Local spin base invariance from a global differential-geometrical point of view

TL;DR

This paper provides a geometric interpretation of local spin base invariance in curved spacetime, relating it to principal bundles and Dirac structures, and discusses its implications for quantum gravity.

Contribution

It establishes a connection between Dirac structures and Spin/Spin^C structures, showing that Dirac structures can serve as a more natural variable for quantum gravity.

Findings

Existence of a Dirac structure implies Spin^C structure.

Spin base invariance allows more physical degrees of freedom.

Dirac structure may be preferable for quantum gravity variables.

Abstract

This article gives a geometric interpretation of the spin base formulation with local spin base invariance of spinors on a curved space-time and in particular of a central element, the global Dirac structure, in terms of principal and vector bundles and their endomorphisms. It is shown that this is intimately related to Spin and Spin^C structures in the sense that the existence of one of those implies the existence of a Dirac structure and allows an extension to local spin base invariance. Vice versa, as a central result, the existence of a Dirac structure implies the existence of a Spin^C structure. Nevertheless, the spin base invariant setting may be considered more general, allowing more physical degrees of freedom. Furthermore, arguments are given that the Dirac structure is a more natural choice as a variable for (quantum) gravity than tetrads/vielbeins.