Einstein metrics on homogeneous superspaces

TL;DR

This paper explores Einstein metrics on homogeneous supermanifolds, providing explicit curvature formulas, constructing examples via Dynkin diagrams, and revealing diverse solution behaviors that challenge classical geometric conjectures.

Contribution

It introduces the study of Einstein equations on supermanifolds, constructs homogeneous supermanifolds using Dynkin diagrams, and presents new examples with varied Einstein metric solutions.

Findings

Existence of supermanifolds with no Einstein solutions.

Presence of discrete and continuous families of Einstein metrics.

Discovery of Ricci-flat invariant metrics on supermanifolds.

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.