Advertisement

Three-Partition Hodge Integrals and the Topological Vertex

  • Toshio NakatsuEmail author
  • Kanehisa Takasaki
Article
  • 19 Downloads

Abstract

We give a new proof of the equivalence between the cubic Hodge integrals and the topological vertex in topological string theory. A central role is played by the notion of generalized shift symmetries in a fermionic realization of the two-dimensional quantum torus algebra. These algebraic relations of operators in the fermionic Fock space are used to convert generating functions of the cubic Hodge integrals and the topological vertex to each other. As a byproduct, the generating function of the cubic Hodge integrals at special values of the parameters \(\overrightarrow{w}\) therein is shown to be a tau function of the generalized KdV (aka Gelfand-Dickey) hierarchies.

Notes

Acknowledgements

This work is supported by JSPS KAKENHI Grant Numbers JP15K04912 and JP18K03350.

References

  1. 1.
    Graber, T., Pandharipande, R.: Localization of virtual classes. Invent. Math. 135, 487–518 (1999)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Kontsevich, M.: Enumeration of rational curves via torus actions, Progr. Math. 129, Birkhäuser, Boston, Boston, MA, pp. 335–368 (1995)Google Scholar
  3. 3.
    Li, J., Liu, C.-C.M., Liu, K., Zhou, J.: A mathematical theory of the topological vertex. Geom. Topol. 13, 527–621 (2009). arXiv:math/0408426v3 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Aganagic, M., Klemm, A., Mariño, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425–478 (2005). arXiv:hep-th/0305132 ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Liu, C.-C., Liu, K., Zhou, J.: On a proof of a conjecture of Mariño-Vafa on Hodge Integrals. Math. Res. Lett. 11(2–3), 259–272 (2004). arXiv:math/0306257 ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Liu, C.-C., Liu, K., Zhou, J.: A proof of a conjecture of Mariño-Vafa on Hodge Integrals. J. Differ. Geom. 65, 289–340 (2003). arXiv:math/0306434 CrossRefGoogle Scholar
  7. 7.
    Liu, C.-C.M., Liu, K., Zhou, J.: A formula of two-partition Hodge integrals. J. Am. Math. Soc. 20(1), 149–184 (2007). arXiv:math/0310272v3 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Okounkov, A., Pandharipande, R.: Hodge integrals and invariants of the unknot. Geom. Topol. 8, 675–699 (2004). arXiv:math/0307209 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mariño, M., Vafa, C.: Framed knots at large N, Contemp. Math. 310 (Amer. Math. Soc., Providence, RI), pp. 185–204, (2002). arXiv:hep-th/0108064
  10. 10.
    Ekedahl, T., Lando, S., Shapiro, M., Vainshtein, A.: Hurwitz numbers and intersections on moduli spaces of curves. Invent. Math. 146, 297–327 (2001). arXiv:math/0004096 ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten Theory and Donaldson–Thomas Theory, I. Compos. Math. 142(5), 1263–1285 (2006). arXiv:math/0312059 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten Theory and Donaldson–Thomas Theory, II. Compos. Math. 142(5), 1286–1304 (2006). arXiv:math/0406092 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Okounkov, A., Reshetikhin, N., Vafa, C.: Quantum Calabi-Yau and classical crystals, The unity of mathematics, pp. 597–618, Progr. Math., 244, Birkhäuser Boston, Boston, MA, (2006). arXiv:hep-th/0309208
  14. 14.
    Maulik, D., Oblomkov, A., Okounkov, A., Pandharipande, R.: Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds. Invent. Math. 186(2), 435–479 (2011). arXiv:0809.3976 ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Nakatsu, T., Takasaki, K.: Melting crystal, quantum torus and Toda hierarchy. Commun. Math. Phys. 285, 445 (2009). arXiv:0710.5339 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Nakatsu, T., Takasaki, K.: Integrable structure of melting crystal model with external potentials. Adv. Stud. Pure Math. vol. 59, pp. 201–223 (Mathematical Society of Japan, Tokyo) (2010). arXiv:0807.4970 [math-ph]
  17. 17.
    Takasaki, K., Nakatsu, T.: Open string amplitudes of closed topological vertex. J. Phys. A Math. Theor. 49, 025403 (2016). arXiv:1507.07053 [math-ph]MathSciNetCrossRefGoogle Scholar
  18. 18.
    Takasaki, K.: Generalized Ablowitz–Ladik hierarchy in topological string theory. J. Phys. A Math. Theor. 47, 165201 (2014). arXiv:1312.7184 [math-ph]ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Takasaki, K., Nakatsu, T.: \(q\)-difference Kac–Schwarz operators in topological string theory. SIGMA 13, 009 (2017). arXiv:1609.00882 [math-ph]MathSciNetzbMATHGoogle Scholar
  20. 20.
    Nakatsu, T., Takasaki, K.: to appearGoogle Scholar
  21. 21.
    Aganagic, M., Dijkgraaf, R., Klemn, A., Mariño, M., Vafa, C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451–516 (2006). arXiv:hep-th/0312085 ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhou, J.: Hodge integrals and integrable hierarchies. Lett. Math. Phys. 93, 55–71 (2010). arXiv:math/0310408 ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  24. 24.
    Iqbal, A., Kzçaz, C., Vafa, C.: The refined topological vertex. JHEP 0910, 069 (2009). arXiv:hep-th/0701156 ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Taki, M.: Refined topological vertex and instanton counting. JHEP 0803, 048 (2008). arXiv:0710.1776 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Dickey, L.A.: Soliton Equations and Hamiltonian System. World Scientific, Singapore (2003)CrossRefGoogle Scholar
  27. 27.
    Goulden, I.P., Jackson, D.M.: Transitive factorizations into transpositions and holomorphic mappings on the sphere. Proc. Am. Math. Soc. 125, 51–60 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for Fundamental SciencesSetsunan UniversityNeyagawaJapan
  2. 2.Department of MathematicsKindai UniversityHigashi-OsakaJapan

Personalised recommendations