Uniformization Theory and 2d Gravity I. Liouville Action and Intersection Numbers

Department of Mathematics Imperial College 180 Queen’s Gate, London SW7 2BZ, U.K. and Department of Physics “G. Galilei” Istituto Nazionale di Fisica Nucleare University of Padova Via Marzolo, 8 35131 Padova, Italy ABSTRACT This is the first part of an investigation concerning the formulation of 2D gravity in the framework of the uniformization theory of Riemann surfaces. As a first step in this direction we show that the classical Liouville action appears in the expression of the correlators of topological gravity. Next we derive an inequality involving the cutoff of 2D gravity and the background geometry. Another result, still related to uniformization theory, concerns a relation between the higher genus normal ordering and the Liouville action. We introduce operators covariantized by means of the inverse map of uniformization. These operators have interesting properties including holomorphicity. In particular they are crucial to show that the chirally split anomaly of CFT is equivalent to the Krichever-Novikov cocycle and vanishes for deformation of the complex structure induced by the harmonic Beltrami differentials. By means of the inverse map we propose a realization of the Virasoro algebra on arbitrary Riemann surfaces and find the eigenfunctions for the holomorphic covariant operators defining higher order cocycles and anomalies which are related to W -algebras. Finally we face the problem of considering the positivity of e, with σ the Liouville field, by proposing an explicit construction for the Fourier modes on compact Riemann surfaces. These functions, whose underlying number theoretic structure seems related to Fuchsian groups and to the eigenvalues of the Laplacian, are quite basic and may provide the building blocks to properly investigate the long-standing uniformization problem posed by Klein, Koebe and Poincaré.