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 $2$ (Part $1$)

TL;DR

This paper classifies irreducible weak modules for the fixed point subalgebra of a lattice vertex algebra, showing that in rank 1, all non-zero modules contain a non-zero submodule from a specific Heisenberg algebra fixed point subalgebra.

Contribution

It proves that for rank 1 lattices, every non-zero weak module over the fixed point subalgebra contains a submodule from the Heisenberg algebra fixed point subalgebra.

Findings

Every non-zero weak $V_{L}^{+}$-module contains a non-zero $M(1)^{+}$-module in rank 1.

Classification of irreducible weak modules for $V_{L}^{+}$.

Connection between modules of $V_{L}^{+}$ and submodules of $V_{L}$ or twisted modules.

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 $\theta$. In this series of papers, 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. In this paper (Part 1), we show that when the rank of $L$ is $1$, every non-zero weak $V_{L}^{+}$-module contains a non-zero $M(1)^{+}$-module, where $M(1)^{+}$ is the fixed point subalgebra of the Heisenberg vertex operator algebra $M(1)$ under the action of $\theta$.