The Geometry of Point Particles

TL;DR

This paper explores a geometric map from point configurations in 3D space to flag manifolds, proposes conjectures supported by numerical evidence, and identifies energy-minimizing configurations related to polyhedral structures in physics.

Contribution

It introduces a novel geometric map linking point configurations to flag manifolds, formulates conjectures with numerical support, and computes energy-minimizing arrangements up to 32 particles.

Findings

Configurations form polyhedral structures dual to physical theories

Numerical evidence supports the conjectured map and energy minimization

Extensions to planar and hyperbolic spaces are explored

Abstract

There is a very natural map from the configuration space of n distinct points in Euclidean 3-space into the flag manifold U(n)/U(1)^n, which is compatible with the action of the symmetric group. The map is well-defined for all configurations of points provided a certain conjecture holds, for which we provide numerical evidence. We propose some additional conjectures, which imply the first, and test these numerically. Motivated by the above map, we define a geometrical multi-particle energy function and compute the energy minimizing configurations for up to 32 particles. These configurations comprise the vertices of polyhedral structures which are dual to those found in a number of complicated physical theories, such as Skyrmions and fullerenes. Comparisons with 2-particle and 3-particle energy functions are made. The planar restriction and the generalization to hyperbolic 3-space are also investigated.