For any n-dimensional compact spin Riemannian manifold M with a given spin structure and a spinor bundle M, and any compact Riemannian manifold N, we show an -regularity theorem for weakly Dirac-harmonic maps (φ, ψ) : M ⊗ M → N ⊗ φ * T N. 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

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... ville theorem for stationary Dirac-harmonic maps in R n . If, in addition, ψ ∈ 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. Introduction The notation of Dirac-harmonic maps is inspired by the supersymmetric nonlinear sigma model from the quantum field theory [8], and is a very natural and interesting extension of harmonic maps. In a series of papers [4, 5], Chen-Jost-Li-Wang recently introduced the subject of Dirac-harmonic maps and studied some analytic aspects of Dirac-harmonic maps from a spin Riemann surface into another Riemannian manifold. In order to review M |dφ| 2 dv g is the Dirichlet energy functional of φ : M → N, and its critical points are harmonic maps that have been extensively studied (see Lin-Wang [20] for relevant references). On the other hand, when φ = constant : M → N is a constant map, L(constant, ψ) = M ψ, Dψ dv g is the Dirac functional of ψ ∈ ( ) k , and its critical points are harmonic spinors ∂ψ = 0 that have also been well studied (see Lawson-Michelsohn [19]). Regularity of Dirac-Harmonic Maps 3 Studying the regularity of weakly Dirac-harmonic maps is one of our main interests. For this purpose, we introduce the natural Sobolev space in which the functional