Convergence of classical conformal blocks

TL;DR

This paper introduces a recursive algebraic method for computing classical conformal blocks in Liouville theory, proves their series convergence, and explores their relation to the Riemann-Hilbert problem.

Contribution

It provides a novel recursive scheme for calculating conformal blocks and rigorously establishes their convergence properties.

Findings

Recursive algebraic computation method for conformal blocks

Proof of finite convergence radius of the series

Discussion on relation to Riemann-Hilbert problem

Abstract

We give a recursive method to compute the classical conformal blocks in Liouville field theory. The values of the expansion coefficients are given by an algebraic scheme which works to all orders. The algebraic expression of the intervening matrices are explicitly given. With regard to the problem of the convergence of the series we rigorously prove that it has a finite (non zero) convergence radius. We then comment on the relation of the conformal block problem with the Riemann-Hilbert problem.