# $G_2$-structure in analogy with supergravity superspace constraints

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There is an evident analogy between (1) closed $G_2$-structure and (2) supergravity superspace constraints. I am wondering if and where in the literature this analogy has been expanded on.

What I mean is this:

(1) On the 7-dimensional Euclidean space $\mathbb{R}^7$ there is a canonical constant 3-form $\phi$, often called the "associative 3-form", and a $G_2$-structure on a 7-manifold $X$ is a 3-form on $X$ which locally looks like this $\phi$.

(2) On the super-Minkowski spacetime $\mathbb{R}^{10,1|32}$ there is a canonical constant super 4-form whose components are $\Gamma_{a b \alpha \beta}$, and an on-shell background of 11-dimensional supergravity is constrained to have super 4-form flux that locally looks like this.

In stating it this way, I am intentionally glossing over some fine print, but not over much.

For the first statement any classical source will do, a good discussion is (see in particular on p. 21) in

• Robert BryantSome remarks on G2-structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 pdf

The second statement originates around

• Gianguido Dall’Agata, Davide Fabbri, Christophe Fraser, Pietro Fré, Piet Termonia, Mario Trigiante,The Osp(8|4) singleton action from the supermembrane, Nucl.Phys.B542:157-194,1999, (arXiv:hep-th/9807115)

This is for 11d supergravity. For 10d supergravity theories similar statements hold, but the formulas have more components and are a little bit less directly analogous to $G_2$-structures.

And of course there is the whole story of compactifying 11d sugra on $G_2$-manifolds. So what I am after here must be well known and this question will just show my ignorance of the literature, but anyway:

Where is an explicit discussion that makes the above anaology manifest, that the associative 3-form on $\mathbb{R}^7$ is a local model for a differential 3-form on a curved $G_2$-manifold in much the same way that the super-4-form with compoments $\Gamma_{a b \alpha \beta}$ on $\mathbb{R}^{10,1|32}$ is the local model for the super-4-form flux on curved solutions to 11-dimensional supergravity?

edited Jan 12, 2015

Maybe I should put this more directly:

A $G_2$-manifold structure is equivalently a 3-form on a Riemannian 7-manifold which is

1. locally of the form $\phi_{a b c}E^a \wedge E^b\wedge E^c$ for $(E^a)$ the vielbein field;
2. covariantly constant.

Analogously, the spinorial part of the 4-form flux in 11-dimensional supegravity is

1. locally of the form $\Gamma_{a b \alpha \beta} E^a \wedge E^b \wedge E^\alpha \wedge E^\beta$ for $(E^a, E^\alpha)$ the super-vielbein;
2. covariantly constant.

Clearly these are two instances of the same pattern. Which article would make this relation?

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