Regularity of Dirac-harmonic maps

TL;DR

This paper establishes regularity, convergence, and Liouville theorems for Dirac-harmonic maps, revealing their smoothness in two dimensions and partial regularity in higher dimensions under certain conditions.

Contribution

It introduces an $ ext{epsilon}$-regularity theorem, a weak convergence theorem, and a Liouville theorem for stationary Dirac-harmonic maps, advancing understanding of their regularity properties.

Findings

Weakly Dirac-harmonic maps are smooth in 2D.

Established a weak convergence theorem for approximate maps in 2D.

Proved a Liouville theorem and partial regularity results in higher dimensions.

Abstract

For any $n$-dimensional compact spin Riemannian manifold $M$ with a given spin structure and a spinor bundle $\Sigma M$, and any compact Riemannian manifold $N$, we show an $\epsilon$-regularity theorem for weakly Dirac-harmonic maps . As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when $n = 2$. For $n \ge 3$, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in $R^n$. If, additions, $\psi\in W^{1,p}$ for some $p>2n/3$, then we obtain an energy monotonicity formula and prove a partial regularity theorem for any such a stationary Dirac-harmonic map.