Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions

TL;DR

This paper explores how gravity in three and four dimensions can be formulated as gauge theories using Cartan connections on symmetric spaces, revealing new insights into topological mass and the Immirzi parameter.

Contribution

It introduces a Cartan geometric formulation of gravity models in 3 and 4 dimensions, unifying various approaches and highlighting the role of symmetric space structures.

Findings

3D gravity as constrained Chern-Simons theory

4D MacDowell-Mansouri gravity with Immirzi parameter

Topological mass and Immirzi parameter linked to Lie algebra non-simplicity

Abstract

Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formulation of topologically massive gravity, with arbitrary cosmological constant, as a single constrained Chern-Simons action. In 4 dimensions the main model of interest is MacDowell-Mansouri gravity, generalized to include the Immirzi parameter in a natural way. I formulate these theories in Cartan geometric language, emphasizing also the role played by the symmetric space structure of the model. I also explain how, from the perspective of these Cartan-geometric formulations, both the topological mass in 3d and the Immirzi parameter in 4d are the result of non-simplicity of the Lorentz Lie algebra so(3,1) and its relatives. Finally, I suggest how the language of Cartan geometry provides a guiding principle for elegantly reformulating any 'gauge theory of geometry'.