The Higgs Model for Anyons and Liouville Action: Chaotic Spectrum, Energy Gap and Exclusion Principle

Geodesic completness 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 DeligneMumford 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) in the Poincaré 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 Poincaré metric (hard-core). This indicates that the exclusion principle has a geometrical intepretation. Finally we give the differential equation satisfied by the generating function for volumes of the configuration space of anyons. Partly Supported by the European Community Research Programme Gauge Theories, applied supersymmetry and quantum gravity, contract SC1-CT92-0789 e-mail: [email protected], mvxpd5::matone 1. Let us begin with the following remark a. The configuration space of n anyons is the space of n unordered points in Ĉ = C∪{∞} Mn = (Ĉ \∆n)/Symm(n), with ∆n the diagonal subset where two or more punctures coincide. b. The Liouville action on the Riemann sphere with n-punctures evaluated on the classical solution is the Kähler potential for the natural metric (the Weil-Petersson metric) on Mn = (Ĉ \∆n)/Symm(n)× PSL(2,C). This remark implies that starting from anyons on Ĉ one can recover the two-form associated to the natural metric on the configuration space by first computing the Poincaré metric e on the punctured sphere and then, after evaluating the Liouville action for φ = φ, computing the curvature two-form of the Hermitian line bundle on Mn defined by the classical action (see below). To understand the physical relevance of this remark we notice that the quantum Hamiltonian for n anyons is proportional to the covariant Laplacian on Mn. In the case of anyons on the thrice punctured Riemann sphere we have H = − 1 2m ∆WP , (1) where ∆WP is the Weil-Petersson Laplacian. A crucial point is the physical requirement of self-adjointness of the Hamiltonian. In our approach this requirement is satisfied by considering the Hamiltonian defined on Mn where ∂Mn = Mn\Mn denotes the Deligne-Mumford compactification of Mn. In particular we recall that the Weil-Petersson metric defines a Hermitian inner product and has a completion to the boundary ∂Mn. Thus (1) describes a well-defined physical problem. In particular the structure of the Deligne-Mumford compactification fixes possible anyon-configurations. We notice that in the degeneration limit punctures never collide (hard-core). This is a consequence of the requirement of stability. On the other hand this can be seen as a consequence of the fact that the Liouville equation has delta-singularities at the punctures (hard-core). The PSL(2,C) group reflects the Möbius symmetry of the Riemann sphere. Thus one can consider Mn as the configuration space for n − 3 anyons on the Riemann sphere with 3-punctures.