Supersymmetry of the Chiral de Rham Complex 2: Commuting Sectors

We construct two commuting N = 2 structures on the space of sections of the chiral de Rham complex (CDR) of a Calabi–Yau manifold. We use this extra supersymmetry to construct a nonlinear automorphism of CDR preserving these N = 2 structures. Finally, we show how to extend these results to construct a commuting pair of N = 4 super vertex algebras in the HyperKähler case.