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Super Riemann surfaces: Uniformization and Teichmüller theory. (English) Zbl 0659.30039
In view of applications in the superstring theory the global structure of the space of super moduli is studied. By employing the Rogers theory of supermanifolds and adopting Friedans global definition of a super Riemann surface the Teichmüller theory of Riemann surfaces is generalized.
The construction of super Teichmüller space follows closely that of the ordinary Teichmüller space. For the cases relevant for superstring applications the results characterizing the structure of super Teichmüller space are proven. It appears to be a complex super-orbifold whose body is the ordinary Teichmüller space of the associated Riemann surfaces with spin structure and has, for \(g>1\), 3g-3 complex even and 2g-2 complex odd dimensions. The important result is that the super modular group is simply the ordinary modular group.
The uniformization theorem for super Riemann surfaces is proved.
The generalization of the representation of Riemann surfaces in terms of Beltrami differentials is given. The super Beltrami equations are derived and their solution is discussed. It is then shown that metrizable super Riemann surfaces can be represented by Schottky supergroups. The notion of universal super Teichmüller space is defined.
Some remarks about the uniformization theorem are super Riemann surfaces with arbitrary topology are also included.
Reviewer: P.Maślanka

MSC:
30F99 Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
83E99 Unified, higher-dimensional and super field theories
81Q99 General mathematical topics and methods in quantum theory
81V99 Applications of quantum theory to specific physical systems
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