Compactifications of Deformed Conifolds, Branes and the Geometry of Qubits

TL;DR

This paper constructs new Einstein metrics with positive cosmological constant on various manifolds, explores their geometric properties, and applies K"ahler geometry to quantum mechanics and qubit entanglement.

Contribution

It introduces three families of explicit Einstein metrics generalizing Ricci-flat metrics to positive cosmological constant cases, and connects these geometries to quantum entanglement analysis.

Findings

New Einstein metrics on complex projective spaces, Grassmannians, and product manifolds.

Application of K"ahler geometry to quantum mechanics and qubit entanglement.

Potential relevance of these metrics in string theory compactifications and AdS/CFT.

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-K\"ahler 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 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 $CP^{n+1}$, and we apply the formalism to study the quantum entanglement of qubits.