A Physicist's Visit to Exotic Spheres

TL;DR

This thesis explores the geometry of exotic 7-spheres using analytical, topological, and computational physics tools, including a novel machine learning algorithm for Einstein metrics.

Contribution

It introduces a new geometric formalism for exotic spheres, derives explicit metrics on the Gromoll-Meyer sphere, and develops a machine learning approach for Einstein metrics.

Findings

Derived a family of Riemannian metrics on the Gromoll-Meyer sphere.

Identified the metric with maximal isometry and studied its curvature properties.

Presented a machine learning algorithm for finding Einstein metrics on manifolds.

Abstract

This thesis discusses exotic 7-spheres, i.e. manifolds that are homeomorphic but not diffeomorphic to the ordinary 7-sphere, using a set of analytical and computational tools from theoretical physics. The theory of fibre bundles and instantons, together with their relation to Yang-Mills theory, are reviewed, before presenting a generalisation of self-duality to twisted self-duality. The formalism required to derive and geometrically interpret some solutions to twisted-self-duality is relevant to the main subject of this thesis: investigating the geometry of the Gromoll-Meyer sphere. Through a Kaluza-Klein ansatz, motivated by bundle-theoretic arguments, an analytic expression for a family of Riemannian metrics on the Gromoll-Meyer sphere is derived. After a detailed study of its geometric constituents, recast as quaternionic-valued objects, the metric with maximal isometry is identified. Its curvature properties are also studied and the associated energy conditions are assessed. Then, an up-to-date and broader overview on the current work concerning exotic spheres and exotic manifolds in general is offered, before focusing again on the Gromoll-Meyer sphere, but this time under the lens of differential topology. Some explicit realisations of the homeomorphism between an exotic 7-sphere and an ordinary one are discussed, together with their possible interpretations in the context of general relativity. Finally, a numerical algorithm for finding Riemannian Einstein metrics on arbitrary manifolds is presented; it is based on machine learning, and highly generalisable in many directions. The current work on implementing its application to exotic spheres is also discussed. The thesis ends with an ample discussion of possible future directions.