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had added to finite group two classical references, Atiyah on group cohomology of finite groups, and Milnor on free actions of finite groups on $n$-spheres.
What I’d really like to know eventually is the degree-3 group cohomology with coefficients in $U(1)$ for the finite subgroups of $SO(3)$.
Is this in a similar context to
In the case when the microscopic gauge group is Higgsed down to a finite group $G$, a complete classification (at least if no further global symmetries are postulated) has been given by Dijkgraaf and Witten [1]. Namely, topological actions in d dimensions are classified by degree-d cohomology classes for G with coefficients in U(1). (Higher symmetry and gapped phases of gauge theories]
[1] R. Dijkgraaf and E. Witten, “Topological Gauge Theories and Group Cohomology,” Commun. Math. Phys. 129, 393 (1990).
Yes. I am still trying, from the other thread, to see if there is a way to find the enhanced gauge symmetry of M-theory on ADE ordifolds directly, without recourse to heterotic duality. Now the extended action functional of the M2-brane is a 2-gerbe connection $\mathbf{L} : X \longrightarrow \mathbf{B}^3 (\mathbb{C}/\Gamma)_{conn}$ and so as we do the higher geometric prequantization of the M2 then the phase space over a 2-surface $\Sigma_2$ is the mapping stack $[\Pi(\Sigma_2), X]$. This is just $X$ itself at all smooth points, reflecting the fact that the critical M2-configurations there are just shrunken to single points. But at each orbifold fixed point in $X$ of the form $\ast / G \simeq \mathbf{B}G$ this phase space becomes that of $G$-Chern-Simons theory on $\Sigma_2$ and specifically of $G$-Dijkgraaf-Witten theory since $G$ is finite. Hence there is “gauge enhancement” at these points in a sense, and I am trying to see if this is the right kind of sense.
Now $\mathbf{L}$ is constrained to be a definite globalization hence its form at these orbifold singularities is not completely arbitrary. One gets that it is of the form $\ast /G \simeq \mathbf{B}G \to \flat \mathbf{B}^3 (\mathbb{C}/\mathbb{Z}) \to \mathbf{B}^3 (\mathbb{C}/\mathbb{Z})_{conn}$. In other words, it is given there by a degree-3 group cocycle on $G$ with coefficients in $\mathbb{C}/\mathbb{Z}$ (a Dijkgraaf-Witten action functional, if you wish, as in the quote that you give).
So what’s the cohomology in degree-3 with coefficients in $\mathbb{C}/\mathbb{Z}$ (or $\mathbb{R}/\mathbb{Z}$) of the finite groups appearing in the ADE – table, the finite subgroups of $SO(3)$?
I see that the icosahedral group has group cohomology with integral coefficients concentrated in degrees $4k$. What exactly is it in degree $4$? Is it $\mathbb{Z}$ there?
So the cyclic groups are apparently known
degree-3 cohomology of Z/n with coefficients in R/Z (with a trivial action of Z/n) is isomorphic to Z/n.
There’s cohomology of dihedral D8 for abelian coefficients here
The same site only has integral homology for the tetrahedral group here. Same for $S_4$ (octohedral) here and SL(2, 5).
Edit: should be $A_5$ not $SL(2, 5)$.
Oh, doesn’t $H^k(B G, R/Z) = H^{k+1}(B G, Z)$, because of the long exact sequence from $0 \to Z \to R \to R/Z \to 0$, and cohomology with coefficients in $R$ being trivial?
Integral cohomology of $S_4$ is on p. 2 here.
Thanks!!
I have added these pointers to dihedral group and tetrahedral group and maybe elsewhere.
Have to run now. Thanks again.
Wait a minute. You’re only considering orientation preserving transformations, as subgroups of $SO(3)$. So you should have tetrahedral group as $A_4$, octohedral group as $S_4$, and icosohedral group as $A_5$.
Are these official names? I see Mathworld has ’tetrahedral group’ down as the full symmetry group with inversions, so $S_4$.
The 2-group extensions of Platonic groups by BU(1) were done by a student of Nora Ganter – I think it was Narthana Epa. Sorry for no more, I’m off to bed
Re #9, that’s in the paper Platonic 2-groups.
@David C., I have added the distinctions to tetrahedral group and octahedral group and symmetric group.
@David R. thanks! I have found it:
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