# nLab Cartan connection

### Context

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

differential cohomology

# Contents

## Idea

The notion of Cartan connection is a special case of that of connection on a bundle which is more general than the notion of affine connection , but more special than the notion of principal connection , in general:

it is an $G$-principal connection subject to the constraint that the connection 1-form linearly identifies each tangent space of the base space with a quotient $\mathfrak{g}/\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ of $G$ by a sub-Lie algebra $\mathfrak{h}$.

The fiber of the bundle underlying a Cartan connection is a homogeneous space. This notion is closely related to Klein geometries.

Cartan connections are also just called Cartan geometries .

## Definition

Let $G$ be a Lie group and $H \hookrightarrow G$ a sub-Lie group. (So that we may think of the coset space $G/H$ as a Klein geometry.) Write $\mathfrak{h} \hookrightarrow \mathfrak{g}$ for the corresponding Lie algebras.

###### Definition

A $(H \hookrightarrow G)$-Cartan connection over a smooth manifold $X$ is;

• a $G$-affine connection $\nabla$ on $X$;

• such that

1. there is a reduction of structure groups along $H \hookrightarrow G$;

2. for each point $x \in X$ the canonical composite (for any local trivialization)

$T_x X \stackrel{\nabla}{\to} \mathfrak{g} \to \mathfrak{g}/\mathfrak{h}$

is an isomorphism.

This appears for instance as (Sharpe, section 5.1).

## Examples

### (pseudo-)Riemannian geometry

Let $G = Iso(d,1)$ be the Poincare group and $H \subset G$ the orthogonal group $O(d,1)$. Then the quotient

$\mathfrak{iso}(d,1)/\mathfrak{so}(d,1) \simeq \mathbb{R}^{d+1}$

is Lorentzian spacetime. Therefore an $(O(d,1)\hookrightarrow Iso(d,1))$-Cartan connection is equivalently an $O(d,1)$-connection on a manifold whose tangent spaces look like Minkowski spacetime: this is equivalently a pseudo-Riemannian manifold from the perspective discussed at first-order formulation of gravity:

the $\mathbb{R}^{d+1}$-valued part of the connection is the vielbein.

## References

Élie Cartan has introduced Cartan connections in his work on the Cartan’s “method of moving frames” (cf. Cartan geometry).

A standard textbook reference is

• R. Sharpe, Differential Geometry – Cartan’s Generalization of Klein’s Erlagen program Springer (1997)