The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order $2$

TL;DR

This paper classifies all irreducible weak modules for the fixed point subalgebra of a vertex algebra associated with a non-degenerate even lattice under an automorphism of order 2, revealing their structure in relation to modules of the original algebra.

Contribution

It provides a complete classification of irreducible weak modules for the fixed point subalgebra, connecting them to modules of the original lattice vertex algebra and its twisted modules.

Findings

All irreducible weak modules are submodules of modules from the original algebra or twisted modules.

The classification clarifies the module structure under the automorphism of order 2.

The results extend understanding of module categories for lattice vertex algebras.

Abstract

Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $\theta$ the automorphism of $V_{L}$ induced from the $-1$ symmetry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $\theta$. We classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $\theta$-twisted $V_{L}$-module.