Mathematische Annalen

, Volume 361, Issue 3–4, pp 611–645 | Cite as

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

  • S. ShadrinEmail author
  • L. Spitz
  • D. Zvonkine


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.



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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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Institut de Mathématiques de Jussieu-Paris Rive Gauche and CNRSParis Cedex 05Paris

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