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



Geoffrey Mason (Department of Mathematics, University of California, Santa Cruz, Calif., U.S.A.)

Kiyokazu Nagatomo (Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, Japan)

Yuichi Sakai (Multiple Zeta Research Center, Kyushu University, Fukuoka, Japan)


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.


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

Published 4 January 2021