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

Establish an ELSV-type formula for one-part double Hurwitz numbers that expresses H_g((d), ν) as a tautological intersection integral on a compactification of the universal Picard stack of the form d · r! ∫ (Λ_0 − Λ_2 + ··· + (−1)^g Λ_{2g}) / ∏(1 − ν_i ψ_i), and identify the required classes Λ_{2i} and ψ_i relations; in particular, prove the strong form presented in equation (eq:elsv).

References

They further conjectured an ELSV-type formula for one-part double Hurwitz numbers H_g((d),ν), i.e. a tautological intersection theoretic expression on some compactification of the universal Picard stack of the form: (eq:elsv). While the Goulden-Jackson-Vakil conjecture in its strong form eq:elsv is still wide open, some results expressed one-part double Hurwitz numbers as intersection numbers on moduli spaces of curves, confirming the predicted polynomial and integrability structure.

eq:elsv:

Hg((d),)=d⋅r!∫Pic‾g,nΛ0−Λ2+…+(−1)gΛ2g∏1−νiψi,H_g((d),) = d\cdot r! \int_{\overline{\mathrm{Pic}}_{g,n}} \frac{\Lambda_0 -\Lambda_2 +\ldots +(-1)^g\Lambda_{2g}}{\prod 1 - \nu_i\psi_i},

— One part leaky covers (Cavalieri et al., 4 Sep 2025) in Context section; equation (eq:elsv)