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Geodesic completness and self-adjointness imply that the Hamiltonian foranyons is the Laplacian with respect to the Weil-Petersson metric. This metricis complete on the Deligne-Mumford compactification of moduli (configuration)space. The structure of this compactification fixes the possible anyonconfigurations. This allows us to identify anyons with singularities (ellipticpoints with ramification $q^{-1}$) in the Poincar\'e metric implying that anyonspectrum is chaotic for $n\ge 3$. Furthermore, the bound on the holomorphicsectional curvature of moduli spaces implies a gap in the energy spectrum. For$q=0$ (punctures) anyons are infinitely separated in the Poincar\'e metric(hard-core). This indicates that the exclusion principle has a geometricalintepretation. Finally we give the differential equation satisfied by thegenerating function for volumes of the configuration space of anyons.Comment: DFPD/93/TH/69, LaTe

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