nLab
bundle gerbe module

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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.

Definition

Definition

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.

References

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.

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

  • 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 July 30, 2021 at 10:26:48. See the history of this page for a list of all contributions to it.