Representations of 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 proves that all weak modules over the fixed point subalgebra of a vertex algebra associated with a non-degenerate even lattice are completely reducible, enhancing understanding of their representation theory.

Contribution

It establishes the complete reducibility of weak modules for the fixed point subalgebra under an automorphism of order 2, a new result in vertex algebra theory.

Findings

All weak $V_{L}^{+}$-modules are completely reducible.

Provides a classification framework for modules over fixed point subalgebras.

Advances understanding of automorphism-induced subalgebra representations.

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 $-1$-isometry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $\theta$. We show that every weak $V_{L}^{+}$-module is completely reducible.