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 $CP^{n+1}$, written in a Stenzel form, whose principal orbits are the Stiefel manifolds $V_2(R ^{n+2})=SO(n+2)/SO(n)$ divided by $Z_2$. The second family are also Einstein-Kähler metrics, now on the Grassmannian manifolds $G_2(R^{n+3})=SO(n+3)/((SO(n+1)\times SO(2))$, whose principal orbits are the Stiefel manifolds $V_2(R^{n+2})$ (with no $Z_2$ factoring in this case). The third family are Einstein metrics on the product manifolds $S^{n+1}\times S^{n+1}$, and are Kähler 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 Kähler geometry of Fubini-Study metrics on $CP^{n+1}$, and we apply the formalism to study the quantum entanglement of qubits.