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nLab > Latest Changes: Cayley form

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 2nd 2019

    added references and various quick remarks, such as that the subspace of Cayley 4-planes

    CAY⊂Gr(4,8) CAY \subset Gr(4,8)

    has this coset space structure:

    CAY≃Spin(7)/(Spin(4).Spin(3)) CAY \;\simeq\; Spin(7)/\big( Spin(4).Spin(3)\big)

    I am guessing we also have

    ⋯≃Spin(8)/(Spin(5).Spin(3)) \cdots \simeq Spin(8)/\big( Spin(5).Spin(3)\big)

    (??)

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 3rd 2019

    There’s an isomorphism between Sp(2)·Sp(1) and Spin(5)·Spin(3), (here, p. 384). But then

    The homogeneous space Spin(8)/Sp(2)·Sp(1) determined by the inclusion μ\mu is diffeomorphic to the Grassmann manifold G 8,3G_{8,3}. (p. 389)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2019

    The homogeneous space Spin(8)/Sp(2)·Sp(1) determined by the inclusion μ\mu is diffeomorphic to the Grassmann manifold G 8,3G_{8,3}. (p. 389)

    Ah, thanks for spotting. So it’s not isomorphic to Spin(7)/Spin(4)⋅Spin(3)≃Gr 7,4Spin(7)/Spin(4)\cdot Spin(3) \simeq Gr_{7,4}. Hm…

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2019

    Have made a note on this at Sp(n).Sp(1), here.

    Next question would be: What is Spin(6)/Spin(3)⋅Spin(3)Spin(6)/ Spin(3)\cdot Spin(3) (hence Spin(6)/Sp(1)⋅Sp(1)Spin(6)/ Sp(1) \cdot Sp(1))

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2019

    Well, by “incomplete induction”, it looks like we might generally have Spin(n 1+n 2)/Spin(n 1)⋅Spin(n 2)≃SO(n 1+n 2)/(SO(n 1)×SO(n 2))≃Gr(n 1,n 1+n 2)Spin(n_1 + n_2) / Spin(n_1)\cdot Spin(n_2) \simeq SO(n_1 + n_2)/(SO(n_1) \times SO(n_2)) \simeq Gr(n_1, n_1 + n_2). (?)

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 3rd 2019
    • (edited Apr 3rd 2019)

    Next question would be: What is Spin(6)/Spin(3)⋅Spin(3)Spin(6)/ Spin(3)\cdot Spin(3) (hence Spin(6)/Sp(1)⋅Sp(1)Spin(6)/ Sp(1) \cdot Sp(1))

    Since Gr 7,4≃Gr 7,3Gr_{7,4} \simeq Gr_{7,3}, and we had earlier Gr 8,3Gr_{8,3}, is the answer as simple as Gr 6,3Gr_{6,3}?

    [Looks like we’re guessing the same thing.]

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2019
    • (edited Apr 3rd 2019)

    Yeah. But then, we want to remember that this may be a misleading way to look at the situation.

    For Spin(7)/(Spin(4)⋅Spin(3))Spin(7) / (Spin(4)\cdot Spin(3)) the point is that in this form, this wants to be identified with the excetional Grassmannian of Cayley-4 planes in 8d (here) and that it’s more a coincidence that this happens to be homeomorphic to the ordinary Grassmannian of 4-planes in 7d.

    From this point of view, the interesting question is: What, if anything, is Spin(6)/(Spin(3)⋅Spin(3))Spin(6) / (Spin(3)\cdot Spin(3)) from the point of view of the exceptional calibration of ℝ 8\mathbb{R}^8?

    Since this spaces covers the space of Cayley 4-planes, it ought to be the moduli space of something like Cayley 4-planes equipped with some extra structure. Which extra structure would that be?

    (Of course this has a tautological answer, but maybe it also has an interesting non-tautological answer.)

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 3rd 2019

    Maybe something useful around pp. 6-7 of hep-th/9608116.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2019

    Just came here to say the same… :-)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2019
    • (edited Apr 3rd 2019)

    Right, so “case 2” on page 8 there says that Spin(6)/(Spin(3)⋅Spin(3))Spin(6)/( Spin(3)\cdot Spin(3) ), which is Spin(6)/(SO(4))Spin(6)/( SO(4) ), is the space of Cayley-4-planes which are also special Lagrangian!

    Excellent.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2019

    added brief mentioning of this:

    Moreover, the coset space of Spin(6) by Spin(3).Spin(3) ≃\simeq SO(4)

    CAY sL≃Spin(6)/(Spin(3).Spin(3))≃SU(4)/SO(4) CAY_{sL} \;\simeq\; Spin(6)/\big( Spin(3).Spin(3)\big) \;\simeq\; SU(4)/SO(4)

    is the Grassmannian of those Cayley 4-planes which are also special Lagrangian submanifolds (BBMOOY 96, p. 8).

    diff, v7, current

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 3rd 2019

    The page number is 8 on the pdf but labelled 7 in the footer. I guess we should go with the latter, no?

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2019

    Okay, I have changed it to “p. 7 (8 of 17)”

    diff, v8, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2023

    added pointer to today’s:

    diff, v10, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2024

    added pointer to last week’s

    diff, v11, current

1 to 15 of 15

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