Accessory parameters for Liouville theory on the torus

TL;DR

This paper derives an implicit equation for the accessory parameter in Liouville theory on the torus, analyzes its convergence, and explores its properties including modular invariance and analyticity, providing explicit solutions in special cases.

Contribution

It introduces a rigorous implicit equation for the accessory parameter, analyzes its convergence and analytic properties, and explicitly solves three special cases.

Findings

Perturbative series converges in a finite disk with a lower bound for the radius.

Accessory parameter is continuous and analytic except at finite points.

Explicit solutions are obtained for three special cases using hypergeometric functions.

Abstract

We give an implicit equation for the accessory parameter on the torus which is the necessary and sufficient condition to obtain the monodromy of the conformal factor. It is shown that the perturbative series for the accessory parameter in the coupling constant converges in a finite disk and give a rigorous lower bound for the radius of convergence. We work out explicitly the perturbative result to second order in the coupling for the accessory parameter and to third order for the one-point function. Modular invariance is discussed and exploited. At the non perturbative level it is shown that the accessory parameter is a continuous function of the coupling in the whole physical region and that it is analytic except at most a finite number of points. We also prove that the accessory parameter as a function of the modulus of the torus is continuous and real-analytic except at most for a zero measure set. Three soluble cases in which the solution can be expressed in terms of hypergeometric functions are explicitly treated.