Cartan Connections and Atiyah Lie Algebroids

TL;DR

This paper develops a framework for Cartan connections using Atiyah Lie algebroids, providing a complete characterization and addressing a key mathematical question in gauge theories and gravity.

Contribution

It introduces a new approach relating two Atiyah Lie algebroids to characterize Cartan connections and clarifies the geometric setting for gauge transformations and diffeomorphisms.

Findings

Established a commutative, exact diagram relating two Atiyah Lie algebroids.

Provided a complete characterization of Cartan connections on principal bundles.

Clarified the geometric-algebraic setting for gauge transformations and diffeomorphisms in gravity.

Abstract

This work extends previous developments carried out by some of the authors on Ehresmann connections on Atiyah Lie algebroids. In this paper, we study Cartan connections in a framework relying on two Atiyah Lie algebroids based on a $H$-principal fiber bundle $\mathcal{P}$ and its associated $G$-principal fiber bundle $\mathcal{Q} := \mathcal{P} \times_H G$, where $H \subset G$ defines the model for a Cartan geometry. The first main result of this study is a commutative and exact diagram relating these two Atiyah Lie algebroids, which allows to completely characterize Cartan connections on $\mathcal{P}$. Furthermore, in the context of gravity and mixed anomalies, our construction answers a long standing mathematical question about the correct geometrico-algebraic setting in which to combine inner gauge transformations and infinitesimal diffeomorphisms.