Analytic definition of spin structure

TL;DR

This paper provides an analytic framework for defining spin structures on Lorentzian 4-manifolds and 3-dimensional Riemannian manifolds, using differential operators and gauge transformations, offering a new perspective beyond traditional topological definitions.

Contribution

It introduces an analytic definition of spin structures on Lorentzian and Riemannian manifolds, connecting geometric concepts with differential operator theory.

Findings

Analytic definition of spin structure on Lorentzian 4-manifolds

Analytic definition of spin structure on 3D Riemannian manifolds

Equivalence to traditional topological notions

Abstract

We work on a parallelizable time-orientable Lorentzian 4-manifold and prove that in this case the notion of spin structure can be equivalently defined in a purely analytic fashion. Our analytic definition relies on the use of the concept of a non-degenerate two-by-two formally self-adjoint first order linear differential operator and gauge transformations of such operators. We also give an analytic definition of spin structure for the 3-dimensional Riemannian case.