超Teichmueller空间的整体解析方法
Global Analytic Approach to Super Teichmueller Spaces
摘要：
在这篇论文中，我们研究了超几何的一个新的形式主义，它关注于超几何理论的范畴性质。这种方法是我们随后研究构建超级Teichmueller空间的全局分析方法的主要工具。这些结果对其他领域，特别是超弦理论和超共形域论都有重要意义。这种新方法实际上是由莫洛特科夫、施瓦茨和沃罗诺夫在80年代中期首先提出的，它是基于对点的函子的后续使用。除了澄清超几何的各种问题，有时仍然模糊的标准（环空间）的方法，它的主要成就是，它使无限维超流形。我们用它来定义一个有限维超流形上的所有殆复结构的超流形，并证明它实际上携带一个复结构本身。此外，我们还成功地给出了有限维超流形的超同构超群的明确定义和构造。然后，我们结合这些结果表明，可以构造一个切片的可积殆复结构的子空间上的光滑闭定向2上的自同态超群的作用|二维超曲面这个切片代表了超Teichmueller空间的局部补丁。我们还研究如何建设的变化，而不是看在N=1超共形结构在这样的表面上，并表明，一个获得类似的结果。
英文摘要：
In this thesis we investigate a new formalism for supergeometry which focuses on the categorical properties of the theory. This approach is our main tool in the subsequent investigation of a global analytic approach to the construction of super Teichmueller spaces. These results should be of importance to various other fields, in particular, superstring theory and superconformal field theory. This new approach, which was actually first proposed by Molotkov and Schwarz and Voronov already in the mid-80s, is based on a consequent use of the functor of points. Apart from clarifying various issues of supergeometry which sometimes remain obscure in the standard (ringed-space) approach, its main achievement is that it makes infinite-dimensional supermanifolds available. We use this to define the supermanifold of all almost complex structures on a given finite-dimensional supermanifold and show that it actually carries a complex structure itself. Moreover we succeed in giving an explicit definition and construction of the diffeomorphism supergroup of a finite-dimensional supermanifold. We then combine these results to show that one can construct a slice for the action of the diffeomorphism supergroup on the subspace of integrable almost complex structures on a smooth closed oriented 2|2-dimensional supersurface. This slice represents a local patch of super Teichmueller space. We also investigate how this construction changes if one instead looks at N=1 superconformal structures on such a surface and show that one obtains an analogous result.
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