%0 Journal Article %J Comm. Math.\\nPhys. 250 (2004) 161-193. %D 2004 %T Virasoro Symmetries of the Extended Toda Hierarchy %A Boris Dubrovin %A Zhang Youjin %X 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. %B Comm. Math.\\nPhys. 250 (2004) 161-193. %G en_US %U http://hdl.handle.net/1963/2544 %1 1575 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-21T12:53:52Z\\nNo. of bitstreams: 1\\n0308152v2.pdf: 341202 bytes, checksum: 306b1f696e6ec01c0eabbeeeca895290 (MD5) %R 10.1007/s00220-004-1084-9