nLab twisted K-theory

Idea

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.

Definition

By sections of associated $K U$-bundles

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

$$B U \times \mathbb{Z} = {\lim_\to}_n B U(n) \times \mathbb{Z}$$

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

$$(K U)_0 \simeq B U \times \mathbb{Z} \simeq Fred(\mathcal{H}) \,.$$

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 $KU$:

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

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 $\alpha \in H^3(X, \mathbb{Z})$.

Definition

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

$$K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} (K U)_0) \,.$$

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

$$\tilde K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} B U) \,.$$

By bundles of Fredholm operators

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

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.

Definition

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

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

(where $\mathcal{B}$ denotes bounded operators and $\mathcal{K}$ denotes compact operators and where $[-,-]$ denotes the graded commutator) and the topology on this set is the subspace topology induced by the embedding

$$Fred^{(n)} \hookrightarrow \mathcal{B}(S_n \otimes H_0) \times \mathcal{K}(S_n \otimes H_0)$$

given by

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

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:

Proposition

For all $n \in \mathbb{N}$ there are natural weak homotopy equivalences

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

and

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

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

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

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

Proposition

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).

Definition

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

$$Fred^{(n)+ \chi} \coloneqq P \underset{PU(H_0)}{\times} Fred^{(n)} \,,$$

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

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

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

By twisted vector bundles (gerbe modules)

Definition

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

$$\tilde K_\alpha(X) := TwBund(X,P)$$

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.)

Proposition

This definition of twisted $K_0$ is equivalent to that of prop. .

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 $Vectr$ 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

$$\rho : \mathbf{B} U(1) \to Vect \hookrightarrow Vectr$$

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

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

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

The entire morphism above deloops

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

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

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:

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

After unwrapping what this means, the result of (Gomi) shows that concordance classes in $\mathbf{H}^c(X,Vectr)$ yield twisted K-theory.

Twists

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.

Related concepts

• K-theory

• topological K-theory, KR-theory

• twisted K-theory

  • twisted equivariant K-theory

    • twisted ad-equivariant K-theory

  • differential K-theory

  • twisted differential K-theory

  • fiber integration in K-theory

• twisted ordinary cohomology

• twisted Bredon cohomology

• twisted Cohomotopy

References

General

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)):

• Ellen Maycock Parker, around Prop. 2.2 of: The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras, Transactions of the American Mathematical Society 308 1 (1988) [jstor:2000953]

reviewed in

• Alan Carey, Bai-Ling Wang, p. 5 of: Thom isomorphism and Push-forward map in twisted K-theory, Journal of K-Theory 1 2 (2008) 357-393 (arXiv:math/0507414, doi:10.1017/is007011015jkt011)

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 (doi:10.1017/S1446788700033097)

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:

• 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)

and for more general twists in

• Daniel Freed, Michael Hopkins, Constantin Teleman, Loop Groups and Twisted K-Theory I , J. Topology, 4 (2011), 737-789 (arXiv:0711.1906)

See also

• Max Karoubi, Twisted bundles and twisted K-theory, in: Guillermo Cortiñas (ed.) Topics in Noncommutative Geometry (arXiv:1012.2512, ISBN:978-0-8218-6864-5)

Textbook accounts:

• Dale Husemöller, Michael Joachim, Branislav Jurčo, Martin Schottenloher, Part IV of: Basic Bundle Theory and K-Cohomology Invariants, Springer Lecture Notes in Physics 726, 2008, (pdf, doi:10.1007/978-3-540-74956-1)

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

• Daniel Freed, Michael Hopkins, Constantin Teleman, Twisted K-theory and loop group representations arXiv:math/0312155; Loop Groups and Twisted K-Theory I (arXiv:0711.1906) ; Loop Groups and Twisted K-Theory II (arXiv:math/0511232)

The result on twisted K-groups has been lifted to an equivalence of categories in

• Daniel Freed, Constantin Teleman, Dirac families for loop groups as matrix factorizations, arxiv/1409.6051

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)

See the references at (infinity,1)-vector bundle for more on this.

Discussion in terms of twisted bundles/bundle gerbe modules is in

• Peter Bouwknegt, Alan Carey, Varghese Mathai, Michael Murray, Danny Stevenson, Twisted K-theory and K-theory of bundle gerbes, Commun Math Phys, 228 (2002) 17-49 (arXiv:hep-th/0106194, doi:10.1007/s002200200646)

but apparently contains a mistake, as pointed out in

• Alan Carey, Bai-Ling Wang, top of p. 10 in Thom isomorphism and Push-forward map in twisted K-theory, Journal of K-Theory 1 2 (2008) 357-393 (arXiv:math/0507414, doi:10.1017/is007011015jkt011)

The generalization of this to groupoid K-theory is in (FHT 07, around p. 26) and

• Jean-Louis Tu, Ping Xu, Camille Laurent-Gengoux, Twisted K-theory of differentiable stacks (arXiv:math/0306138)

(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:

• Fabian Hebestreit, Steffen Sagave, Homotopical and operator algebraic twisted K-theory, Mathematische Annalen 378 (2020) 1021-1059 (arXiv:1904.01872, doi:10.1007/s00208-020-02066-6)

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:

• Jonathan Rosenberg, Twisted cohomology, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2401.03966]

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

• Kiyonori Gomi, Freed-Moore K-theory (arXiv:1705.09134)

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):

• Daniel Grady, Hisham Sati, Ramond-Ramond fields and twisted differential K-theory (arXiv:1903.08843)

Discussion of twisted differential orthogonal K-theory and its relation to D-brane charge in type I string theory (on orientifolds):

• Daniel Grady, Hisham Sati, Twisted differential KO-theory (arXiv:1905.09085)

Higher twists

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]

see also:

• Ulrich Pennig, Equivariant Higher Twisted K-Theory of $SU(n)$ via Exponential Functors, talk at CQTS (20 Sep 2023) [video:YT]

Discussion of the twisted Chern character for higher twisted K-theory:

• Lachlan Macdonald, Varghese Mathai, Hemanth Saratchandran, On the Chern character in Higher Twisted K-theory and spherical T-duality, Commun. Math. Phys. 385, 331-368 (2021) (arXiv:2007.02507)