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nLab > Latest Changes: Spin(11,3)

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2021

    I am giving this group a stub entry just to have a decent place to record today’s

    For this entry not to be all too lonely I made Spin(10,2) point to D=12 supergravity, for the moment. In the long run all this derserves to be expanded on, clearly.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 6th 2021
    • (edited Apr 6th 2021)

    This times with a series of talks he’s running - Octonions and the standard model, recordings.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 6th 2021

    Added link to D=14 supersymmetry.

    diff, v3, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2021

    Ah, thanks for making that link! I had almost forgotten about the entry D=14 supersymmetry. Good that we have a wiki! :-)

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 6th 2021

    I await your ’bosonic Hypothesis H’ for bosonic M-theory to find out more about the role of the octonions in physics.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2021
    • (edited Apr 6th 2021)

    Right.

    It’s pretty clear what the bosonic Hypothesis H wants to be: replace the role of the quaternionic Hopf fibration throughout by the octonionic Hopf fibration. This way it’s even clear how the “bosonic M” version of Hypothesis H will reduce to the 11d M-theory version: in the way analogous to how the latter further reduces to the complex Hopf fibration on M5-branes.

    What remains missing is more insight into what actually happens in bosonic M-theory. One day it will fall into place.

    In the meantime, there is no lack of octonions and standard model hints in Hypothesis H.

    (Flavour sector, though. I saw in these Perimeter octonion talks the suggestion that the “standard model is not really as messy as often claimed” – followed by full focus on the color sector and full disregard of the flavor sector.)

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 7th 2021

    replace the role of the quaternionic Hopf fibration throughout by the octonionic Hopf fibration.

    Is this suggesting there isn’t an analogous twistor factorization to S 7→ℂP 3→S 4S^7 \to \mathbb{C} P^3 \to S^4 for S 15→S 7S^{15} \to S^7 through ℍP 3\mathbb{H} P^3?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2021
    • (edited Apr 7th 2021)

    Oh, that could be. But it would make sense: The factorization of the quaternionic Hopf fibration through ℂP 3\mathbb{C}P^3 corresponds, when regarded as coefficients for M-brane charge cohomology, to passage from plain to (Horava-Witten-)heterotic M-theory (here). But there is not supposed to be an “HW/heterotic” version of bosonic M-theory. In fact there cannot be if it really is bosonic.

    Maybe some stand-in does exist, one might hope, but it’s plausible that it’s not straightforward.

    Anyways, all these thoughts about “bosonic M-theory” are lacking some anchor point at which one could nail down the fabric of the theory, to proceed ironing it out from there.

    When we proved (here) how the passage to fixed loci in the equivariantized quaternionic Hopf fibration yields the restriction to the M5-brane, I thought it would now be immediate to apply the analogous construction to the octonionic Hopf fibration and see the 11d M-theory spacetime similarly arise as the localized brane of a larger structure in yet higher dimensional bosonic M-theory spacetime. This is roughly what the folklore suggests, and something like this certainly comes out from the equivariantized ocotnionic Hopf fibration. But then I didn’t see the dimensions work out to match to existing speculations; and before I could further plummet into that rabbit hole, I was gently guided to instead do something more productive. I think that’s when we started to write M/F-theory as Mf-theory, instead.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2021
    • (edited Dec 7th 2021)

    Re #6,

    This way it’s even clear how the “bosonic M” version of Hypothesis H will reduce to the 11d M-theory version: in the way analogous to how the latter further reduces to the complex Hopf fibration on M5-branes.

    Is this latter reduction to the complex Hopf fibration what you’re pointing to in #8

    When we proved here how the passage to fixed loci in the equivariantized quaternionic Hopf fibration yields the restriction to the M5-brane ?

    In ’Mysterious Triality’, they make great play of thinking of the duality between the algebraic-geometric side (del Pezzo sufaces) and the algebraic topology side (iterated cyclic loop spaces) via Hopf fibrations

    The transition is from the complex Hopf fibration, which can be viewed as part of the process of blowing up, to the quaternionic Hopf fibration (viewed as a topological, i.e., in some sense, real) (p. 7)

    If you have a physics-based process to mediate between the Hopf fibrations, that surely should feature in the explanation of mysterious duality.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2021

    Maybe Remark 3.28 of M/F-Theory as M f-Theory for the reduction of one Hopf fibration to the other.

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