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Virasoro Symmetries of the Extended Toda Hierarchy

Boris Dubrovin·Youjin Zhang
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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 $L_m$, $m\geq -1$ 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 $CP^1$ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.

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Boris Dubrovin, Youjin Zhang. 2003-09-10. Virasoro Symmetries of the Extended Toda Hierarchy. https://doi.org/10.1007/s00220-004-1084-9

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