Spin structures on compact homogeneous pseudo-Riemannian manifolds

TL;DR

This paper classifies spin structures on compact homogeneous pseudo-Riemannian manifolds, especially flag manifolds and C-spaces, providing conditions for their existence and identifying those with invariant spin structures.

Contribution

It offers a classification of spin and metaplectic structures on flag manifolds and C-spaces, including necessary and sufficient conditions for the existence of invariant spin structures.

Findings

Classified spin structures on flag manifolds of compact simple Lie groups.

Provided conditions for the existence of spin structures on C-spaces.

Identified all C-spaces fibered over spin flag manifolds that are spin.

Abstract

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M=G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin.