  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.
