Classification of some vertex operator algebras of rank 3
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 $\mathbf{VB}^\natural_{(0)}$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2019
- DOI:
- 10.48550/arXiv.1905.07500
- arXiv:
- arXiv:1905.07500
- Bibcode:
- 2019arXiv190507500F
- Keywords:
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- Mathematics - Quantum Algebra;
- Mathematical Physics;
- Mathematics - Number Theory
- E-Print:
- 52 pages