Super Riemann Surfaces: Uniformization and Teichmiiller Theory

Teichmiiller theory for super Riemann surfaces is rigorously developed using the supermanifold theory" of Rogers. In the case of trivial topology in the soul directions, relevant for superstring applications, the following results are proven. The super Teichmiiller space is a complex super-orbifold whose body is the ordinary Teichmiiller space of the associated Riemann surfaces with spin structure. For genus g > 1 it has 3g-3 complex even and 29-2 complex odd dimensions. The super modular group which reduces super Teichmfiller space to super moduli space is the ordinary modular group; there are no new discrete modular transformations in the odd directions. The boundary of super Teichmiiller space contains not only super Riemann surfaces with pinched bodies, but Rogers supermanifolds having nontrivial topology in the odd dimensions as well. We also prove the uniformization theorem for super Riemann surfaces and discuss their representation by discrete supergroups of Fuchsian and Schottky type and by Beltrami differentials. Finally we present partial results for the more difficult problem of classifying super Riemann surfaces of arbitrary topology.