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

Xavier Blot

Geometry & Topology 26 (2022) 1669–1743

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.

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moduli space of curves, double ramification cycle, quantum KdV, quantum tau function, Hurwitz numbers
Mathematical Subject Classification
Primary: 05A99, 53D55
Secondary: 14H10
Received: 24 April 2020
Revised: 13 May 2021
Accepted: 18 June 2021
Published: 28 October 2022
Proposed: Paul Seidel
Seconded: Jim Bryan, Leonid Polterovich
Xavier Blot
Weizmann Institute of Science