nLab
spherical space form

Idea

A spherical space form is a quotient space $S^n/G$ of a round Riemannian n-sphere ($n \geq 2$) by a subgroup $G$ of its isometry group, which acts freely and properly discontinuously.

Equivalently, a spherical space form is a Riemannian manifold of constant positive sectional curvature (an elliptic geometry) which is connected and geodesically complete (see e.g. Gadhia 07, Lemma 5).

The spherical space forms are one of three classical examples of Clifford-Klein space forms.

Their classification was raised as an open problem by Killing 1891 and a complete solution was finally compiled by (Wolf 74) (reviewed in Gadhia 07, section 2.2). A re-proof of the classification is claimed in (Allock 15).

For more on this see at group actions on n-spheres.

Examples

7-dim spherical space forms with $Spin$-structure

For $n =7$ all the groups $G$ in Wolf’s classification act as subgroups of SO(8), the latter equipped with its defining action on $\mathbb{R}^8$ restricted to the action on $S^7 = S(\mathbb{R}^8) \subset \mathbb{R}^8$.

Therefore one may consider the lift $\widehat{G}$ of these subgroups to subgroups of the spin group $Spin(8) \to SO(8)$ through the double cover-projection. Such a lift corresponds to a choice of spin structure on the spherical space form $S^7/G$. These Spin-lifts $\widehat{G}$, have been classified in (Gadhia 07).

Given any such lift $\widehat{G} \subset Spin(8)$, one may consider its action on the two irreducible real spin representations $\mathbf{8}_\pm$ of $Spin(8)$. Write

$$N_\pm \;\coloneqq\; dim_{\mathbb{R}}\left( \left(\mathbf{8}_{\pm}\right)^{\widehat{G}} \right)$$

for the dimension of the subspace of spinors that are fixed by the action of $\widehat{G}$. For $\widehat{G}$ non-trivial, we have

$$(N_+, N_-) = (N,0) \phantom{AA} \text{or} \phantom{AA} (N_+, N_-) = (0,N)$$

and hence up to a choice of orientation there is a unique

$$N \in \{0,1,\cdots, 8\}$$

associated with each 7-dimensional spherical space form equipped with spin structure.

Hence this allows to stratify Wolf’s classification of 7-dimensional spherical space forms, first into the cases that do and that do not admit any spin structure, and then the former further into the dimension $N$ of the space of Killing spinors that they carry.

It turns out that the resulting sub-classification for $N \geq 4$ demands $\widehat{G}$ to be a finite subgroup of SU(2); hence this is an ADE classification (MFFGME 09, Sections 3-7):

• $N = 8$: here $\widehat{G} = \mathbb{Z}/2$, the cyclic group of order 2;

• $N = 7$: does not occur;

• $N = 6$: here $\widehat{G} = \mathbb{Z}/k$ (for $k \gt 2)$, a cyclic group;

• $N = 5$: here $\widehat{G} =$ a non-cyclic finite subgroup of SU(2), hence a binary dihedral group or the binary tetrahedral group or binary octahedral group or binary icosahedral group, acting diagonally on $\mathbb{R}^8 \simeq \mathbb{H} \oplus \mathbb{H}$;

• $N = 4$: here $\widehat{G} =$ any finite subgroup of SU(2) except the binary tetrahedral groups, but acting via the diagonal action composed with a suitable non-trivial outer automorphism of $\widehat{G}$ on one of the two sides.

(In the last case, while there is one nontrivial outer automorphism of the binary tetrahedral group, its twisted action yields $N =5$, hence is equivalent to one of the previous cases (MFFGME 09, section 7.3).)

$N$ on $S^7/\widehat{G}$	$\phantom{AA}\widehat{G} =$	-lift of of of	on with $AdS_4 \times S^7/\widehat{G}$
$\phantom{AA}N = 8\phantom{AA}$	$\phantom{AA}\mathbb{Z}_2$
$\phantom{AA}N = 7\phantom{AA}$	—	—	—
$\phantom{AA}N = 6\phantom{AA}$	$\phantom{AA}\mathbb{Z}_{k\gt 2}$
$\phantom{AA}N = 5\phantom{AA}$	$\phantom{AA}2 D_{k+2}$ $2 T$, $2 O$, $2 I$	, , ,	(HLLLP 08a, BHRSS 08)
$\phantom{AA}N = 4\phantom{AA}$	$\phantom{A}2 D_{k+2}$ $2 O$, $2 I$	, ,	(HLLLP 08b, Chen-Wu 10)

• et al. 2009 (arXiv:0909.0163, pdf slides)

This analysis applies to the classification of the near horizon geometry of smooth (i.e. non-orbifold) $\geq \tfrac{1}{2}$ BPS black M2-brane-solutions of the equations of motion of 11-dimensional supergravity:

These are the Cartesian product $AdS_4 \times (S^7/G)$ of 4-dimensional anti de Sitter spacetime with a 7-dimensional spherical space form $S^7/G$ with spin structure and $N \geq 4$, as above (MFFGME 09).

References

• Wilhelm Killing, Ueber die Clifford-Klein’schen Raumformen, Math. Ann. 39 (1891), 257–278

• Joseph Wolf, Spaces of constant curvature, Publish or Perish, Boston, Third ed., 1974

• Paul de Medeiros, José Figueroa-O'Farrill, Sunil Gadhia, Elena Méndez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP 0910:038,2009 (arXiv:0909.0163, pdf slides)

• Sunil Gadhia, Supersymmetric quotients of M-theory and supergravity backgrounds, PhD thesis, School of Mathematics, University of Edinburgh, 2007 (spire:1393845)

• Ian Hambleton, Topological spherical space forms, Handbook of Group Actions (Vol. II), ALM 32 (2014), 151-172. International Press, Beijing-Boston (arXivL:1412.8187)

• Daniel Allcock, Spherical space forms revisited (arXiv:1509.00906)