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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeMay 7th 2015

    I made Cayley plane a little less stubby.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 17th 2019

    Added a couple of properties.

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 27th 2020

    Added Hisham’s paper

    Hmm, what’s this about “We have provided evidence for some relations between M-theory and an octonionic version of Kreck-Stolz elliptic homology.”

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2020
    • (edited Nov 27th 2020)

    Oh, I was offline and now just about to do the same. Okay, great.

    (I don’t know about that octonionic elliptic cohomology. But I think we are beginning to see now how plain elliptic cohomological really fits in…)

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2020

    have expanded this citation now as follows:

    Indications that M-theory in 10+1 dimensions may be understood as the KK-compactification on Cayley-plane fibers of some kind of bosonic M-theory in 26+1 dimensions:

    diff, v8, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 28th 2020

    But I think we are beginning to see now how plain elliptic cohomological really fits in…

    I look forward to seeing that.

    Out of interest, is F-theory a target? I see at F-theory

    A series of articles arguing for a relation between the elliptic fibration of F-theory and elliptic cohomology…

    naming a paper by Hisham.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2020
    • (edited Nov 28th 2020)

    Yes. So here is the idea:

    The equivariance group of the quaternionic Hopf fibration is Sp(2).Sp(1)Spin(8)Sp(2).Sp(1) \subset Spin(8), hence J-twists for Cohomotopy in joint degree 7 and 4 requires Sp(2).Sp(1)Sp(2).Sp(1)-structure folds X 8X^8.

    Up to homotopy this may be thought of as stand-ins for 11-manifolds 2,1×X 8 \mathbb{R}^{2,1} \times X^8, as considered in M-theory on 8-manifolds. But can we see any non-trivial longitudinal geometry other than 2,1\mathbb{R}^{2,1} ?

    Consider, for simplicity, the case that X 8=( 8{0})/GX^8 = (\mathbb{R}^8 \setminus \{0\})/G, for GSp(2).Sp(1)G \subset Sp(2).Sp(1) a finite subgroup, the near horizon geometry of a black M2-brane. Then J-twisted 4-Cohomotopy on X 8X^8 is GG-equivariant 4-Cohomotopy of S 7S^7.

    To analyze this, choose a multiplicative GG-equivariant cohomology theory EE to detect the resulting C-field fluxes. The latter trivialize on the S 7S^7, but so we record how it trivializes (i.e. the H 3H_3-flux). This is equivalently the choice of a quaternionic orientation to-second-stage on EE.

    In this way second-stage quaternionic orientations on cohomology theories EE serve to define EE-valued character maps on twistorial Cohomotopy in which to detect C-field flux in the vicinity of M2-branes.

    Now one way to obtain quaternionic orientations p 1 Ep_1^E is to choose a complex orientation c 1 Ec_1^E and set p 1 Ec 2 Ep_1^E \coloneqq c_2^E. Notice that, via Hypothesis H, this equates the shifted C-field flux G 4 EG_4^E as seen in EE-theory with the second EE-Chern class of a gauge bundle – which “is” again the Horava-Witten Green-Schwarz mechanism (since in the present situation of finite GG-quotients of flat space, no gravitational instanton contribution is present).

    In conclusion then, the C-field flux seen in EE-theory after a choice of complex orientation is the second Conner-Floyd EE-Chern class of the complex vector bundle underlying the tautological quaternionic line bundle over P 2\mathbb{H}P^2 (all GG-equivariantly).

    If we take EE to be Landweber exact, then the resulting flux is a function on a group scheme. If it’s moreover a Calabi-Yau cohomology theory this function is a wave function of a higher self-dual gauge theory on some “hidden” variety (hidden in that it’s not part of the 8-manifold X 8X^8 we started with). In particular, if EE is elliptic, the flux we measure depends, besides the visible 8-geometry, on an elliptic curve Σ\Sigma.

    Hence this way we end up cohomologically detecting an effective spacetime geometry of the form 0,1×X 8×Σ\mathbb{R}^{0,1} \times X^8 \times \Sigma.

    This “is” the F-theory setup of M-theory (on 8-manifolds) compactified on an elliptic curve. Much as in Connes’s NCG formulation of KK-compactification, the fiber space Σ\Sigma has shrunken to the point that it is classically invisible, but reveals itself through cohomological effects.

    Something like this.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 28th 2020
    • (edited Nov 28th 2020)

    This will take some digesting, but thanks!

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 30th 2020
    • (edited Nov 30th 2020)

    At M-theory on 8-manifolds we have

    If the 8-dimensional fibers themselves are elliptic fibrations, then M-theory on these 8-manifolds is supposedly T-dual to F-theory KK-compactified to 3+1 spacetime-dimensions.

    But at the end of #7 you’re talking about a further compactification on an elliptic curve, so in total on the 10-manifold X 8×ΣX^8 \times \Sigma?

    I don’t recall mention of compactification on a 10-manifold, but there is a mention in Twisted K-Theory from Monodromies

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2020

    We were lacking a page on that, true. Starting one here now: F/M-theory on elliptically fibered Calabi-Yau 5-folds.

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