Induced current by a cosmic string and a brane in high-dimensional AdS spacetime

In this paper we investigate the bosonic current induced by a brane and a magnetic flux running along the idealized cosmic string in a (D+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(D+1)$$\end{document}-dimensional anti-de Sitter (AdS) background. We consider the brane is parallel to the AdS boundary and the cosmic string is orthogonal to them. Moreover, we assume that on the brane the charged bosonic field obeys the Robin boundary condition. The brane divides the space into two regions with different properties of the vacuum state. We show that the only nonzero component of the current density is along the azimuthal direction in both regions. In order to develop this analysis we calculate, for both regions, the positive frequency Wightman functions. Both functions present a part associated with the AdS in presence of a cosmic string only, and the other part induced by the brane. In this paper we consider only the contributions induced by the brane. We show that in both regions the azimuthal current densities are odd functions of the magnetic flux along the string. Different analytic and numerical analysis are performed and an application of our results is provided for the Randall-Sundrum braneworld model with a single brane.