On the Goulden–Jackson–Vakil conjecture for double Hurwitz numbers

Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of CP1 with prescribed ramification profile over ∞, unique preimage 0, simple branching elsewhere. This led them to conjecture the existence moduli spaces tautological classes whose intersection theory produces an analogue celebrated ELSV formula for single numbers. In this paper, we present three formulas that express numbers as on certain spaces. The first involves Hodge stable maps classifying spaces; second Chiodo r-spin curves; third curves. We proceed discuss merits these against list desired properties enunciated by Vakil. Our lead non-trivial relations between curves hints at further classes. paper concludes generalisations our results context