Geometries with Killing Spinors and Supersymmetric <Emphasis Type="Italic">AdS</Emphasis> Solutions

The seven and nine dimensional geometries associated with certain classes of supersymmetric AdS ₃ and AdS ₂ solutions of type IIB and D = 11 supergravity, respectively, have many similarities with Sasaki-Einstein geometry. We further elucidate their properties and also generalise them to higher odd dimensions by introducing a new class of complex geometries in 2n + 2 dimensions, specified by a Riemannian metric, a scalar field and a closed three-form, which admit a particular kind of Killing spinor. In particular, for n ≥ 3, we show that when the geometry in 2n + 2 dimensions is a cone we obtain a class of geometries in 2n + 1 dimensions, specified by a Riemannian metric, a scalar field and a closed two-form, which includes the seven and nine-dimensional geometries mentioned above when n = 3, 4, respectively. We also consider various ansätze for the geometries and construct infinite classes of explicit examples for all n.