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# Communications in Number Theory and Physics

## Volume 15 (2021)

### Number 1

### Vertex operator algebras of rank $2$: The Mathur–Mukhi–Sen theorem revisited

Pages: 59 – 90

DOI: https://dx.doi.org/10.4310/CNTP.2021.v15.n1.a2

#### Authors

#### Abstract

Let $V$ be a strongly regular vertex operator algebra and let $\mathfrak{ch}_V$ be the space spanned by the characters of the irreducible $V$-modules. It is known that $\mathfrak{ch}_V$ is the space of solutions of a so-called modular linear differential equation (MLDE). In this paper we obtain a classification of those $V$ for which the corresponding MLDE is irreducible and monic of order $2$. It turns out that $V$ is either one of seven affine Kac–Moody algebras of level $1$, or the Yang–Lee Virasoro VOA of central charge $c = - 22/5$. Our proof establishes new connections between the characters of $V$ and Gauss hypergeometric series, and as a Corollary of our classification we complete the work of Mathur, Mukhi and Sen who considered a closely related problem thirty years ago.

#### Keywords

strongly regular vertex operator algebra, modular linear differential equation, hypergeometric series

#### 2010 Mathematics Subject Classification

Primary 17B69. Secondary 33C05.

The first-named author was supported by the Simons Foundation #427007.

The second-named author was supported in part by JSPS KAKENHI #JP17K05171, the Max-Planck Institute für Mathematik, Germany, and the International Center for Theoretical Physics, Italy.

The third-named author was supported in part by JSPS KAKENHI #18K03215 and 16H06336.

Received 17 August 2018

Accepted 7 September 2020