Special and general types
Twisted K-theory is a twisted cohomology version of (topological) K-theory.
The most famous twist is by a class in degree 3 ordinary cohomology (geometrically a -bundle gerbe or circle 2-group-principal 2-bundle), but there are various other twists.
By sections of associated -bundles
Write for the spectrum of complex topological K-theory. Its degree-0 space is, up to weak homotopy equivalence, the space
or the space of Fredholm operators on some separable Hilbert space .
The ordinary topological K-theory of a topological space is
The projective unitary group (a topological group) acts canonically by automorphisms on . Therefore for any -principal bundle, we can form the associated bundle .
Since the homotopy type of is that of an Eilenberg-MacLane space , there is precisely one isomorphism class of such bundles representing a class .
The twisted K-theory with twist is the set of homotopy-classes of sections of such a bundle
Similarily the reduced -twisted K-theory is the subset
By twisted vector bundles (gerbe modules)
Let be a class in degree-3 integral cohomology and let be any cocycle representative, which we may think of either as giving a circle 2-bundle or a bundle gerbe.
Write for the groupoid of twisted bundles on with twist given by . Then let
be the set of isomorphism classes of twisted bundles. Call this the twisted K-theory of with twist .
(Some technical details need to be added for the non-torsion case.)
This definition of twisted is equivalent to that of prop. 1.
This is (CBMMS, prop. 6.4, prop. 7.3).
By KK-theory of twisted convolution algebras
A circle 2-group principal 2-bundle is also incarnated as a centrally extended Lie groupoid. The corresponding twisted groupoid convolution algebra has as its operator K-theory the twisted K-theory of the base space (or base-stack). See at KK-theory for more on this.
Let be the stack of vectorial bundles. (If we just take vector bundles we get a notion of twisted K-theory that only allows twists that are finite order elements in their cohomology group).
There is a canonical morphism
coming from the standard representation of the group .
Let be the delooping of with respect to the tensor product monoidal structure (not the additive structure).
Then we have a fibration sequence
of (infinity,1)-categories (instead of infinity-groupoids).
The entire morphism above deloops
being the standard representation of the 2-group .
From the general nonsense of twisted cohomology this induces canonically now for every -cocycle (for instance given by a bundle gerbe) a notion of -twisted -cohomology:
After unwrapping what this means, the result of (Gomi) shows that concordance classes in yield twisted K-theory.
By the general discussion of twisted cohomology the moduli space for the twists of periodic complex K-theory is the Picard ∞-group in . The “geometric” twists among these have as moduli space the non-connected delooping of the ∞-group of units of .
A model for this in 4-truncation is given by super line 2-bundles. For the moment see there for further discussion and further references.
An original article is
- Peter Donovan, Max Karoubi, Graded Brauer groups and -theory with local coefficients, Publications Mathématiques de l’IHÉS, 38 (1970), p. 5-25 (numdam)
which discusses twists of and over some by elements in .
The formulation in terms of sections of Fredholm bundles seems to go back to
- Jonathan Rosenberg, Continuous-trace algebras from the bundle theoretic point of view , J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368-381.
A comprehensive account of twisted K-theory with twists in is in
The seminal result on the relation to loop group representations, now again with twists in , is in the series of articles
Discussion in terms of Karoubi K-theory/Clifford module bundles is in
- Max Karoubi, Clifford modules and twisted K-theory, Proceedings of the International Conference on Clifford algebras (ICCA7) (arXiv:0801.2794)
The perspective of twisted K-theory by sections of a -bundle of spectra is discussed for instance in section 22 of
- May, Sigurdsson, Parametrized homotopy theory (pdf) AMS Lecture notes 132
See the references at (infinity,1)-vector bundle for more on this.
Discussion in terms of twisted bundles/bundle gerbe modules is in
and for generalization to groupoid K-theory also (FHT 07, around p. 26) and
(which establishes the relation to KK-theory).
Discussion in terms of vectorial bundles is in
- Kiyonori Gomi und Yuji Terashima, Chern-Weil Construction for Twisted K-Theory Communication ins Mathematical Physics, Volume 299, Number 1, 225-254,
The twisted version of differential K-theory is discussed in
- Alan Carey, Differential twisted K-theory and applications ESI preprint (pdf)
Twists of -theory relevant for orientifolds are discussed in