论文标题
用于理性加权的Hurwitz号码的星座和$τ$ functions
Constellations and $τ$-functions for rationally weighted Hurwitz numbers
论文作者
论文摘要
加权星座给出了Riemann球体加权分支覆盖物的图形表示。引入了他们的介绍，以提供对超几何类型的$ 2 $ d toda $τ$ functions的组合解释，在多项式重量生成功能的情况下，用作加权Hurwitz数字的生成功能。给定加权星座的所有顶点和边缘权重的产品，总结在所有配置上，都重现了$τ$功能。在目前的工作中，这被概括为星座，在该星座中，加权参数取决于理性权重生成函数。相关的$τ$功能可以表示为双重标记的加权星座的权重的总和，其中两种类型的加权参数与每个等价类别的分支覆盖物相关联。分支点的双重标记，称为“颜色”和“风味”指数，这是因为在泰勒（Taylor）扩展重量生成函数的事实中，分母参数中的特定颜色可能会出现多重，而风味标签则表明这种多重性。
Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $τ$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $τ$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rational weight generating function. The associated $τ$-function may be expressed as a sum over the weights of doubly labelled weighted constellations, with two types of weighting parameters associated to each equivalence class of branched coverings. The double labelling of branch points, referred to as "colour" and "flavour" indices, is required by the fact that, in the Taylor expansion of the weight generating function, a particular colour from amongst the denominator parameters may appear multiply, and the flavour labels indicate this multiplicity.
