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This paper reviews the investigations on the vacuum expectation value of thecurrent density for a charged scalar field in spacetimes with toroidallycompactified spatial dimensions. As background geometries locally Minkowskian(LM), locally de Sitter (LdS) and locally anti-de Sitter (LAdS) spacetimes areconsidered. Along compact dimensions quasiperiodicity conditions are imposed onthe field operator and the presence of a constant gauge field is assumed. Thevacuum current has non-zero components only along compact dimensions. Thosecomponents are periodic functions of the magnetic flux enclosed by compactdimensions with the period equal to the flux quantum. For LdS and LAdSgeometries and for small values of the length of a compact dimension, comparedwith the curvature radius, the leading term in the expansion of the the vacuumcurrent along that dimension coincides with that for LM bulk. In this limit thedominant contribution to the mode sum for the current density comes from thevacuum fluctuations with wavelength smaller than the curvature radius and theinfluence of the gravitational field is weak. The effects of the gravitationalfield are essential for lengths of compact dimensions larger than the curvatureradius. In particular, instead of the exponential suppression of the currentdensity in LM bulk one can have power law decay in LdS and LAdS spacetimes.

Vacuum currents for a scalar field in models with compact dimensions