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Induced fermionic currents in de Sitter spacetime in the presence of a compactified cosmic string

We investigate the vacuum fermionic currents in the geometry of a compactified cosmic string on background of de Sitter spacetime. The currents are induced by magnetic fluxes running along the cosmic string and enclosed by the compact dimension. We show that the vacuum charge and the radial component of the current density vanish. By using the Abel-Plana summation formula, the azimuthal and axial currents are explicitly decomposed into two parts: the first one corresponds to the geometry of a straight cosmic string and the second one is induced by the compactification of the string along its axis. For the axial current the first part vanishes and the corresponding topological part is an even periodic function of the magnetic flux along the string axis and an odd periodic function of the flux enclosed by the compact dimension with the periods equal to the flux quantum. The azimuthal current density is an odd periodic function of the flux along the string axis and an even periodic function of the flux enclosed by the compact dimension with the same period. Depending on the magnetic fluxes, the planar angle deficit can either enhance or reduce the azimuthal and axial currents. The influence of the background gravitational field on the vacuum currents is crucial at distances from the string larger than the de Sitter curvature radius. In particular, for the geometry of a straight cosmic string and for a massive fermionic field, we show that the decay of the azimuthal current density is damping oscillatory with the amplitude inversely proportional to the fourth power of the distance from the string. This behavior is in clear contrast with the case of the string in Minkowski bulk where the current density is exponentially suppressed at large distances.