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Equivalence of ELSV and Bouchard–Mariño conjectures for \(r\)-spin Hurwitz numbers

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Abstract

We propose two conjectures on Hurwitz numbers with completed \((r+1)\)-cycles, or, equivalently, on certain relative Gromov-Witten invariants of the projective line. The conjectures are analogs of the ELSV formula and of the Bouchard–Mariño conjecture for ordinary Hurwitz numbers. Our \(r\)-ELSV formula is an equality between a Hurwitz number and an integral over the space of \(r\)-spin structures, that is, the space of stable curves with an \(r\)th root of the canonical bundle. Our \(r\)-BM conjecture is the statement that \(n\)-point functions for Hurwitz numbers satisfy the topological recursion associated with the spectral curve \(x = -y^r + \log y\) in the sense of Chekhov, Eynard, and Orantin. We show that the \(r\)-ELSV formula and the \(r\)-BM conjecture are equivalent to each other and provide some evidence for both.

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Notes

  1. This has nothing to do with stable curves.

  2. The standard definition involves certain additive constants that we have dropped to simplify the expression, since these constants play no role here.

References

  1. Abramovich, D., Jarvis, T.J.: Moduli of twisted spin curves. Proc. Am. Math. Soc. 131(3), 685–699 (2003). math.AG/0104154

  2. Borot, G., Eynard, B., Mulase, M., Safnuk, B.: A matrix model for simple Hurwitz numbers, and topological recursion. J. Geom. Phys. 61(2), 522–540 (2011). arXiv:0906.1206

  3. Bouchard, V., Mariño, M.: Hurwitz numbers, matrix models and enumerative geometry. From Hodge theory to integrability and TQFT tt*-geometry. Proceedings of Symposium on Pure Mathematics, vol. 78, pp. 263–283. American Mathematical Society (2008). arXiv:0709.1458

  4. Caporaso, L., Casagrande, C., Cornalba, M.: Moduli of roots of line bundles on curves. Trans. Am. Math. Soc. 359(8), 3733–3768 (2007). math.AG/0404078

  5. Chekhov, L., Eynard, B.: Matrix eigenvalue model: Feynman graph technique for all genera. J. High Energy Phys. 12(026), 29 (2006). math-ph/0604014

  6. Chiodo, A.: Stable twisted curves and their \(r\)-spin structures. Ann. Inst. Fourier (Grenoble) 58(5), 1635–1689 (2008). math.AG/0603687

  7. Chiodo, A.: Towards an enumerative geometry of the moduli space of twisted curves and rth roots. Compos. Math. 144(6), 1461–1496 (2008). math.AG/0607324v2

  8. Chiodo, A.: Witten’s top Chern class via K-theory. J. Algebr. Geom. 15(4), 681–707 (2006). math.AG/0210398

  9. Chiodo, A., Ruan, Y.: Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations. Invent. Math. 182(1), 117–165 (2010). arXiv:0812.4660

  10. Chiodo, A., Zvonkine, D.: Twisted Gromov–Witten r-spin potential and Givental’s quantization. Adv. Theor. Math. Phys. 13(5), 1335–1369 (2009). arXiv:0711.0339

  11. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W-function. Adv. Comput. Math. 5, 329–359 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dunin-Barkowski, P., Orantin, N., Shadrin, S., Spitz, L.: Identification of the Givental formula with the spectral curve topological recursion procedure. Commun. Math. Phys. (to appear). arXiv:1211.4021

  13. Dunin-Barkowski, P., Shadrin, S., Spitz, L.: Givental graphs and inversion symmetry. Lett. Math. Phys. 103(5), 533–557 (2013). arxiv:1201:4930

  14. Dumitrescu, O., Mulase, M., Safnuk, B., Sorkin, A.: The spectral curve of the Eynard–Orantin recursion via the Laplace transform. arXiv:1202.1159

  15. Ekedahl, T., Lando, S.K., Shapiro, M., Vainshtein, A.: Hurwitz numbers and intersections on moduli spaces of curves. Invent. Math. 146, 297–327 (2001). math.AG/0004096

  16. Eynard, B.: Invariants of spectral curves and intersection theory of moduli spaces of complex curves. arXiv:1110.2949

  17. Eynard, B., Mulase, M., Safnuk, B.: The Laplace transform of the cut-and-join equation and the Bouchard–Mario conjecture on Hurwitz numbers. Publ. Res. Inst. Math. Sci. 47(2), 629670 (2011). arXiv:0907.5224

  18. Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2), 347–452 (2007). arXiv:math-ph/0702045

