Equivalence of ELSV and Bouchard-Mario conjectures for -spin Hurwitz numbers

We propose two conjectures on Hurwitz numbers with completed -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-Mario conjecture for ordinary Hurwitz numbers. Our -ELSV formula is an equality between a Hurwitz number and an integral over the space of -spin structures, that is, the space of stable curves with an th root of the canonical bundle. Our -BM conjecture is the statement that -point functions for Hurwitz numbers satisfy the topological recursion associated with the spectral curve in the sense of Chekhov, Eynard, and Orantin. We show that the -ELSV formula and the -BM conjecture are equivalent to each other and provide some evidence for both.