Metrics and Gravitinos

Abstract

Let i: |M|→ M be an embedding of an underlying even manifold into a super Riemann surface M. In this chapter we are concerned with the structure induced on |M|. We will show that a given $\operatorname {\mathrm {U}}(1)$-structure on M induces a metric g, a spinor bundle S and a differential form χ with values in S, called gravitino, on |M|. Different $\operatorname {\mathrm {U}}(1)$-structures on M induce metrics and gravitinos which differ only by conformal and super Weyl transformations. Furthermore, the triple (g, S, χ) on |M| is sufficient to reconstruct the super Riemann surface M. Supersymmetry of metric and gravitino are interpreted as an infinitesimal change of the embedding i. From this point of view we are able to give a description of the infinitesimal deformations of a super Riemann surface in terms of metric and gravitino.