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 – 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 for a given subgroup inclusion is data that identifies all the tangent spaces of with the tangent space of the coset space Klein geometry , such that the choice of these identifications is transported along compatibly.
In yet a little bit more detail, an -Cartan connection on is a -principal connection on equipped with a reduction of its structure group along and such that the connection 1-form linearly identifies each tangent space of with the tangent space of the coset space.
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 -Cartan connection is a -Ehresmann connection on a -principal bundle equipped with an -principal subbundle , such that the restriction of the connection form along this inclusion yields a form that determines an isomorphism of each tangent space of with .
A -Cartan connection over a smooth manifold is;
More explicitly for various component-realizations of principal connections, this means the following:
In terms of Ehresmann connection-data def. 1 says that an -Cartan connection is an Ehresmann connection form on a -principal bundle together with an -principal bundle and an -equivariant map such that yields an isomorphism .
In this form the definition is due to (Ehresmann 50), recalled in (Marle 14, def. 4). Often this definition is stated by describing directly without mentioning of , e.g. (Sharpe 97, section 5.3, def. 3.1, Cap-Slovák 09, 1.5.1). Beware that is a principal connection but is not.
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).
The last clause in def. 1 says that the tangent space of at any point is being identified with the tangent space of the homogeneous space at the base point . This may be visualized by imagining that “tangentially touches” at and .
But by homogeneity, all the tangent spaces of 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 touching at “rolls” along paths (infinitesimal paths, vectors) through . This picture of “rolling” is particularly vivid for the case that is the inclusion of orthogonal groups, which gives that 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.
for the sheaf (on the site of formal smooth manifolds) of Lie algebra valued differential forms, regarded as the smooth moduli space of -differential forms (as explained at geometry of physics in the chapter on differential forms)
for the canonical morphism.
where is the -Maurer-Cartan form modulo .
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 on is exhibited by the forgetful map and since the action of on is the restricted action, the square on the right of
is a homotopy pullback. From this the pasting law implies that in the top left corner we have indeed , this being the homotopy fiber of . Similar considerations show that the top left map is the abstractly defined Maurer-Cartan form.
We need this and one more ingredient for synthetically formalizing Cartan connections:
Let be the first-oder infinitesimal neighbourhood of a point in a manifold . This being first order means that every differential p-form for vanishes on . In particular therefore every principal connection restricted to becomes a flat connection and hence is indeed gauge equivalent to the trivial connection. In particular every map
has a null-homotopy, hence fits into a square of the form
The following is a synthetic formulation of Cartan connections, def. 1.
Let be a smooth set. Then an -Cartan connection on is
We discuss here some weakining of the above definition of Cartan connection that have their uses.
|gauge group||stabilizer subgroup||local model space||local geometry||global geometry||differential cohomology||first order formulation of gravity|
|general||Lie group/algebraic group||subgroup (monomorphism)||quotient (“coset space”)||Klein geometry||Cartan geometry||Cartan connection|
|examples||Euclidean group||rotation group||Cartesian space||Euclidean geometry||Riemannian geometry||affine connection||Euclidean gravity|
|Poincaré group||Lorentz group||Minkowski space||Lorentzian geometry||pseudo-Riemannian geometry||spin connection||Einstein gravity|
|super Poincaré group||spin group||super Minkowski spacetime||Lorentzian supergeometry||supergeometry||superconnection||supergravity|
|linear algebraic group||parabolic subgroup/Borel subgroup||flag variety||parabolic geometry|
|orthochronous Lorentz group||conformal geometry||conformal connection||conformal gravity|
|general||smooth 2-group||2-monomorphism||homotopy quotient||Klein 2-geometry||Cartan 2-geometry|
|cohesive ∞-group||∞-monomorphism (i.e. any homomorphism)||homotopy quotient of ∞-action||higher Klein geometry||higher Cartan geometry||higher Cartan connection|
|examples||extended super Minkowski spacetime||extended supergeometry||higher supergravity: type II, heterotic, 11d|
is Lorentzian spacetime. Therefore an -Cartan connection is equivalently an -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 -valued part of the connection is the vielbein.
More generally, G-structures equipped with compatible principal connections are given by Cartan connections. (We will speak of “-structure” here, since the reudced structure will correspond to the group denoted above, while what is denoted above will be the semidirect product of with the translation group).
Textbook accounts include
R. Sharpe, Differential Geometry – Cartan’s Generalization of Klein’s Erlagen program Springer (1997)
Discussion with an eye towards torsion constraints in supergravity is in
Further discussion of Cartan connections as models for the first order formulation of gravity is in