classification results of VOAs. After this we summarize known classification results in rank two, emphasizing the geometric theory of vector-valued modular forms as a means for simplifying the discussion. We conclude by summarizing some known examples, and by providing some new examples, in higher ranks. In particular, the paper contains a number of potential character vectors that could plausibly correspond to a VOA, but such that the existence of a corresponding hypothetical VOA is presently unknown.

Key words: vertex operator algebras; conformal field theory; modular forms.

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