  The vacuum expectation value of the fermionic current is evaluated for a
massive spinor field in spacetimes with an arbitrary number of toroidally
compactified spatial dimensions in presence of a constant gauge field. By using
the Abel-Plana type summation formula and the zeta function technique we
present the fermionic current in two different forms. Non-trivial topology of
the background spacetime leads to the Aharonov-Bohm effect on the fermionic
current induced by the gauge field. The current is a periodic function of the
magnetic flux with the period equal to the flux quantum. In the absence of the
gauge field it vanishes for special cases of untwisted and twisted fields.
Applications of the general formulae to Kaluz-Klein type models and to
cylindrical and toroidal carbon nanotubes are given. In the absence of magnetic
flux the total fermionic current in carbon nanotubes vanishes, due to the
cancellation of contributions from two different sublattices of the graphene
hexagonal lattice.
