TY - JOUR
T1 - Virasoro Symmetries of the Extended Toda Hierarchy
JF - Comm. Math.\\nPhys. 250 (2004) 161-193.
Y1 - 2004
A1 - Boris Dubrovin
A1 - Zhang Youjin
AB - 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.
UR - http://hdl.handle.net/1963/2544
U1 - 1575
U2 - Mathematics
U3 - Mathematical Physics
ER -