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Free field realization of twisted Heisenberg-Virasoro algebra at level zero and $W(2,2)$-algebra

We present a free field representation of twisted Heisenberg- Virasoro algebra at level zero H. We consider a screening operator Q acting on a rank two Heisenberg algebra and associated Fock space such that ker Q extends Heisenberg-Virasoro vertex algebra. In this way we obtain a large family of highest weight representations, explicit formulas for singular vectors in Verma modules, and fusion rules for an interesting subcategory of these modules. In order to realize the missing representations we consider a deformed action of H on a Fock space and on Whittaker modules. In this process we construct logarithmic modules with different types of highest weight modules as subquotients. In particular we show that there exists a non-split self-extension of irreducible self-dual module which is a logarithmic module of rank two. Furthermore, we consider a W(2, 2) algebra which can (as a vertex algebra) be embedded in H. We provide branching rules and a screening operator whose kernel is a W(2, 2) VOA. This is a joint work with Dražen Adamović.

vertex algebras, Heisenberg-Virasoro algebra, intertwining operators, screening operator

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