Virasoro Symmetries of the Extended Toda Hierarchy

Abstract

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 $L_m$, $m\ge -1$ 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 $C{P}^1$ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.

arXiv commentA remark at the end of Section 5 is added; more detailed explanations in Appendix; references added

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• Gromov-Witten classes, quantum cohomology, and enumerative geometry
• Quantum Cohomology and Virasoro Algebra
• Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation
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Cited by in corpus (29)

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• Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations
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• On Equivariant Gromov--Witten Invariants of Resolved Conifold with Diagonal and Anti-Diagonal Actions
• The Sigma Form for the PII Hierarchy
• Darboux transformations and Fay identities for the extended bigraded Toda hierarchy
• Dressing operators in equivariant Gromov-Witten theory of ${\mathbb{CP}}^1$
• A strongly coupled extended Toda hierarchy and its Virasoro symmetry
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