The functional of super Riemann surfaces -- a "semi-classical" survey

TL;DR

This paper explores the functional of super Riemann surfaces through classical differential geometry, clarifying its role in super Teichmüller space and highlighting geometric challenges in super geometry.

Contribution

It offers a semi-classical, symmetry-based perspective on the super Riemann surface functional, bridging classical and super differential geometry.

Findings

Clarifies the borderline between classical and super differential geometry.

Highlights geometric issues in analyzing the super Riemann surface functional.

Provides insights into the role of the functional in super Teichmüller space.

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 Teichm\"uller 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.