  We study a new set of coupled field equations motivated by the non-linear
supersymmetric sigma model of quantum field theory. These equations couple a
map into a Riemannian manifold controlled by a harmonic map like action with a
spinor field along that map. We study the solutions which we call
Dirac-harmonic maps from a Riemann surface to a sphere $\S^n$. We show that a
weakly Dirac-harmonic map is in fact smooth, and prove that the energy identity
holds during the blow-up process.
