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.
Highlights
This paper is motivated by the following problem from physics: In [BMS19] we showed how a bundle gerbe with connection on Rdt on (Rd) gives rise to a 3-cocycle on the translation group Rdt of Rd
We study bundle gerbes on manifolds M that carry an action of a connected Lie group G
Starting from a basic gerbe on a compact -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
Summary
This paper is motivated by the following problem from physics: In [BMS19] we showed how a bundle gerbe with connection on Rd gives rise to a 3-cocycle on the translation group Rdt of Rd. In the case M = Rd , where G = Rdt is the translation group of Rd , and where G = IB is a trivial gerbe on Rd with a connection B ∈ 2(Rd ) corresponding to a magnetic field, we show that the extension SymRdt (I) −→ Rdt reproduces the 3-cocycles we obtained in [BMS19] We achieve this by choosing a certain global section of the path fibration of Rdt and implicitly pass through DesL in the computation. We show that the parallel transport we defined implements nonassociative magnetic translations on the sections of the gerbe, whereas the 2-group extension SymRdt (I) −→ Rdt allows us to understand the algebraic structure of nonassociative magnetic translations even without making any reference to sections The latter is useful in cases where there is no good notion of sections, such as when the Dixmier–Douady class of G is non-torsion. We defer some technical results on categories fibred in groupoids and on principal 2-bundles to Appendix A
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