Opers and nonabelian Hodge: numerical studies

TL;DR

This paper conducts numerical experiments to test conjectures relating opers, the nonabelian Hodge correspondence, and Hitchin's hyperk"ahler metric, providing evidence supporting the theoretical predictions in this geometric setting.

Contribution

It provides the first numerical validation of conjectural formulas for monodromy, Stokes data, and Hitchin metric in the context of opers and nonabelian Hodge theory.

Findings

Numerical results support the conjecture's predictions.

Validated formulas for Stokes data and Hitchin metric.

Enhanced understanding of the geometric structures involved.

Abstract

We present numerical experiments that test the predictions of a conjecture of Gaiotto-Moore-Neitzke and Gaiotto concerning the monodromy map for opers, the nonabelian Hodge correspondence, and the restriction of Hitchin's hyperk\"ahler metric to the Hitchin section. These experiments are conducted in the setting of polynomial holomorphic differentials on the complex plane, where the predictions take the form of conjectural formulas for the Stokes data and the Hitchin metric tensor. Overall, the results of our experiments support the conjecture.