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Virasoro Symmetries of the Extended Toda Hierarchy

Abstract

We prove that the extended Toda hierarchy of \cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators LmL_m, m1m\geq -1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the CP1CP^1 Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.Comment: A remark at the end of Section 5 is added; more detailed explanations in Appendix; references adde

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    Last time updated on 26/02/2019