Old and New Fields on Super Riemann Surfaces

TL;DR

This paper demonstrates that the recently introduced superconformal functions on N=1 super Riemann surfaces are equivalent to Abelian differentials on associated N=2 surfaces, clarifying their structure and properties.

Contribution

It establishes the equivalence between superconformal functions and Abelian differentials, and clarifies their algebraic structure on super Riemann surfaces.

Findings

Superconformal functions coincide with Abelian differentials plus constants.

They do not form a super vector space as originally defined.

The relationship between N=1 and N=2 super Riemann surfaces is clarified.

Abstract

The ``new fields" or ``superconformal functions" on $N=1$ super Riemann surfaces introduced recently by Rogers and Langer are shown to coincide with the Abelian differentials (plus constants), viewed as a subset of the functions on the associated $N=2$ super Riemann surface. We confirm that, as originally defined, they do not form a super vector space.