THE HIGGS-MODEL FOR ANYONS AND LIOUVILLE ACTION - CHAOTIC SPECTRUM, ENERGY-GAP AND EXCLUSION-PRINCIPLE

The requirements of geodesic completeness and self-adjointness imply t hat the Hamiltonian for anyons is the Laplacian with respect to the We il-Petersson metric. This metric is complete on the Deligne-Mumford co mpactification of moduli (configuration) space. The structure of this compactification fixes the possible anyon configurations. This allows us to identify anyons with singularities (elliptic points with ramific ation q-1) in the Poincare metric implying that anyon spectrum is chao tic for n greater-than-or-equal-to 3. Furthermore, the bound on the ho lomorphic sectional curvature of moduli spaces implies a gap in the en ergy spectrum. For q = 0 (punctures) anyons are infinitely separated i n the Poincare metric (hard core). This indicates that the exclusion p rinciple has a geometrical interpretation. Finally we give the differe ntial equation satisfied by the generating function for volumes of the configuration space of anyons.