Orbifolds of lattice vertex algebras under an isometry of order two

TL;DR

This paper classifies irreducible modules of orbifold vertex algebras formed by lattice automorphisms of order two, providing explicit examples involving root lattices and Dynkin diagram automorphisms.

Contribution

It offers a complete classification of irreducible modules for orbifold vertex algebras under order two lattice automorphisms, including detailed examples.

Findings

Classification of irreducible modules for $V_Q^\sigma$

Identification of modules as submodules of twisted or untwisted $V_Q$-modules

Explicit examples with root lattices and Dynkin automorphisms

Abstract

Every isometry $\sigma$ of a positive-definite even lattice $Q$ can be lifted to an automorphism of the lattice vertex algebra $V_Q$. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the $\sigma$-invariant subalgebra $V_Q^\sigma$ of $V_Q$, known as an orbifold. In the case when $\sigma$ is an isometry of $Q$ of order two, we classify the irreducible modules of the orbifold vertex algebra $V_Q^\sigma$ and identify them as submodules of twisted or untwisted $V_Q$-modules. The examples where $Q$ is a root lattice and $\sigma$ is a Dynkin diagram automorphism are presented in detail.