# Exact WKB and abelianization for the $T_3$ equation

## TL;DR

This paper explores the exact WKB method and abelianization for the $T_3$ equation, revealing new Darboux coordinates on moduli spaces of flat connections and verifying their properties numerically.

## Contribution

It introduces a novel application of exact WKB and abelianization to the $T_3$ equation, including the construction of new Darboux coordinates on moduli spaces.

## Key findings

- Numerical verification of monodromy asymptotics in new coordinates
- Development of a Darboux coordinate system for the $T_3$ equation
- Extension of abelianization techniques to higher-order equations

## Abstract

We describe the exact WKB method from the point of view of abelianization, both for Schr\"odinger operators and for their higher-order analogues (opers). The main new example which we consider is the "$T_3$ equation," an order $3$ equation on the thrice-punctured sphere, with regular singularities at the punctures. In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate system on a moduli space of flat $\mathrm{SL}(3)$-connections. We give the simplest example of such a coordinate system, and verify numerically that in these coordinates the monodromy of the $T_3$ equation has the expected asymptotic properties. We also briefly revisit the Schr\"odinger equation with cubic potential and the Mathieu equation from the point of view of abelianization.