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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2015

    added to E7 the statement of the decomposition of the smallest fundamental rep under SL(8,)SL(8,\mathbb{R}) and SL(7,)SL(7,\mathbb{R}) (here) and used this then to expand the existing paragraph on As U-duality group of 4d SuGra

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 30th 2015

    I have further been adding and briefly commenting further references at E7 and at U-duality, also at 4-dimensional supergravity and maybe elsewhere.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 13th 2015
    • (edited Aug 13th 2015)

    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.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 13th 2015

    Are we supposed to imagine that table continuing ?

    E10 1d supergravity

    E11 0d supergravity

    Whatever that might mean.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 13th 2015

    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)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)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 10E_{10} and E 11E_{11} it is maybe just some much smaller Borcherds algebra that does the job.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 19th 2015

    I added a paper that described E 7E_7 via a 28-dimensional quaternionic representation.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2015
    • (edited Aug 19th 2015)


    (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.
    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 19th 2015

    Cool, thanks, and thanks for the tip.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2015
    • (edited Aug 19th 2015)

    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

    𝔢 7/𝔰𝔲(8)( 7( 7) *) symtraceless 3( 7) * 6( 7) *. \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. (?)

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 19th 2015

    So what is the space E 7/SU(8)E_7/SU(8)?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2015
    • (edited Aug 19th 2015)

    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))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 10,1 3,1 7\mathbb{R}^{10,1} \simeq \mathbb{R}^{3,1} \oplus \mathbb{R}^7, then extending the second summand to the 56\mathbf{56} fundamental representation of E 7(7)E_{7(7)} and finally reducing the structure group of this rank-56 vector bundle from E 7(7)E_{7(7)} to SU(8)/ 2SU(8)/\mathbb{Z}_2. The moduli space of such reductions is, locally, again the coset E 7(7)/(SU(8)/ 2)E_{7(7)}/(SU(8)/\mathbb{Z}_2).

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 19th 2015

    Hmm, interesting!

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2015

    For the classical construction of E 7E_7 via that quartic form on GL(56)GL(56) I have added pointer to

    • Eli Cartan, Thesis, in Oeuvres complètes T1, Part I, Gauthier-Villars, Paris 1952

    But is there a better way to cite this? Is there any online trace of this?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2023

    added pointers here for the the “intermediate Lie algebra” E 712E_{7 \tfrac{1}{2}}:

    (clearly this deserves an entry of its own, eventually, but for the moment to record them somewhere)

    diff, v26, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2024

    added pointer to today’s

    diff, v27, current