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 \( \mathbb{C}{\mathrm{\mathbb{P}}}^{n+1} \), written in a Stenzel form, whose principal orbits are the Stiefel manifolds \( {V}_2\left({\mathbb{R}}^{n+2}\right)=\mathrm{SO}\left(n+2\right)/\mathrm{SO}(n) \) divided by \( {\mathbb{Z}}_2 \). The second family are also Einstein-Kähler metrics, now on the Grassmannian manifolds \( {G}_2\left({\mathbb{R}}^{n+3}\right)=\mathrm{SO}\left(n+3\right)/\left(\left(\mathrm{SO}\left(n+1\right)\times \mathrm{SO}(2)\right)\right. \), whose principal orbits are the Stiefel manifolds \( {V}_2\left({\mathbb{R}}^{n+2}\right) \) (with no \( {\mathbb{Z}}_2 \) factoring in this case). The third family are Einstein metrics on the product manifolds S n+1 × 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 \( \mathbb{C}{\mathrm{\mathbb{P}}}^{n+1} \), and we apply the formalism to study the quantum entanglement of qubits.
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Cvetič, M., Gibbons, G.W. & Pope, C.N. Compactifications of deformed conifolds, branes and the geometry of qubits. J. High Energ. Phys. 2016, 135 (2016). https://doi.org/10.1007/JHEP01(2016)135
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DOI: https://doi.org/10.1007/JHEP01(2016)135