Supersymmetry of AdS and flat backgrounds in M-theory

We give a systematic description of all warped AdSn and ℝn−1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb{R}}^{n-1,1} $$\end{document} backgrounds of M-theory and identify the a priori number of supersymmetries that these backgrounds preserve. In particular, we show that AdSn backgrounds preserve N=2n2k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N={2}^{\left[\frac{n}{2}\right]}k $$\end{document} for n ≤ 4 and N=2n2+1k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N={2}^{\left[\frac{n}{2}\right]+1}k $$\end{document} for 4 < n ≤ 7 supersymmetries while ℝn−1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb{R}}^{n-1,1} $$\end{document} backgrounds preserve N=2n2k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N={2}^{\left[\frac{n}{2}\right]}k $$\end{document} for n ≤ 4 and N=2n+12k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N={2}^{\left[{\scriptscriptstyle \frac{n+1}{2}}\right]}k $$\end{document} for 4 < n ≤ 7, supersymmetries. Furthermore for AdSn backgrounds that satisfy the requirements for the maximum principle to hold, we show that the Killing spinors can be identified with the zero modes of Dirac-like operators on M11−n coupled to fluxes thus establishing a new class of Lichnerowicz type theorems. We also demonstrate that the Killing spinors of generic warped AdSn backgrounds do not factorize into products of Killing spinors on AdSn and Killing spinors on the transverse space.