Laughlin States Change Under Large Geometry Deformations and Imaginary Time Hamiltonian Dynamics

We study the dependence of the Laughlin states on the geometry of the sphere and the plane, for one-parameter Mabuchi geodesic families of curved metrics with Hamiltonian S^1 -symmetry. For geodesics associated with convex functions of the symmetry generator, as the geodesic time goes to infinity, the geometry of the sphere becomes that of a thin cigar collapsing to a line and the Laughlin states become concentrated on a discrete set of S^1 -orbits, corresponding to Bohr–Sommerfeld orbits of geometric quantization.