bundle gerbe module





Special and general types

Special notions


Extra structure





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

If a central extension AG^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.



If YXY \to X is the surjective submersion relative to which the bundle gerbe cc is defined, and if

LY× 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

EY E \to Y

equipped with an action

ρ:π 2 *ELπ 1 *E \rho : \pi_2^* E \otimes L \to \pi_1^* E

(where π 1,π 2:Y× XYY\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 *Lπ 23 *Lπ 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 iX}\{U_i \to X\} the above almost manifestly reproduces the explicit description of twisted bundles given there.


The concept of bundle gerbe modules was introduced in

for modelling twisted K-theory by twisted bundles.

A splitting principle for bundle gerbe modules is discussed in

  • Atsushi Tomoda, On the splitting principle of bundle gerbe modules, Osaka J. Math. Volume 44, Number 1 (2007), 231-246. (Euclid, talk slides pdf)

For more see at twisted vector bundle.

Last revised on February 20, 2018 at 10:05:19. See the history of this page for a list of all contributions to it.