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

TL;DR

This paper proves the real analyticity of accessory parameters in Liouville field theory with both elliptic and parabolic sources, extending previous results and including cases on the torus and higher genus surfaces.

Contribution

It extends the proof of real analyticity of accessory parameters to include parabolic sources and higher genus surfaces, using a novel regulator method.

Findings

Proved real analyticity of accessory parameters with parabolic sources.

Extended the method to the case of the torus.

Discussed potential extension to higher genus surfaces.

Abstract

We give the proof of the real analyticity of the accessory parameters in Liouville field theory as a function of the position of the sources in the case in which in addition to elliptic sources, parabolic sources are present. The method is a non trivial extension of the elliptic case as it requires in an intermediate step the introduction of a regulator. The treatment holds also in the case of the torus. A discussion is given of the extension to higher genus surfaces.