nLab
bundle gerbe module

Idea

For $A$ an abelian Lie group (often taken to be the circle group $U(1)$), a bundle gerbe on $X$ is a representation of a cocycle $c$ in $H(X,B^{2}A)$.

If a central extension $A\to \hat{G}\to G$ is given (often taken to be $U(1)\to U(n)\to PU(n)$) there is a notion of $\hat{G}$-twisted bundles with twist given by $c$.

A bundle gerbe module is the presentation of such a $\hat{G}$-twisted bundle corresponding to the presentation of the $B^{2}A$-cocycle by a bundle gerbe.

Definition

Definition

If $Y\to X$ is the surjective submersion relative to which the bundle gerbe $c$ is defined, and if

$$L\to Y{\times }_{X}Y$$L \to Y \times_X Y

is the transition line bundle of the bundle gerbe, then a bundle gerbe module for $c$ is a Hermitean vector bundle

$$E\to Y$$E \to Y

equipped with an action

$$\rho :{\pi }_2^{*}E\otimes L\to {\pi }_1^{*}E$$\rho : \pi_2^* E \otimes L \to \pi_1^* E

(where ${\pi }_1,{\pi }_2:Y{\times }_{X}Y\to Y$ are the two projections out of the fiber product)

that respects the bundle gerbe product

$$\mu :{\pi }_{12}^{*}L\otimes {\pi }_{23}^{*}L\to {\pi }_{13}^{*}L$$\mu : \pi_{12}^* L \otimes \pi_{2 3}^* L \to \pi_{1 3}^* L

in the obvious way.

When $Y={\coprod }_i{U}_i$ comes form an an open cover $\{U_i\to X\}$ the above almost manifestly reproduces the explicit description of twisted bundles given there.

References

Bundle gerbe modules were apparently introduced 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)

for modelling twisted K-theory by twisted bundles.