Moduli spaces of real algebraic \(N=2\) supercurves
In this article, the moduli space of real \(N=2\) supercurves is studied. \(N=2\) complex supercurves are obtained by factoring out a super-Fuchsian group from the \(N=2\) super-half-plane. Real supercurves are given by a complex supercurve with an antiholomorphic involution. The real points of the curve are the fixed points of the involution. A complete set of topological invariants of the curve is given by the genus, the number of ovals, and by connectedness (resp. non-connectedness) of the complex curve with the real points removed. The connected components of the moduli space are explicitly described and a complete set of topological invariants for the components are given. Each of the components is represented as \(\mathbb R^{(m| n)}/\text{Mod}\) of a linear superspace by a discrete group \(\text{Mod}\). The dimensions \(m\) and \(n\) are given. Depending on whether the two Arf functions coincide or not, one obtains \((m| n)=(4g-3| 4g-4)\), resp. \((4g-4| 4g-4)\). Real \(N=2\) supercurves can be mapped to real spinor bundles of rank 2 over real algebraic curves. Hence a description of the moduli space of spinor bundles of rank 2 over a real curve is obtained.
scientific article; zbMATH DE number 1053618
scientific article