nLab
higher Cartan geometry

Context

Geometry

\infty-Chern-Weil theory

Differential cohomology

Contents

Idea

Higher Cartan geometry is the generalization of Cartan geometry to higher geometry.

It is the globalized version of higher Klein geometry.

As Cartan geometry is a special case of the theory of principal connections, so higher Cartan geometry is a special case of the theory of ∞-connections on principal ∞-bundles.

Definition

Let 𝔥𝔤\mathfrak{h} \to \mathfrak{g} be a morphism of L-∞ algebras. Write i:BHBGi : \mathbf{B}H \to \mathbf{B}G for a Lie integration to a morphism of smooth ∞-groups. Notice that this defines a higher Klein geometry

G//HBHBG. G//H \to \mathbf{B}H \to \mathbf{B}G \,.

Let XX be a smooth ∞-groupoid. For x:*Xx : * \to X any point, write T xXT_x X for its tangent L L_\infty-algebra.

An \infty-Cartan geometry over XX with respect to ii is

  • a GG-principal ∞-bundle PXP \to X whose structure group reduces to HH, hence such that there is morphism g:XBHg : X \to \mathbf{B}H and a fiber sequence

    PXigBG, P \to X \stackrel{i g}{\to} \mathbf{B}G \,,
  • equipped with an 𝔤\mathfrak{g} ∞-connection \nabla;

  • such that for every point x:*Xx : * \to X any any local trivialization, the canonical composite

    T xX𝔤𝔤/𝔥 T_x X \stackrel{\nabla}{\to} \mathfrak{g} \stackrel{}{\to} \mathfrak{g}/\mathfrak{h}

    (of the ∞-Lie algebra valued differential form of the connection at that point) with the quotient projection is an equivalence.

Examples

Higher Cartan super-Poincaré-geometry

Notice that ordinary gravity can be understood as the theory of (O(d,1)Iso(d,1))(O(d,1) \hookrightarrow Iso(d,1))-Cartan geometry, where Iso(d,1)Iso(d,1) is the Poincare group and O(d,1)O(d,1) 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 spaceglobal geometrydifferential cohomologyfirst order formulation of gravity
generalKlein geometryCartan geometryCartan connection
examplesEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
Lorentzian supergeometysupergeometrysuperconnectionsupergravity
generalKlein 2-geometryCartan 2-geometry
higher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

Revised on September 12, 2013 02:00:18 by Urs Schreiber (145.116.131.249)