Connecting Through Obstruction; Relating Gauge Gravity and String Theory

TL;DR

This paper explores the geometric and topological foundations of gravity as a gauge theory using composite bundles, revealing that spacetime manifolds in this framework admit both spin and string structures, extending previous global structure results.

Contribution

It provides a detailed account of the composite bundle formalism for gravity, linking gauge bundles to tangent bundles and demonstrating the existence of string structures on spacetime manifolds.

Findings

Spacetime manifolds in this formalism admit spin structures.

Spacetime manifolds also admit string structures.

The composite bundle approach clarifies the global geometric structure of gravity.

Abstract

In this article we provide a more detailed account of the geometry and topology of the composite bundle formalism introduced by Tresguerres in Phys. Rev. D 66 (2002) 064025 [1] to accommodate gravitation as a gauge theory. In the first half of the article we identify a global structure required by the composite construction which not only exposes how the ordinary frame and tangent bundle expected in general relativity arise but provides the link or connection between these ordinary bundles and the gauge bundles of the composite bundle construction. In the second half of the article we discuss implications of this method of constructing gravity as a fiber bundle for the global structure of spacetime. We find that the underlying manifold of the composite bundle construction is expected to admit, not only a spin structure but also a string structure. As a consequence of our work we are able to extend past work on global structures of physically reasonable spacetime manifolds. It has been shown that in four spacetime dimensions, that if an oriented, Lorentzian, four dimensional manifold is stably casual that it is parallizable, and hence admits a spin structure which allows for chiral spinors. We may now add to this that such a manifold also admits a string structure.