On G/H geometry and its use in M-theory compactifications

The Riemannian geometry of cost spaces is reviewed, with emphasis on its applications to supergravity and M-theory compactifications. Formulae for the connection and curvature of rescaled coset manifolds are generalized to the case of nondiagonal Killing metrics. The example of the N-010 spaces is discussed in detail. These are a subclass of the coset manifolds N-pqr = G H = SU(3) x U(1) U(1) x U(1), the integers p,q,r characterizing the embedding of H in G. We study the realization of N-010 as G H = SU(3) x SU(2) U(1) x SU(2) (with diagonal embedding of the SU(2)is an element ofH into G). For a particular G-symmetric rescaling there exist three Killing spinors, implying N = 3 supersymmetry in the AdS(4) x N-010 compactitications of D = 11 supergravity. This rescaled N-010 space is of particular interest for the AdS(4) CFT3 correspondence, and its SU(3) x SU(2) isometric realization is essential for the OSp(4/3) classification of the Kaluza Klein modes. (C) 2001 Academic Press.