higher geometry / derived geometry
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The coset space of Spin(5) by its subgroup SU(2) is diffeomorphic to the standard 7-sphere:
This is however not an isometry to the standard Riemannian manifold-structure (“round n-sphere”), whence one speaks of a squashed n-sphere.
The identification (1) follows via the exceptional isomorphisms
and
as a special case of the general statement
(see this Prop.).
coset space-structures on n-spheres:
standard: | |
---|---|
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$ | this Prop. |
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$ | this Prop. |
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$ | this Prop. |
exceptional: | |
$S^7 \simeq_{diff} Spin(7)/G_2$ | Spin(7)/G2 is the 7-sphere |
$S^7 \simeq_{diff} Spin(6)/SU(3)$ | since Spin(6) $\simeq$ SU(4) |
$S^7 \simeq_{diff} Spin(5)/SU(2)$ | since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere |
$S^6 \simeq_{diff} G_2/SU(3)$ | G2/SU(3) is the 6-sphere |
$S^15 \simeq_{diff} Spin(9)/Spin(7)$ | Spin(9)/Spin(7) is the 15-sphere |
see also Spin(8)-subgroups and reductions
homotopy fibers of homotopy pullbacks of classifying spaces:
(from FSS 19, 3.4)
Discussion of this squashed 7-sphere coset space as a fiber for KK-compactification of 11-dimensional supergravity:
M. A. Awada, Mike Duff, Christopher Pope, $N=8$ Supergravity Breaks Down to $N=1$, Phys. Rev. Lett. 50, 294 – Published 31 January 1983 (doi:10.1103/PhysRevLett.50.294)
Mike Duff, Bengt Nilsson, Christopher Pope, Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere, Phys. Rev. Lett. 50, 2043 – Published 27 June 1983; Erratum Phys. Rev. Lett. 51, 846 (doi:10.1103/PhysRevLett.50.2043)
Last revised on April 29, 2019 at 04:36:47. See the history of this page for a list of all contributions to it.