The general mechanism here is the supergeometric analog of what is known as Cartan geometry:
given an inclusion of Lie groups H↪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∈X the tangent space TxX with the quotient g/h, for h,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)↪Iso(d,1). The quotient Iso(d,1)/O(d,1)≃Rd,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