A Superconformal Action Functional for Super Riemann Surfaces

The harmonic action functional on Riemann surfaces is a tool to study the moduli space of Riemann surfaces. In 2D-Supergravity and string theory one considers a supersymmetric extension of the action functional, where both the field and the metric get a superpartner. The appropriate geometric setting for this action functional is supergeometry and in particular super Riemann surfaces. In this talk we are going to see how the geometric properties of super Riemann surfaces lead to the known symmetries of the supersymmetric action functional and that the supersymmetric action functional can be used as a tool to study the moduli space of super Riemann surfaces. It will be shown that the gravitino, which is the superpartner of the metric, encodes information about the odd dimensions of this moduli space, similar to the idea of Teichmüller spaces, where metrics encode information on the moduli space of Riemann surfaces.