The Higgs Model for Anyons and Liouville Action

We connect Liouville theory, anyons and Higgs model in a purely geometrical way. 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 The Higgs model for anyons The main aim of this paper is to show that the Higgs model for anyons and Liouville theory are strictly related. The crucial point is based on the following remark a. The configuration space of n anyons is the manifold of n unordered points in Ĉ = C ∪ {∞} Mn = (Ĉ \∆n)/Symm(n), (1.1) 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). (1.2) 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 ecl on the punctured sphere and then, after evaluating the Liouville action for φ = φcl, computing the curvature two-form of the Hermitian line bundle defined by classical action. To understand the physical relevance of this remark we recall that the quantum Hamiltonian for n anyons is proportional to the covariant Laplacian on Mn. Actually, the connection between anyons and Liouville arises also in considering the critically coupled abelian Higgs model in (2+1)-dimensions where the space Mn plays a crucial role in the analysis of n topological sector of the theory. Remarkably f = 2Re log φ, with φ the Higgs field, satisfies the modified (non covariant) Liouville equation fzz̄ = e f − 1, (1.3) which is the second Bogomol’nyi equation. Notice that the non covariance (the −1 in (1.3)) is due to the Higgs mass. Let us consider the Lagrangian of the Higgs model (see [1–3] for details) L = 1 2 DμφDφ− 1 4 FμνF μν − 1 8 ( |φ| − 1 )2 , φ = φ1 + iφ2, (1.4) 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 (for example at 0, 1 and ∞).