Compactifications of Deformed Conifolds, Branes and the Geometry of Qubits

Abstract

We present three families of exact, cohomogeneity-one Einstein metrics in (2n+2) dimensions, which are generalizations of the Stenzel construction of Ricci-flat metrics to those with a positive cosmological constant. The first family of solutions are Fubini-Study metrics on the complex projective spaces CPn+1, written in a Stenzel form, whose principal orbits are the Stiefel manifolds V2(R n+2)=SO(n+2)/SO(n) divided by Z2. The second family are also Einstein-K\"ahler metrics, now on the Grassmannian manifolds G2(Rn+3)=SO(n+3)/((SO(n+1)× SO(2)), whose principal orbits are the Stiefel manifolds V2(Rn+2) (with no Z2 factoring in this case). The third family are Einstein metrics on the product manifolds Sn+1× Sn+1, and are K\"ahler only for n=1. Some of these metrics are believed to play a role in studies of consistent string theory compactifications and in the context of the AdS/CFT correspondence. We also elaborate on the geometric approach to quantum mechanics based on the K\"ahler geometry of Fubini-Study metrics on CPn+1, and we apply the formalism to study the quantum entanglement of qubits.