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 KU 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 suitable topological space is given by the set of homotopy classes of maps from (the suspension spectrum of) to :
The projective unitary group (a topological group) acts canonically by automorphisms on . (This follows by the identificatioon of with the space of Fredholm operators, see below) 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 bundles of Fredholm operators
The following is due to (Atiyah-Singer 69, Atiyah-Segal 04).
For , the topological space of Fredholm operators on is the set
(where denotes bounded operators and denotes compact operators and where denotes the graded commutator) and the topology on this set is the subspace topology induced by the embedding
where is equipped with the compact-open topology and with the norm topology.
(Atiyah-Singer 69, p. 7, Atiyah-Segal 04, p. 21, Freed-Hopkins-Teleman 11, def. A.40)
These spaces indeed form a model for the KU spectrum:
For all there are natural weak homotopy equivalences
between the spaces of graded Fredholm operators of def. 2 and their loop spaces.
(Atiyah-Singer 69, theorem B(k), Atiyah-Segal 04 (4.2), Freed-Hopkins-Teleman 11, below def. A.40)
Regard the stable unitary group as equipped with the subspace topology induced by the inclusion
from the compact-open topology on the bounded linear operators.
This follows with (Atiyah-Segal 04, prop. A1.1).
Given a class represented by a -bundle with associated Fredholm bundle
then the corresponding -twisted cohomology spectrum is that consisting of the spaces of sections
(Freed-Hopkins-Teleman 11, def. 3.14)
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.
The concept of twisted K-theory originates in
Max Karoubi, Algèbres de Clifford et K-théorie. Ann. Sci. Ecole Norm. Sup. (4), pp. 161-270 (1968).
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
Michael Atiyah, Isadore Singer, Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’Institut des Hautes Études Scientifiques January 1969, Volume 37, Issue 1, pp 5-26 (pdf)
Michael Atiyah, Graeme Segal, Twisted K-theory (arXiv:math/0407054)
Michael Atiyah, Graeme Segal, Twisted K-theory and cohomology (arXiv:math/0510674)
and for more general twists in
The seminal result on the relation to loop group representations, now again with twists in , is in the series of articles
The result on twisted K-groups has been lifted to an equivalence of categories in
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 (parameterized spectra) is discussed in
May, Sigurdsson, section 22 of Parametrized homotopy theory (pdf) AMS Lecture notes 132
Matthew Ando, Andrew Blumberg, David Gepner, section 7 of Twists of K-theory and TMF, in Jonathan Rosenberg et al. (eds.), Superstrings, Geometry, Topology, and -algebras, volume 81 of Proceedings of Symposia in Pure Mathematics, 2009 (arXiv:1002.3004)
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, Twisted K-theory and finite-dimensional approximation (arXiv)
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
Discussion of twisted K-homology: