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
and a quantum
parameter .
When
,
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
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
to
with a complete ramification
over
, 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
