Classification of some vertex operator algebras of rank 3

Franc, Cameron; Mason, Geoffrey

doi:10.2140/ant.2020.14.1613

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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

DOI: 10.2140/ant.2020.14.1613

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 ${VB}_{(0)}^{♮}$ 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

Milestones

Received: 28 May 2019

Revised: 5 November 2019

Accepted: 10 February 2020

Published: 30 July 2020

Authors

Cameron Franc
University of Saskatchewan Saskatoon, SK Canada

Geoffrey Mason
University of California at Santa Cruz Santa Cruz, CA United States

Vol. 14, No. 6, 2020