#
nLab

first-order formulation of gravity

# Contents

## Idea

The action functional of gravity was originally conceived as a functional on the space of pseudo-Riemannian metrics of a manifold $X$. 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.

The field strength of gravity – the Riemann tensor – is the curvature of Levi-Civita connection. Typically this is referred to a spin connection in this context.

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 |
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differential geometry | Lie group/algebraic group $G$ | subgroup (monomorphism) $H \hookrightarrow G$ | quotient (“coset space”) $G/H$ | Klein geometry | Cartan geometry | Cartan connection | |

examples | Euclidean group $Iso(d)$ | rotation group $O(d)$ | Cartesian space $\mathbb{R}^d$ | Euclidean geometry | Riemannian geometry | affine connection | Euclidean gravity |

| Poincaré group $Iso(d-1,1)$ | Lorentz group $O(d-1,1)$ | Minkowski spacetime $\mathbb{R}^{d-1,1}$ | Lorentzian geometry | pseudo-Riemannian geometry | spin connection | Einstein 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 group | parabolic subgroup/Borel subgroup | flag variety | parabolic geometry | | | |

| conformal group $O(d,t+1)$ | conformal parabolic subgroup | Möbius space $S^{d,t}$ | | conformal geometry | conformal connection | conformal gravity |

supergeometry | super Lie group $G$ | subgroup (monomorphism) $H \hookrightarrow G$ | quotient (“coset space”) $G/H$ | super Klein geometry | super Cartan geometry | Cartan superconnection | |

examples | super Poincaré group | spin group | super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$ | Lorentzian supergeometry | supergeometry | superconnection | supergravity |

| super anti de Sitter group | | super anti de Sitter spacetime | | | | |

higher differential geometry | smooth 2-group $G$ | 2-monomorphism $H \to G$ | homotopy quotient $G//H$ | Klein 2-geometry | Cartan 2-geometry | | |

| cohesive ∞-group | ∞-monomorphism (i.e. any homomorphism) $H \to G$ | homotopy quotient $G//H$ of ∞-action | higher Klein geometry | higher Cartan geometry | higher Cartan connection | |

examples | | | extended super Minkowski spacetime | | extended supergeometry | | higher supergravity: type II, heterotic, 11d |

## References

A decent introduction is in sections 4 and 5 of

A detailed account is in section I.4.1 of

Revised on March 11, 2015 16:50:12
by

Urs Schreiber
(95.82.152.166)