Einstein Metrics on Homogeneous Superspaces

Abstract

This paper initiates the study of the Einstein equation on homogeneous supermanifolds. First, we produce explicit curvature formulas for graded Riemannian metrics on these spaces. Next, we present a construction of homogeneous supermanifolds by means of Dynkin diagrams, resembling the construction of generalised flag manifolds in classical (non-super) theory. We describe the Einstein metrics on several classes of spaces obtained through this approach. Our results provide examples of compact homogeneous supermanifolds on which the Einstein equation has no solutions, discrete families of solutions, and continuous families of Ricci-flat solutions among invariant metrics. These examples demonstrate that the finiteness conjecture from classical homogeneous geometry fails on supermanifolds, and challenge the intuition furnished by Bochner’s vanishing theorem.