Connections on Cahen-Wallach spaces

TL;DR

This paper systematically studies connections on the spinor bundle of Cahen-Wallach spaces, revealing a quadratic Clifford algebra relation linked to the space's defining map, and provides a complete classification of solutions under certain algebraic conditions.

Contribution

It introduces a comprehensive analysis of connections on Cahen-Wallach spaces, establishing a quadratic Clifford algebra relation and classifying all solutions satisfying specific algebraic criteria.

Findings

Identified a quadratic relation on Clifford algebras associated with Cahen-Wallach spaces.

Provided a complete list of solutions under particular algebraic conditions.

Linked the solutions to the symmetric linear map defining the space.

Abstract

We systematically discuss connections on the spinor bundle of Cahen-Wallach symmetric spaces. A large class of these connections is closely connected to a quadratic relation on Clifford algebras. This relation in turn is associated to the symmetric linear map that defines the underlying space. We present various solutions of this relation. Moreover, we show that the solutions we present provide a complete list with respect to a particular algebraic condition on the parameters that enter into the construction.