实代数曲线的二维拓扑域论的非交换扩张与Hurwitz数
Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves
摘要：
众所周知，经典的二维拓扑域论与交换Frobenius代数是一一对应的。开闭二维拓扑域论是经典二维拓扑域论的一个重要推广。本文将开闭二维拓扑域论推广到不可定向曲面。我们称之为克莱因拓扑域论（KTFT）。我们证明了KTFT双射对应于代数与某些额外的结构，称为结构代数。对半单结构代数进行了分类。从任意有限群出发，构造了一个结构代数，并证明了它是半单的。定义了实代数曲线的类似Hurwitz数，证明了它们是KTFT的分解子.该KTFT的结构代数是对称群的结构代数。
英文摘要：
It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend open-closed two-dimensional topological field theories to nonorientable surfaces. We call them Klein topological field theories(KTFT). We prove that KTFTs bijectively correspond to algebras with certain additional structures, called structure algebras. Semisimple structure algebras are classified. Starting from an arbitrary finite group, we construct a structure algebra and prove that it is semisimple. We define an analog of Hurwitz numbers for real algebraic curves and prove that they are correlators of a KTFT. The structure algebra of this KTFT is the structure algebra of the symmetric group.
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