Line bundles on super Riemann surfaces

Abstract

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. We give the elements of a theory of line bundles, their classification, and their connections on super Riemann surfaces. There are several salient departures from the classical case. For example, the dimension of the Picard group is not constant, and there is no natural hermitian form on Pic. Furthermore, the bundles with vanishing Chern number aren't necessarily flat, nor can every such bundle be represented by an antiholomorphic connection on the trivial bundle. Nevertheless the latter representation is still useful in investigating questions of holomorphic factorization. We also define a subclass of all connections, those which are compatible with the superconformal structure. The compatibility conditions turn out to be constraints on the curvature 2-form.12/87.

Supercurves, their Jacobians, and super KP equations

“…Moduli of invertible sheaves. In this subsection we will discuss some facts about invertible sheaves on super curves and their moduli spaces, see also [RSV88,GN88b]. An invertible sheaf on (X, O X ) is determined by transition functions g αβ on overlaps U α ∩U β , and so isomorphism classes of invertible sheaves are classified by the cohomology group H 1 (X, O × X,ev ).…”

D-modules on 1|1 supercurves

“…We can cancel dθ , so dα = 0. This result seems to be new in this generality, although it is known for super Riemann surfaces where X =X [12,4].…”

Old and new fields on super Riemann surfaces

“…It has been known for some time that the globally defined holomorphic functions on a generic super Riemann surface (SRS) with odd spin structure do not form a super vector space [1]. Neither do the holomorphic superconformal tensors of weights −1 through 2 [2].…”