A structure theorem of Dirac-harmonic maps between spheres

TL;DR

This paper investigates the zeros of spinor fields in Dirac-harmonic maps between surfaces, revealing a connection between zero order, surface genus, and classifying nontrivial maps from sphere to sphere.

Contribution

It provides a structure theorem for Dirac-harmonic maps between spheres, clarifying their zeros and classifying nontrivial cases from S^2 to S^2.

Findings

Zeros of $|C6|$ relate to surface genus

All nontrivial Dirac-harmonic maps from S^2 to S^2 are characterized

Bochner formulas are used to analyze zeros

Abstract

For an arbitrary Dirac-harmonic map $(\phi,\psi)$ between compact oriented Riemannian surfaces, we shall study the zeros of $|\psi|$. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of $|\psi|$ and the genus of $M$ and $N$. On the basis, we could clarify all of nontrivial Dirac-harmonic maps from $S^2$ to $S^2$.