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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 is diffeomorphic to the Grassmann manifold G8,3. (p. 389)
The homogeneous space Spin(8)/Sp(2)·Sp(1) determined by the inclusion μ is diffeomorphic to the Grassmann manifold G8,3. (p. 389)
Ah, thanks for spotting. So it’s not isomorphic to Spin(7)/Spin(4)⋅Spin(3)≃Gr7,4. Hm…
Have made a note on this at Sp(n).Sp(1), here.
Next question would be: What is Spin(6)/Spin(3)⋅Spin(3) (hence Spin(6)/Sp(1)⋅Sp(1))
Well, by “incomplete induction”, it looks like we might generally have Spin(n1+n2)/Spin(n1)⋅Spin(n2)≃SO(n1+n2)/(SO(n1)×SO(n2))≃Gr(n1,n1+n2). (?)
Next question would be: What is Spin(6)/Spin(3)⋅Spin(3) (hence Spin(6)/Sp(1)⋅Sp(1))
Since Gr7,4≃Gr7,3, and we had earlier Gr8,3, is the answer as simple as Gr6,3?
[Looks like we’re guessing the same thing.]
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)) 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)) from the point of view of the exceptional calibration of ℝ8?
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.)
Maybe something useful around pp. 6-7 of hep-th/9608116.
Just came here to say the same… :-)
Right, so “case 2” on page 8 there says that Spin(6)/(Spin(3)⋅Spin(3)), which is Spin(6)/(SO(4)), is the space of Cayley-4-planes which are also special Lagrangian!
Excellent.
added brief mentioning of this:
Moreover, the coset space of Spin(6) by Spin(3).Spin(3) ≃ SO(4)
CAYsL≃Spin(6)/(Spin(3).Spin(3))≃SU(4)/SO(4)is the Grassmannian of those Cayley 4-planes which are also special Lagrangian submanifolds (BBMOOY 96, p. 8).
The page number is 8 on the pdf but labelled 7 in the footer. I guess we should go with the latter, no?
added pointer to today’s:
added pointer to last week’s
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