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Free field realization and weight representation theory of the twisted Heisenberg-Virasoro algebra

In this talk we discuss weight representations of the twisted Heisenberg-Virasoro Lie algebra at level zero H. By using free field realization of the Heisenberg-Virasoro vertex algebra LH(cL, cL, I) we construct highest weight H-modules and present explicit formulas for singular vectors in Verma modules over H. These formulas are crucial for proving irreducibility of certain tensor product modules. As a result, we get the fusion rules for an interesting subcategory of irreducible highest weight H-modules. In the end, we give a nontrivial homomorphism between the universal vertex-algebra LW (2, 2) (cW , cL) associated to Lie algebra W(2, 2) and LH(cL, cL, I). As a consequence, highest weight H-modules become W(2, 2)-modules. This talk is based on a recent joint work with Dražen Adamović.

vertex algebras, singular vectors, Heisenberg-Virasoro, tensor product modules

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