Induced fermionic current in toroidally compactified spacetimes with applications to cylindrical and toroidal nanotubes

The vacuum expectation value of fermionic current is evaluated for a massive spinor field in spacetimes with an arbitrary number of toroidally compactified spatial dimensions in the 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. Nontrivial topology of the background spacetime leads to the Aharonov-Bohm effect for 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 gauge field it vanishes for special cases of untwisted and twisted fields. Applications of general formulas to Kaluza-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 hexagonal lattice of graphene.