Communications in Mathematical Physics

, Volume 250, Issue 1, pp 161–193

Virasoro Symmetries of the Extended Toda Hierarchy

  • Boris Dubrovin
  • Youjin Zhang


We prove that the extended Toda hierarchy of [1] admits a nonabelian Lie algebra of infinitesimal symmetries isomorphic to 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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Boris Dubrovin
    • 1
  • Youjin Zhang
    • 2
  1. 1.SISSATriesteItaly
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingP.R. China

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