The quantum Witten–Kontsevich series and one-part double Hurwitz numbers

Abstract

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 $𝜖$ and a quantum parameter $\hslash$. When $\hslash =0$, 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 $𝜖=0$ 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 $g$ to ${\mathbb{P}}^1$ with a complete ramification over $0$, a prescribed ramification profile over $\infty$ 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 $\infty$. 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.