Topological invariants and moduli of hyperbolic $n=2$ Riemann supersurfaces

This article contains an investigation of $N=2$ Riemann supersurfaces arising in models of field theory. It is proved that the topological invariants of $N=2$ supersurfaces consist of the invariants of the underlying space (genus, number of holes and punctures) and the topological invariants of a pair of induced spinor forms. For each set of topological invariants a corresponding moduli space of supersurfaces is constructed. It is represented in the form $T/\mathrm{Mod}$, where $T$ is a linear superspace, and $\mathrm{Mod}$ is a discrete group. In passing, a classification is obtained for two-dimensional spinor bundles, along with an imbedding of the space of $N=1$ supersurfaces in the space of $N=2$ supersurfaces.