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2022 The quantum Witten–Kontsevich series and one-part double Hurwitz numbers
Xavier Blot
Geom. Topol. 26(4): 1669-1743 (2022). DOI: 10.2140/gt.2022.26.1669


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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Xavier Blot. "The quantum Witten–Kontsevich series and one-part double Hurwitz numbers." Geom. Topol. 26 (4) 1669 - 1743, 2022.


Received: 24 April 2020; Revised: 13 May 2021; Accepted: 18 June 2021; Published: 2022
First available in Project Euclid: 11 November 2022

Digital Object Identifier: 10.2140/gt.2022.26.1669

Primary: 05A99 , 53D55
Secondary: 14H10

Keywords: double ramification cycle , Hurwitz numbers , moduli space of curves , quantum KdV , quantum tau function

Rights: Copyright © 2022 Mathematical Sciences Publishers


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Vol.26 • No. 4 • 2022
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