We show that the Aharonov-Bohm effect finds a natural description in the
setting of QFT on curved spacetimes in terms of superselection sectors of local
observables. The extension of the analysis of superselection sectors from
Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the
presence of a new quantum number labeling charged superselection sectors. In
the present paper we show that this "topological" quantum number amounts to the
presence of a background flat potential which rules the behaviour of charges
when transported along paths as in the Aharonov-Bohm effect. To confirm these
abstract results we quantize the Dirac field in presence of a background flat
potential and show that the Aharonov-Bohm phase gives an irreducible
representation of the fundamental group of the spacetime labeling the charged
sectors of the Dirac field. We also show that non-Abelian generalizations of
this effect are possible only on space-times with a non-Abelian fundamental
group
