Casimir effect for scalar current densities in topologically nontrivial spaces

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.