We study the quantum Witten–Kontsevich series introduced by Buryak, Dubrovin, Guéré and Rossi (2020) as the logarithm of a quantum tau function for the quantum KdV hierarchy. This series depends on a genus parameter and a quantum parameter . When , this series restricts to the Witten–Kontsevich generating series for intersection numbers of psi classes on moduli spaces of stable curves.
We establish a link between the part of the quantum Witten–Kontsevich series and one-part double Hurwitz numbers. These numbers count the number of nonequivalent holomorphic maps from a Riemann surface of genus to with a complete ramification over , a prescribed ramification profile over and a given number of simple ramifications elsewhere. Goulden, Jackson and Vakil (2005) proved that these numbers have the property of being polynomial in the orders of ramification over . We prove that the coefficients of these polynomials are the coefficients of the quantum Witten–Kontsevich series.
We also present some partial results about the full quantum Witten–Kontsevich power series.
"The quantum Witten–Kontsevich series and one-part double Hurwitz numbers." Geom. Topol. 26 (4) 1669 - 1743, 2022. https://doi.org/10.2140/gt.2022.26.1669