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

The requirements of geodesic completeness and self-adjointness imply that the Hamiltonian for anyons is the Laplacian with respect to the Weil-Petersson metric. This metric is complete on the Deligne-Mumford compactification 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 ramification q-1) in the Poincare metric implying that anyon spectrum is chaotic for n≥3. Furthermore, the bound on the holomorphic sectional curvature of moduli spaces implies a gap in the energy spectrum. For q=0 (punctures) anyons are infinitely separated in the Poincare metric (hard core). This indicates that the exclusion principle has a geometrical interpretation. Finally we give the differential equation satisfied by the generating function for volumes of the configuration space of anyons.