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Hirota quadratic equations for the extended Toda hierarchy. (English) Zbl 1188.14037

Integrable hierarchies have many interesting applications in algebraic geometry, and in particular for purposes of computing Gromov-Witten (GW) invariants of curves. It is known that vertex operators are instrumental for writing Hirota equation of the KdV hierachy satisfied by the tau-function. It is also known (due to E. Getzler [in: Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14–18, 2000. Singapore: World Scientific. 51–79 (2001; Zbl 1047.37046)] and Y. Zhang [J. Geom. Phys. 40, 215–232 (2002; Zbl 1001.37066)]) that extended Toda hierarchy (ETH) governs GW-invariants of the projective line. The aim of the paper is to present a (new) vertex operator approach for computing Hirota quadratic equations for the mentioned ETH. The main results are Theorem 1.1 and Corollary 1.2.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
17B69 Vertex operators; vertex operator algebras and related structures
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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References:

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