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

Twisted Heisenberg-Virasoro vertex algebra $\mathcal{; ; H}; ; $ is an important example of vertex algebra with many applications in the representation theory and conformal field theory. If the level of the corresponding Heisenberg vertex subalgebra is non-zero, $\mathcal{; ; H}; ; $ is isomorphic to the tensor product of Heisenberg vertex algebra and the (universal or simple) Virasoro vertex algebra. The study of the twisted Heisenberg-Virasoro vertex algebra at the level zero was initiated by Y.\ Billig motivated by applications to the toroidal Lie algebras. It turns out that level zero has another interesting application - in realization of a class of vertex algebras which appear in physics literature under the name of Galilean algebra. I shall present a joint work with D.\ Adamovi\'{; ; c}; ; on bosonic realization of this algebra and its highest weight representations. Furthermore I will discuss on some of many applications of this realization.

vertex algebras, Heisenberg-Virasoro algebra, Galilean algebras

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