2008/07/17 by Katrina Barron, Barron, Katrina
Mathematics · Physics and Astronomy · #17A70 #32C11 #32Q30 #51P05 #53Z05 #58A50 #81T40 #81T60 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #hep-th #math-ph #math.DG #math.MP #msc:17A70 #msc:32C11 #msc:32Q30 #msc:51P05 #msc:53Z05 #msc:58A50 #msc:81T40 #msc:81T60
paper · pdf · doi:10.48550/arxiv.0807.2826
A mistake in the statement of one of the main theorems (Theorem 4.1) and a corollary (Corollary 4.2) is corrected. The other results are unchanged. Some typos are corrected. References are added. Additional expository comments are added. Parts of Sections 4 and 5 are rearranged, rewritten and simplified
openalex publication_date 2008/07/17 · arxiv created 2011/07/26 · arxiv updated 2011/07/27 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
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.