Abstract
The generating function of cubic Hodge integrals satisfying the local Calabi–Yau condition is conjectured to be a tau function of a new integrable system which can be regarded as a fractional generalization of the Volterra lattice hierarchy, so we name it the fractional Volterra hierarchy. In this paper, we give the definition of this integrable hierarchy in terms of Lax pair and Hamiltonian formalisms, construct its tau functions, and present its multi-soliton solutions.
Similar content being viewed by others
References
Buryak, A.: Dubrovin–Zhang hierarchy for the Hodge integrals. Commun. Number Theory Phys. 9, 239–272 (2015)
Buryak, A.: ILW equation for the Hodge integrals revisited. Math. Res. Lett. 23, 675–683 (2016)
Carlet, G.: The extended bigraded Toda hierarchy. J. Phys. A 39, 9411–9435 (2006)
Dubrovin, B., Yang, D.: Private communications (2016)
Dubrovin, B., Liu, S.-Q., Yang, D., Zhang, Y.: Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs. Adv. Math. 293, 382–435 (2016)
Dubrovin, B., Liu, S.-Q., Yang, D., Zhang, Y.: Hodge-GUE correspondence and the discrete KdV equation, arXiv:1612.02333 [math-ph]
Ekedahl, T., Lando, S., Shapiro, M., Vainshtein, A.: Hurwitz numbers and intersections on moduli spaces of curves. Invent. Math. 146, 297–327 (2001)
Gu, C., Hu, H., Zhou, Z.: Darboux Transformations in Integrable Systems. Theory and Their Applications to Geometry. Mathematical Physics Studies, vol. 26. Springer, Dordrecht (2005)
Hall, B.: Lie Groups, Lie Algebras, and Representations. An Elementary Introduction. Graduate Texts in Mathematics, vol. 222, 2nd edn. Springer, Cham (2015)
Hu, X.-B., Zhao, J.-X., Li, C.-X.: Matrix integrals and several integrable differential-difference systems. J. Phys. Soc. Jpn. 75, 054003 (2006)
Inoue, R., Hikami, K.: Construction of soliton cellular automaton from the vertex model-the discrete 2D Toda equation and the Bogoyavlensky lattice. J. Phys. A 32, 6853–6868 (1999)
Kupershmidt, B.A.: Discrete Lax equations and differential-difference calculus. Astérisque No. 123 (1985)
Liu, C.-C.M., Liu, K., Zhou, J.: A proof of a conjecture of Mariño–Vafa on Hodge integrals. J. Differ. Geom. 65, 289–340 (2003)
Liu, S.-Q., Petrov, F., Dotsenko, V.: Two combinatorial identities. Math Overflow. http://mathoverflow.net/questions/261414
Mariño, M., Vafa, C.: Framed knots at large N. In: Adam, A., Morava, J., Ruan, Y. (eds) Orbifolds in Mathematics and Physics: Proceedings of a Conference on Mathematical Aspects of Orbifold String Theory May 4–8, 2001, University of Wisconsin, Madison, WI. Contemporary Mathematics, vol. 310, pp. 185–204. American Mathematical Society, Providence, RI (2002)
Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer, Berlin (1991)
Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Artin, M., Tate, J. (eds) Arithmetic and Geometry. Progress in Mathematics, vol. 36, pp. 271–328. Birkhäuser, Boston, MA (1983)
Okounkov, A., Pandharipande, R.: Hodge integrals and invariants of the unknot. Geom. Topol. 8, 675–699 (2004)
Smirnov, S.V.: Semidiscrete Toda lattices. Theor. Math. Phys. 172, 1217–1231 (2012)
Ueno, K., Takasaki, K.: Toda lattice hierarchy. In: Okamoto, K. (ed) Group Representations and Systems of Differential Equations. Advanced Studies in Pure Mathematics, vol. 4. North-Holland/Kinokuniya, Amsterdam (1984)
Acknowledgements
We are grateful to Boris Dubrovin and Di Yang for sharing with us their discovery of the relation of the special cubic Hodge integrals with Eq. (1.8) and for helpful discussions. We would also like to thank Fedor Petrov and Vladimir Dotsenko for their proof of the four identities given in the end of Sect. 3, and the referee for the suggestion to simplify some proofs of the paper. This work is supported by NSFC No. 11371214 and No. 11471182.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, SQ., Zhang, Y. & Zhou, C. Fractional Volterra hierarchy. Lett Math Phys 108, 261–283 (2018). https://doi.org/10.1007/s11005-017-1006-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-017-1006-3