Energy identity and necklessness for $$\alpha $$ α -Dirac-harmonic maps into a sphere

Let $$(\phi _\alpha , \psi _\alpha )$$ be a sequence of $$\alpha $$ -Dirac-harmonic maps from a Riemann surface M to a compact Riemannian manifold N with uniformly bounded energy. If the target N is a sphere $$S^{K-1}$$ , we show that the energy identity and necklessness hold during the interior blow-up process as $$\alpha \searrow 1$$ for such a sequence .