Regularity of Dirac-harmonic maps with $$\lambda $$-curvature term in higher dimensions

In this paper, we will study the partial regularity for stationary Dirac-harmonic maps with $$\lambda $$λ-curvature term. For a weakly stationary Dirac-harmonic map with $$\lambda $$λ-curvature term $$(\phi ,\psi )$$(ϕ,ψ) from a smooth bounded open domain $$\Omega \subset {\mathbb {R}}^m$$Ω⊂Rm with $$m\ge 2$$m≥2 to a compact Riemannian manifold N, if $$\psi \in W^{1,p}(\Omega )$$ψ∈W1,p(Ω) for some $$p>\frac{2m}{3}$$p>2m3, we prove that $$(\phi , \psi )$$(ϕ,ψ) is smooth outside a closed singular set whose $$(m-2)$$(m-2)-dimensional Hausdorff measure is zero. Furthermore, if the target manifold N does not admit any harmonic sphere $$S^l$$Sl, $$l=2,\ldots , m-1$$l=2,…,m-1, then $$(\phi ,\psi )$$(ϕ,ψ) is smooth.