nLab
twisted K-theory

Idea

Twisted K-theory is a twisted cohomology version of K-theory, where the twist is given by a cocycle in degree 3 ordinary cohomology.

Definition

By sections of associated $KU$-bundles

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

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

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

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

The ordinary topological K-theory of a topological space $X$ is

$$K(X{)}_{•}\simeq [X,(KU{)}_{•}].$$K(X)_\bullet \simeq [X, (K U)_\bullet] \,.

The projective unitary group $PU(\mathscr{H})$ (a topological group) acts canonically by automorphisms on $(KU{)}_0$. Therefore for $P\to X$ any $\mathrm{PU}(\mathscr{H})$-principal bundle, we can form the associated bundle $P{\times }_{PU(\mathscr{H})}(KU{)}_0$.

Since the homotopy type of $PU(\mathscr{H})$ 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

$$K_{\alpha }(X{)}^0:={\Gamma }_X(P^{\alpha }{\times }_{PU(\mathscr{H})}(KU{)}_0).$$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

$${\tilde{K}}_{\alpha }(X{)}^0:={\Gamma }_X(P^{\alpha }{\times }_{PU(\mathscr{H})}BU).$$\tilde K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} B U) \,.

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 $P\in H^3(X,B^{2}U(1))$ be any cocycle representative, which we may think of either as giving a circle 2-bundle or a bundle gerbe.

Write $\mathrm{TwBund}(X,P)$ for the groupoid of twisted bundles on $X$ with twist given by $P$. Then let

$${\tilde{K}}_{\alpha }(X):=\mathrm{TwBund}(X,P)$$\tilde K_\alpha(X) := TwBund(X,P)

be the set of isomorphism classes of twisted bundles. Call this the twisted K-theory of $X$ 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. 1.

This is (CBMMS, prop. 6.4, prop. 7.3).

Other constructions

Let $\mathrm{Vectr}$ 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

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

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

Let $B_{\otimes }\mathrm{Vectr}$ be the delooping of $\mathrm{Vectr}$ with respect to the tensor product monoidal structure (not the additive structure).

Then we have a fibration sequence

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

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

The entire morphism above deloops

$$B\rho :B^{2}U(1)\to B_{\otimes }\mathrm{Vect}\hookrightarrow B_{\otimes }\mathrm{Vectr}$$\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 $BU(1)$.

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

$$\begin{array}{ccc}{H}^c(X,\mathrm{Vectr})& \to & *\\ \downarrow & & {\downarrow }^{B\rho \circ c}\\ *& \to & H(X,B_{\otimes }\mathrm{Vectr})\end{array}.$$\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 $H^c(X,\mathrm{Vectr})$ yield twisted K-theory.

Related concepts

• K-theory

• topological K-theory

• twisted K-theory

• differential K-theory

• twisted differential K-theory

References

The perspective of twisted K-theory by sections of a $KU$-bundle of spectra is discussed for instance in section 22 of

• May, Sigurdsson, Parametrized homotopy theory (pdf) AMS Lecture notes 132

Discussion in terms of twisted bundles (bundle gerbe modules) is in

• Alan Carey, Peter Bouwknegt, Varghese Mathai, Michael Murray and Danny Stevenson, K-theory of bundle gerbes and twisted K-theory , Commun Math Phys, 228 (2002) 17-49 (arXiv)

• Max Karoubi, Twisted bundles and twisted K-theory, arxiv/1012.2512

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 $Kℝ$-theory relevant for orientifolds are discussed in

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