Types of quantum field thories
The action functional of gravity was originally conceived as a functional on the space of pseudo-Riemannian metrics of a manifold . Later on it was realized that it may alternatively naturally be thought of as a functional on the space of connections with values in the Poincaré Lie algebra – essentially the Levi-Civita connection – subject to the constraint that the component in the translation Lie algebra defines a vielbein field. Mathematically this means that the field of gravity is modeled as a Cartan connection for the Lorentz group inside the Poincaré group. In physics this is known as the first order formalism or the Palatini formalism for gravity.
Promoting this perspective form the Poincaré group to the super Poincaré group yields supergravity. Promoting it further to the Lie n-algebra extensions of the super Poincaré group (from the brane scan/brane bouquet) yields type II supergravity, heterotic supergravity and 11-dimensional supergravity in higher Cartan geometry-formulation (D'Auria-Fré formulation of supergravity).
|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|
A decent introduction is in sections 4 and 5 of
A detailed account is in section I.4.1 of