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

Here is an evident proposal. Needs to be refined somewhat.

Let $𝔥\to 𝔤$ be a morphism of L-∞ algebras. Write $i:BH\to BG$ for a Lie integration to a morphism of smooth ∞-groups. Notice that this defines a higher Klein geometry

Let $X$ be a smooth ∞-groupoid. For $x:*\to X$ any point, write $T_{x}X$ for its tangent $L_{\mathrm{\infty }}$-algebra.

An $\mathrm{\infty }$-Cartan geometry over $X$ with respect to $i$ is

• a $G$-principal ∞-bundle $P\to X$ whose structure group reduces to $H$, hence such that there is morphism $g:X\to BH$ and a fiber sequence

  $$P\to X\stackrel{ig}{\to }BG,$$P \to X \stackrel{i g}{\to} \mathbf{B}G \,,

• equipped with an $𝔤$ ∞-connection $\nabla$;

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

  $$T_{x}X\stackrel{\nabla }{\to }𝔤\stackrel{}{\to }𝔤/𝔥$$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 $(O(d,1)\hookrightarrow \mathrm{Iso}(d,1))$-Cartan geometry, where $\mathrm{Iso}(d,1)$ is the Poincare group and $O(d,1)$ 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.

Related concepts