Super Riemann surfaces, metrics and gravitinos

The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of non-linear super symmetric sigma models, the action functional underlying string theory, can be obtained from a geometric action functional on super Riemann surfaces. All invariances of the super symmetric action functional are explained in super geometric terms and the action functional is a functional on the moduli space of super Riemann surfaces.

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We wish to thank Ron Donagi for helpful comments on earlier versions of this paper and for reminding us that smooth functions can not in general be extended. We are grateful to the anonymous referee for constructive criticism and, in particular, for pointing us at the importance of an invariant definition of the Berezin integral. The second author wants to thank the International Max Planck Research School Mathematics in the Sciences for financial support. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement nº 267087.

Published 8 March 2018