Equivalence of ELSV and Bouchard-Mari\~no conjectures for $r$-spin Hurwitz numbers

S. Shadrin; L. Spitz; D. Zvonkine

doi:10.1007/s00208-014-1082-y

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arXiv · 1306.6226

Equivalence of ELSV and Bouchard-Mari\~no conjectures for $r$-spin Hurwitz numbers

S. Shadrin·L. Spitz·D. Zvonkine

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Abstract

We propose two conjectures on Huwritz 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-Mari\~no 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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S. Shadrin, L. Spitz, D. Zvonkine. 2013-06-26. Equivalence of ELSV and Bouchard-Mari\~no conjectures for $r$-spin Hurwitz numbers. https://doi.org/10.1007/s00208-014-1082-y

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