Differential Operators on Surfaces and Rational WKB Method

TL;DR

This paper introduces a geometric and formal approach to understanding the character variety of surface groups into SL_n(C) using a modified WKB method called rational WKB, linking it to higher complex structures.

Contribution

It provides a new geometric interpretation of the character variety via the rational WKB method and higher complex structures, generalizing known results for SL_2(C).

Findings

Character variety parametrized by cotangent bundle of higher complex structures

Generalizes the moduli space of flat SL_2(C)-connections

Introduces rational WKB method for geometric analysis

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.