Chiral de Rham Complex on Riemannian Manifolds and Special Holonomy

Interpreting the chiral de Rham complex (CDR) as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model, we suggest a setup for the study of CDR on manifolds with special holonomy. We show how to systematically construct global sections of CDR from differential forms, and investigate the algebra of the sections corresponding to the covariantly constant forms associated with the special holonomy. As a concrete example, we construct two commuting copies of the Odake algebra (an extension of the N = 2 superconformal algebra) on the space of global sections of CDR of a Calabi-Yau threefold and conjecture similar results for G₂ manifolds. We also discuss quasi-classical limits of these algebras.