Geometrical Foundations of Cartan Gauge Gravity

TL;DR

This paper explores the geometric structure of Cartan gauge gravity, showing how Cartan connections unify key variables in general relativity and clarifying the relationship between gauge fixing and symmetry preservation.

Contribution

It provides a detailed geometric analysis of Cartan connections in gravity, highlighting their role in unifying variables and understanding gauge symmetry breaking.

Findings

Cartan connections unify spin connection and soldering form.

Gauge fixing corresponds to bundle reduction and partial symmetry breaking.

Internal translational invariance is preserved as external diffeomorphism invariance.

Abstract

We use the theory of Cartan connections to analyze the geometrical structures underpinning the gauge-theoretical descriptions of the gravitational interaction. According to the theory of Cartan connections, the spin connection $\omega$ and the soldering form $\theta$ that define the fundamental variables of the Palatini formulation of general relativity can be understood as different components of a single field, namely a Cartan connection $A=\omega+\theta$. In order to stress both the similarities and the differences between the notions of Ehresmann connection and Cartan connection, we explain in detail how a Cartan geometry $(P_{H}\rightarrow M, A)$ can be obtained from a $G$-principal bundle $P_{G}\rightarrow M$ endowed with an Ehresmann connection (being the Lorentz group $H$ a subgroup of $G$) by means of a bundle reduction mechanism. We claim that this reduction must be understood as a partial gauge fixing of the local gauge symmetries of $P_{G}$, i.e. as a gauge fixing that leaves "unbroken" the local Lorentz invariance. We then argue that the "broken" part of the symmetry--that is the internal local translational invariance--is implicitly preserved by the invariance under the external diffeomorphisms of $M$.