nLab Cartan connection

Context

Differential cohomology

differential cohomology

Contents

Idea

A Cartan connection is a principal connection on a smooth manifold equipped with a certain compatibility condition with the tangent bundle of the manifold. It combines the concept of G-structure with that of soldering form. This combination allows us to express various types of geometric structures on $X$ – such as notably (pseudo-)Riemannian geometry, conformal geometry and many more (see below) – in terms of connection data, i.e. in terms of nonabelian differential cohomology-data. In particular the first order formulation of gravity in terms of Cartan connections has been one of the historical motivations (Cartan 23).

In a little bit more detail, a Cartan connection on a manifold $X$ for a given subgroup inclusion $H \hookrightarrow G$ is data that identifies all the tangent spaces $T_x X$ of $X$ with the tangent space $\mathfrak{g}/\mathfrak{h} = T_{e H} (G/H)$ of the coset space Klein geometry $G/H$, such that the choice of these identifications is transported along compatibly.

Therefore a manifold equipped with a Cartan connection is also called a Cartan geometry (see also there), a generalization (globalization) of the concept of Klein geometry.

In yet a little bit more detail, an $(H \hookrightarrow G)$-Cartan connection on $X$ is a $G$-principal connection on $X$ equipped with a reduction of its structure group along $H \to G$ and such that the connection 1-form linearly identifies each tangent space $T_x X$ of $X$ with the tangent space $\mathfrak{g}/\mathfrak{h} = T_{e H} (G/H)$ of the coset space.

History

The concept essentially originates around (Cartan 23), but the formulation in terms of principal connections and in fact the terminology “Cartan connection” is due to Charles Ehresmann who formulated principal connections as what, in turn, today are called Ehresmann connections (Ehresmann 50).

In (Ehresmann 50) Cartan’s ideas are formalized (see Marle 14, page 9, 10 for review) by saying that an $(G \hookrightarrow H)$-Cartan connection is a $G$-Ehresmann connection on a $G$-principal bundle $P$ equipped with an $H$-principal subbundle $Q$, such that the restriction of the connection form along this inclusion yields a form that determines an isomorphism of each tangent space of $Q$ with $\mathfrak{g}$.

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.

There are various equivalent forms of the definition of Cartan connections. The following one characterizes it as a $G$-principal connection equipped with extra structure and property.

Definition

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

• a $G$-principal 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.

Remark

More explicitly for various component-realizations of principal connections, this means the following:

1. In terms of Ehresmann connection-data def. 1 says that an $(H\hookrightarrow G)$-Cartan connection is an Ehresmann connection form $A$ on a $G$-principal bundle $P$ together with an $H$-principal bundle $Q$ and an $H$-equivariant map $i \colon Q\to P$ such that $i^\ast A$ yields an isomorphism $T Q \simeq Q\times \mathfrak{g}$.

In this form the definition is due to (Ehresmann 50), recalled in (Marle 14, def. 4). Often this definition is stated by describing $i^\ast A$ directly without mentioning of $A$, e.g. (Sharpe 97, section 5.3, def. 3.1, Cap-Slovák 09, 1.5.1). Beware that $A$ is a principal connection but $i^\ast A$ is not.

2. In terms of Cech cocycle data, def. 1 says that an $(H\hookrightarrow G)$-Cartan connection is a cover $\{U_i \to X\}$ equipped with 1-forms $A_i \in \Omega^1(U_i, \mathfrak{g})$ and with transition functions $h_{i j} \in C^\infty(U_i \cap U_j, h)$ such that

• $h_{i j} h_{j k} = h_{i k}$ on $U_i \cap U_j \cap U_k$;

• $A_j = h_{i j}^{-1}(A_i + \mathbf{d})h_{i j}$ on $U_i \cap U_j$;

• $A_i(-)_x \colon T_x U_i \stackrel{\simeq}{\longrightarrow} \mathfrak{g}/\mathfrak{h}$.

In this form, the definition appears in (Sharpe 97, section 5.1 def. 1.3 together with section 5.2, Cap-Slovák 09, 1.5.4).

