group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
If $Y \to X$ is the surjective submersion relative to which the bundle gerbe $c$ is defined, and if
is the transition line bundle of the bundle gerbe, then a bundle gerbe module for $c$ is a Hermitean vector bundle
equipped with an action
(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
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.
The concept is due to
recognized as equivalent to earlier discussion of twisted bundles in
both motivated by modelling twisted K-theory in terms of Grothendieck groups of twisted bundles.
For more see also the references at twisted vector bundle.
The Cech cocycle-incarnation of bundle gerbe modules (effectively due to Lupercio & Uribe 2001, Def. 7.2.1) was then also considered in:
A splitting principle for bundle gerbe modules is discussed in
On bundle gerbe modules with higher G-structures (including String structure, etc.) and the corresponding twisted Chern character:
Last revised on February 5, 2023 at 10:51:09. See the history of this page for a list of all contributions to it.