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

Xavier Blot

Geometry & Topology 26 (2022) 1669–1743
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 . When = 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 1 with a complete ramification over 0, 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.

Keywords
moduli space of curves, double ramification cycle, quantum KdV, quantum tau function, Hurwitz numbers
Mathematical Subject Classification
Primary: 05A99, 53D55
Secondary: 14H10
References
Publication
Received: 24 April 2020
Revised: 13 May 2021
Accepted: 18 June 2021
Published: 28 October 2022
Proposed: Paul Seidel
Seconded: Jim Bryan, Leonid Polterovich
Authors
Xavier Blot
Weizmann Institute of Science
Rehovot
Israel
https://sites.google.com/view/xavier-blot