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Abstract

We study bundle gerbes on manifolds $M$ that carry an action of a connected
Lie group $G$. We show that these data give rise to a smooth 2-group extension
of $G$ by the smooth 2-group of hermitean line bundles on $M$. This 2-group
extension classifies equivariant structures on the bundle gerbe, and its
non-triviality poses an obstruction to the existence of equivariant structures.
We present a new global approach to the parallel transport of a bundle gerbe
with connection, and use it to give an alternative construction of this smooth
2-group extension in terms of a homotopy-coherent version of the associated
bundle construction. We apply our results to give new descriptions of
nonassociative magnetic translations in quantum mechanics and the
Faddeev-Mickelsson-Shatashvili anomaly in quantum field theory. We also propose
a definition of smooth string 2-group models within our geometric framework.
Starting from a basic gerbe on a compact simply-connected Lie group $G$, we
prove that the smooth 2-group extensions of $G$ arising from our construction
provide new models for the string group of $G$.