曲面的正则覆盖的Hurwitz数与有限群的Cardy-Frobenius代数
Hurwitz numbers for regular coverings of surfaces by seamed surfaces and Cardy-Frobenius algebras of finite groups
摘要：
经典的Hurwitz数的一个新的定义是在有缝曲面对带标记点的曲面的正则覆盖的工作中。曲面类包括任何亏格和可定向的曲面，有或没有边界;覆盖在边界和标记点上可能具有某些奇点。前面介绍的接缝曲面实际上不是曲面。接缝表面的一个简单例子是书状接缝表面：几个矩形的边缘粘合在一起，就像书中的纸张。本文证明了一类正则覆盖的Hurwitz数定义了Klein拓扑域论（KTFT）的一个新例子，其中覆盖空间上的固定有限群$G$的作用使得泛点的稳定子与固定子群$K\subset G$共轭.已知KTFT与在Cardy-Frobenius代数中被称为的某类代数一一对应。我们构造了一类广泛的Cardy-Frobenius代数，特别是包含了有限群的所有Hecke代数.本文用群$G$及其子群描述了与缝合曲面对曲面的正则覆盖相对应的Cardy-Frobenius代数。因此，我们给出了引入Hurwitz数的一个代数公式.
英文摘要：
Analogue of classical Hurwitz numbers is defined in the work for regular coverings of surfaces with marked points by seamed surfaces. Class of surfaces includes surfaces of any genus and orientability, with or without boundaries; coverings may have certain singularities over the boundary and marked points. Seamed surfaces introduced earlier are not actually surfaces. A simple example of seamed surface is book-like seamed surface: several rectangles glued by edges like sheets in a book. We prove that Hurwitz numbers for a class of regular coverings with action of fixed finite group $G$ on cover space such that stabilizers of generic points are conjugated to a fixed subgroup $K\subset G$ defines a new example of Klein Topological Field Theory (KTFT). It is known that KTFTs are in one-to-one correspondence with certain class of algebras, called in the work Cardy-Frobenius algebras. We constructed a wide class of Cardy-Frobenius algebras, including particularly all Hecke algebras for finite groups. Cardy-Frobenius algebras corresponding to regular coverings of surfaces by seamed surfaces are described in terms of group $G$ and its subgroups. As a result, we give an algebraic formula for introduced Hurwitz numbers.
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