Teleparallel Gravity as a Higher Gauge Theory

TL;DR

This paper reformulates general relativity as a higher gauge theory using the teleparallel 2-group, highlighting the role of torsion and coframe fields within a categorical framework.

Contribution

It introduces a novel higher gauge theory perspective on general relativity via the teleparallel 2-group, connecting torsion and coframe fields to 2-connections.

Findings

Reformulates GR as a 2-connection theory with teleparallel 2-group

Constructs principal 2-bundles with Poincare 2-group structure

Shows equivalence of flat 2-connections and GR in teleparallel form

Abstract

We show that general relativity can be viewed as a higher gauge theory involving a categorical group, or 2-group, called the teleparallel 2-group. On any semi-Riemannian manifold M, we first construct a principal 2-bundle with the Poincare 2-group as its structure 2-group. Any flat metric-preserving connection on M gives a flat 2-connection on this 2-bundle, and the key ingredient of this 2-connection is the torsion. Conversely, every flat strict 2-connection on this 2-bundle arises in this way if M is simply connected and has vanishing 2nd deRham cohomology. Extending from the Poincare 2-group to the teleparallel 2-group, a 2-connection includes an additional piece: a coframe field. Taking advantage of the teleparallel reformulation of general relativity, which uses a coframe field, a flat connection and its torsion, this lets us rewrite general relativity as a theory with a 2-connection for the teleparallel 2-group as its only field.