Geometric Models of Matter

TL;DR

This paper introduces a geometric framework modeling static particles as special 4-manifolds with properties reflecting electric charge and baryon number, linking topology and geometry to particle physics.

Contribution

It proposes a novel geometric model of particles using Riemannian 4-manifolds with self-dual Weyl tensor, connecting topology with physical particle properties.

Findings

Taub-NUT manifold models the electron

Atiyah-Hitchin manifold models the proton

CP^2 models the neutron

Abstract

Inspired by soliton models, we propose a description of static particles in terms of Riemannian 4-manifolds with self-dual Weyl tensor. For electrically charged particles, the 4-manifolds are non-compact and asymptotically fibred by circles over physical 3-space. This is akin to the Kaluza-Klein description of electromagnetism, except that we exchange the roles of magnetic and electric fields, and only assume the bundle structure asymptotically, away from the core of the particle in question. We identify the Chern class of the circle bundle at infinity with minus the electric charge and the signature of the 4-manifold with the baryon number. Electrically neutral particles are described by compact 4-manifolds. We illustrate our approach by studying the Taub-NUT manifold as a model for the electron, the Atiyah-Hitchin manifold as a model for the proton, CP^2 with the Fubini-Study metric as a model for the neutron, and S^4 with its standard metric as a model for the neutrino.