有理加权Hurwitz数的星座和$τ$ -函数
Constellations and $τ$-functions for rationally weighted Hurwitz numbers
摘要：
加权星座给出黎曼球面的加权分支覆盖的图形表示。它们被引入到提供组合解释的$2$ D户田$τ$ -函数的超几何类型作为生成函数的加权Hurwitz数的情况下，多项式的重量生成函数。在所有配置上求和的给定加权星座的所有顶点和边权重上的乘积再现了$τ$函数。在目前的工作中，这是广义的星座，其中的加权参数是由一个合理的权重生成函数。相关联的$τ$函数可以表示为双标记加权星座的权重之和，其中两种类型的加权参数与分支覆盖的每个等价类相关联。分支点的双重标记，称为“颜色”和“风味”指数，是需要的事实，即在泰勒展开的权重生成函数，一个特定的颜色从分母参数中可能会出现乘法，和风味标签指示这种多样性。
英文摘要：
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.
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