Filtered Stokes G-local Systems in Nonabelian Hodge Theory on Curves

TL;DR

This paper constructs the moduli space of filtered Stokes G-local systems in nonabelian Hodge theory on curves, establishing a G-version of the irregular Riemann-Hilbert correspondence and exploring specific examples like the Eguchi-Hanson space.

Contribution

It introduces the Betti moduli space for filtered Stokes G-local systems and proves the G-version of the irregular Riemann-Hilbert correspondence on curves.

Findings

Construction of the Betti moduli space reduces to wild character variety with trivial weights.

Examples include Eguchi-Hanson space and the Airy equation.

Established the correspondence among irregular G-connections, Stokes G-local systems, and G-representations.

Abstract

In the wild nonabelian Hodge correspondence on curves, filtered Stokes G-local systems are regarded as the objects on the Betti side. In this paper, we demonstrate a construction of the moduli space of them, called the Betti moduli space, and it reduces to the wild character variety when the Betti weights are trivial. We study some particular examples including Eguch-Hanson space and the Airy equation together with the corresponding moduli spaces. Furthermore, we provide a proof of the correspondence among irregular singular G-connections, Stokes G-local systems, and Stokes G-representations. This correspondence can be viewed as the G-version of irregular Rieman-Hilbert correspondence on curves.