Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painlevé VI

Abstract

We present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral. The Schwarz–Christoffel accessory parameter problem, relevant when all sides have zero curvature, is also captured within our approach. The method exploits the isomonodromic tau function associated with the Painlevé VI equation. Recently, these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the Young diagrams. After showing how to extract the monodromy data associated with the target domain, we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem. The viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides.