Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: Cayley form

Bottom of Page

1 to 15 of 15

  1.  
    • 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

  1.