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

Blot, Xavier

doi:10.2140/gt.2022.26.1669

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

Xavier Blot

Geometry & Topology 26 (2022) 1669–1743

DOI: 10.2140/gt.2022.26.1669

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 $ℏ$ . When $ℏ = 0$ , 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 $ℙ^{1}$ 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.

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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

Volume 26, issue 4 (2022)