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 in \cite{buryak2016integrable} as the logarithm of a quantum tau function for the quantum KdV hierarchy. This series depends on a genus parameter $\epsilon$ 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 $\epsilon =0$ part of the quantum Witten-Kontsevich series and one-part double Hurwitz numbers. These numbers count the number non-equivalent holomorphic maps from a Riemann surface of genus $g$ to ${\mathbb{P}}^1$ with a prescribe ramification profile over $0$, a complete ramification over $\infty$ and a given number of simple ramifications elsewhere. Goulden, Jackson and Vakil proved in \cite{goulden2005towards} that these numbers have the property to be polynomial in the orders of ramification over $0$. 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.

References in corpus (2)

• Double ramification cycles and integrable hierarchies
• Tautological and non-tautological cohomology of the moduli space of curves

Cited by in corpus (4)

• On the spectral problem of the quantum KdV hierarchy
• Quantum KdV hierarchy and shifted symmetric functions
• Stable tree expressions with Omega-classes and Double Ramification cycles
• Relations for quadratic Hodge integrals via stable maps