Holomorphic Supercurves and Supersymmetric Sigma Models

TL;DR

This paper generalizes holomorphic curves to supermanifold morphisms called holomorphic supercurves, linking them to supersymmetric sigma models and showing they are extrema of a supersymmetric action functional.

Contribution

It introduces holomorphic supercurves as a new concept, extending classical holomorphic curves to the supermanifold setting with a supersymmetric interpretation.

Findings

Holomorphic supercurves satisfy a generalized Cauchy-Riemann condition.

They are identified as extrema of a supersymmetric action functional.

The framework connects supergeometry with supersymmetric field theories.

Abstract

We introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds, referred to as holomorphic supercurves. More precisely, supercurves are morphisms from a Riemann surface, endowed with the structure of a supermanifold which is induced by a holomorphic line bundle, to an ordinary almost complex manifold. They are called holomorphic if a generalised Cauchy-Riemann condition is satisfied. We show, by means of an action identity, that holomorphic supercurves are special extrema of a supersymmetric action functional.