The general mechanism here is the supergeometric analog of what is known as *Cartan geometry*:

given an inclusion of Lie groups $H \hookrightarrow G$ a Cartan connection on some spacetime $X$ is a $G$-principal connection -- a $G$-gauge field -- satisfying the constraint that it identifies on each point $x \in X$ the tangent space $T_x X$ with the quotient $\mathfrak{g}/\mathfrak{h}$, for $\mathfrak{h}, \mathfrak{g}$ the Lie algebras of $H$ and $G$, respectively.

Consider this for the case of the inclusion of the orthogonal group (Lorentz group) into the Poincaré group $O(d,1) \hookrightarrow Iso(d,1)$. The quotient $Iso(d,1)/O(d,1) \simeq \mathbb{R}^{d,1}$ is Minkowski spacetime and a Cartan connection for this inclusion of gauge groups is equivalently

a choice of vielbein field

its Levi-Civita connection

on spacetime, hence is equivalently a field configuration of gravity, exhibited in first order formalism as a (constrained) gauge theory.

The analogous story goes through with the Poincaré group extended to the super Poincaré group. Now a Cartan connection for the inclusion of the super Lorentz group into the super Poincaré group is equivalently a field configuration of supergravity on a supermanifold spacetime, exhibited in first order formulation as a configuration of a super-gauge theory.

This is a standard story, but here is something interesting: of course higher dimensional supergravity theories (such as 11d sugra/M-theory, and 10d heterotic and type II supergravity) famously tend to have more fields than just the graviton and the gravitino: they also contain higher degree form fields.

Interestingly, this can also be described by Cartan gauge connections, but now in higher gauge theoretic generalization: *higher Cartan connections*. Here the super-Poincaré Lie algebra is generalized to super Lie n-algebras such as the supergravity Lie 3-algebra and the supergravity Lie 6-algebra.

For instance 11-dimensional supergravity has been shown (somewhat implicitly) to be a higher Cartan gauge theory for the supergravity Lie 6-algebra by Riccardo D'Auria, Pietro Fre. This is really the content of the textbook

These authors speak of the "FDA method". These "FDAs" however are just the dg-algebras dual to the above super Lie $n$-algebras (their "Chevalley-Eilenberg algebras"). This is explained a bit in the entry

There is much more that flows from this. For instance the complete and exact super $p$-brane content of string/M-theory is induced from the extension theory of these super Lie $n$-algebras, hence from the theory of "reduction of higher gauge groups" for the higher extensions of the super-Poincaré Lie group/algebra. This is indicated in our notes here:

This post imported from StackExchange Physics at 2014-03-09 16:11 (UCT), posted by SE-user Urs Schreiber