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.


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.)
Full Text: DOI arXiv


[1] G. Carlet, B. Dubrovin, and Y. Zhang, The extended Toda hierarchy , Mosc. Math. J. 4 , no. 2 (2004), 313–332. · Zbl 1076.37055
[2] B. Dubrovin and Y. Zhang, Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants , preprint,\arxivmath/0108160v1[math.DG]
[3] I. M. Gel’Fand [Gelfand] and L. A. Dikiĭ [Dickey], Fractional powers of operators, and Hamiltonian systems (in Russian), Funct. Anal. i Priložen 10 , no. 4 (1976), 13–29.; English translation in Funct. Anal. Appl. 10 (1976), 259–273. · Zbl 0346.35085
[4] E. Getzler, “The Toda conjecture” in Symplectic Geometry and Mirror Symmetry (Seoul, 2000) , World Sci., River Edge, N.J., 2001, 51–79. · Zbl 1047.37046
[5] A. Givental, \(A_n-1\) singularities and nKdV hierarchies , Mosc. Math. J. 3 , no. 2 (2003), 475–505. · Zbl 1054.14067
[6] V. G. Kac, Infinite-Dimensional Lie Algebras , 3rd ed., Cambridge Univ. Press, Cambridge, 1990. · Zbl 0716.17022
[7] K. Ueno and K. Takasaki, “Toda lattice hierarchy” in Group Representations and Systems of Differential Equations (Tokyo, 1982) , Adv. Stud. Pure Math. 4 , North-Holland, Amsterdam, 1984, 1–95.
[8] P. Van Moerbeke, “Integrable foundations of String theory” in Lectures on Integrable Systems (Sophia-Antipolis, France, 1991) , World Sci., River Edge, N.J., 1994, 163–267.
[9] Y. Zhang, On the CP \(^1\) topological sigma model and the Toda lattice hierarchy, J. Geom. Phys. 40 (2002), 215–232. · Zbl 1001.37066 · doi:10.1016/S0393-0440(01)00036-5
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