  We study the change of the Laughlin states under large deformations of the
geometry of the sphere and the plane, associated with Mabuchi geodesics on the
space of metrics with Hamiltonian $S^1$-symmetry.
  For geodesics associated with the square 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.
  The lifting of the Mabuchi geodesics to the bundle of quantum states, to
which the Laughlin states belong, is achieved via generalized coherent state
transforms, which correspond to the KZ parallel transport of Chern-Simons
theory.
