# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 24th 2018

Presently this entry has much overlap with Clifford-Klein space form and group actions on spheres. Eventually the three will diverge.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 25th 2018

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).

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 25th 2018
• (edited Apr 25th 2018)

finally spelled out the full statement (here) of the ADE-classification of 7d spherical space forms equipped with spin structures admitting $N \geq 4$ Killing spinors

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeApr 25th 2018

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?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 25th 2018
• (edited Apr 25th 2018)

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.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeApr 25th 2018

Thanks!

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 25th 2018

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