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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 is diffeomorphic to the Grassmann manifold . (p. 389)
The homogeneous space Spin(8)/Sp(2)·Sp(1) determined by the inclusion is diffeomorphic to the Grassmann manifold . (p. 389)
Ah, thanks for spotting. So it’s not isomorphic to . Hm…
Have made a note on this at Sp(n).Sp(1), here.
Next question would be: What is (hence )
Well, by “incomplete induction”, it looks like we might generally have . (?)
Next question would be: What is (hence )
Since , and we had earlier , is the answer as simple as ?
[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 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 from the point of view of the exceptional calibration of ?
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 , which is , 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)
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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