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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 $\mu$ is diffeomorphic to the Grassmann manifold $G_{8,3}$. (p. 389)
The homogeneous space Spin(8)/Sp(2)·Sp(1) determined by the inclusion $\mu$ is diffeomorphic to the Grassmann manifold $G_{8,3}$. (p. 389)
Ah, thanks for spotting. So it’s not isomorphic to $Spin(7)/Spin(4)\cdot Spin(3) \simeq Gr_{7,4}$. Hm…
Have made a note on this at Sp(n).Sp(1), here.
Next question would be: What is $Spin(6)/ Spin(3)\cdot Spin(3)$ (hence $Spin(6)/ Sp(1) \cdot Sp(1)$)
Well, by “incomplete induction”, it looks like we might generally have $Spin(n_1 + n_2) / Spin(n_1)\cdot Spin(n_2) \simeq SO(n_1 + n_2)/(SO(n_1) \times SO(n_2)) \simeq Gr(n_1, n_1 + n_2)$. (?)
Next question would be: What is $Spin(6)/ Spin(3)\cdot Spin(3)$ (hence $Spin(6)/ Sp(1) \cdot Sp(1)$)
Since $Gr_{7,4} \simeq Gr_{7,3}$, and we had earlier $Gr_{8,3}$, is the answer as simple as $Gr_{6,3}$?
[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 $Spin(7) / (Spin(4)\cdot Spin(3))$ 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 $Spin(6) / (Spin(3)\cdot Spin(3))$ from the point of view of the exceptional calibration of $\mathbb{R}^8$?
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.)
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 $Spin(6)/( Spin(3)\cdot Spin(3) )$, 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) $\simeq$ 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).
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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