Semiclassical analysis of M2-brane in AdS4×S7/Zk

We start from the classical action describing a single M2-braneon \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {\text{Ad}}{{\text{S}}_4} \times {{{{{\text{S}}^7}}} \left/ {{{\mathbb{Z}_k}}} \right.} $\end{document} and consider semiclassical fluctuaitions around a static, 1/2 BPS configuration whose shape is AdS2×S1. The internal manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {{{{{\text{S}}^7}}} \left/ {{{\mathbb{Z}_k}}} \right.} $\end{document} is described as a U(1) fibration over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathbb{C}{{\text{P}}^3} $\end{document} and the static configuration is wrapped on the U(1) fiber. Then the configuration is reduced to an AdS2 world-sheet of type IIA string on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {\text{Ad}}{{\text{S}}_4} \times \mathbb{C}{{\text{P}}^3} $\end{document} through the Kaluza-Klein reduction on S1. It is shown that the fluctuations form an infinite set of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} = 1 $\end{document} supermultiplets on AdS2 for k = 1, 2. The set is invariant under SO(8) which may be consistent with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} = 8 $\end{document} supersymmetry on AdS2. We discuss the behavior of the fluctuations around the boundary of AdS2 and its relation to deformations of Wilson loop operator.