Simple restricted modules over the Heisenberg-Virasoro algebra as VOA modules

TL;DR

This paper classifies all simple restricted modules over certain Heisenberg-Virasoro algebras, explores their VOA module structures, and provides examples of non-tensor product modules, advancing understanding of their representation theory.

Contribution

It provides a complete classification of simple restricted modules over mirror and twisted Heisenberg-Virasoro algebras, including VOA module interpretations and novel examples.

Findings

Classified all simple restricted modules over ${rak{D}}$ and $ar{rak{D}}$.

Characterized simple Whittaker and highest weight modules.

Presented examples of non-tensor product simple modules.

Abstract

In this paper, we determine all simple restricted modules over the mirror Heisenberg-Virasoro algebra ${\mathfrak{D}}$, and the twisted Heisenberg-Virasoro algebra $\bar\mathfrak{D}$ with nonzero level. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak{D}}$. A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg-Virasoro vertex operator algebras $\mathcal V^{c} \cong V_{Vir}^{c}\otimes M(1)$. We also present a few examples of simple restricted ${\mathfrak{D}}$-modules and $\bar\mathfrak{D}$-modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak{D}$ and $\bar\mathfrak{D}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.