论文标题
Quantum Witten-Kontsevich系列和一部分双Hurwitz编号
The quantum Witten-Kontsevich series and one-part double Hurwitz numbers
论文作者
论文摘要
我们研究了Buryak，Dubrovin，Guéré和Rossi在\ cite {Buryak2016Integrable}中引入的量子Witten-Kontsevich系列，作为量子tau函数的对数。该系列取决于属参数$ε$和量子参数$ \ hbar $。当$ \ hbar = 0 $时，该系列将限制在稳定曲线模量空间上PSI类的交点的Witten-Kontsevich生成系列。我们建立了$ε= 0 $ witten-kontsevich系列的$ε= 0 $之间的链接。这些数字计算了数字从$ g $属的riemann表面到$ \ mathbb {p}^{1} $的数字非等效的全态图，其规定超过$ 0 $，对$ \ infty $的全面分支和给定的简单分支的完全分支。 Goulden，Jackson和Vakil在\ cite {goulden2005towards}中证明了这些数字在$ 0 $ 0 $的后果的顺序中具有多项式。我们证明这些多项式的系数是量子witten-kontsevich系列的系数。我们还提供了有关完整量子Witten-Kontsevich Power系列的一些部分结果。
We study the quantum Witten-Kontsevich series introduced by Buryak, Dubrovin, Guéré and Rossi in \cite{buryak2016integrable} 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 $\hbar$. When $\hbar=0$, 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 non-equivalent holomorphic maps from a Riemann surface of genus $g$ to $\mathbb{P}^{1}$ with a prescribe ramification profile over $0$, a complete ramification over $\infty$ and a given number of simple ramifications elsewhere. Goulden, Jackson and Vakil proved in \cite{goulden2005towards} that these numbers have the property to be polynomial in the orders of ramification over $0$. 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.
