nLab twisted ad-equivariant K-theory

Idea

The twisted equivariant K-theory of suitable compact Lie groups equivariant with respect to the adjoint action of the group on itself.

By FHT this is equivalent to the Verlinde ring of positive energy representations of the corresponding loop group.

(…)

Related concepts

• twisted ad-equivariant Tate K-theory

References

On twisted ad-equivariant K-theory of compact Lie groups and the identification with the Verlinde ring of positive energy representations of their loop group:

• Daniel S. Freed, The Verlinde algebra is twisted equivariant K-theory, Turk. J. Math 25 (2001) 159-167 (arXiv:math/0101038)

• Daniel S. Freed, Michael Hopkins, Constantin Teleman,

  1. Loop Groups and Twisted K-Theory I,

     J. Topology, 4 (2011), 737-789

     (arXiv:0711.1906, doi:10.1112/jtopol/jtr019)

  2. Loop Groups and Twisted K-Theory II,

     J. Amer. Math. Soc. 26 (2013), 595-644

     (arXiv:0511232, doi:10.1090/S0894-0347-2013-00761-4)

  3. Loop Groups and Twisted K-Theory III,

     Annals of Mathematics, Volume 174 (2011) 947-1007

     (arXiv:math/0312155, doi:10.4007/annals.2011.174.2.5)

• Daniel S. Freed, Constantin Teleman,

  Dirac families for loop groups as matrix factorizations,

  Comptes Rendus Mathematique, Volume 353, Issue 5, May 2015, Pages 415-419

  (arxiv/1409.6051, doi:10.1016/j.crma.2015.02.011)

Discussion in terms of differentiable stacks:

• Ulrich Bunke, Ingo Schroeder, Twisted K-Theory and TQFT (arXiv:math/0504458, spire:681555)

Review:

• Dale Husemöller, Michael Joachim, Branislav Jurčo, Martin Schottenloher, Section 22 of: Basic Bundle Theory and K-Cohomology Invariants, Springer Lecture Notes in Physics 726, 2008, (pdf, doi:10.1007/978-3-540-74956-1)

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

Generalization to KR-theory:

• Chi-Kwong Fok, The Real K-theory of compact Lie groups, SIGMA 10 (2014), 022, (arXiv:1308.3871, doi:10.3842/SIGMA.2014.022)

• Chi-Kwong Fok, Equivariant twisted Real K-theory of compact Lie groups, Journal of Geometry and Physics 124 (2018) 325-349 (arXiv:1503.00957, doi:10.1016/j.geomphys.2017.11.013)