There is an evident analogy between (1) closed G2-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 R7 there is a canonical constant 3-form ϕ, often called the "associative 3-form", and a G2-structure on a 7-manifold X is a 3-form on X which locally looks like this ϕ.
(2) On the super-Minkowski spacetime R10,1|32 there is a canonical constant super 4-form whose components are Γabαβ, 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
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Robert Bryant, Some remarks on G2-structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 pdf
The second statement originates around
see also for instance section 3.1 of
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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 G2-structures.
And of course there is the whole story of compactifying 11d sugra on G2-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 R7 is a local model for a differential 3-form on a curved G2-manifold in much the same way that the super-4-form with compoments Γabαβ on R10,1|32 is the local model for the super-4-form flux on curved solutions to 11-dimensional supergravity?