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nLab > Latest Changes: twisted equivariant K-theory

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2020
    • (edited Nov 3rd 2020)

    finally splitting this off, for ease of organizing references. Not much here yet…

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2020

    added this pointer:

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2021

    added pointer to the general definition in terms of equivariant sections of equivariant bundles of equivariant classifying spaces for equivariant K-theory:

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2021

    added pointer to:

    diff, v5, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2021

    added pointer to:

    diff, v6, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2021

    added pointer to:

    diff, v7, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2021

    added pointer to:

    diff, v9, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2022

    For GG a finite group (at least),

    it is known that classifying GG-space for rational GG-equivariant KU-theory assigns the rationalized complex representation rings

    G/H↦∏kB 2kℚ⊗R ℂ(H) G/H \;\mapsto\; \underset{k}{\prod} B^{2k} \; \mathbb{Q} \otimes R_{\mathbb{C}}(H)

    and that the equivariant fundamental group of B GPU(ℋ)B_G PU(\mathcal{H}) assigns the character group

    G/H↦Hom(H,U(1)). G/H \;\mapsto\; Hom\big(H, U(1)\big) \,.

    Since the character group has a canonical action on the representation ring, it ought to be the case for the classifying coefficient GG-bundle for 3-twisted equivariant K-theory, that the action of the equivariant fundamental group of the base on the homotopy fibers is, at each stage G/HG/H, the canonical action of the HH-character group on the HH-representation ring.

    Is this proven anywhere?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2022
    • (edited Jan 11th 2022)

    a reference item to add to twisted equivariant K-theory once the edit-functionality is back:

    • Alejandro Adem, José Cantarero, José Manuel Gómez, Twisted equivariant K-theory of compact Lie group actions with maximal rank isotropy, J. Math. Phys. 59 113502 (2018) (arXiv:1709.00989, doi:10.1063/1.5036647)
    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 28th 2022
    • (edited Jan 28th 2022)

    It is

    • well-known that the connected components of the HH-fixed locus of the classifying space for equivariant K-theory is the representation ring R(H)R(H)

    • known that the possible twistings of equivariant K-theory over an HH-fixed point include – in addition to the notorious “gerbe” – a complex line bundle with structure group H 2(G;ℤ)≃H 1(G;U(1))⊂U(1)H^2(G; \mathbb{Z}) \simeq H^1\big(G; \mathrm{U}(1)\big) \,\subset \, \mathrm{U}(1) (aka “local system”).

    But, in describing how this degree-1 twist actually acts on the classifying space, all authors I have seen (where “all” is no more than 2 or 3 groups, apparently) pass to the perspective of “delocalized” cohomology.

    While the delocalized picture has some clear virtues, it does a fair bit of violence to the classifying picture of K-theory. In the latter picture, there is an evident guess for how the 1-twist acts: It ought to be the canonical operation of tensoring representations with group characters regarded as 1d reps:

    Hom(G;U(1))×R(G) R(G) (κ,ρ) ↦ κ⊗ρ \array{ Hom(G; U(1)) \,\times\, R(G) && R(G) \\ (\kappa, \rho) &\mapsto& \kappa \otimes \rho }

    Because what else can it be. But also because I think I have proven this now. (It follows from the observation mentioned in another thread, here.)

    I keep wondering, though, if this has not been discussed elsewhere, before?

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