Vacuum currents in elliptic pseudosphere tubes

We examine the effects of spatial topology, curvature, and magnetic flux on the vacuum expectation value (VEV) of the current density for a charged scalar field in (2+1)-dimensional spacetime. The elliptic pseudosphere is considered as an exactly solvable background geometry. The topological contribution is separated in the Hadamard function for general phases in the periodicity condition along the compact dimension. Two equivalent expressions are provided for the component of the current density in that direction. The corresponding VEV is a periodic function of the magnetic flux with a period equal to the flux quantum. In the flat spacetime limit, we recover the result for a conical space with a general value of the planar angle deficit. Near the origin of the elliptic pseudosphere, the effect of the spatial curvature on the vacuum current density is weak. The same applies for small values of the length of the compact dimension. Using the conformal relations between the elliptic pseudosphere and the (2+1)-dimensional de Sitter spacetime with a planar angle deficit, we determine the current densities for a conformally coupled massless scalar field in the static and hyperbolic vacuum states of locally de Sitter spacetime.