Towards the geometry of double Hurwitz numbers

Double Hurwitz numbers count branched covers of the projective line with fixed branch points, with simple branching required over all but two points 0 and infinity, and the branching over 0 and infinity specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers have a rich structure, explored by authors in many fields. The ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves, and has led to many consequences. We determine the structure of double Hurwitz numbers using geometry, algebra, and representation theory. Our motivation is geometric: we give strong evidence that double Hurwitz numbers are top intersections on a universal Picard variety. In particular, we prove a piecewise-polynomiality result analogous to that implied by the ELSV formula. In the case m=1 and n is arbitrary, we conjecture an ELSV-type formula, and show it to be true in genus 0 and 1. The corresponding Witten-type correlation function has a better structure than that for single Hurwitz numbers, and it satisfies many geometric properties, such as the string and dilaton equations, and a genus expansion ansatz analogous to that of Itzykson and Zuber. We give a symmetric function description of the double Hurwitz generating series, which leads to explicit formulae for double Hurwitz numbers with given m and n. In the case where m is fixed but not necessarily 1, we prove a topological recursion on the corresponding generating series, which leads to closed-form expressions for double Hurwitz numbers and an analogue of the Goulden-Jackson polynomiality conjecture (an early conjectural variant of the ELSV formula).