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Virasoro Symmetries of the Extended Toda Hierarchy

TitleVirasoro Symmetries of the Extended Toda Hierarchy
Publication TypeJournal Article
Year of Publication2004
AuthorsDubrovin, B, Youjin, Z
JournalComm. Math.\\nPhys. 250 (2004) 161-193.

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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