Classifying Spinor Structures

TL;DR

This paper provides a comprehensive classification of spinor structures in Riemannian geometry, linking principal fibre bundles, covering spaces, and spinor calculus, with implications for the Dirac equation in physics.

Contribution

It offers a complete, constructive classification of spinor structures, extending previous results and connecting geometric ideas with physical applications.

Findings

Classification of spinor structures using principal fibre bundles and covering spaces

Comparison of different spinor connection types

Impact of spinor structure choice on the Dirac equation

Abstract

I begin by explaining how Riemannian geometry can be understood in terms of principal fibre bundles and connections thereon. I then introduce and motivate the definition of a spinor structure in terms of familiar geometrical ideas. The central result of this thesis is a complete and constructive classification of spinor structures, generalising some earlier results. I will explain how principal fibre bundles and covering spaces provide the key ingredients to the proof. A different type of classification can also be attempted, in terms of the underlying principal fibre bundle, and this allows us to compare `spinor connections'. The final part indicates how spinor structures for Lorentzian manifolds provide the natural setting for the `spinor calculus', and so for the Dirac equation for the electron. The effect of the choice of spinor structure on the Dirac equation is investigated.