Convergence of classical conformal blocks

TL;DR

The paper tackles the convergence of classical conformal blocks in Liouville theory by presenting two complementary computational schemes—the algebraic approach with nested upper-triangular matrices and the Green function monodromy method. It proves that the expansion of the accessory parameter $C(x)$ in the modulus $x$ is convergent within a finite radius near $x=0$ and that $C(x)$ is analytic in that disk, with consistent results between the two methods and a procedure to bound the radius. It discusses the relationship to the Riemann–Hilbert problem, clarifying how solvability and potential apparent singularities affect analytic structure and uniqueness. Together, the results provide a practical all-orders algorithm for conformal blocks and a rigorous foundation for their local analytic behavior around the four-p punctured sphere configuration.

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.

Figures (1)

• Figure 1: The monodromy contour