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January 2023 Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra
Sven Möller, Nils Scheithauer
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Ann. of Math. (2) 197(1): 221-288 (January 2023). DOI: 10.4007/annals.2023.197.1.4

Abstract

We prove a dimension formula for the weight-$1$ subspace of a vertex operator algebra $V^{\mathrm{orb}}(g)$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with a finite-order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $\mathrm{Aut}(V)$.

Then we show that the orbifold construction defines a bijection between the generalised deep holes of the Leech lattice vertex operator algebra $V_\Lambda$ with non-trivial fixed-point Lie subalgebra and the strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-$1$ space. This provides a uniform construction of these vertex operator algebras and naturally generalises the correspondence between the deep holes of the Leech lattice $\Lambda$ and the 23 Niemeier lattices with non-vanishing root system found by Conway, Parker, Sloane and Borcherds.

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Sven Möller. Nils Scheithauer. "Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra." Ann. of Math. (2) 197 (1) 221 - 288, January 2023. https://doi.org/10.4007/annals.2023.197.1.4

Information

Published: January 2023
First available in Project Euclid: 22 November 2022

Digital Object Identifier: 10.4007/annals.2023.197.1.4

Subjects:
Primary: 11F11 , 11F27 , 17B69

Keywords: deep holes , dimension formula , Eisenstein series , Leech lattice , modular forms , orbifold construction , vertex algebras , Weil representation

Rights: Copyright © 2023 Department of Mathematics, Princeton University

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Vol.197 • No. 1 • January 2023
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