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

By Sergey Shadrin, L. Spitz, Dimitri Zvonkine. 2013

47 citations recorded by the source corpus.

We propose two conjectures on Hurwitz numbers with completed $$(r+1)$$(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 $$r$$r-ELSV formula is an equality between a Hurwitz number and an integral over the space of $$r$$r-spin structures, that is, the space of stable curves with an $$r$$rth root of the canonical bundle. Our $$r$$r-BM conjecture is the statement that $$n$$n-point functions for Hurwitz numbers satisfy the topological recursion associated with the spectral curve $$x = -y^r + \log y$$x=-yr+logy in the sense of Chekhov, Eynard, and Orantin. We show that the $$r$$r-ELSV formula and the $$r$$r-BM conjecture are equivalent to each other and provide some evidence for both.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/da68da3563418fbb785beed75218fbba7f4057a7