On uniformization of N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces

TL;DR

This paper establishes a comprehensive uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, linking their classification to cohomology groups and holomorphic line bundles.

Contribution

It provides new criteria for classifying super-Riemann surfaces via cohomology and holomorphic line bundles, extending uniformization results to supergeometric contexts.

Findings

Classifies genus-zero and genus-one super-Riemann surfaces up to superconformal equivalence.

Connects superconformal equivalence classes to holomorphic line bundles over the body Riemann surface.

Provides conditions under which super-Riemann surfaces are equivalent to ringed-space supermanifolds.

Abstract

We prove a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, showing that in general an N=2 superconformal (resp. N=1 superanalytic) DeWitt super-Riemann surface is N=2 superconformally (resp., N=1 superanalytically) equivalent to a manifold with transition functions containing no odd functions of the even variable if and only if a certain cohomology group is trivial, namely the first Cech cohomology group of the body Riemann surface with coefficients in the sheaf consisting of the reciprocal of a line bundle tensor the holomorphic vector fields over the body. In particular, this gives a general criteria for when a DeWitt N=1 superanalytic super-Riemann surface is N=1 superanalytically equivalent to a ringed-space (1,1)-supermanifold, as studied in the algebro-geometric setting. This general classification result implies there is a countably infinite family of N=2 superconformal equivalence classes of N=2 superconformal DeWitt super-Riemann surfaces with genus-zero compact body, and N=2 superconformal DeWitt super-Riemann surfaces with simply connected body are classified up to N=2 superconformal equivalence by conformal equivalence classes of holomorphic line bundles over the underlying body Riemann surface. In addition, N=2 superconformal DeWitt super-Riemann surfaces with compact genus-one body and transition functions which correspond to the trivial cocycle in the first Cech cohomology group of the body Riemann surface with coefficients in the reciprocal of a line bundle tensor the sheaf of holomorphic vector fields over the body are classified up to N=2 superconformal equivalence by holomorphic line bundles over the torus modulo conformal equivalence. The corresponding results for the uniformization of N=1 superanalytic DeWitt super-Riemann surfaces of genus zero or one are presented.