The continuation method and the real analyticity of the accessory parameters: the general elliptic case

TL;DR

This paper proves that accessory parameters in Liouville theory depend real-analytically on source positions for elliptic singularities, extending the method to complex surfaces and discussing potential generalizations.

Contribution

It applies the Le Roy-Poincaré continuation method to establish real analyticity of accessory parameters in the general elliptic case, including higher genus surfaces.

Findings

Proves real analyticity of accessory parameters for elliptic sources.

Extends the method to tori with multiple elliptic singularities.

Discusses potential extensions to parabolic singularities and higher genus surfaces.

Abstract

We apply the Le Roy-Poincar\'e continuation method to prove the real analytic dependence of the accessory parameters on the position of the sources in Liouville theory in presence of any number of elliptic sources. The treatment is easily extended to the case of the torus with any number of elliptic singularities. A discussion is given of the extension of the method to parabolic singularities and higher genus surfaces.