Simple weak modules for the fixed point subalgebra of the Heisenberg vertex operator algebra of rank $1$ by an automorphism of order $2$ and Whittaker vectors

TL;DR

This paper classifies certain simple weak modules over the fixed point subalgebra of the rank 1 Heisenberg vertex operator algebra, revealing they are either modules over the original algebra or its twisted version.

Contribution

It provides a classification of simple weak modules with specific properties for the fixed point subalgebra of the rank 1 Heisenberg VOA under an order 2 automorphism.

Findings

Any such module is isomorphic to a simple weak module of the original or twisted VOA.

The classification includes modules with specific eigenvector properties for the Virasoro element.

The result extends understanding of module structures for fixed point subalgebras in VOA theory.

Abstract

Let $M(1)$ be the vertex operator algebra with the Virasoro element $\omega$ associated to the Heisenberg algebra of rank $1$ and let $M(1)^{+}$ be the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of order $2$. We classify the simple weak $M(1)^{+}$-modules with a non-zero element $w$ such that for some integer $s\geq 2$, $\omega_i w\in{\mathbb C}w$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $\omega_{s}w\in{\mathbb C}^{\times}w$, and $\omega_i w=0$ for all $i>s$. The result says that any such simple weak $M(1)^{+}$-module is isomorphic to some simple weak $M(1)$-module or to some $\theta$-twisted simple weak $M(1)$-module.