higher Cartan geometry



\infty-Chern-Weil theory

Differential cohomology



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.


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.


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.

gauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
generalLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group SO(d)SO(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz groupMinkowski space d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
super Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
orthochronous Lorentz groupconformal geometryconformal connectionconformal gravity
generalsmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher 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 (