nLab bundle gerbe module

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

The concept is due to

• Peter Bouwknegt, Alan Carey, Varghese Mathai, Michael Murray, Danny Stevenson, Section 4 of: K-theory of bundle gerbes and twisted K-theory , Commun Math Phys, 228 (2002) 17-49 (arXiv:hep-th/0106194, doi:10.1007/s002200200646)

recognized as equivalent to earlier discussion of twisted bundles in

• Ernesto Lupercio, Bernardo Uribe, Prop. 7.2.2 (from v2 on) in: Gerbes over Orbifolds and Twisted K-theory, Comm. Math. Phys. 245(3): 449-489. (arXiv:math/0105039, doi:10.1007/s00220-003-1035-x)

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:

• Marco Mackaay, A note on the holonomy of connections in twisted bundles, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 44 (2003) no. 1, pp. 39-62. (arXiv:math/0106019, numdam:CTGDC_2003__44_1_39_0)

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)

On bundle gerbe modules with higher G-structures (including String structure, etc.) and the corresponding twisted Chern character:

• Fei Han, Ruizhi Huang, Varghese Mathai, Fractional structures on bundle gerbe modules and fractional classifying spaces [arXiv:2203.14439]