Liouville theory and uniformization of four-punctured sphere

TL;DR

This paper develops a method to compute the uniformizing map and group for four-punctured spheres using classical Liouville action, building on Zamolodchikovs' conjecture relating 4-point and 3-point actions.

Contribution

It introduces a new computational approach for the uniformization problem of four-punctured spheres based on classical Liouville theory and Zamolodchikovs' conjecture.

Findings

Explicit construction of the uniformizing map for four-punctured spheres.

Validation of Zamolodchikovs' conjecture in the context of uniformization.

Potential applications to conformal field theory and complex analysis.

Abstract

Few years ago Zamolodchikov and Zamolodchikov proposed an expression for the 4-point classical Liouville action in terms of the 3-point actions and the classical conformal block. In this paper we develop a method of calculating the uniformizing map and the uniformizing group from the classical Liouville action on n-punctured sphere and discuss the consequences of Zamolodchikovs conjecture for an explicit construction of the uniformizing map and the uniformizing group for the sphere with four punctures.