Monopoles of Twelve Types in 3-Body Problems
Abstract
We consider twelve different ways of modelling the 3-body problem in dimension ≥ 2. These can be viewed as models of classical and quantum background independence. We show that a different type of monopole is realized in each's relational space: a type of reduced configuration space. 8 cases occur in 2-d, and 4 distinct ones in 3-d; these reflect counts of non-equivalent subgroup actions of S3 × C2 and S3 respectively. The S3 acts on particle labels; the extra C2 corresponds to the purely 2-d option of whether or not to identify mirror images. The non-equivalent realization is due to a suite of subgroup, orbit space and stratification features. Our 2-d monopoles include 4 known ones: a realization of Dirac's monopole in relational space rather than its more habitual setting of space, the 2-d version of Iwai's monopole, and indistinguishable particle monopoles with and without mirror image identification. The 4 new ones are indistinguishable under a 2-particle label switch or under even permutations, in each case with optional mirror image identification. Our 4 3-d monopoles are 2 known ones: the actual Iwai monopole and its already-announced indistinguishable-particles counterpart, and 2 new ones: the two-particle label switch and even permutation cases. All 4 3-d cases are stratified. The three even-permutation cases are orbifolds, two with boundary, the 3-d case's boundary constituting a separate stratum, giving a stratified orbifold. We document each of the 12 cases' underlying shape space and relational space, and each monopole's Hopf mathematics, global-section versus topological quantization dichotomy, Dirac string positioning, and Chern integral concordance with topological contributions form of Gauss--Bonnet Theorem.