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

. We study the WKB expansion of 2 × 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 [10] 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.

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