Geometry of the Aharonov-Bohm Effect

Abstract

We show that the connection responsible for any abelian or non abelian Aharonov-Bohm effect with $n$ parallel ``magnetic'' flux lines in $\R^3$, lies in a trivial $G$-principal bundle $P\to M$, i.e. $P$ is isomorphic to the product $M\times G$, where $G$ is any path connected topological group; in particular a connected Lie group. We also show that two other bundles are involved: the universal covering space $\tilde{M}\to M$, where path integrals are computed, and the associated bundle $P\times_G \C^m \to M$, where the wave function and its covariant derivative are sections.