Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 037, 18 pages      arXiv:2110.03317

Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals

Si-Qi Liu, Zhe Wang and Youjin Zhang
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China

Received October 28, 2021, in final form May 15, 2022; Published online May 18, 2022

We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the intermediate long wave hierarchy, and the remaining flows coincide with a certain limit of the flows of the fractional Volterra hierarchy which controls the special cubic Hodge integrals.

Key words: integrable hierarchy; limit fractional Volterra hierarchy; intermediate long wave hierarchy.

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  1. Bouchard V., Mariño M., Hurwitz numbers, matrix models and enumerative geometry, in From Hodge Theory to Integrability and TQFT tt*-Geometry, Proc. Sympos. Pure Math., Vol. 78, Amer. Math. Soc., Providence, RI, 2008, 263-283, arXiv:0709.1458.
  2. Brini A., The local Gromov-Witten theory of $\mathbb{C}\mathbb{P}^1$ and integrable hierarchies, Comm. Math. Phys. 313 (2012), 571-605, arXiv:1002.0582.
  3. Buryak A., Dubrovin-Zhang hierarchy for the Hodge integrals, Commun. Number Theory Phys. 9 (2015), 239-272, arXiv:1308.5716.
  4. Buryak A., Rossi P., Simple Lax description of the ILW hierarchy, SIGMA 14 (2018), 120, 7 pages, arXiv:1809.00271.
  5. Carlet G., Dubrovin B., Zhang Y., The extended Toda hierarchy, Mosc. Math. J. 4 (2004), 313-332, arXiv:nlin.SI/0306060.
  6. Dubrovin B., Liu S.-Q., Yang D., Zhang Y., Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs, Adv. Math. 293 (2016), 382-435, arXiv:1409.4616.
  7. Getzler E., The Toda conjecture, in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, 51-79, arXiv:math.AG/0108108.
  8. Gosper Jr. R.W., Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA 75 (1978), 40-42.
  9. Liu C.-C.M., Liu K., Zhou J., A proof of a conjecture of Mariño-Vafa on Hodge integrals, J. Differential Geom. 65 (2003), 289-340, arXiv:math.AG/0306434.
  10. Liu C.-C.M., Liu K., Zhou J., Mariño-Vafa formula and Hodge integral identities, J. Algebraic Geom. 15 (2006), 379-398, arXiv:math.AG/0308015.
  11. Liu S.-Q., Yang D., Zhang Y., Zhou C., The Hodge-FVH correspondence, J. Reine Angew. Math. 775 (2021), 259-300, arXiv:1906.06860.
  12. Liu S.-Q., Zhang Y., Zhou C., Fractional Volterra hierarchy, Lett. Math. Phys. 108 (2018), 261-283, arXiv:1702.02840.
  13. Mariño M., Vafa C., Framed knots at large $N$, in Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math., Vol. 310, Amer. Math. Soc., Providence, RI, 2002, 185-204, arXiv:hep-th/0108064.
  14. Mulase M., Zhang N., Polynomial recursion formula for linear Hodge integrals, Commun. Number Theory Phys. 4 (2010), 267-293, arXiv:0908.2267.
  15. Mumford D., Towards an enumerative geometry of the moduli space of curves, in Arithmetic and Geometry, Vol. II, Progr. Math., Vol. 36, Birkhäuser Boston, Boston, MA, 1983, 271-328.
  16. Okounkov A., Pandharipande R., The equivariant Gromov-Witten theory of ${\bf P}^1$, Ann. of Math. 163 (2006), 561-605, arXiv:math.AG/0207233.
  17. Takasaki K., Toda hierarchies and their applications, J. Phys. A: Math. Theor. 51 (2018), 203001, 35 pages, arXiv:1801.09924.
  18. Takasaki K., Cubic Hodge integrals and integrable hierarchies of Volterra type, in Integrability, Quantization, and Geometry. I. Integrable Systems, Proc. Sympos. Pure Math., Vol. 103, Amer. Math. Soc., Providence, RI, 2021, 481-502, arXiv:1909.13095.
  19. Toda M., Vibration of a chain with nonlinear interaction, J. Phys. Soc. Japan 22 (1967), 431-436.
  20. Ueno K., Takasaki K., Toda lattice hierarchy, in Group Representations and Systems of Differential Equations (Tokyo, 1982), Adv. Stud. Pure Math., Vol. 4, North-Holland, Amsterdam, 1984, 1-95.

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