Vacuum polarization in high-dimensional AdS space-time in the presence of a cosmic string and a compactified extra dimension

In this paper, we analyze the vacuum expectation values (VEVs) of the field squared and the energy–momentum tensor associated with a charged and massive scalar quantum field in a generalized (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 space in the presence of a cosmic string, admitting a magnetic flux running along the string’s core. In addition, we admit that an extra coordinate is compactified to a circle and presents an extra magnetic flux running along its center. This compactification is implemented by imposing a quasiperiodic condition on the field with an arbitrary phase. The calculation of the VEVs of the field squared and the energy–momentum tensor are developed using the positive-energy Wightman function. The latter is constructed by the mode sum over the complete set of normalized bosonic wave functions. Due to the compactification, the Wightman function is presented as the sum of two distinct contributions. The first one corresponding to the idealized cosmic string, and the second is induced by the compactification. The latter goes to zero for an infinite length of the compactification. As a consequence of the general structure of the Wightman function, both VEVs present also this decomposition. Moreover, due to the Aharonov–Bohm type of interaction between the field with the magnetic fluxes, the VEVs depend on the fractional part of the ration between the total flux and the quantum one.