The Automorphism Group of the Vertex Operator Algebra $V_L^+$ for an even lattice $L$ without roots

TL;DR

This paper investigates the automorphism group of a specific class of vertex operator algebras derived from even lattices without roots, identifying its structure for several notable lattice cases.

Contribution

It determines the shape of the automorphism group of $V_L^+$ for certain even lattices, including unimodular and root lattices, expanding understanding of their symmetries.

Findings

Automorphism group shape for unimodular lattices without roots

Automorphism group shape for lattices of the form √2R with R of type ADE

Automorphism group shape for the Barnes-Wall lattice of rank 16

Abstract

The automorphism group of the vertex operator algebra $V_L^+$ is studied by using its action on isomorphism classes of irreducible $V_L^+$-modules. In particular, the shape of the automorphism group of $V_L^+$ is determined when $L$ is isomorphic to an even unimodular lattice without roots, $\sqrt2R$ for an irreducible root lattice $R$ of type $ADE$ and the Barnes-Wall lattice of rank 16.