On Free Field Realizations of $W(2,2)$-Modules

TL;DR

This paper explores the relationship between modules of the twisted Heisenberg-Virasoro algebra and the $W(2,2)$-algebra, establishing conditions for irreducibility and constructing a screening operator to identify the $W(2,2)$ algebra within a vertex algebra framework.

Contribution

It demonstrates when irreducible highest weight modules of the twisted Heisenberg-Virasoro algebra remain irreducible as $W(2,2)$-modules and constructs a screening operator for the $W(2,2)$ algebra.

Findings

Irreducible highest weight modules are irreducible as $W(2,2)$-modules if and only if they are typical.

Constructed a screening operator whose kernel is the $W(2,2)$ vertex algebra.

Established a method to realize $W(2,2)$ within the Heisenberg-Virasoro vertex algebra.

Abstract

The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra $\mathcal H$ at level zero as modules for the $W(2,2)$-algebra by using construction from [J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:1405.1707]. We prove that the irreducible highest weight ${\mathcal H}$-module is irreducible as $W(2,2)$-module if and only if it has a typical highest weight. Finally, we construct a screening operator acting on the Heisenberg-Virasoro vertex algebra whose kernel is exactly $W(2,2)$ vertex algebra.