Hyperbolic deformation of the strip-equation and the accessory parameters for the torus

TL;DR

This paper introduces a hyperbolic deformation approach to compute accessory parameters for the torus with one source, providing an expansion in the modular parameter q and confirming consistency with known limits.

Contribution

It presents a novel method for calculating accessory parameters via hyperbolic deformation and expands the analysis to higher orders in q, improving precision in uniformization problems.

Findings

At order q^0, the method reproduces known accessory parameter equations.

The correction to the conformal weight parameter is computed up to order q^2.

Unwanted O(q) contributions to the accessory parameter cancel exactly.

Abstract

By applying an hyperbolic deformation to the uniformization problem for the infinite strip, we give a method for computing the accessory parameter for the torus with one source as an expansion in the modular parameter q. At O(q^0) we obtain the same equation for the accessory parameter and the same value of the semiclassical action as the one obtained from the b -> 0 limit of the quantum one point function. The procedure can be carried over to the full O(q^2) or even higher order corrections although the procedure becomes somewhat complicated. Here we compute to order q^2 the correction to the weight parameter intervening in the conformal factor and it is shown that the unwanted contribution O(q) to the accessory parameter equation cancel exactly.