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

We investigate the free field realization of the twisted Heisenberg–Virasoro algebra HH 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 the tensor products of irreducible highest weight modules with intermediate series. We also determine the fusion rules for an interesting subcategory of HH-modules. Finally, as an application we present a free-field realization of the W(2,2)W(2,2)-algebra and interpret the W(2,2)W(2,2)-singular vectors as HH-singular vectors in Verma modules.