2003/08/31 by Boris Dubrovin, Youjin Zhang · 1 citation
Mathematics · Physics and Astronomy · #math.DG #math-ph #math.MP
paper · pdf · doi:10.1007/s00220-004-1084-9
A remark at the end of Section 5 is added; more detailed explanations in Appendix; references added
arxiv created 2003/09/10 · arxiv updated 2009/12/01
We prove that the extended Toda hierarchy of \citeCDZ admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators Lm, m≥ -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 CP1 Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.