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We evaluate the Hadamard function and the vacuum expectation value (VEV) ofthe current density for a charged scalar field in the region between twoco-dimension one branes on the background of locally AdS spacetime with anarbitrary number of toroidally compactified spatial dimensions. Along compactdimensions periodicity conditions are considered with general values of thephases and on the branes Robin boundary conditions are imposed for the fieldoperator. In addition, we assume the presence of a constant gauge field. Thelatter gives rise to Aharonov-Bohm type effect on the vacuum currents. Thereexists a range in the space of the Robin coefficients for separate branes wherethe vacuum state becomes unstable. Compared to the case of the standard AdSbulk, in models with compact dimensions the stability condition imposed on theparameters is less restrictive. The current density has nonzero componentsalong compact dimensions only. These components are decomposed into thebrane-free and brane-induced contributions. The component along a given compactdimension is a periodic function of the gauge field flux, enclosed by thatdimension, with the period of the flux quantum. An important feature, thatdistinguishes the current density from the expectation values of the fieldsquared and energy-momentum tensor, is its finiteness on the branes. Inparticular, for Dirichlet boundary condition the current density vanishes onthe branes. We show that, depending on the constants in the boundaryconditions, the presence of the branes may either increase or decrease thecurrent density compared with that for the brane-free geometry. Applicationsare given to the Randall--Sundrum 2-brane model with extra compact dimensions.

Hadamard function and the vacuum currents in braneworlds with compact dimensions: Two-branes geometry