Definition

Given a Cartan connection $\nabla$, def. 1, its torsion of a Cartan connection is the image of its curvature under the projection $\mathfrak{g} \to \mathfrak{g}/\mathfrak{h}$.

Remark

In the case of vanishing torsion, the resulting flat parallel transport with values in $G/H$ identifies an open neighbourhood of each point of $X$ with an open neighbourhood in $G/H$.

Remark

The last clause in def. 1 says that the tangent space of $X$ at any point $x$ is being identified with the tangent space of the homogeneous space $G/H$ at the base point $e H$. This may be visualized by imagining that $X$ “tangentially touches” $G/H$ at $x\in X$ and $e H \in G/H$.

But by homogeneity, all the tangent spaces of $G/H$ are isomorphic, and canonically so by left translation. Hence by the path-lifting property of principal connections, one may (at least for vanishing torsion, def. 2) visualize the Cartan connection as describing how the $G/H$ touching $X$ at $x$ “rolls” along paths (infinitesimal paths, vectors) through $x$. This picture of “rolling” is particularly vivid for the case that $(H \hookrightarrow G) = (O(n)\hookrightarrow O(n+1))$ is the inclusion of orthogonal groups, which gives that $G/H = S^n$ is the n-sphere (for more on this see at conformal connection).

This picture of model spaces rolling along was influential in the historical development of the concept of Cartan geometry in the spirit of Klein geometry.

Synthetically in terms of differential cohesion

We discuss a synthetic formulation of Cartan connections in terms of differential cohesion.

under construction

Write

Proposition

There is a homotopy fiber sequence of smooth groupoids

$\array{ G/H &\stackrel{\theta/H}{\longrightarrow}& \Omega^1(-,\mathfrak{g})//H \\ && \downarrow^{\mathrlap{\mathbf{J}}} \\ && \mathbf{B}G_{conn} } \,,$

where $\theta/H$ is the $G$-Maurer-Cartan form modulo $H$.

Proof

A detailed proof for the statement as given is spelled out at this proposition.

But this statement holds generally in cohesive (∞,1)-toposes and an argument at this generality proceeds as follows: via the discussion at ∞-action the action of $G$ on $\Omega^1(-,\mathfrak{g})$ is exhibited by the forgetful map $\mathbf{B}G_{conn}\to \mathbf{B}G$ and since the action of $H$ on $\Omega^1(-,\mathfrak{g})$ is the restricted action, the square on the right of

$\array{ G/H &\stackrel{\theta/H}{\longrightarrow}& \Omega^1(-,\mathfrak{g})//H &\longrightarrow& \mathbf{B}H \\ \downarrow && \downarrow^{\mathrlap{\mathbf{J}}} && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G_{conn} &\longrightarrow& \mathbf{B}G }$

is a homotopy pullback. From this the pasting law implies that in the top left corner we have indeed $G/H$, this being the homotopy fiber of $\mathbf{B}H \to \mathbf{B}G$. Similar considerations show that the top left map is the abstractly defined Maurer-Cartan form.

Example

For $G$ a semisimple compact Lie group and $H = T\hookrightarrow G$ a maximal torus, then prop. 1 plays a central role in the stacky formulation of the orbit method. See there at this proposition.

We need this and one more ingredient for synthetically formalizing Cartan connections:

Remark

Let $\mathbb{D}^d_x \hookrightarrow X$ be the first-oder infinitesimal neighbourhood of a point in a manifold $X$. This being first order means that every differential p-form for $p \geq 2$ vanishes on $\mathbb{D}^d$. In particular therefore every principal connection restricted to $\mathbb{D}^d$ becomes a flat connection and hence is indeed gauge equivalent to the trivial connection. In particular every map

$\mathbb{D}^d_x \longrightarrow \Omega^1(-,\mathfrak{g})//H \to \mathbf{B}G_{conn}$

has a null-homotopy, hence fits into a square of the form

$\array{ \mathbb{D}^d_x &\stackrel{\nabla_H}{\longrightarrow}& \Omega^1(-,\mathfrak{g})//H \\ \downarrow &\swArrow_{\mathrlap{\simeq}}& \downarrow^{\mathrlap{\mathbf{J}}} \\ \ast &\longrightarrow& \mathbf{B}G_{conn} } \,.$

It follows by prop. 1 that $\nabla_H$ here factors through the Maurer-Cartan form

$\nabla_H|_{\mathbb{D}^d_x} \;\colon\; \mathbb{D}^d_x \stackrel{}{\longrightarrow} G/H \stackrel{\theta/H}{\longrightarrow} \Omega^1(-,\mathfrak{g})//H \,.$

The following is a synthetic formulation of Cartan connections, def. 1.

