group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
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 identification 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
The following is due to (Atiyah-Singer 69, Atiyah-Segal 04).
Write
for the complexification of the Clifford algebra of the Cartesian space with its standard inner product;
for its -graded irreducible module (see at spin representation);
for the -graded separable Hilbert space whose even and odd part are both infinite-dimensional.
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
given by
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
and
between the spaces of graded Fredholm operators of def. 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.
The conjugation action of the stable unitary group on , def. , is continuous.
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)
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 is (CBMMS, prop. 6.4, prop. 7.3).
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.
twisted K-theory
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) (numdam:ASENS_1968_4_1_2_161_0)
Peter Donovan, Max Karoubi, Graded Brauer groups and -theory with local coefficients, Publications Mathématiques de l’IHÉS, 38 (1970), p. 5-25 (numdam:PMIHES_1970__38__5_0)
which discusses twists of and over some by elements in .
Including the twist in degree 1 (see also at PU(ℋ)):
reviewed in
The formulation in terms of sections of Fredholm bundles seems to go back to
and is expanded on in:
Daniel Freed, Michael Hopkins, Constantin Teleman, Diagram (2.6) in: Twisted equivariant K-theory with complex coefficients, Journal of Topology, Volume 1, Issue 1, 2007 (arXiv:math/0206257, doi:10.1112/jtopol/jtm001)
Michael Atiyah, Graeme Segal, Twisted K-theory, Ukrainian Math. Bull. 1 3 (2004) [arXiv:math/0407054, journal page, published pdf]
Peter May, Johann Sigurdsson, Section 22.3 of: Parametrized Homotopy Theory, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, arXiv:math/0411656, pdf)
Michael Atiyah, Graeme Segal, Twisted K-theory and cohomology (arXiv:math/0510674)
Matthew Ando, Andrew Blumberg, David Gepner, sections 2.1 and 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, doi:10.1090/pspum/081)
A comprehensive account of twisted K-theory with twists in is in:
January 1969, Volume 37, Issue 1, pp 5-26 (pdf)
and for more general twists in
See also
Textbook accounts:
Discussion of twisted ad-equivariant K-theory and relation to loop group representations and the Verlinde ring, 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
See the references at (infinity,1)-vector bundle for more on this.
Discussion in terms of twisted bundles/bundle gerbe modules is in
but apparently contains a mistake, as pointed out in
The generalization of this to groupoid K-theory is in (FHT 07, around p. 26) and
(which establishes the relation to KK-theory).
Max Karoubi, Twisted bundles and twisted K-theory, arxiv/1012.2512
Ulrich Pennig, Twisted K-theory with coefficients in -algebras, (arXiv:1103.4096)
Comparison between homotopy theoretic and operator algebraic constructions:
Discussion in terms of vectorial bundles is in
Kiyonori Gomi, Twisted K-theory and finite-dimensional approximation (arXiv:0803.2327)
Kiyonori Gomi, Yuji Terashima, Chern-Weil Construction for Twisted K-Theory, Communication ins Mathematical Physics, Volume 299, Number 1, 225-254 (doi:10.1007/s00220-010-1080-1)
Discussion of combined twisted equivariant KR-theory on orbi- orientifolds:
El-kaïoum M. Moutuou, Twistings of KR for Real groupoids (arXiv:1110.6836)
El-kaïoum M. Moutuou, Graded Brauer groups of a groupoid with involution, J. Funct. Anal. 266 (2014), no.5 (arXiv:1202.2057)
Daniel Freed, Lectures on twisted K-theory and orientifolds, lectures at ESI Vienna, 2012 (pdf)
Daniel Freed, Gregory Moore, Section 7 of: Twisted equivariant matter, Ann. Henri Poincaré (2013) 14: 1927 (arXiv:1208.5055)
Kiyonori Gomi, Freed-Moore K-theory (arXiv:1705.09134, spire:1601772)
Review:
Discussion of twisted K-homology:
Bai-Ling Wang, Gemometric cycles, index theory and twisted K-homology (arXiv:0710.1625)
Eckhard Meinrenken, Twisted K-homology and group-valued moment maps, International Mathematics Research Notices 2012 (20) (2012), 4563–4618 (arXiv:1008.1261)
Bei Liu, Twisted K-homology,Geometric cycles and T-duality (arXiv:1411.1575)
Discussion of combined twisted and equivariant and real K-theory
Discussion of twisted differential K-theory and its relation to D-brane charge in type II string theory (see at K-theory classification of D-brane charge):
Discussion of twisted differential orthogonal K-theory and its relation to D-brane charge in type I string theory (on orientifolds):
Discussion of higher twists of K-theory (above degree 3):
Ib Madsen, Victor Snaith, Jørgen Tornehave, Infinite loop maps in geometric topology, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 81, Issue 3, (1977)(doi:10.1017/S0305004100053482)
Constantin Teleman, Section 3 of: K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra (arXiv:math/0306347), in Ulrike Tillmann (ed.) Topology, Geometry and Quantum Field Theory, Cambridge University Press 2004 (doi:10.1017/CBO9780511526398)
Via -algebras:
Ulrich Pennig, A noncommutative model for higher twisted K-Theory, J Topology (2016) 9 (1): 27-50 (arXiv:1502.02807)
Marius Dadarlat, Ulrich Pennig, A Dixmier-Douady theory for strongly self-absorbing -algebras, J. Reine Angew. Math. 718 (2016) 153-181 (arXiv:1302.4468, doi:10.1515/crelle-2014-0044)
Marius Dadarlat, Ulrich Pennig, Unit spectra of K-theory from strongly self-absorbing -algebras, Algebr. Geom. Topol. 15 (2015) 137-168 [arXiv:1306.2583, doi:10.2140/agt.2015.15.137]
David Brook, Computations in higher twisted K-theory (arXiv:2007.08964A)
David Brook, Higher twisted K-theory (dspace:2440/125740)
David E. Evans, Ulrich Pennig, Spectral Sequence Computation of Higher Twisted K-Groups of [arXiv:2307.00423]
see also:
Discussion of the twisted Chern character for higher twisted K-theory:
Last revised on February 9, 2024 at 16:30:31. See the history of this page for a list of all contributions to it.