Simple Whittaker modules over free bosonic orbifold vertex operator algebras

TL;DR

This paper constructs and classifies simple weak modules over the fixed-point subalgebra of a higher rank free bosonic vertex operator algebra, extending previous results to a higher rank setting.

Contribution

It introduces a higher rank generalization of Whittaker modules over $M(1)^+$ and proves their simplicity and classification as Virasoro modules.

Findings

Constructed simple weak modules from higher rank Whittaker modules.

Calculated the Whittaker type for Virasoro modules.

Showed all Virasoro Whittaker modules arise from this construction.

Abstract

We construct weak (i.e. non-graded) modules over the vertex operator algebra $M(1)^+$, which is the fixed-point subalgebra of the higher rank free bosonic (Heisenberg) vertex operator algebra with respect to the $-1$ automorphism. These weak modules are constructed from Whittaker modules for the higher rank Heisenberg algebra. We prove that the modules are simple as weak modules over $M(1)^+$ and calculate their Whittaker type when regarded as modules for the Virasoro Lie algebra. Lastly, we show that any Whittaker module for the Virasoro Lie algebra occurs in this way. These results are a higher rank generalization of some results by Tanabe.