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Mathematics, Algebraic Geometry, Mathematical Physics
Abstract:
Goulden, Jackson and Vakil observed a polynomial structure underlying
one-part double Hurwitz numbers, which enumerate branched covers of
$\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique
preimage over 0, and simple branching elsewhere. This led them to conjecture
the existence of moduli spaces and tautological classes whose intersection
theory produces an analogue of the celebrated ELSV formula for single Hurwitz
numbers.
In this paper, we present three formulas that express one-part double Hurwitz
numbers as intersection numbers on certain moduli spaces. The first involves
Hodge classes on moduli spaces of stable maps to classifying spaces; the second
involves Chiodo classes on moduli spaces of spin curves; and the third involves
tautological classes on moduli spaces of stable curves. We proceed to discuss
the merits of these formulas against a list of desired properties enunciated by
Goulden, Jackson and Vakil. Our formulas lead to non-trivial relations between
tautological intersection numbers on moduli spaces of stable curves and hints
at further structure underlying Chiodo classes. The paper concludes with
generalisations of our results to the context of spin Hurwitz numbers.
