Scalar Vacuum Densities on Beltrami Pseudosphere

We investigate the combined effects of spatial curvature and topology on the properties of the vacuum state for a charged scalar field localized on the (2 + 1)-dimensional Beltrami pseudosphere, assuming that the field obeys the quasiperiodicity condition with constant phase. As important local characteristics of the vacuum state, the vacuum expectation values (VEVs) of the field squared and energy–momentum tensor are evaluated. The contributions in the VEVs coming from geometry with an uncompactified azimuthal coordinate are divergent, whereas the compact counterparts are finite and are analyzed both numerically and asymptotically. For small values of the proper radius of the compactified dimension, the leading terms of topological contributions are independent of the field mass and curvature coupling parameter, increasing by a power law. In the opposite limit, the VEVs decay following a power law in the general case. In the special case of a conformally coupled massless field, the behavior is different. Unlike the VEV of field squared and vacuum energy density, the radial and azimuthal stresses are increasing by absolute value. As a consequence, the effects of nontrivial topology are strong for the stresses, in this case, at small values of the radial coordinate.