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 forulated in super Cartan geometry. 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).
|geometric context||gauge group||stabilizer subgroup||local model space||local geometry||global geometry||differential cohomology||first order formulation of gravity|
|differential geometry||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 spacetime||Lorentzian geometry||pseudo-Riemannian geometry||spin connection||Einstein gravity|
|anti de Sitter group||anti de Sitter spacetime||AdS gravity|
|de Sitter group||de Sitter spacetime||deSitter gravity|
|linear algebraic group||parabolic subgroup/Borel subgroup||flag variety||parabolic geometry|
|conformal group||conformal parabolic subgroup||Möbius space||conformal geometry||conformal connection||conformal gravity|
|supergeometry||super Lie group||subgroup (monomorphism)||quotient (“coset space”)||super Klein geometry||super Cartan geometry||Cartan superconnection|
|examples||super Poincaré group||spin group||super Minkowski spacetime||Lorentzian supergeometry||supergeometry||superconnection||supergravity|
|super anti de Sitter group||super anti de Sitter spacetime|
|higher differential geometry||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|
Discussion in the physics literature traditionally tends to ignore the global structure of spacetime manifolds and pretends that a vielbein field may be chosen globally, hence that spacetime admits a framing.
In general that is only valid locally, but it so happens that in the archetypical case of interest, namely for 4-dimensional globally hyperbolic spacetimes with orientable spatial slices, it is valid globally. See this remark at framed manifold for more.
See also at teleparallel gravity.
A decent introduction is in sections 4 and 5 of
A detailed account is in section I.4.1 of