Generating function of monodromy symplectomorphism for $2\times 2$ Fuchsian systems and its WKB expansion

TL;DR

This paper investigates the WKB expansion of 2x2 Fuchsian systems, focusing on the generating function of monodromy symplectomorphism and its connection to tau-functions, providing explicit expansions and links to Bergman tau-function.

Contribution

It computes the first three terms of the WKB expansion of the generating function and establishes its relation to the Bergman tau-function, advancing understanding of monodromy in Fuchsian systems.

Findings

First three terms of WKB expansion computed

Established link to Bergman tau-function

Enhanced understanding of monodromy symplectomorphism

Abstract

We study the WKB expansion of $2\times 2$ system of linear differential equations with four fuchsian singularities. The main focus is on the generating function of the monodromy symplectomorphism which, according to a recent paper is closely related to the Jimbo-Miwa tau-function. We compute the first three terms of the WKB expansion of the generating function and establish the link to the Bergman tau-function.