Supercoset CFT’s for String Theories on Non-compact Special Holonomy Manifolds

It is known that an arbitrary (m − 1)-dimensional Einstein space Xm−1 possesses a Ricci flat metric on its m-dimensional cone C(Xm−1) of the form (1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ds^{2} = dr^{2} + r^{2}ds^{2}_{X_{m-1}}$$\end{document}, where r is the radial coordinate, and the special holonomies on C(Xm − 1) originate from the “weak special holonomies” on Xm − 1. To be more precise, the SU(n), Sp(n), G2 and Spin(7) holonomies on the cone C(Xm − 1) are in one to one correspondence with the Sasaki-Einstein (m = 2n) , tri-Sasakian (m = 4n) , nearly Kähler (m = 7) and weak G2 (m = 8) structures on Xm − 1, respectively. This fact is very useful to systematically construct special holonomy manifolds with conical singularities, because the Einstein homogeneous spaces Xm − 1 = G/H endowed with these geometrical structures are well known from the old days of Kaluza-Klein supergravity (SUGRA).