Dirac-harmonic maps with trivial index

TL;DR

This paper introduces a new invariant to establish the existence of Dirac-harmonic maps with trivial index, extending the theory to cases previously limited to constructive solutions and uncoupled solutions.

Contribution

It defines a new homotopy-invariant quantity and proves its invariance, enabling the existence of Dirac-harmonic maps in trivial index cases, including from Riemann surfaces to Kähler manifolds.

Findings

Proves homotopy invariance of the new quantity.

Establishes existence of Dirac-harmonic maps in trivial index cases.

Provides short-time existence of the $eta$-Dirac-harmonic map flow.

Abstract

For a homotopy class $[u]$ of maps between a closed Riemannian manifold $M$ and a general manifold $N$, we want to find a Dirac-harmonic map with the map component in the given homotopy class. Most known results require the index to be nontrivial. When the index is trivial, the few known results are all constructive and produce uncoupled solutions. In this paper, we define a new quantity. As a byproduct of proving the homotopy invariance of this new quantity, we find a new simple proof for the fact that all Dirac-harmonic spheres in surfaces are uncoupled. More importantly, by using the homotopy invariance of this new quantity, we prove the existence of Dirac-harmonic maps from manifolds in the trivial index case. In particular, when the domain is a closed Riemann surface, we prove the short-time existence of the $\alpha$-Dirac-harmonic map flow in the trivial index case. Together with the density of the minimal kernel, we get an existence result for Dirac-harmonic maps from closed Riemann surfaces to K\"ahler manifolds, which extends the previous result of the first and third authors. This establishes a general existence theory for Dirac-harmonic maps in the context of trivial index.