Reconstruction phases in the planar three- and four-vortex problems

Pure reconstruction phases, geometric and dynamic, are computed in the $N$-point-vortex model in the plane, for the cases $N=3$ and $N=4$. The phases are computed relative to a metric-orthogonal connection on appropriately defined principal fiber bundles. The metric is similar to the kinetic energy metric for point masses but with the masses replaced by vortex strengths. The geometric phases are shown to be proportional to areas enclosed by the closed orbit on the symmetry reduced spaces. More interestingly, simple formulae are obtained for the dynamic phases, analogous to Montgomery's result for the free rigid body, which show them to be proportional to the time period of the symmetry reduced closed orbits. For the case $N = 3$ a non-zero total vortex strength is assumed. For the case $N = 4$ the vortex strengths are assumed equal.