Noncommutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves
Noncommutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves
复制标题
二维拓扑场论和实代数曲线的赫尔维茨数的非交换扩展
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
S. Natanzon
中科院分区：
文献类型：
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作者：
A. Alexeevski;S. Natanzon
众所周知，经典的二维拓扑场论与交换的Frobenius代数一一对应。开闭二维拓扑场论是经典二维拓扑场论的一个重要扩展。
本文将开闭二维拓扑场理论推广到不可定向曲面。我们称之为克莱因拓扑场论(KTFT)。
我们证明了KTFT双射对应于具有某些附加结构的代数，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。
我们定义了实代数曲线的一个Hurwitz数的模拟，并证明了它们是KTFT的相关子。这个KTFT的结构代数是对称群的结构代数。
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.