Celebratio Mathematica

M. F. Atiyah and N. S. Manton: “Skyrmions from instantons,” Phys. Lett. B 222 : 3–4 (1989), pp. 438–442. MR 1001325 article

M. F. Atiyah and N. S. Manton: “Geometry and kinematics of two skyrmions,” Comm. Math. Phys. 153 : 2 (1993), pp. 391–422. MR 1218307 Zbl 0778.53051 article

In Skyrme’s soliton model of baryons, a single Skyrmion has six degrees of freedom, so it is expected that two-Skyrmion dynamics at modest energies can be modelled by motion on a 12-dimensional space of Skyrme fields. A candidate for this space is generated by the gradient flow of the potential energy function, descending from the unstable, baryon number two, hedgehog solutions of the Skyrme field equation. An apparently very similar space is obtained by restricting the gradient flow to the Skyrme fields derived from \( \mathit{SU}(2) \) Yang–Mills instantons of charge two. On both of these spaces, one may quotient out by the group of translations and isospin rotations. Hartshome’s geometrical description of charge two instantons leads us to a conjecture for the global structure of the 6-dimensional quotient space. The conjectured structure is that of complex projective 3-space, with complex conjugate points on one projective plane identified and the real points in this plane removed.

M. F. Atiyah, N. S. Manton, and B. J. Schroers: “Geometric models of matter,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 : 2141 (2012), pp. 1252–1279. MR 2910348 Zbl 1364.53046 ArXiv 1108.5151 article

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, at least provisionally, 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–Newman, Unti, Tamburino (Taub–NUT) manifold as a model for the electron, the Atiyah–Hitchin manifold as a model for the proton, \( \mathbb{C}P^2 \) with the Fubini–Study metric as a model for the neutron and \( S^4 \) with its standard metric as a model for the neutrino.

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