The Functional of Super Riemann Surfaces – A “Semi-Classical” Survey

2015/11/16 by Enno Keßler, Jürgen Tolksdorf
Mathematics · #Action (physics) #Algebra over a field #Algebraic Geometry and Number Theory #Computer science #Differential (mechanical device) #Differential geometry #Geometric Analysis and Curvature Flows #Geometric function theory #Geometry #Mathematical analysis #Mathematics #Mathematics and Applications #Physics #Point (geometry) #Pure mathematics #Riemann hypothesis #Riemann surface #Space (punctuation) #Symmetry (geometry) #math.DG #msc:14H55 #msc:32G15 #msc:58A50

paper · pdf · doi:10.1007/s10013-016-0183-1

published as Vietnam Journal of Mathematics 44 (2016) 215-229

arxiv created 2015/11/16 · openalex publication_date 2016/02/18 · arxiv updated 2016/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

This article provides a brief discussion of the functional of super Riemann surfaces from the point of view of classical (i.e., not “super-”) differential geometry. The discussion is based on symmetry considerations and aims to clarify the “borderline” between classical and super differential geometry with respect to the distinguished functional that generalizes the action of harmonic maps and is expected to play a basic role in the discussion of “super Teichmüller space”. The discussion is also motivated by the fact that a geometrical understanding of the functional of super Riemann surfaces from the point of view of super geometry seems to provide serious issues to treat the functional analytically.

Citations

• Dirac-harmonic maps
• Harmonic Mappings of Riemannian Manifolds
• A locally supersymmetric and reparametrization invariant action for the spinning string
• The geometry of string perturbation theory
• The Einstein-Hilbert action with cosmological constant as a functional of generic form
• Riemannian Geometry and Geometric Analysis