  Vacuum expectation values (VEVs) of the current densities for charged scalar
and Dirac spinor fields are investigated in (D+1)-dimensional de Sitter (dS)
spacetime with toroidally compactified spatial dimensions. Along compact
dimensions we impose quasiperiodicity conditions with arbitrary phases. In
addition, the presence of a classical constant gauge field is assumed. The VEVs
of the charge density and of the components for the current density along
noncompact dimensions vanish. The gauge field leads to Aharonov-Bohm-like
oscillations of the components along compact dimensions as functions of the
magnetic flux. For small values of the comoving length of a compact dimension,
compared with the dS curvature scale, the current density is related to the
corresponding current in the Minkowski spacetime by a conformal relation. For
large values of the comoving length and for a scalar field, depending on the
mass of the field, two different regimes are realized with the monotonic and
oscillatory damping of the current density. For a massive spinor field, the
decay of the current density is always oscillatory. In supersymmetric models on
the background of Minkowski spacetime with equal number of scalar and fermionic
degrees of freedom and with the same phases in the periodicity conditions, the
total current density vanishes due to the cancellation between the scalar and
fermionic parts. The background gravitational field modifies the current
densities for scalar and fermionic fields in different ways and for massive
fields there is no cancellation in the dS spacetime.
