Doctoral thesis 2/2009
Global Analytic Approach to Super Teichmüller Spaces
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Submission date: 18. Feb. 2009
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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 Teichmüller 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.
In the last part of the thesis we show that, unlike the case of Riemann surfaces, not every almost complex structure on a super Riemann surface is integrable. We then combine this result with the above constructions 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 Teichmüller 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.