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added to E7 the statement of the decomposition of the smallest fundamental rep under $SL(8,\mathbb{R})$ and $SL(7,\mathbb{R})$ (here) and used this then to expand the existing paragraph on As U-duality group of 4d SuGra
I have further been adding and briefly commenting further references at E7 and at U-duality, also at 4-dimensional supergravity and maybe elsewhere.
I had added a definition to E7.
In the course of this I also touched entries related to this in the context of strings, such as KK-monopole, dual graviton, M9-brane.
Are we supposed to imagine that table continuing ?
E10 1d supergravity
E11 0d supergravity
Whatever that might mean.
That’s at least the hypothesis and the research programs of Hermann Nicolai and of Peter West, respectively.
Nicolai and coauthors have shown in a long series of articles that at least much of 11-dimensional supergravity with “M-brane corrections” may equivalently be encoded in a 1-dimensional field theory with fields in $E_{10(10)}/K_{10(10)}$. West’s conjecture amounts to saying that in some way this in turn is equivalent to picking single points in $E_{11(11)}/K_{11(11)}$, for some way of making sense of these symbols.
Both these proposals share the property that on the one hand there is a lot of evidence for them, while at the same time a lot of questions remain unsolved.
The references listed here (you had highlighted them yourself just recently) suggest that instead of the huge $E_{10}$ and $E_{11}$ it is maybe just some much smaller Borcherds algebra that does the job.
I added a paper that described $E_7$ via a 28-dimensional quaternionic representation.
Thanks!
(You had added it to the section References–U-duality. I moved it up to References–General, now here)
By the way, regarding adding the anchors to reference items, I used to do this wrongly myself, allow me to highlight this: the anchor needs to go at the beginning of the bullet item text, not at the end. If it is at the end then it seems to work well some time, but sometimes not. Since I realized this I always code like so:
* {#AuthorNameYear} AuthorName, _Title_, etc.
Cool, thanks, and thanks for the tip.
added a section on the adjoint representation of E7.
I am concluding this with the observation that, as a consequence of the previous statements that are given with citation, we have the linear isomorphism
$\mathfrak{e}_7/\mathfrak{su}(8) \simeq (\mathbb{R}^7\otimes (\mathbb{R}^7)^\ast)_{sym-traceless} \oplus \wedge^3 (\mathbb{R}^7)^\ast \oplus \wedge^6 (\mathbb{R}^7)^\ast \,.$This seems to be a crucial statement for the use of this representation theory in exceptional generalized geometry, but presently I don’t see it made explicit in the references that I am looking at. (?)
So what is the space $E_7/SU(8)$?
Way back in Cremmer-Julia 79 it was noticed that after compactifying 11d SuGra to 4d its moduli space of (pseudo-)scalar fields is this coset $E_{7(7)}/(SU(8))$. This was a miracle back then. Ever since there is the quest to see the origin of this structure already in the full 11d theory, before KK-compactification.
The idea of Hull 07 and Pacheco-Waldram 08 was to understand this as the moduli of generalized vielbein fields in close analogy to Hitchin’s generalized complex geometry/type II geometry which would unify (locally) the metric (graviton) of 11dSugra with the 3-form field into an “exceptional generalized metric”. Specifically this was suggested to be encoded by splitting the typical fiber of the tangent bundle as $\mathbb{R}^{10,1} \simeq \mathbb{R}^{3,1} \oplus \mathbb{R}^7$, then extending the second summand to the $\mathbf{56}$ fundamental representation of $E_{7(7)}$ and finally reducing the structure group of this rank-56 vector bundle from $E_{7(7)}$ to $SU(8)/\mathbb{Z}_2$. The moduli space of such reductions is, locally, again the coset $E_{7(7)}/(SU(8)/\mathbb{Z}_2)$.
Hmm, interesting!
For the classical construction of $E_7$ via that quartic form on $GL(56)$ I have added pointer to
But is there a better way to cite this? Is there any online trace of this?
added pointers here for the the “intermediate Lie algebra” $E_{7 \tfrac{1}{2}}$:
J. M. Landsberg, L. Manivel, The sextonions and $E_{7\tfrac{1}{2}}$, Advances in Mathematics 201 1 (2006) 143-179 [arXiv:math/0402157, doi:10.1016/j.aim.2005.02.001]
Kimyeong Lee, Kaiwen Sun, Haowu Wang, On intermediate Lie algebra $E_{7 \tfrac{1}{2}}$ [arXiv:2306.09230]
(clearly this deserves an entry of its own, eventually, but for the moment to record them somewhere)
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