twisted K-theory




Special and general types

Special notions


Extra structure





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)U(1)-bundle gerbe or circle 2-group-principal 2-bundle), but there are various other twists.


By sections of associated KUK U-bundles

Write KU for the spectrum of complex topological K-theory. Its degree-0 space is, up to weak homotopy equivalence, the space

BU×=lim nBU(n)× B U \times \mathbb{Z} = {\lim_\to}_n B U(n) \times \mathbb{Z}

or the space Fred()Fred(\mathcal{H}) of Fredholm operators on some separable Hilbert space \mathcal{H}.

(KU) 0BU×Fred(). (K U)_0 \simeq B U \times \mathbb{Z} \simeq Fred(\mathcal{H}) \,.

The ordinary topological K-theory of a suitable topological space XX is given by the set of homotopy classes of maps from (the suspension spectrum of) XX to KUKU:

K(X) [X,(KU) ]. K(X)_\bullet \simeq [X, (K U)_\bullet] \,.

The projective unitary group PU()P U(\mathcal{H}) (a topological group) acts canonically by automorphisms on (KU) 0(K U)_0. (This follows by the identificatioon of KU 0KU_0 with the space of Fredholm operators, see below) Therefore for PXP \to X any PU()PU(\mathcal{H})-principal bundle, we can form the associated bundle P× PU()(KU) 0P \times_{P U(\mathcal{H})} (K U)_0.

Since the homotopy type of PU()P U(\mathcal{H}) is that of an Eilenberg-MacLane space K(,2)K(\mathbb{Z},2), there is precisely one isomorphism class of such bundles representing a class αH 3(X,)\alpha \in H^3(X, \mathbb{Z}).


The twisted K-theory with twist αH 3(X,)\alpha \in H^3(X, \mathbb{Z}) is the set of homotopy-classes of sections of such a bundle

K α(X) 0:=Γ X(P α× PU()(KU) 0). K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} (K U)_0) \,.

Similarily the reduced α\alpha-twisted K-theory is the subset

K˜ α(X) 0:=Γ X(P α× PU()BU). \tilde K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} B U) \,.


By bundles of Fredholm operators

The following is due to (Atiyah-Singer 69, Atiyah-Segal 04).



For nn \in \mathbb{N}, the topological space Fred (n)Fred^{(n)} of Fredholm operators on S nH 0S_n \otimes H_0 is the set

Fred (n){F(S nH 0)|Fodd,F *=F,F 21𝒦(S nH 0),[F,γ]=0forγCl n} Fred^{(n)} \coloneqq \left\{ F \in \mathcal{B}(S_n \otimes H_0) \;|\; F \, odd\,, F^\ast = F \,, F^2 - 1 \in \mathcal{K}(S_n \otimes H_0)\,, [F,\gamma] = 0 \, for\, \gamma \in Cl_n \right\}

(where \mathcal{B} denotes bounded operators and 𝒦\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

Fred (n)(S nH 0)×𝒦(S nH 0) Fred^{(n)} \hookrightarrow \mathcal{B}(S_n \otimes H_0) \times \mathcal{K}(S_n \otimes H_0)

given by

F(F,F 21), F\mapsto (F, F^2 - 1) \,,

where \mathcal{B} is equipped with the compact-open topology and 𝒦\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 nn \in \mathbb{N} there are natural weak homotopy equivalences

Fred (n+1)ΩFred (n) Fred^{(n+1)} \stackrel{\simeq}{\longrightarrow} \Omega Fred^{(n)}


Fred (n+2)Fred (n) Fred^{(n+2)} \stackrel{\simeq}{\longrightarrow} Fred^{(n)}

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 U(H 0)U(H_0) as equipped with the subspace topology induced by the inclusion

U(H 0)(id,() 1)(H 0)×(H 0) U(H_0) \stackrel{(id,(-)^{-1})}{\hookrightarrow} \mathcal{B}(H_0)\times\mathcal{B}(H_0)

from the compact-open topology on the bounded linear operators.


