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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.

Regularity of Dirac-Harmonic Maps