Exotic Spheres' Metrics and Solutions via Kaluza-Klein Techniques

TL;DR

This paper constructs explicit metrics on exotic seven-spheres using Kaluza-Klein techniques, and derives Einstein-Yang-Mills solutions that distinguish between standard and exotic spheres through instanton winding numbers.

Contribution

It provides explicit coordinate expressions for metrics on exotic spheres and applies Kaluza-Klein reduction to connect these geometries with four-dimensional Einstein-Yang-Mills solutions.

Findings

Explicit metrics for exotic spheres are obtained.

Solutions to Einstein-Yang-Mills equations are derived for both standard and exotic spheres.

Differences in solutions are characterized by instanton winding numbers.

Abstract

By applying an inverse Kaluza-Klein procedure, we provide explicit coordinate expressions for Riemannian metrics on two homeomorphic but not diffeomorphic spheres in seven dimensions. We identify Milnor's bundles, among which ten out of the fourteen exotic seven-spheres appear (ignoring orientation), with non-principal bundles having homogeneous fibres. Then, we use the techniques in \cite{10.1063/1.525753} to obtain a general ansatz for the coordinate expression of a metric on the total space of any Milnor's bundle. The ansatz is given in terms of a metric on $S^4$, a metric on $S^3$ (which can smoothly vary throughout $S^4$), and a connection on the principal $SO(4)$-bundle over $S^4$. As a concrete example, we present explicit formulae for such metrics for the ordinary sphere and the Gromoll-Meyer exotic sphere. Then, we perform a non-abelian Kaluza-Klein reduction to gravity in seven dimensions, according to (a slightly simplified version of) the metric ansatz above. We obtain the standard four-dimensional Einstein-Yang-Mills system, for which we find solutions associated with the geometries of the ordinary sphere and of the exotic one. The two differ by the winding numbers of the instantons involved.