Some examples of Dirac-harmonic maps

TL;DR

This paper examines a method for constructing Dirac-harmonic maps using harmonic and twistor spinors, clarifies when the method applies, and demonstrates the existence of solutions in specific cases.

Contribution

It clarifies the conditions under which the existing construction method for Dirac-harmonic maps applies, especially for isometric immersions into spaceforms, and provides examples of solutions.

Findings

Conditions for the method's applicability are fulfilled only under special assumptions.

Existence of solutions is demonstrated in several cases.

The method mainly applies to codimension 1 maps into constant curvature spaces.

Abstract

We discuss a method to construct Dirac-harmonic maps developed by J.~Jost, X.~Mo and M.~Zhu in J.~Jost, X.~Mo, M.~Zhu, \emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. \textbf{59} (2009), no. 11, 1512--1527.The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmonic maps of codimension $1$ with target spaces of constant sectional curvature.Before the present article, it remained unclear when the conditions of the theorems in J.~Jost, X.~Mo, M.~Zhu, \emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. \textbf{59} (2009), no. 11, 1512--1527, were fulfilled. We show that for isometric immersions into spaceforms, these conditions are fulfilled only under special assumptions.In several cases we show the existence of solutions.