Fourvolutions and automorphism groups of orbifold lattice vertex operator algebras

TL;DR

This paper determines the automorphism groups of orbifold lattice vertex operator algebras associated with even positive definite lattices with no roots, focusing on cases with specific order 4 automorphisms.

Contribution

It explicitly computes the automorphism group of $V_L^{ ilde{g}}$ for a class of orbifold VOAs, extending understanding of their symmetries.

Findings

Automorphism group is isomorphic to a normalizer quotient for most lattices.

Special cases include lattices isomorphic to $ frac{1}{ oot2}E_8$ or $BW_{16}$.

Provides a classification of automorphism groups for these orbifold VOAs.

Abstract

Let $L$ be an even positive definite lattice with no roots, i.e., $L(2)=\{x\in L\mid (x|x)=2\}=\emptyset$. Let $g\in O(L)$ be an isometry of order $4$ such that $g^2=-1$ on $L$. In this article, we determine the full automorphism group of the orbifold vertex operator algebra $V_L^{\hat{g}}$. As our main result, we show that $Aut(V_L^{\hat{g}})$ is isomorphic to $N_{Aut(V_L)}(\langle \hat{g}\rangle)/ \langle\hat{g}\rangle $ unless $L\cong \sqrt{2}E_8$ or $BW_{16}$.