Vertex operator algebras of rank 2 - the Mathur-Mukhi-Sen theorem revisited

Mason, Geoffrey; Nagatomo, Kiyokazu; Sakai, Yuichi

doi:10.4310/CNTP.2021.v15.n1.a2

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Vertex operator algebras of rank 2 - the Mathur-Mukhi-Sen theorem revisited

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Nagatomo, Kiyokazu
Max Planck Institute for Mathematics, Max Planck Society;

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https://dx.doi.org/10.4310/CNTP.2021.v15.n1.a2
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https://doi.org/10.48550/arXiv.1803.11281
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Citation

Mason, G., Nagatomo, K., & Sakai, Y. (2021). Vertex operator algebras of rank 2 - the Mathur-Mukhi-Sen theorem revisited. Communications in Number Theory and Physics, 15(1), 59-90. doi:10.4310/CNTP.2021.v15.n1.a2.

Cite as: https://hdl.handle.net/21.11116/0000-0008-15B8-1

Abstract

Let $V$ be a strongly regular vertex operator algebra and let $\frak{ch}_V$ be the space spanned by the characters of the irreducible $V$-modules.\ It is known that $\frak{ch}_V$ 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.