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

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.