Regularity of Dirac-Harmonic Maps

Abstract

For any n-dimensional compact spin Riemannian manifold M with a given spin structure and a spinor bundle Σ M, and any compact Riemannian manifold N, we show an ϵ-regularity theorem for weakly Dirac-harmonic maps (ϕ, ψ):M ⊗ Σ M → N ⊗ ϕ * TN. 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 ≥ 3, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in . If, in addition, ψ ∈ W 1,p for some , then we obtain an energy monotonicity formula and prove a partial regularity theorem for any such a stationary Dirac-harmonic map.