bundle gerbe module in nLab
If a central extension A→G^→GA \to \hat G \to G is given (often taken to be U(1)→U(n)→PU(n)U(1) \to U(n) \to P U (n)) there is a notion of G^\hat G-twisted bundles with twist given by cc.
A bundle gerbe module is the presentation of such a G^\hat G-twisted bundle corresponding to the presentation of the B 2A\mathbf{B}^2 A-cocycle by a bundle gerbe.
Definition
If Y→XY \to X is the surjective submersion relative to which the bundle gerbe cc is defined, and if
L→Y× XY L \to Y \times_X Y
is the transition line bundle of the bundle gerbe, then a bundle gerbe module for cc is a Hermitean vector bundle
E→Y E \to Y
equipped with an action
ρ:π 2 *E⊗L→π 1 *E \rho : \pi_2^* E \otimes L \to \pi_1^* E
(where π 1,π 2:Y× XY→Y\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
μ:π 12 *L⊗π 23 *L→π 13 *L \mu : \pi_{12}^* L \otimes \pi_{2 3}^* L \to \pi_{1 3}^* L
in the obvious way.
When Y=∐ iU iY = \coprod_i U_i comes form an an open cover {U i→X}\{U_i \to X\} the above almost manifestly reproduces the explicit description of twisted bundles given there.