  19. Faber, C., Shadrin, S., Zvonkine, D.: Tautological relations and the \(r\)-spin Witten conjecture. Ann. Sci. Éc. Norm. Supér. (4) 43(4), 621–658 (2010). math.AG/0612510

  20. Givental, A.: Gromov–Witten invariants and quantization of quadratic hamiltonians. Mosc. Math. J. 1(4), 551–568 (2001). math/0108100

  21. Gukov, S., Sułkowski, P.: A-polynomial, B-model, and quantization. J. High Energy Phys. 2(070), 56 (2012). arXiv:1108.0002v1

  22. Itzykson, C., Zuber, J.-B.: The planar approximation II. J. Math. Phys. 21(3), 411–421 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jarvis, T.J.: Geometry of moduli of higher spin curves. Int. J. Math. 11, 637–663 (2000). arXiv:math/9809138

  24. Kazarian, M., Lando, S.: An algebro-geometric proof of Witten’s conjecture. J. Am. Math. Soc. 20(4), 1079–1089 (2007). arXiv:math/0601760

  25. Kerov, S., Olshanski, G.: Polynomial functions on the set of Young diagrams. C. R. Acad. Sci. Paris Sér. I Math 319(2), 121–126 (1994)

    MATH  MathSciNet  Google Scholar 

  26. Luke, Y.L.: The special functions and their approximations, vol. I. Academic Press, New York (1969)

    MATH  Google Scholar 

  27. Milanov, T.: The Eynard–Orantin recursion for the total ancestor potential. arXiv:1211.5847

  28. Mulase, M., Shadrin, S., Spitz, L.: The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures. Commun. Number Theory Phys. (to appear) arXiv:1301.5580

  29. Mumford, D.: Arithmetics and geometry. In: Artin, M., Tate, J. (eds.) Towards enumerative geometry on the moduli space of curves, pp. 271–328. Birkhäuser, Basel (1983)

    Google Scholar 

  30. Okounkov, A.: Toda equations for Hurwitz numbers. Math. Res. Lett. 7(4), 447–453 (2000). math.AG/0004128

  31. Okounkov, A., Pandharipande, R.: Gromov–Witten theory, Hurwitz theory, and completed cycles. Ann. Math. (2). 163(2), 517–560 (2006). math.AG/0204305

  32. Pixton, A., Pandharipande, R., Zvonkine, D.: Relations on \({\overline{\cal M}}\) via 3-spin structures. arXiv:1303.1043

  33. Polishchuk, A., Vaintrob, A.: Algebraic construction of Witten’s top Chern class. Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Contemporary Mathematics, vol. 276, pp. 229–249, American Mathematical Society, Providence (2001). math.AG/0011032

  34. Shadrin, S.: BCOV theory via Givental group action on cohomological fields theories. Mosc. Math. J. 9(2), 411–429 (2009). arXiv:0810.0725

  35. Shadrin, S., Zvonkine, D.: Intersection numbers with Witten’s top Chern class. Geom. Topol. 12(4), 713–745 (2008). math.AG/0601075

  36. Vakil, R.: The moduli space of curves and Gromov–Witten theory. Enumerative invariants in algebraic geometry and string theory. Lecture Notes in Mathematics, pp. 143–198. Springer, Berlin, 2008 (1947). math.AG/0602347

  37. Witten, E.: Algebraic geometry associated with matrix models of two-dimensional gravity. Topological methods in modern mathematics (Stony Brook, NY, 1991), pp. 235–269, Publish or Perish, Houston, TX (1993)

  38. Zvonkine, D.: A preliminary text on the \(r\)-ELSV formula. Preprint (2006)

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Acknowledgments

L. S. and S. S. were supported by the Netherlands Organization for Scientific Research. D. Z. was supported by the grant ANR-09-JCJC-0104-01. The \(r\)-ELSV formula was conjectured by D. Z. [38] who had important discussions about it with many people before this paper appeared. We are particularly grateful to Alessandro Chiodo, Maxim Kazarian, Sergey Lando, Sergei Natanzon, Christian Okonek, Rahul Pandharipande, Dmitri Panov, Mathieu Romagny, Mikhail Shapiro, and Claire Voisin. We also thank Gaëtan Borot, Leonid Chekhov, Petr Dunin-Barkowski, Bertrand Eynard, Motohico ulase, and Nicolas Orantin for plenty of helpful discussions on closely related topics and teaching us the methods of spectral curve topological recursion.

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Shadrin, S., Spitz, L. & Zvonkine, D. Equivalence of ELSV and Bouchard–Mariño conjectures for \(r\)-spin Hurwitz numbers. Math. Ann. 361, 611–645 (2015). https://doi.org/10.1007/s00208-014-1082-y

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