Line Bundles on Super Riemann Surfaces

We give the elements of a theory of line bundles, their classification, and their connec-tions on super Riemann surfaces. There are several salient departures from the classicalcase. For example, the dimension of the Picard group is not constant, and there is nonatural hermitian form on Pic. Furthermore, the bundles with vanishing Chern numberaren’t necessarily flat, nor can every such bundle be represented by an antiholomorphicconnection on the trivial bundle. Nevertheless the latter representation is still useful ininvestigating questions of holomorphic factorization. We also define a subclass of all con-nections, those which are compatible with the superconformal structure. The compatibilityconditions turn out to be constraints on the curvature 2-form.