Differential Operators on Surfaces and Rational WKB Method

Abstract

In this paper, we give a simple and geometric, but formal, description of an open subset of the character variety of surface groups into $\mathrm{SL}_n(\mathbb{C})$. The main ingredient is a modified version of the WKB method, which we call rational WKB method. The geometric interpretation uses higher complex structures introduced by Vladimir Fock and the author. More precisely, the character variety is parametrized by the cotangent bundle of the moduli space of higher complex structures. This generalizes the well-known description of the moduli space of flat $\mathrm{SL}_2(\mathbb{C})$-connections by the cotangent bundle of Teichm\"uller space.