A multiparametric family of 2D Toda $τ$-functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of branched coverings of the Riemann sphere and paths in the Cayley graph of $S_n$. The coefficients $F^{c_1, ..., c_l}_{d_1, ..., d_m}(μ, ν)$ in their series expansion over products $P_μP'_ν$ of power sum symmetric functions in the two sets of Toda flow parameters and powers of the $l+m$ auxiliary parameters are shown to enumerate $|μ|=|ν|=n$ fold branched covers of the Riemann sphere with specified ramification profiles $ μ$ and $ν$ at a pair of points, and two sets of additional branch paints, satisfying certain additional conditions on their ramification profile lengths. The first group consists of $l$ branch points, with ramification profile lengths fixed to be the numbers $(n-c_1, ..., n- c_l)$; the second consists of $m$ further groups of "coloured" branch points, of variable number, for which the sums of the complements of the ramification profile lengths within the groups are fixed to equal the numbers $(d_1, ..., d_m)$. The latter are counted with sign determined by the parity of the total number of such branch points. The coefficients $F^{c_1, ..., c_l}_{d_1, ..., d_m}(μ, ν)$ are also shown to enumerate paths in the Cayley graph of the symmetric group $S_n$ generated by transpositions, starting, as in the usual double Hurwitz case, at an element in the conjugacy class of cycle type $μ$ and ending in the class of type $ν$, but with the first $l$ consecutive subsequences of $(c_1, ..., c_l)$ transpositions strictly monotonically increasing, and the subsequent subsequences of $(d_1, ..., d_m)$ transpositions weakly increasing.
Generalized Brézin-Gross-Witten tau-function as a hypergeometric solution of the BKP hierarchy
本文证明了广义Brézin-Gross-Witten τ-函数是具有简单权母函数的BKP族的超几何解。我们声称，它描述了一个自旋版本的严格单调赫维茨数。本文研究了一类BKP族的超几何τ-函数族，它对应于有理权母函数。特别地，构造了割-并算子，并导出了BKP Sato Grassmannian点的显式描述。这个家族的代表可以与自旋赫维茨数的有趣家族相关联，包括单调赫维茨数的自旋版本。
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）
Generalisations of the Harer-Zagier recursion for 1-point functions
Harer和Zagier在他们关于曲线的模空间的欧拉特征的工作中证明了一个递归来枚举导致可定向亏格$g$曲面的$2d$ -边形的胶合。类似的结果已经发现其他枚举问题，所以很自然地提出以下问题：有多大的家庭的问题，这些所谓的1点递归存在？本文证明了一类具有Schur函数展开式的计数问题的1-点递归的存在性。特别是，我们恢复Harer-Zagier递归，但我们的方法也适用于枚举的dessins d 'enfant，Bousquet-Mélou-Schaeffer数，单调Hurwitz数，等等。另一方面，我们证明，没有1点递归，治理简单的Hurwitz数。我们的结果是有效的，在这个意义上，可以明确计算特定的1点递归的实例，我们提供了几个例子。最后，我们简要的讨论和猜想有关的1点递归理论的拓扑递归的文件。
