#### Vol. 14, No. 6, 2020

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Classification of some vertex operator algebras of rank 3

### Cameron Franc and Geoffrey Mason

Vol. 14 (2020), No. 6, 1613–1667
##### Abstract

We discuss the classification of strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy. Our main theorem provides a classification of all such VOAs in the form of one infinite family of affine VOAs, one individual affine algebra and two Virasoro algebras, together with a family of eleven exceptional character vectors and associated data that we call the $U$-series. We prove that there are at least $15$ VOAs in the $U$-series occurring as commutants in a Schellekens list holomorphic VOA. These include the affine algebra ${E}_{8,2}$ and Höhn’s baby monster VOA ${VB}_{\left(0\right)}^{♮}$ but the other $13$ seem to be new. The idea in the proof of our main theorem is to exploit properties of a family of vector-valued modular forms with rational functions as Fourier coefficients, which solves a family of modular linear differential equations in terms of generalized hypergeometric series.

##### Keywords
vertex operator algebras, vector-valued modular forms, modular linear differential equations
##### Mathematical Subject Classification 2010
Primary: 17B69
Secondary: 11F03, 17B65