The partition function for quantum weighted double Hurwitz numbers can be interpreted in terms of the energy distribution of a quantum Bose gas with vanishing fugacity. We compute the leading term of the partition function and the quantum weighted Hurwitz numbers in the zero temperature limit $T \rightarrow 0$, as well as the next order corrections. The leading term is shown to reproduce the case of uniformly weighted Hurwitz numbers of Belyi curves. In particular, the KP or Toda $τ$-function serving as generating function for the quantum Hurwitz numbers is shown in the limit to give the one for Belyi curves and, with suitable scaling, the same holds true for the partition function, the weights and the expectations of Hurwitz numbers.
Weighted Hurwitz numbers and hypergeometric $τ$-functions: an overview
This is an overview of recent results on the use of 2D Toda $τ$ -functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an isomorphism between the zero charge sector of the Fermionic Fock space and the direct sum of the centers of the group algebra of the symmetric groups $S_n$ . Specializing the fermionic formula to the case of diagonal group elements gives $τ$ -functions of hypergeometric type, for which the expansion over products of Schur functions is diagonal, with coefficients of…
A new family of weighted double Hurwitz numbers and a new ELSV-type formula with $Ω$-classes
Center for Geometry and Physics, Institute for Basic Science (IBS)（基本科学研究所几何与物理中心）；University of Haifa（海法大学）；HSE University（高等经济学院）；Skoltech（斯科尔科沃科学技术研究所）；NRC “Kurchatov Institute” – ITEP（库尔恰托夫国家研究中心-物理技术研究所）；I.M. Krichever Scientific School, Skoltech（斯科尔科沃 I.M.克里切弗科学学校）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）
Integer moments of complex Wishart matrices and Hurwitz numbers
本文给出了复Wishart矩阵（LUE）和逆Wishart矩阵（逆LUE）的累积量公式。它们的大- $N$展开式是双（严格和弱）单调Hurwitz数的生成函数，它在对称群中计数约束因子分解。这两个展开式可以用[F]中证明的对偶关系进行比较和组合。D. Cunden，F. Mezzadri，N. O 'Connell和N. J. Simm，arXiv：1805.08760]得到：i）亏格为零的LUE和逆LUE的矩之间的反射公式的组合证明; ii）单调和严格单调Hurwitz数的生成函数之间的新的函数关系。主要结果解决了[F. D. Cunden，F. Mezzadri，N. J. Simm and P. Vivo，J. Phys. A 49（2016）] on the time delay cumulants in quantum chaotic transport.本文给出的累积量的精确组合描述可能对随机矩阵与半经典理论的一致性有新的认识。
Triply mixed coverings of arbitrary base curves: Quasimodularity, quantum curves and a mysterious topological recursions
简单的Hurwitz数枚举具有固定分歧数据的黎曼曲面之间的分支态射。近年来，在文献中出现了几种关于属$0$基曲线的这一概念的变体。其中有所谓的单调Hurwitz数，它与随机矩阵理论中的HCIZ积分有关，以及严格单调Hurwitz数，它计数某些Grothendieck dessins d 'enfants。我们概括的概念，Hurwitz数简单，单调和严格单调的Hurwitz数之间的插值到任何属和任意数量的任意，但固定的衍生配置文件。这产生了几个已知的赫尔维茨数的结果的概括。当目标曲面的亏格为1时，我们证明了这些插值Hurwitz数的生成序列是拟模形式。在所有分歧是简单的情况下，我们通过将这个序列写为对应于由Gromov-Witten不变量加权的热带覆盖的拟模形式的和来改进这个结果。此外，我们推导出一个量子曲线的单调和Grothendieck dessins d 'enfants Hurwitz数为任意的一般和一个任意的，但固定的衍生物配置文件。因此，我们通过半经典极限得到谱曲线作为CEO拓扑递归的输入数据。令人惊讶的是，我们发现严格单调Hurwitz数的亏格$1$谱曲线的CEO拓扑递归计算亏格$0$中的单调Hurwitz数。从而给出了单调Hurwitz数满足CEO拓扑递归的一个新的证明。这表明这些枚举数之间存在未知的关系。最后，专门针对目标表面$\mathbb{P}^1$，我们发现递…
