nLab
twisted bundle

Idea

A twisted principal-bundle is the object classified by a cocycle in twisted cohomology the way an ordinary principal bundle is the object classified by a cocycle in plain cohomology (generally in nonabelian cohomology).

For $\hat{G}$ a group, a $\hat{G}$-principal bundle is classified in degree 1 nonabelian cohomology with coefficients in the delooped groupoid $B\hat{G}$.

Given a realization of $\hat{G}$ as an abelian extension

$$A\to \hat{G}\to G$$A \to \hat G \to G

of groups, i.e. given a fibration sequence

$$BA\to B\hat{G}\to BG$$\mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G

of groupoids such that $BA$ is once deloopable so that the fibration sequence continues to the right at least one step as

$$B\hat{G}\to BG\to B^{2}A$$\mathbf{B}\hat G \to \mathbf{B}G \to \mathbf{B}^2 A

the general nonsense of twisted cohomology induces a notion of twisted $\hat{G}$-cohomology. The fibrations classified by this are the twisted $\hat{G}$-bundles.

Definition

For every $B^{2}A$-cocycle $c\in H(X,B^{2}A)$ the $c$-twisted $\hat{G}$-cohomology $H^c(X,G\hat{G})$ is the homotopy pullback

$$\begin{array}{ccc}{H}^c(X,B\hat{G})& \to & *\\ \downarrow & & {\downarrow }^c\\ H(X,BG)& \stackrel{}{\to }& H(X,B^{2}A)\end{array}.$$\array{ \mathbf{H}^c(X,\mathbf{B}\hat G) &\to& {*} \\ \downarrow && \downarrow^{c} \\ \mathbf{H}(X,\mathbf{B}G) &\stackrel{}{\to}& \mathbf{H}(X,\mathbf{B}^2 A) } \,.

The cocycles in $H^c(X,B\hat{G})$ are called the (representatives of) $c$-twisted $\hat{G}$-principal bundles.

Details

One may unwrap this abstract definition to obtain a concrete cocycle formula for twisted bundles.

For that purpose the above homotopy pullback is conveniently computed as an ordinary pullback after making the fibrant replacement

$$\begin{array}{ccccc}H(X,B(A\to \hat{G}))& \to & H(X,B^{2}A)& \leftarrow & *\\ {\downarrow }^{\simeq }& & {\downarrow }^{\mathrm{Id}}& & {\downarrow }^{\mathrm{Id}}\\ H(X,BG)& \to & H(X,B^{2}A)& \leftarrow & *\end{array},$$\array{ \mathbf{H}(X,\mathbf{B}(A \to \hat G)) &\to & \mathbf{H}(X,\mathbf{B}^2 A) &\leftarrow& {*} \\ \downarrow^{\simeq} && \downarrow^{Id} && \downarrow^{Id} \\ \mathbf{H}(X,\mathbf{B}G) &\to & \mathbf{H}(X,\mathbf{B}^2 A) &\leftarrow& {*} } \,,

where $(A\to \hat{G})$ denotes the 2-group corresponding to the crossed module obtained from the central extension $\hat{G}$.

Similarly identifying $B^{2}A=B(A\to 1)$ as the delooping of the 2-group coming from the crossed module $(A\to 1)$ the morphism $H(X,B(A\to G))\to H(X,B^{2}A)$ is now induced from the obvious projection of crossed modules

$$(A\to G)\to (A\to 1).$$(A \to G) \to (A \to 1) \,.

This then says that $c$-twisted $B\hat{G}$-cocycles are precisely those $B(A\to \hat{G})$-cocycles whose projection to an $B(A\to 1)$-cocycle is the prescribed twisting cocycle $c$.

We may consider a Čech cocycle relative to a cover $\{U_i\to X\}$, i.e. an anafunctor

$$\begin{array}{ccc}C(U)& \stackrel{{\hat{g}}_{\mathrm{tw}}}{\to }& B(A\to \hat{G})\\ \downarrow \\ X\end{array}$$\array{ C(U) &\stackrel{\hat g_{tw}}{\to}& \mathbf{B}(A \to \hat G) \\ \downarrow \\ X }

out of a Čech nerve $C(U)$ and notice that the functor ${\hat{g}}_{\mathrm{tw}}$ is an assignment

$${\hat{g}}_{\mathrm{tw}}:\begin{array}{ccc}& & (x,j)\\ & \nearrow & \Downarrow & \searrow \\ (x,i)& & \to & & (x,k)\end{array}\mapsto \begin{array}{ccc}& & •\\ & {}^{{\hat{g}}_{ij}(x)}\nearrow & {\Downarrow }^{c_{ijk}(x)}& {\searrow }^{{\hat{g}}_{jk}(x)}\\ •& & \stackrel{{\hat{g}}_{ik}(x)}{\to }& & •\end{array}$$\hat g_{tw} \;\; : \;\; \array{ && (x,j) \\ & \nearrow &\Downarrow& \searrow \\ (x,i) &&\to&& (x,k) } \;\;\;\; \mapsto \;\;\;\; \array{ && \bullet \\ & {}^{\hat g_{i j}(x)}\nearrow &\Downarrow^{c_{i j k}(x)}& \searrow^{\hat g_{j k}(x)} \\ \bullet &&\stackrel{\hat g_{i k}(x)}{\to}&& \bullet }

As already indicated by the notation, the further projection

$$C(U)\stackrel{{\hat{g}}_{\mathrm{tw}}}{\to }B(A\to \hat{G})\to B^{2}A$$C(U) \stackrel{\hat g_{tw}}{\to} \mathbf{B}(A \to \hat G) \to \mathbf{B}^2 A

is constrained to be the cocycle $c$. Using the rules for crossed modules one reads off that the existence of the triangular cell on the right in $B(A\to \hat{G})$ is equivalent to the equation

$${\hat{g}}_{ij}(x){\hat{g}}_{jk}(x)={\hat{g}}_{ik}(x)c_{ijk}(x).$$\hat g_{i j}(x) \hat g_{j k}(x) = \hat g_{i k}(x) c_{i j k}(x) \,.

In this cocycle equation form twisted bundles traditionally appear in the literature. Alternatively, allowing a general surjective submersion instead of an open cover, this yields the description of twisted bundles as prefered in the literature on bundle gerbes, where they are called bundle gerbe modules: the $B^{2}A$-cocycle $c$ represents the $A$-bundle gerbe.

Examples

twisted K-theory

Just as vector bundles model cocycles in K-theory, twisted vector bundles model cocycles in twisted K-theory.

For twists $c$ that are torsion class (i.e. have finite order as group elements in the cohomology group $H(X,B^{2}A)$ ) this was realized in

• P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray, D. Stevenson, Twisted K-theory and K-theory of bundle gerbes (arXiv)

which also, apparently, is the source where gerbe modules as such were first introduced.

The generalization of this construction to non-torsion twists requires using vectorial bundles instead of plain vector bundles. Full twisted K-theory in terms of twisted vectorial bundles was realized in

• Kiyonori Gomi, Twisted K-theory and finite-dimensional approximation (arXiv)

There the twisted cocycle equation discussed above appears on the bottom of page 7.