derived smooth geometry
It is the globalized version of higher Klein geometry.
An -Cartan geometry over with respect to is
equipped with an ∞-connection ;
such that for every point any any local trivialization, the canonical composite
Notice that ordinary gravity can be understood as the theory of -Cartan geometry, where is the Poincare group and the orthogonal group of Minkowski space. This is called the first order formulation of gravity.
One can read the D'Auria-Fre formulation of supergravity as saying that higher dimensional supergravity is analogously given by higher Cartan supergeometry. See there and see the examples at higher Klein geometry for more on this.
|local model space||global geometry||differential cohomology||first order formulation of gravity|
|general||Klein geometry||Cartan geometry||Cartan connection|
|examples||Euclidean geometry||Riemannian geometry||affine connection||Euclidean gravity|
|Lorentzian geometry||pseudo-Riemannian geometry||spin connection||Einstein gravity|
|general||Klein 2-geometry||Cartan 2-geometry|
|higher Klein geometry||higher Cartan geometry||higher Cartan connection|
|examples||extended super Minkowski spacetime||extended supergeometry||higher supergravity: type II, heterotic, 11d|