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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2015
    • (edited Aug 19th 2015)

    had added to finite group two classical references, Atiyah on group cohomology of finite groups, and Milnor on free actions of finite groups on nn-spheres.

    What I’d really like to know eventually is the degree-3 group cohomology with coefficients in U(1)U(1) for the finite subgroups of SO(3)SO(3).

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2015

    Is this in a similar context to

    In the case when the microscopic gauge group is Higgsed down to a finite group GG, 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).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2015
    • (edited Aug 19th 2015)

    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 L:XB 3(/Γ) conn\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 Σ 2\Sigma_2 is the mapping stack [Π(Σ 2),X][\Pi(\Sigma_2), X]. This is just XX 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 XX of the form */GBG\ast / G \simeq \mathbf{B}G this phase space becomes that of GG-Chern-Simons theory on Σ 2\Sigma_2 and specifically of GG-Dijkgraaf-Witten theory since GG 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 L\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 */GBGB 3(/)B 3(/) conn\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 GG 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)SO(3)?

    I see that the icosahedral group has group cohomology with integral coefficients concentrated in degrees 4k4k. What exactly is it in degree 44? Is it \mathbb{Z} there?

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2015
    • (edited Aug 19th 2015)

    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 4S_4 (octohedral) here and SL(2, 5).

    Edit: should be A 5A_5 not SL(2,5)SL(2, 5).

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2015

    Oh, doesn’t H k(BG,R/Z)=H k+1(BG,Z)H^k(B G, R/Z) = H^{k+1}(B G, Z), because of the long exact sequence from 0ZRR/Z00 \to Z \to R \to R/Z \to 0, and cohomology with coefficients in RR being trivial?

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2015

    Integral cohomology of S 4S_4 is on p. 2 here.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2015

    Thanks!!

    I have added these pointers to dihedral group and tetrahedral group and maybe elsewhere.

    Have to run now. Thanks again.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2015

    Wait a minute. You’re only considering orientation preserving transformations, as subgroups of SO(3)SO(3). So you should have tetrahedral group as A 4A_4, octohedral group as S 4S_4, and icosohedral group as A 5A_5.

    Are these official names? I see Mathworld has ’tetrahedral group’ down as the full symmetry group with inversions, so S 4S_4.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 19th 2015

    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

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2015

    Re #9, that’s in the paper Platonic 2-groups.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2015
    • (edited Aug 19th 2015)

    @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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