Free field realization of the twisted Heisenberg-Virasoro algebra at level zero and its applications

TL;DR

This paper develops a free field realization of the twisted Heisenberg-Virasoro algebra at level zero, describing its representations, singular vectors, and fusion rules, with applications to the $W(2,2)$-algebra.

Contribution

It provides a complete free field realization of the algebra, constructs singular vectors via vertex algebra methods, and solves the irreducibility and fusion problems for modules.

Findings

Complete structure of Fock representations described

Singular vectors constructed as Schur polynomials

Fusion rules for a subcategory determined

Abstract

We investigate the free fields realization of the twisted Heisenberg-Virasoro algebra $\mathcal{H}$ at level zero. We completely describe the structure of the associated Fock representations. Using vertex-algebraic methods and screening operators we construct singular vectors in certain Verma modules as Schur polynomials. We completely solve the irreducibility problem for tensor product of irreducible highest weight modules with intermediate series. We also determine the fusion rules for an interesting subcategory of $\mathcal{H}$-modules. Finally, as an application we present a free field realization of the $W(2,2)$-algebra and interpret the $W(2,2)$-singular vectors as $\mathcal{H}$-singular vectors in Verma modules.