Processing math: 100%
Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: twisted equivariant K-theory

Bottom of Page

1 to 10 of 10

  1.  
    • 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 G a finite group (at least),

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

    G/H↦∏kB2kℚ⊗Rℂ(H)

    and that the equivariant fundamental group of BGPU(ℋ) assigns the character group

    G/H↦Hom(H,U(1)).

    Since the character group has a canonical action on the representation ring, it ought to be the case for the classifying coefficient G-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/H, the canonical action of the H-character group on the H-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 H-fixed locus of the classifying space for equivariant K-theory is the representation ring R(H)

    • known that the possible twistings of equivariant K-theory over an H-fixed point include – in addition to the notorious “gerbe” – a complex line bundle with structure group H2(G;ℤ)≃H1(G;U(1))⊂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

    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?

1 to 10 of 10

  1.