$P=W$ conjecture in lowest degree for rank $2$ over the $5$-punctured sphere

TL;DR

This paper investigates the large-scale behavior of Fenchel--Nielsen coordinates on the moduli space of rank 2 Higgs bundles over a 5-punctured sphere, proving parts of the $P=W$ conjecture using abelianization and Hitchin WKB analysis.

Contribution

It provides the first proof of the lowest degree weighted pieces of the $P=W$ conjecture for rank 2 Higgs bundles on a 5-punctured sphere.

Findings

Confirmed the lowest degree weighted pieces of the $P=W$ conjecture in this setting.

Analyzed the large-scale behavior of Fenchel--Nielsen coordinates.

Solved the related Hitchin WKB problem.

Abstract

We use abelianization of Higgs bundles away from the ramification divisor and fiducial solutions to analyze the large scale behaviour of Fenchel--Nielsen co-ordinates on the moduli space of rank $2$ Higgs bundles on the Riemann sphere with $5$ punctures. We solve the related Hitchin WKB problem and prove the lowest degree weighted pieces of the $P=W$ conjecture in this case.