On supersymmetric Einstein-Weyl spaces

TL;DR

This paper classifies Lorentzian Einstein-Weyl geometries that admit weighted parallel spinors, revealing they are either conformally related to geometries with parallel spinors or to specific Kundt spacetimes, with detailed results in 4 and 6 dimensions.

Contribution

It provides a classification of supersymmetric Lorentzian Einstein-Weyl spaces admitting weighted parallel spinors, linking them to known geometries and Kundt spacetimes.

Findings

Geometries are either conformally related to parallel spinor geometries or Kundt spacetimes.

Full classification achieved for 4 and 6 dimensions.

Weighted parallel spinors imply Einstein-Weyl structure in Lorentzian geometry.

Abstract

We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor, implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to supergravity theories to characterise those Lorentzian EW geometries that allow for a weighted parallel spinor, calling the resulting geometries supersymmetric. The overall result is that they are either conformally related to ordinary geometries admitting parallel spinors (w.r.t. the Levi-Civita connection) or are conformally related to certain Kundt spacetimes. A full characterisation is obtained for the 4 and 6 dimensional cases.