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  Relation between the 11d "hidden supergroup" and Exceptional generalized geometry

+ 4 like - 0 dislike

Threre is the famous proposal that -- in analogy to the role of generalized (doubled) geometry in type II string theory -- there should be exceptional generalized geometry for M-theory, with generalized tangent bundle not of the form \(T \oplus T^\ast\) (for momenta and string charges) but of the form

 \(T \oplus \wedge^2 T^\ast \oplus \wedge^5 T^\ast \oplus \cdots\) 

(for momenta and M2-brane charges and M5-brane charges) on which U-duality groups such as at least \(E_{k \leq 7}\) are naturally represented.

This proposal is due to 

and it has excellent motivation and plausibility checks, even though a comprehensive development of exceptional generalized geometry (EGG) as a theory is still in the making.

What I am wondering is, and that's my question here, if anyone had ever noticed (in print or in some citable form) that there exists a more systematic mathematical derivation of this structure of the M-theory generalized tangent bundle which predates the EGG proposal by a margin, but was maybe not put in proper connection with it yet.

Namely the main result (last section) of 

of which a more comprehensive analysis later appeared in

is the identification of a  super Lie algebra extension of 11d super Minkowski spacetime on which the M2-brane 4-cocycle  (the one that makes the brane scan).trivializes. (That's not quite how these authors phrase it, but this is exactly what they do, when phrased super Lie-theoretically.) And the bosonic body of that extension turns out to be none other than...

 \(\mathbb{R}^{10,1} \oplus \wedge^2 (\mathbb{R}^{10,1})^\ast \oplus \wedge^5 (\mathbb{R}^{10,1})^\ast\) 

hence none other than the fiber space of the exceptional tangent bundle.

It is natural to wonder about what this means. I am aware of speculations that this extended spacetime super Lie algebra itself is a contraction limit of the orthosymplectic \(\mathfrak{osp}(1|32)\) or the like, for instance in

J.J. FernandezJ.M. IzquierdoM.A. del Olmo, Contractions from osp(1|32)⊕osp(1|32) to the M-theory superalgebra extended by additional fermionic generators (arXiv:1504.05946)

but what I have not see yet is anyone making the relation between this "hidden supergroup" of 11-dimensional supegravity and the generalized tangent bundle of exceptional generalized geometry.

Mainly I am just asking for a citable reference, if any, that might make the relation. But of course if you have any other insights to share, I'd be interested,too.

asked Jul 30, 2015 in Theoretical Physics by Urs Schreiber (6,095 points) [ revision history ]
edited Jul 30, 2015 by Urs Schreiber

@UrsSchreiber, I am not an expert but your question immediately make me think in the Witten`s papers "D-Branes And K-Theory " and "A Derivation of K-Theory from M-Theory " given that the generalized tangent bundle could be described by K-Theory.  Do you agree?

1 Answer

+ 2 like - 0 dislike

Possibly the only article to ever make the connection that I am asking about is

This points out (on its p. 14) that the bosonic body \(\mathbb{R}^{10,1} \oplus \wedge^2 (\mathbb{R}^{10,1})^\ast \oplus \wedge^5 (\mathbb{R}^{10,1})^\ast\) of the D'Auria-Fre "hidden" super Lie algebra is just the level-2 truncation of the \(l_1\)-representation of  \(E_{11}\), according to sections 4 and 5 of 

I am chatting a bit abut this in the note From higher to exceptional geometry.

But Vaula's article does not go beyond making this observation, and the observation seems not to have been picked up by anyone yet. As far as I see.

(Is Vaula still active? The email addresses on all her articles seem to have expired. (?))

answered Aug 3, 2015 by Urs Schreiber (6,095 points) [ revision history ]

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