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.
Highlights
The theory of conformal mapping has a long history with perennial interest in it due to its role as an invaluable2018 The Authors
As we will see in the following, the problem we address is the original version of the Riemann–Hilbert problem (RHp): how to find the accessory parameters in (1.4), or equivalently (1.2), given the monodromy data of the solutions
This paper has shown how to use the isomonodromic tau function associated with the Painlevé VI equation to determine these two parameters
Summary
Applications of the Jimbo–Miwa–Ueno tau functions, closer in spirit to the topic of this paper, include the connection problem for the Heun differential equation, which was used to study scattering of scalar fields in black hole backgrounds [18,19] as well as the quantization of the Rabi model in quantum optics [20] Another connection to our work is the observation that the tau function for the Painlevé VI transcendent coincides with the Fourier transform of a particular 4-point Virasoro conformal block for c = 1, relating the Painlevé transcendents with the representation theory of the Virasoro algebra, and to (quantum) Liouville field theory.
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