cohomology

# Contents

## 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 $\mathbf{H}(X,\mathbf{B}^2 A)$.

If a central extension $A \to \hat G \to G$ is given (often taken to be $U(1) \to U(n) \to P U (n)$) 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 $\mathbf{B}^2 A$-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

$L \to Y \times_X Y$

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

$E \to Y$

equipped with an action

$\rho : \pi_2^* E \otimes L \to \pi_1^* E$

(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

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

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

for modelling twisted K-theory by twisted bundles.

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