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Virasoro Symmetries of the Extended Toda Hierarchy

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Abstract

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 L m , 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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Authors and Affiliations

  1. SISSA, Via Beirut 2–4, 34014, Trieste, Italy

    Boris Dubrovin

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China

    Youjin Zhang

Authors
  1. Boris Dubrovin
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  2. Youjin Zhang
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Communicated by L. Takhtajan

Acknowledgements.The research of B.D. was partially supported by Italian Ministry of Education research grant Cofin2001 “Geometry of Integrable Systems”. The research of Y.Z. was partially supported by the Chinese National Science Fund for Distinguished Young Scholars grant No.10025101 and the Special Funds of Chinese Major Basic Research Project “Nonlinear Sciences”. Y.Z. thanks Abdus Salam International Centre for Theoretical Physics and SISSA where part of the work was done for their hospitality. The authors are grateful to the referee for the suggested improvements of the paper.

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Dubrovin, B., Zhang, Y. Virasoro Symmetries of the Extended Toda Hierarchy. Commun. Math. Phys. 250, 161–193 (2004). https://doi.org/10.1007/s00220-004-1084-9

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