Uniformization theory and 2D gravity I. Liouville action and intersection numbers

TL;DR

This paper explores the connection between uniformization theory and 2D gravity, revealing how the Liouville action relates to topological gravity correlators, anomalies, and algebraic structures on Riemann surfaces.

Contribution

It establishes new links between Liouville theory, uniformization, and algebraic structures like the Virasoro algebra on Riemann surfaces.

Findings

Liouville action appears in topological gravity correlators

Derived an inequality involving 2D gravity cutoff and background geometry

Connected chiral anomaly to Krichever-Novikov cocycle

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, always related to uniformization theory, concerns a relation between the higher genus normal ordering and the Liouville action. Furthermore, we show that the chirally split anomaly of CFT is equivalent to the Krichever-Novikov cocycle. By means of the inverse map of uniformization we give a realization of the Virasoro algebra on arbitrary Riemann surfaces and find the eigenfunctions for {\it holomorphic} covariant operators defining higher order cocycles and anomalies which are related to $W$-algebras. Finally we attack the problem of considering the positivity of $e^\sigma$, with $\sigma$ the Liouville field, by proposing an explicit construction for the Fourier modes on compact Riemann surfaces.