Obtaining supergravity from gauging global supersymmetry

On page 92, my still favorite supersymmetry book says, by making the global infinitisimal parameter of a SUSY tranformation spacetitime dependent (gauging) it forces one to introduce a new gauge field which turns out to have the properties of a graviton and one obtains supergravity. This is analogue to obtaining electromagnetism from gauging the global $U(1)$ symmetry of the Dirac Lagrangian.

This approach seems to me a nice way to learn more about supergravity, but unfortunately it was only mentioned as an aside comment in that book without explaining it any further. So can somebody explain (or outline) in a bit more detail how this works?

Obtaining a graviton isn't specific to SUSY: it's about gauging the Poincaré group. When you gauge an ordinary global symmetry, you take its current $J_\mu$ and couple it to a gauge field $A^\mu$. But a spacetime symmetry, like the Poincaré group, has a more complicated conserved current. Essentially, because the momentum generator $P_\mu$ already has a Lorentz index, the corresponding conserved current will not be a vector like $J_\mu$ but a tensor with more indices; it turns out to be the stress tensor $T_{\mu\nu}$. So gauging the Poincaré group turns out to mean introducing a graviton $g^{\mu\nu}$ coupling to $T_{\mu\nu}$.

In the SUSY context, your SUSY generators $Q_\alpha, Q^\dagger_{\dot \alpha}$ have an algebra that includes the Poincaré generators as a subalgebra. So when you gauge them, you automatically get gravity. But because you also have the new SUSY generators, you get a bit more structure too: it turns out you need a gravitino, which is the superpartner of the graviton.

This is a sketch. You can find the details in many review articles or textbooks, like the classic book by Wess and Bagger. Daniel Freedman has recently written a textbook that's all about supergravity.

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

1. a choice of vielbein field

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:

• Is it correct to say that the (super)3-Lie algebra corresponds to a 2-membrane as (super) 2-Lie algebra corresponds to a string, so adding dimensions, a M$_{n}$ brane would correspond to a (super)$n+1$ Lie algebra (well maybe only M$_2$ and M$_5$). If so, easily speaking (for simple minds), and physically speaking, what does it means ?? – Trimok Aug 10 '13 at 17:49
• Yes, Lie $n$-algebras and their Lie $n$-groups correspond to Lie algebras and Lie groups as $(n-1)$-branes correspond to 0-branes = point particles. The easiest example is the circle 2-group ncatlab.org/nlab/show/circle+n-group $\mathbf{B}U(1)$. A gauge field for this 2-group is a B-field ncatlab.org/nlab/show/Kalb-Ramond+field and this couples to a string = 1-brane in direct anlogy of how a $U(1)$-gauge field couples to a point particle = 0-brane (by what is called the "WZW term" ncatlab.org/nlab/show/Wess-Zumino-Witten+model). Next up the ladder is... – Urs Schreiber Aug 10 '13 at 21:54
• ... next up the ladder is the circle 3-group $\mathbf{B}^2 U(1)$. A gauge field for this is a $C$-field ncatlab.org/nlab/show/supergravity+C-field and this couples analogously to the membrane = 2-brane. In fact, in both these cases really the circle $n$-group is just one component of a more complicated nonabelian higher group. For instance the circle 2-group is part of what is called the "String 2-group" ncatlab.org/nlab/show/string+2-group and its variant the "String^c 2-group" ncatlab.org/nlab/show/string%5Ec+2-group. A gauge field for this is twisted heterotic B-field. – Urs Schreiber Aug 10 '13 at 21:58
• But apart from their role as higher gauge groups under which higher branes are charged, higher Lie groups also appear as "higher orbispace" target spaces on which higher branes may propagate. This is the second role mentioned in the above reply. For instance the supergravity Lie 3-algebra is also the target "higher super-orbispace" which is such that a sigma-model map into is a combination of a map to spacetime and a 2-form on the worldvolume. This way it serves as a higher geometric target space that renders the 5-brane a genuine, albeit "higher" sigma model. – Urs Schreiber Aug 10 '13 at 22:01