nLab twisted equivariant K-theory

Idea

Twisted equivariant K-theory is the special case of twisted equivariant cohomology for the case that the cohomology theory is K-theory (often understood to be topological K-theory, by default), hence the joint generalization of twisted K-theory and equivariant K-theory.

This is often considered in the special case of twisted ad-equivariant K-theory.

Related concepts

• twisted equivariant Chern character

• universal equivariant $PU(\mathcal{H})$-bundle

• orbifold K-theory

• twisted ad-equivariant K-theory

• twisted ad-equivariant Tate K-theory

• twisted differential K-theory

• twisted equivariant differential K-theory

• K-theory classification of D-brane charge

• K-theory classification of topological phases of matter, topological phases of matter

References

In a context of orbifold K-theory and for torsion class twists (“discrete torsion”):

• Alejandro Adem, Yongbin Ruan, Twisted Orbifold K-Theory, Commun. Math. Phys. 237 (2003) 533-556 (arXiv:math/0107168, doi:10.1007/s00220-003-0849-x)

With an eye towards twisted ad-equivariant K-theory:

• Daniel Freed, Michael Hopkins, Constantin Teleman, Twisted equivariant K-theory with complex coefficients, Journal of Topology 1 1 (2007) [arXiv:math/0206257, doi:10.1112/jtopol/jtm001]

In terms of bundle gerbes on differentiable stacks:

• Jean-Louis Tu, Ping Xu, Camille Laurent-Gengoux, Twisted K-theory of differentiable stacks, Annales Scientifiques de l’École Normale Supérieure Volume 37, Issue 6, November–December 2004, Pages 841-910 (arXiv:math/0306138, doi:10.1016/j.ansens.2004.10.002)

General definition in terms of equivariant sections of equivariant bundles of equivariant classifying spaces for equivariant K-theory:

• Michael Atiyah, Graeme Segal, Section 6 of: Twisted K-theory, Ukrainian Math. Bull. 1, 3 (2004) (arXiv:math/0407054, journal page, published pdf)

• Jose Cantarero, Equivariant K-theory, groupoids and proper actions, Thesis 2009 (ubctheses:1.0068026, pdf)

Reviewed in:

• Valentin Zakharevich, Section 2.2 of: K-Theoretic Computation of the Verlinde Ring, thesis 2018 (hdl:2152/67663, pdf, pdf)

Alternatively, in terms of Grothendieck groups of equivariant twisted vector bundles:

• Christopher Dwyer, Twisted equivariant K-theory for proper actions of discrete groups, K-Theory v. 38, n. 2 (January, 2008): 95-111 (arXiv:0710.2136, doi:10.1007/s10977-007-9016-z)

On the cohomology ring-structure:

• Jean-Louis Tu, Ping Xu, The ring structure for equivariant twisted K-theory, J. Reine Angew. Math. 635 (2009), 97–148 (arXiv:math/0604160, doi:10.1515/CRELLE.2009.077)

On 3-twisted equivariant K-theory via the universal equivariant $PU(\mathcal{H})$-bundle:

• Noé Bárcenas, Jesús Espinoza, Michael Joachim, Bernardo Uribe, Universal twist in Equivariant K-theory for proper and discrete actions, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) (arXiv:1202.1880, doi:10.1112/plms/pdt061)

• Jesus Espinoza, Bernardo Uribe, Topological properties of spaces of projective unitary representations, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 40 (2016), no. 155, 337-352 (arXiv:1511.06785, scielo:S0370-39082016000200013, doi:10.18257/raccefyn.317)

• Bernardo Uribe, Wolfgang Lück, Equivariant principal bundles and their classifying spaces, Algebr. Geom. Topol. 14 (2014) 1925-1995 (arXiv:1304.4862, doi:10.2140/agt.2014.14.1925)

The corresponding version of the Atiyah-Hirzebruch spectral sequence:

• Noé Bárcenas, Jesús Espinoza, Bernardo Uribe, Mario Velásquez, Section 3.2 of: Segal’s spectral sequence in twisted equivariant K-theory for proper and discrete actions, Proceedings of the Edinburgh Mathematical Society 61 1 (2018) (arXiv:1307.1003, doi:10.1017/S0013091517000281)

Relation to projective representations:

• José Manuel Gómez, Johana Ramírez, A Decomposition of twisted equivariant K-theory (arXiv:2001.02164)