Supersymmetry of the chiral de Rham complex

We present a superfield formulation of the chiral de Rham complex (CDR), as introduced by Malikov, Schechtman and Vaintrob in 1999, in the setting of a general smooth manifold, and use it to endow CDR with superconformal structures of geometric origin. Given a Riemannian metric, we construct an N = I structure on CDR, (action of the N = 1 super-Virasoro, or Neveu-Schwarz, algebra). If the metric is Kahler, and the manifold Ricci-flat, this is augmented to all N = 2 structure. Finally, if the manifold is hyperkahler, we obtain an N = 4 structure. The superconformal structures are constructed directly from the Levi-Civita connection. These structures provide an analog for CDR, of the extended supersymmetries of nonlinear sigma-models.