Coset for Hopf fibration and Squashing

TL;DR

This paper derives metrics for geometrical deformations like Hopf fibration, squashing, and Z_k quotient, crucial for AdS4/CFT3 studies, providing a unified approach for complex and quaternion cosets.

Contribution

It offers a simple, general formula for metrics of Hopf fibrations and clarifies how squashing and Z_k quotient affect geometry and topology.

Findings

Derived a general metric formula for Hopf fibrations.

Showed squashing as a similarity transformation preserving isometry.

Described Z_k quotient as a topology-changing lens space.

Abstract

We provide a simple derivation of metrics for fundamental geometrical deformations such as Hopf fibration, squashing and Z_k quotient which play essential roles in recent studies on the AdS4/CFT3. A general metric formula of Hopf fibrations for complex and quaternion cosets is presented. Squashing is given by a similarity transformation which changes the metric preserving the isometry symmetry of the projective space. On the other hand Z_k quotient is given as a lens space which changes the topology preserving the "local" metric.