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 E8,2 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.

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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