# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1904.04915