Supersymmetry of the chiral de Rham complex

Abstract

We present a superfield formulation of the chiral deRham 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=1 structure on CDR (action of the N=1 super-Virasoro, or NeveuSchwarz, algebra). If the metric is Khler, and the manifold Ricci-flat, this is augmented to an N=2 structure. Finally, if the manifold is hyperkhler, 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 -models. © 2008 Foundation Compositio Mathematica.