Definition

Let $X$ be a smooth set. Then an $(H \hookrightarrow G)$-Cartan connection on $X$ is

1. a $G$-principal connection

$\nabla \colon X \longrightarrow \mathbf{B}G_{conn}$
2. equipped with a reduction of structure groups given by a lift through $\mathbf{J}$ in prop. 1

$\array{ && \Omega^1(-,\mathfrak{g})//H \\ & {}^{\mathllap{\nabla^H}}\nearrow & \downarrow^{\mathrlap{\mathbf{J}}} \\ X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}G_{conn} }$
3. such that over each first-order infinitesimal neighbourhood $\mathbb{D}^d_x \hookrightarrow X$ any induced factorization, via remark 4,

$\mathbb{D}^d_x \stackrel{}{\longrightarrow} G/H$

Weaker definitions (pre- and semi-Cartan geometry)

We discuss here some weakining of the above definition of Cartan connection that have their uses.

…(Kuranishi 95)…

Examples

Table of example

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$Klein geometryCartan geometryCartan connection
examplesEuclidean group $Iso(d)$rotation group $O(d)$Cartesian space $\mathbb{R}^d$Euclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group $Iso(d-1,1)$Lorentz group $O(d-1,1)$Minkowski spacetime $\mathbb{R}^{d-1,1}$Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group $O(d-1,2)$$O(d-1,1)$anti de Sitter spacetime $AdS^d$AdS gravity
de Sitter group $O(d,1)$$O(d-1,1)$de Sitter spacetime $dS^d$deSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group $O(d,t+1)$conformal parabolic subgroupMöbius space $S^{d,t}$conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$super Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group $G$2-monomorphism $H \to G$homotopy quotient $G//H$Klein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) $H \to G$homotopy quotient $G//H$ of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

(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.

$G$-Structures

More generally, G-structures equipped with compatible principal connections are given by Cartan connections. (We will speak of “$H$-structure” here, since the reudced structure will correspond to the group denoted $H$ above, while what is denoted $G$ above will be the semidirect product of $H$ with the translation group).

Let $H \to GL(\mathbb{R}^n)$ be a Lie group homomorphism, so that reduction of the structure group of the frame bundle of a manifold of dimension $n$ along this map is an H-structure on the manifold. Then write

$G \coloneqq \mathbb{R}^n \rtimes H$

for the semidirect product of $H$ with the translation group $\mathbb{R}^n$, given via the induced action of $H$ on $\mathbb{R}^n$ via the canonical action of the general linear group $GL(\mathbb{R}^n)$.

With this an $(H \hookrightarrow G)= (H \hookrightarrow \mathbb{R}^n \rtimes H)$-Cartan connection is equivalently an H-structure equipped with a vielbein field and with an $H$-principal connection.

With this identification the torsion of a Cartan connection maps into the torsion of a G-structure.

References

The idea originates in Élie Cartan’s “method of moving frames” (cf. Cartan geometry).

• Élie Cartan Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie). Annales scientifiques de l’École Normale Supérieure, Sér. 3, 40 (1923), p. 325-412 (NUMDAM)

The formalization in terms of principal connections (in their incarnation as Ehresmann connections) is due to

• Charles Ehresmann, Les connexions infinitesimales dans un espace fibre diff´erentiable, Colloque de topologie de Bruxelles, 1950, p. 29–55.

reviewed in

Textbook accounts include

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

• Andreas ?ap?, Jan Slovák, chapter 1 of Parabolic Geometries I – Background and General Theory, AMS 2009

Discussion with an eye towards torsion constraints in supergravity is in

• John Lott, The Geometry of Supergravity Torsion Constraints Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125)