  Dirac-harmonic maps are critical points of a fermionic action functional,
generalizing the Dirichlet energy for harmonic maps. We consider the case where
the source manifold is a closed Riemann surface with the canonical
Spin^c-structure determined by the complex structure and the target space is a
Kaehler manifold. If the underlying map f is a J-holomorphic curve, we
determine a space of spinors on the Riemann surface which form Dirac-harmonic
maps together with f. For suitable complex structures on the target manifold
the tangent bundle to the moduli space of J-holomorphic curves consists of
Dirac-harmonic maps. We also discuss the relation to the A-model of topological
string theory.
