## The geometry of point particles.(English)Zbl 1010.58015

Summary: 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 that are dual to those found in a number of complicated physical theories, such as Skyrmions and fullerenes. Comparisons with 2- and 3-particle energy functions are made. The planar restriction and the generalization to hyperbolic 3-space are also investigated.

### MSC:

 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences 55R80 Discriminantal varieties and configuration spaces in algebraic topology 81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory

### Keywords:

point particles; geometry; energy minimization; polyhedra
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