The conjugation action of the stable unitary group U(H 0)U(H_0) on Fred (n)Fred^{(n)}, def. 2, is continuous.

This follows with (Atiyah-Segal 04, prop. A1.1).


Given a class χH 3(X,)\chi \in H^3(X,\mathbb{Z}) represented by a PU(H 0)PU(H_0)-bundle PXP \to X with associated Fredholm bundle

Fred (n)+χP×PU(H 0)Fred (n), Fred^{(n)+ \chi} \coloneqq P \underset{PU(H_0)}{\times} Fred^{(n)} \,,

then the corresponding χ\chi-twisted cohomology spectrum is that consisting of the spaces of sections

Γ(X,Fred (n)+χ). \Gamma(X, Fred^{(n)+ \chi}) \,.

(Freed-Hopkins-Teleman 11, def. 3.14)

By twisted vector bundles (gerbe modules)


Let αH 3(X,)\alpha \in H^3(X, \mathbb{Z}) be a class in degree-3 integral cohomology and let PH 3(X,B 2U(1))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)TwBund(X, P) for the groupoid of twisted bundles on XX with twist given by PP. Then let

K˜ α(X):=TwBund(X,P) \tilde K_\alpha(X) := TwBund(X,P)

be the set of isomorphism classes of twisted bundles. Call this the twisted K-theory of XX with twist α\alpha.

(Some technical details need to be added for the non-torsion case.)


This definition of twisted K 0K_0 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.

Other constructions

Let VectrVectr 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

ρ:BU(1)VectVectr \rho : \mathbf{B} U(1) \to Vect \hookrightarrow Vectr

coming from the standard representation of the group U(1)U(1).

Let B Vectr\mathbf{B}_{\otimes} Vectr be the delooping of VectrVectr with respect to the tensor product monoidal structure (not the additive structure).

Then we have a fibration sequence

Vectr*B Vectr Vectr \to {*} \to \mathbf{B}_\otimes Vectr

of (infinity,1)-categories (instead of infinity-groupoids).

The entire morphism above deloops

Bρ:B 2U(1)B VectB Vectr \mathbf{B}\rho : \mathbf{B}^2 U(1) \to \mathbf{B}_\otimes Vect \hookrightarrow \mathbf{B}_{\otimes} Vectr

being the standard representation of the 2-group BU(1)\mathbf{B}U(1).

From the general nonsense of twisted cohomology this induces canonically now for every B 2U(1)\mathbf{B}^2 U(1)-cocycle cc (for instance given by a bundle gerbe) a notion of cc-twisted VectrVectr-cohomology:

H c(X,Vectr) * Bρc * H(X,B Vectr). \array{ \mathbf{H}^c(X, Vectr) &\to& {*} \\ \downarrow && \downarrow^{\mathbf{B}\rho \circ c} \\ {*} &\to& \mathbf{H}(X,\mathbf{B}_\otimes Vectr) } \,.

After unwrapping what this means, the result of (Gomi) shows that concordance classes in H c(X,Vectr)\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 KUKU is the Picard ∞-group in KUModKU Mod. The “geometric” twists among these have as moduli space the non-connected delooping bgl 1 *(KU)bgl_1^\ast(KU) of the ∞-group of units of KUKU.

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 KK-theory with local coefficients, Publications Mathématiques de l’IHÉS, 38 (1970), p. 5-25 (numdam)

which discusses twists of KOKO and KUKU over some XX by elements in H 0(X, 2)×H 1(X, 2)×H 3(X,)H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z}).

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 H 3(X,)H^3(X, \mathbb{Z}) is in

and for more general twists in

The seminal result on the relation to loop group representations, now again with twists in H 0(X, 2)×H 1(X, 2)×H 3(X,)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

  • 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 KUK U-bundle of spectra (parameterized spectra) is discussed in

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 KK \mathbb{R}-theory relevant for orientifolds are discussed in

  • El-kaïoum M. Moutuou, Twistings of KR for Real groupoids (arXiv:1110.6836)

Discussion of twisted K-homology:

Revised on April 5, 2017 09:38:00 by Urs Schreiber (