Skip to main content
Log in

The Remodeling Conjecture and the Faber–Pandharipande Formula

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this note, we prove that the free energies F g constructed from the Eynard–Orantin topological recursion applied to the curve mirror to \({\mathbb{C}^3}\) reproduce the Faber–Pandharipande formula for genus g Gromov–Witten invariants of \({\mathbb{C}^3}\) . This completes the proof of the remodeling conjecture for \({\mathbb{C}^3}\) .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Aganagic M., Klemm A., Mariño M., Vafa C.: The topological vertex. Commun. Math. Phys. 254, 425–478 (2005) arXiv:hep-th/0305132

    Article  ADS  MATH  Google Scholar 

  2. Bouchard V., Klemm A., Mariño M., Pasquetti S.: Remodeling the B-model. Commun. Math. Phys. 287, 117 (2009) arXiv:0709.1453 [hep-th]

    Article  ADS  MATH  Google Scholar 

  3. Bouchard V., Klemm A., Mariño M., Pasquetti S.: Topological open strings on orbifolds. Commun. Math. Phys. 296, 589 (2010) arXiv:0807.0597 [hep-th]

    Article  ADS  MATH  Google Scholar 

  4. Bouchard, V., Mariño, M.: Hurwitz numbers, matrix models and enumerative geometry. In: From Hodge Theory to Integrability and tQFT: tt*-geometry. In: Proceedings of Symposia in Pure Mathematics. AMS (2008). arXiv:0709.1458v2 [math.AG]

  5. Bouchard, V., Sułkowski, P.: Topological recursion and mirror curves. arXiv: 1105.2052v1 [hep-th]

  6. Chen, L.: Bouchard-Klemm-Mariño-Pasquetti Conjecture for C**3. arXiv:0910.3739 [math.AG]

  7. Eynard, B.: All orders asymptotic expansion of large partitions. J. Stat. Mech. P07023 (2008). arXiv:0804.0381v2 [math-ph]

  8. Eynard, B.: A matrix model for plane partitions and (T)ASEP. J. Stat. Mech. 0910, P10011 (2009). arXiv:0905.0535 [math-ph]

  9. Eynard, B.: Intersection numbers of spectral curves. arXiv:1104.0176v2 [math-ph]

  10. Eynard, B., Kozcaz, C.: Mirror of the refined topological vertex from a matrix model. arXiv:1107.5181v1 [hep-th]

  11. Eynard, B., Mulase, M., Safnuk, B.: The Laplace transform of the cut-and-join equation and the Bouchard-Mariño conjecture on Hurwitz numbers. arXiv:0907.5224v3 [math.AG]

  12. Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Numb. Theor. Phys. 1, 347–452 (2007). arXiv:math-ph/0702045v4

    Google Scholar 

  13. Eynard, B., Orantin, N.: Algebraic methods in random matrices and enumerative geometry. arXiv:0811.3531v1 [math-ph]

  14. Faber, C., Pandharipande, R.: Hodge integrals and Gromov–Witten theory. Invent. Math. 139, 174–199 (2000). arXiv:math/9810173v1 [math.AG

    Google Scholar 

  15. Fay, J.: Theta functions on Riemann surfaces. In: Lecture Notes in Mathematics, vol. 352. Springer, Berlin (1970)

  16. Hori, K., Vafa, C.: Mirror symmetry. arXiv:hep-th/0002222

  17. 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/0408426 [math.AG]

    Google Scholar 

  18. Liu, C.-C.M., Liu, K., Zhou, J.: Mariño-Vafa formula and Hodge integral identities. J. Algebraic Geom. 15, 379–398 (2006). arXiv:math/0308015v2 [math.AG]

    Google Scholar 

  19. Mariño M.: Open string amplitudes and large order behavior in topological string theory. JHEP 0803, 060 (2008) arXiv:hep-th/0612127

    Article  ADS  Google Scholar 

  20. Ooguri, H., Sułkowski, P., Yamazaki, M.: Wall crossing as seen by matrix models. Commun. Math. Phys. (2011). arXiv:1005.1293 [hep-th]

  21. Sułkowski P.: Refined matrix models from BPS counting. Phys. Rev. D 83, 085021 (2011) arXiv:1012.3228 [hep-th]

    Article  ADS  Google Scholar 

  22. Witten E.: Two dimensional gravity and intersection theory on moduli space. Surveys Differ. Geom. 1, 243–310 (1991)

    MathSciNet  Google Scholar 

  23. Zhou, J.: Local Mirror Symmetry for One-Legged Topological Vertex. arXiv:0910.4320 [math.AG]

  24. Zhu, S.: The Laplace transform of the cut-and-join equation of Mariño-Vafa formula and its applications. arXiv:1001.0618 [math.AG]

  25. Zhu, S.: On a proof of the Bouchard-Sulkowski conjecture. arXiv:1108.2831v1 [math.AG]

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vincent Bouchard.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bouchard, V., Catuneanu, A., Marchal, O. et al. The Remodeling Conjecture and the Faber–Pandharipande Formula. Lett Math Phys 103, 59–77 (2013). https://doi.org/10.1007/s11005-012-0588-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-012-0588-z

Mathematics Subject Classification (2010)

Keywords

Navigation