Vertex Operator Algebras with Two Simple Modules - the Mathur-Mukhi-Sen Theorem Revisited

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 \emph{modular linear differential equation (MLDE)}.\ In this paper we obtain a near-classification of those $V$ for which the corresponding MLDE is irreducible and monic of order $2$.\ As a consequence we derive the complete classification when $V$ has exactly two simple modules.\ It turns out that $V$ is either one of four affine Kac-Moody algebras of level $1$, or the Yang-Lee Virasoro model of central charge $-22/5$.\ Our proof establishes new connections between the characters of $V$ and Gauss hypergeometric series, and puts the finishing touches to work of Mathur, Mukhi and Sen who first considered this problem forty years ago.

References in corpus (1)

• Modular invariance of trace functions in orbifold theory

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