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Presently this entry has much overlap with Clifford-Klein space form and group actions on spheres. Eventually the three will diverge.
Fine-tuned and corrected (hopefully) the description in the Idea-section at spherical space form. Right now it reads as follows:
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).
Is the expectation that all of those 7d spherical space forms partake in the AdS4/CFT3-correspondence? That ABJM theory just concerns the A-type singularities?
Yes, that was the motivation for the classification by Figueroa-O’Farrill et al., stated in the first paragraphs of their MFFME 09 and MFFGME 09.
The $N = 5$-analogs of the ABJM model (hence for the dihedral and exceptional finite groups acting diagonally on $\mathbb{H}^2 \simeq \mathbb{R}^8$) have been identified (I have added pointers here).
I am not sure about the state of the discussion of the $N =4$-case. But clearly one expects this to exist.
Thanks!
ah, the $N=4$-case is discussed in section 4.3 of Bagger-Lambert-Mukhi-Papageorgakis 13. I am not sure, though, whether corresponding singularity structure is discussed
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