Equivalence of ELSV and Bouchard-Mariño conjectures for $r$-spin Hurwitz numbers

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-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.

32 pages

References in corpus (4)

• Hurwitz numbers, matrix models and enumerative geometry
• Twisted Gromov-Witten r-spin potential and Givental's quantization
• Tautological relations via r-spin structures
• Witten's top Chern class via K-theory

Cited by in corpus (22)

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• Pixton's double ramification cycle relations
• Chiodo formulas for the r-th roots and topological recursion
• Higher Airy structures and topological recursion for singular spectral curves
• Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson-Pandharipande-Tseng formula
• Towards an orbifold generalization of Zvonkine's $r$-ELSV formula
• Double Hurwitz numbers: polynomiality, topological recursion and intersection theory
• Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials
• Spectral curves and $W$-representations of matrix models
• Special cases of the orbifold version of Zvonkine's $r$-ELSV formula
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• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• On the Goulden-Jackson-Vakil conjecture for double Hurwitz numbers
• The Moduli Space of Stables Maps with Divisible Ramification
• Generalized $\tilde{W}$ algebras
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
• The $r$-ELSV Formula via Localisation on the Moduli Space of Stable Maps with Divisible Ramification
• Topological recursion on transalgebraic spectral curves and Atlantes Hurwitz numbers
• Any topological recursion on a rational spectral curve is KP integrable