不可定向曲面上的Yang-Mills联络
Yang-Mills Connections on Nonorientable Surfaces
摘要：
在“The Yang-Mills equations over Riemann surfaces”中，Atiyah和Bott从莫尔斯理论的角度研究了Riemann曲面上的Yang-Mills泛函。我们将他们的研究推广到所有封闭的、紧致的、连通的、可能不可定向曲面。我们引入了曲面的基本群的“超中心扩张”的概念。它是曲面可定向时的中心延伸。我们建立了Yang-Mills联络与超中心扩张表示之间的精确对应。知道了这个确切的对应关系，我们就主要在有限维的表象变体的层次上而不是在无限维的层次上工作。
英文摘要：
In "The Yang-Mills equations over Riemann surfaces", Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory. We generalize their study to all closed, compact, connected, possibly nonorientable surfaces. We introduce the notion of "super central extension" of the fundamental group of a surface. It is the central extension when the surface is orientable. We establish a precise correspondence between Yang-Mills connections and representations of super central extension. Knowing this exact correspondence, we work mainly at the level of representation varieties which are finite dimensional instead of the level of strata which are infinite dimensional.
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