cohomology

# Contents

## Idea

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

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

A bundle gerbe module is the presentation of such a $\stackrel{^}{G}$-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

$L\to Y{×}_{X}Y$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 ${\pi }_{1},{\pi }_{2}:Y{×}_{X}Y\to Y$ 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 $\left\{{U}_{i}\to X\right\}$ the above almost manifestly reproduces the explicit description of twisted bundles given there.

## References

Bundle gerbe modules were apparently introduced in

for modelling twisted K-theory by twisted bundles.

Revised on April 18, 2011 14:23:28 by Urs Schreiber (89.204.137.106)