To obtain a generating function of the most general form for Hurwitz numbers with an arbitrary base surface and arbitrary ramification profiles, we consider a matrix model constructed according to a graph on an oriented connected surface $$\Sigma$$ with no boundary. The vertices of this graph, called stars, are small discs, and the graph itself is a clean dessin d’enfants. We insert source matrices in boundary segments of each disc. Their product determines the monodromy matrix for a given star, whose spectrum is called the star spectrum. The surface $$\Sigma$$ consists of glued maps, and each map corresponds to the product of random matrices and source matrices. Wick pairing corresponds to gluing the set of maps into the surface, and an additional insertion of a special tau function in the integration measure corresponds to gluing in Mobius strips. We calculate the matrix integral as a Feynman power series in which the star spectral data play the role of coupling constants, and the coefficients of this power series are just Hurwitz numbers. They determine the number of coverings of $$\Sigma$$ (or its extensions to a Klein surface obtained by inserting Mobius strips) for any given set of ramification profiles at the vertices of the graph. We focus on a combinatorial description of the matrix integral. The Hurwitz number is equal to the number of Feynman diagrams of a certain type divided by the order of the automorphism group of the graph.

中文翻译：

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费曼图中的赫尔维茨数

为了获得具有任意基面和任意分支轮廓的 Hurwitz 数的最一般形式的生成函数，我们考虑根据无边界的有向连接表面 $$\Sigma$$ 上的图构建的矩阵模型。这个图的顶点称为星星，是小圆盘，图本身是一个干净的 dessin d'enfants。我们在每个圆盘的边界段插入源矩阵。他们的乘积决定了给定恒星的单向矩阵，其光谱称为恒星光谱。表面 $$\Sigma$$ 由粘合图组成，每个图对应于随机矩阵和源矩阵的乘积。灯芯配对对应于将一组贴图粘合到表面上，在积分测量中额外插入一个特殊的 tau 函数对应于莫比乌斯带的粘合。我们将矩阵积分计算为一个费曼幂级数，其中恒星光谱数据起到耦合常数的作用，而这个幂级数的系数就是赫维茨数。他们确定了图顶点处任何给定的一组分枝剖面的 $$\Sigma$$（或通过插入莫比乌斯带获得的克莱因表面的扩展）的覆盖数。我们专注于矩阵积分的组合描述。赫尔维茨数等于某类费曼图的个数除以图的自同构群的阶数。并且这个幂级数的系数只是 Hurwitz 数。他们确定了图顶点处任何给定的一组分枝剖面的 $$\Sigma$$（或通过插入莫比乌斯带获得的克莱因表面的扩展）的覆盖数。我们专注于矩阵积分的组合描述。赫尔维茨数等于某类费曼图的个数除以图的自同构群的阶数。并且这个幂级数的系数只是 Hurwitz 数。他们确定了图顶点处任何给定的一组分枝剖面的 $$\Sigma$$（或通过插入莫比乌斯带获得的克莱因表面的扩展）的覆盖数。我们专注于矩阵积分的组合描述。赫尔维茨数等于某类费曼图的个数除以图的自同构群的阶数。