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Supersymmetry of the Chiral de Rham Complex

We present a superfield formulation of the chiral de Rham complex (CDR) of Malikov-Schechtman-Vaintrob 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=1 structure on CDR (action of the N=1 super--Virasoro, or Neveu--Schwarz, algebra). If the metric is K"ahler, and the manifold Ricci-flat, this is augmented to an N=2 structure. Finally, if the manifold is hyperk"ahler, 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.