On Sasakian quasi-Killing spinors in three-dimensions

TL;DR

This paper explores the properties of Sasakian quasi-Killing spinors in three-dimensional pseudo-Riemannian Sasakian space-forms, revealing their geometric significance and solutions to Einstein-Dirac systems with Maxwell fields.

Contribution

It provides a detailed study of SqK-spinors, linking them to Reeb vector fields, particle motion, and Einstein-Dirac-Maxwell solutions in three-dimensional Sasakian manifolds.

Findings

Reeb vector field described by a specific SqK-spinor

Solutions to Einstein-Dirac system with non-zero cosmological constant

Explicit formulae for SqK-spinors in elementary functions

Abstract

A Sasakian quasi-Killing spinor (SqK-spinor), which is a generalization of a Killing spinor on Sasakian manifolds, was defined in \cite{Kim Friedrich 2000}.The purpose of this paper is to study in detail SqK-spinors on three-dimensional pseudo-Riemannian Sasakian space-form.We briefly review some results on SqK-spinors and then investigate some geometric properties.First, we demonstrate that the Reeb vector field is described by a specific SqK-spinor. Then we establish that the motion of a charged particle in the presence of a contact Maxwell field can be depicted using an SqK-spinor. urthermore, we find that almost all SqK-spinors provide solutions to the Einstein-Dirac system with a non-zero cosmological constant.Additionally, we reveal that a particular SqK-spinor in conjunction with a contact Maxwell field satisfies the Einstein-Dirac-Maxwell systems.Finally, we show explicit formulae of SqK-spinors in terms of elementary functions with respect to a certain frame.