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

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 rth 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=−yr+logy 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.