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 $U(1)$-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 $Fred(\mathcal{H})$ of Fredholm operators on some separable Hilbert space $\mathcal{H}$.
The ordinary topological K-theory of a suitable topological space $X$ is given by the set of homotopy classes of maps from (the suspension spectrum of) $X$ to $KU$:
The projective unitary group $P U(\mathcal{H})$ (a topological group) acts canonically by automorphisms on $(K U)_0$. (This follows by the identification of $K U_0$ with the space of Fredholm operators, see below) Therefore for $P \to X$ any $PU(\mathcal{H})$-principal bundle, we can form the associated bundle $P \times_{P U(\mathcal{H})} (K U)_0$.
Since the homotopy type of $P U(\mathcal{H})$ is that of an Eilenberg-MacLane space $K(\mathbb{Z},2)$, there is precisely one isomorphism class of such bundles representing a class $\alpha \in H^3(X, \mathbb{Z})$.
The twisted K-theory with twist $\alpha \in H^3(X, \mathbb{Z})$ is the set of homotopy-classes of sections of such a bundle
Similarily the reduced $\alpha$-twisted K-theory is the subset
The following is due to (Atiyah-Singer 69, Atiyah-Segal 04).
Write
$Cl_n \coloneqq Cl^{\mathbb{C}}(\mathbb{R}^n,\langle -,-\rangle)$
for the complexification of the Clifford algebra of the Cartesian space $\mathbb{R}^n$ with its standard inner product;
$S_n$ for its $\mathbb{Z}/2\mathbb{Z}$-graded irreducible module (see at spin representation);
$H_0$ for the $\mathbb{Z}/2\mathbb{Z}$-graded separable Hilbert space whose even and odd part are both infinite-dimensional.
For $n \in \mathbb{N}$, the topological space $Fred^{(n)}$ of Fredholm operators on $S_n \otimes H_0$ is the set
(where $\mathcal{B}$ denotes bounded operators and $\mathcal{K}$ 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 $\mathcal{B}$ is equipped with the compact-open topology and $\mathcal{K}$ 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 $n \in \mathbb{N}$ 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 $U(H_0)$ 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 $U(H_0)$ on $Fred^{(n)}$, def. , is continuous.
This follows with (Atiyah-Segal 04, prop. A1.1).
Given a class $\chi \in H^3(X,\mathbb{Z})$ represented by a $PU(H_0)$-bundle $P \to X$ with associated Fredholm bundle
then the corresponding $\chi$-twisted cohomology spectrum is that consisting of the spaces of sections
(Freed-Hopkins-Teleman 11, def. 3.14)
Let $\alpha \in H^3(X, \mathbb{Z})$ be a class in degree-3 integral cohomology and let $P \in \mathbf{H}^3(X, \mathbf{B}^2 U(1))$ be any cocycle representative, which we may think of either as giving a circle 2-bundle or a bundle gerbe.
Write $TwBund(X, P)$ for the groupoid of twisted bundles on $X$ with twist given by $P$. Then let
be the set of isomorphism classes of twisted bundles. Call this the twisted K-theory of $X$ with twist $\alpha$.
(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 $Vectr$ 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 $U(1)$.
Let $\mathbf{B}_{\otimes} Vectr$ be the delooping of $Vectr$ 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 $\mathbf{B}U(1)$.
From the general nonsense of twisted cohomology this induces canonically now for every $\mathbf{B}^2 U(1)$-cocycle $c$ (for instance given by a bundle gerbe) a notion of $c$-twisted $Vectr$-cohomology:
After unwrapping what this means, the result of (Gomi) shows that concordance classes in $\mathbf{H}^c(X,Vectr)$ yield twisted K-theory.
By the general discussion of twisted cohomology the moduli space for the twists of periodic complex K-theory $KU$ is the Picard ∞-group in $KU Mod$. The “geometric” twists among these have as moduli space the non-connected delooping $bgl_1^\ast(KU)$ of the ∞-group of units of $KU$.
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 $K$-theory with local coefficients, Publications Mathématiques de l’IHÉS, 38 (1970), p. 5-25 (numdam:PMIHES_1970__38__5_0)
which discusses twists of $KO$ and $KU$ over some $X$ by elements in $H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z})$.
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 $C^\ast$-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 $H^3(X, \mathbb{Z})$ 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 $H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z})$, 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 $C^\ast$-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)
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 $C^\ast$-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 $C^\ast$-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 $C^\ast$-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 $SU(n)$ [arXiv:2307.00423]
Discussion of the twisted Chern character for higher twisted K-theory:
Last revised on September 18, 2023 at 17:53:52. See the history of this page for a list of all contributions to it.