On G/H Geometry and its use in m-theory compactifications

The Riemannian geometry of coset 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 N⁰¹⁰ spaces is discussed in detail. These are a subclass of the coset manifolds N^pqr=G/H=SU(3)×U(1)/U(1)×U(1), the integers p, q, r characterizing the embedding of H in G. We study the realization of N⁰¹⁰ as G/H=SU(3)×SU(2)/U(1)×SU(2) (with diagonal embedding of the SU(2)∈H into G). For a particular G-symmetric rescaling there exist three Killing spinors, implying N=3 supersymmetry in the AdS₄×N⁰¹⁰ compactification of D=11 supergravity. This rescaled N⁰¹⁰ space is of particular interest for the AdS₄/CFT₃ correspondence, and its SU(3)×SU(2) isometric realization is essential for the OSp(43) classification of the Kaluza-Klein modes.