Some classes of spinorial connections and their holonomy

Abstract

We study certain classes of spinorial connections and their holonomy algebras from two different perspectives. Firstly, given a spinorial connection we want to determine its spin holonomy algebra and investigate if the latter admits fixed spinors. This can be achieved for some classes of examples in flat space. For a more general class of connections we can show that the spin holonomy algebra must be either trivial or semisimple. Secondly, given a spin holonomy algebra we want to investigate two things, namely what a spinorial connection inducing this holonomy algebra can Iook like, and what geometries this gives rise to. We will be able to give a complete answer in case when the spin holonomy is trivial for a particular class of spinorial connections. If the holonomy algebra is abelian we can derive some necessary conditions.