Classification of irreducible modules of the vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of an arbitrary rank

TL;DR

This paper classifies all irreducible modules of the vertex algebra $V_L^+$ for negative definite and nondegenerate even lattices of arbitrary and finite rank, revealing their structure as submodules of twisted modules.

Contribution

It provides a comprehensive classification of irreducible modules of $V_L^+$ for various lattices, extending previous results to arbitrary rank and nondegenerate cases.

Findings

Irreducible $V_L^+$-modules are submodules of twisted $V_L$-modules.

Complete classification for negative definite lattices of arbitrary rank.

Extension of classification to nondegenerate even lattices of finite rank.

Abstract

In this paper, we first classify all irreducible modules of the vertex algebra $V_L^+$ when $L$ is a negative definite even lattice of arbitrary rank. In particular, we show that any irreducible $V_L^+$-module is isomorphic to a submodule of an irreducible twisted $V_L$-module. We then extend this result to a vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of finite rank.