  We evaluate the Hadamard function and the vacuum expectation value (VEV) of
the current density for a charged scalar field, induced by flat boundaries in
spacetimes with an arbitrary number of toroidally compactified spatial
dimensions. The field operator obeys the Robin conditions on the boundaries and
quasiperiodicity conditions with general phases along compact dimensions. In
addition, the presence of a constant gauge field is assumed. The latter induces
Aharonov-Bohm-type effect on the VEVs. There is a region in the space of the
parameters in Robin boundary conditions where the vacuum state becomes
unstable. The stability condition depends on the lengths of compact dimensions
and is less restrictive than that for background with trivial topology. The
vacuum current density is a periodic function of the magnetic flux, enclosed by
compact dimensions, with the period equal to the flux quantum. It is explicitly
decomposed into the boundary-free and boundary-induced contributions. In sharp
contrast to the VEVs of the field squared and the energy-momentum tensor, the
current density does not contain surface divergences. Moreover, for Dirichlet
condition it vanishes on the boundaries. The normal derivative of the current
density on the boundaries vanish for both Dirichlet and Neumann conditions and
is nonzero for general Robin conditions. When the separation between the plates
is smaller than other length scales, the behavior of the current density is
essentially different for non-Neumann and Neumann boundary conditions. In the
former case, the total current density in the region between the plates tends
to zero. For Neumann boundary condition on both plates, the current density is
dominated by the interference part and is inversely proportional to the
separation.
