Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere

TL;DR

This paper constructs solutions to self-duality equations for Higgs fields on a 4-punctured sphere, analyzing their limits and revealing connections to hyper-Kähler geometry and multiple polylogarithms, advancing the understanding of the non-abelian Hodge correspondence.

Contribution

It introduces a complex analytic construction of self-duality solutions for parabolic Higgs bundles on a 4-punctured sphere and explores their limit moduli space using twistor methods.

Findings

Identification of the limit moduli space as the completion of a nilpotent orbit with Eguchi-Hanson metric

Explicit Taylor expansions of the holomorphic symplectic form in terms of multiple polylogarithms

Discovery of new identities among multiple polylogarithms from geometric properties

Abstract

The non-abelian Hodge correspondence is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions to the self-duality equations. In this paper we construct self-duality solutions for strongly parabolic $\mathfrak{sl}(2,\mathbb C)$ Higgs fields on a $4$-punctured sphere with parabolic weights $t \sim 0$ using complex analytic methods. We identify the rescaled limit hyper-K\"ahler moduli space $\mathcal M_t$ at $t=0$ to be the completion of the nilpotent orbit in $\mathfrak{sl}(2, \mathbb C)$ modulo a $\mathbb Z_2\times\mathbb Z_2$ action, equipped with the Eguchi-Hanson metric. Our methods and computations are based on the twistor approach to the self-duality equations using Deligne and Simpson's $\lambda$-connections interpretation. By construction we can compute the Taylor expansions of the holomorphic symplectic form $\varpi_t$ on $\mathcal M_t$ at $t=0$ which turn out to have closed form expressions in terms of multiple polylogarithms (MPLs). The geometric properties of $\mathcal M_t$ lead to some identities of certain MPLs which we believe deserve further investigations.