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Article

Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson–Pandharipande–Tseng formula

Journal of London Mathematical Society. 2015. Vol. 92. No. 3. P. 547-565.
Dunin-Barkowski P., Lewanski D., Popolitov A., Shadrin S.

In this paper, we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of Chekhov, Eynard and Orantin, where the main new step compared to the existing proofs is a direct combinatorial proof of their quasi-polynomiality. Spectral curve topological recursion leads to a formula for the orbifold Hurwitz numbers in terms of the intersection theory of the moduli space of curves, which, in this case, appears to coincide with a special case of the Johnson–Pandharipande–Tseng formula.