Are all Dirac-harmonic maps uncoupled?

TL;DR

This paper investigates Dirac-harmonic maps, showing that under certain minimality conditions, these maps are necessarily uncoupled, meaning the map component is harmonic.

Contribution

It proves that Dirac-harmonic maps on closed domains are uncoupled under minimality assumptions, clarifying the relationship between the map and spinor components.

Findings

Dirac-harmonic maps are uncoupled under minimality conditions

On closed domains, the map component is harmonic when the map is Dirac-harmonic

Provides conditions ensuring uncoupling of Dirac-harmonic maps

Abstract

Dirac-harmonic maps $(f,\phi)$ consist of a map $f:M\to N$ and a twisted spinor $\phi\in\Gamma(\Sigma M\otimes f^*TN)$ and they are defined as critical points of the super-symmetric energy functional. A Dirac-harmonic map is called \emph{uncoupled}, if $f$ is a harmonic map. We show that under some minimality assumption Dirac-harmonic maps defined on a closed domain are uncoupled.