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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
I am meaning to write out (here) a detailed proof that for $\Gamma$ an $n$-truncated topological group (e.g. PU(ℋ) for $n \geq 2$) and $S^{n+2}/G$ a spherical space form of dimension $n + 2$, the comparison morphism
$ʃ \, Map \Big( \mathbf{B}G ,\, \mathbf{B}\Gamma \Big) \xrightarrow {\;\;\; ʃ \, Map(p/\!\!/G,\,\mathbf{B}\Gamma) \;\;\;} ʃ \, Map \Big( S^{n+2}/G ,\, \mathbf{B}\Gamma \Big)$is an equivalence of $\infty$-groupoids.
The idea is simple: By the truncation condition we have 1. that $\Gamma$-principal bundles on $S^{n+2}/G$ are isomorphic to those pulled back from $\ast \sslash G$ and 2. that all gauge transformations are concordant to those pulled back from $\ast \sslash G$.
Carefully writing out this simple idea into a formal proof is becoming a little lengthy. So far the entry shows most of the argument for the iso on $\pi_0$. Once this is stated satisfactorily, the generalization to $\pi_n$ should be immediate.
So I am not done yet, but need to grab some late lunch and some coffee now.
now this Lemma has a proof
It dawned on me that my proof strategy only makes sense if I first show that the simplicial sets in question are Kan complexes.
So I have now added a lemma (here) showing in detail the existence of 2-horn fillers.
The filling of the higher horns “clearly” follows by the same mechanism. However, once again, it seems a bit of a pain to turn this evident idea into a fully formal proof.
I have given the lemma regarding surjectivity on $\pi_1$ a big diagram that shows the construction of the required homotopy in some detail (here).
But I got stuck on proving that this homotopy really fixes both endpoints (it’s clear for the left one, but subtle for the right one), without further assumption.
So I added now one more assumption on the structure group $\Gamma$, namely that $Shp(\Gamma)$ be braided, hence that $B \Gamma$ has group structure itself. (This is a harmless assumption in the intended applications, where $Shp(\Gamma)$ is a truncation of a connective spectrum).
With this assumption, also the space of concordances inherits $\infty$-group-structure (using the smooth Oka principle, now this Remark) which implies that we may compute the fundamental group equivalently in any connected component. But in the connected component of the trivial cocycle the above issue goes away.
With that, I think I have now typed out detailed proof that the comparison map is an iso on homotopy groups in degrees $\leq 1$, which “obviously” generalizes to all higher homotopy groups.
Next I should add some subsections to disentangle all the lemmas from the main claim to make it discernible. Will do…
I have now considerably relaxed the running Assumption (here) by appealing to the Madsen-Thomas-Wall theorem (here):
Now $\Gamma$ is just required to be $n$-truncated for any $n$, no longer depending on $G$ (as the MTW theorem says that if $G$ acts freely and smoothly on any $d$-sphere, then it does so on one whose dimension exceeds any given bound $n + 2$).
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