Hidden symmetries and Lie algebra structures from geometric and supergravity Killing spinors

TL;DR

This paper explores the algebraic structures formed by Killing spinors and their bilinears in geometric and supergravity contexts, revealing Lie algebra and superalgebra structures in various dimensions.

Contribution

It introduces new Lie algebra and superalgebra structures derived from geometric and supergravity Killing spinors and their bilinears across different spacetime dimensions.

Findings

Killing-Yano forms form a Lie superalgebra in constant curvature spacetimes.

Dirac currents of geometric Killing spinors satisfy a Lie algebra under certain conditions.

Supergravity Killing forms form Lie algebras in six and ten dimensions, with additional conditions in five and eleven dimensions.

Abstract

We consider geometric and supergravity Killing spinors and the spinor bilinears constructed out of them. The spinor bilinears of geometric Killing spinors correspond to the antisymmetric generalizations of Killing vector fields which are called Killing-Yano forms. They constitute a Lie superalgebra structure in constant curvature spacetimes. We show that the Dirac currents of geometric Killing spinors satisfy a Lie algebra structure up to a condition on 2-form spinor bilinears. We propose that the spinor bilinears of supergravity Killing spinors give way to different generalizations of Killing vector fields to higher degree forms. It is also shown that those supergravity Killing forms constitute a Lie algebra structure in six and ten dimensional cases. For five and eleven dimensional cases, the Lie algebra structure depends on an extra condition on supergravity Killing forms.