Branes at conical singularities and holography

For supergravity solutions which are the product of an anti-de Sitter space with an Einstein space X, we study the relation between the amount of supersymmetry preserved and the geometry of X. Depending on the dimension and the amount of supersymmetry, the following geometries for X are possible, in addition to the maximally supersymmetric spherical geometry: Einstein-Sasaki in dimension 2fc-hl, 3-Sasaki in dimension 4fc+3, 7-dimensional manifolds of weak G2 holonomy and 6-dimensional nearly Kahler manifolds.Many new examples of such manifolds are presented which are not homogeneous and hence have escaped earlier classification efforts.String or M theory in these vacua are conjectured to be dual to superconformal field theories.The brane solutions interpolating between these anti-de Sitter near-horizon geometries and the product of Minkowski space with a cone over X lead to an interpretation of the dual superconformal field theory as the world-volume theory for branes at a conical singularity (cone branes).We propose a description of those field theories whose associated cones are obtained by (hyper-)Kahler quotients.