Goulden–Jackson–Vakil ELSV-type formula for one-part double Hurwitz numbers

Establish the Goulden–Jackson–Vakil conjecture asserting that, for every genus g and partition data with one ramification profile equal to (d) over 0 and a partition (ν1,…,νn) over ∞ with ν1+…+νn=d, the one-part double Hurwitz number H_g((d),(ν1,…,νn)) equals d·r! times an intersection integral on a suitable compactification of the universal Picard stack over M_{g,n}, namely H_g((d),ν)=d·r!·∫_{overline{Pic}_{g,n}}(Λ0−Λ2+⋯+(−1)^gΛ2g)/∏_{i=1}^n(1−νiψi), where r is the number of simple branch points, the ψi are cotangent line classes pulled back from M_{g,n}, and the Λ2i are tautological classes of degree 2i on the compactified universal Picard stack.

Background

One-part double Hurwitz numbers count genus g covers of the projective line with a single part (d) over 0 and a partition (ν1,…,νn) over ∞, with simple branching elsewhere. Goulden, Jackson, and Vakil proved piecewise polynomiality and, for the one-part chamber, explicit polynomial formulas in low genus.

They further proposed an ELSV-type formula expressing H_g((d),ν) as a tautological integral on an appropriate compactification of the universal Picard stack, involving cotangent classes ψi and classes Λ2i. Such a formula would parallel the classical ELSV formula for single Hurwitz numbers and would explain structural properties (degree, parity, vanishing ranges) of the one-part double Hurwitz polynomials.

In this paper’s context overview, the authors recall the conjectural formula and explicitly note that it remains unresolved in its strong form, situating their results on leaky and descendant variants against this longstanding open problem.