2022/05/24 by Lynn Heller, Heller, Lynn, Sebastian Heller +3 · 1 citation
Mathematics · Physics and Astronomy · #14H60 #14H70 #53A10 #53C42 #53C43 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2205.12106
openalex publication_date 2022/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathfraksl(2,\mathbb C) Higgs fields on a 4-punctured sphere with parabolic weights t ∼ 0 using complex analytic methods. We identify the rescaled limit hyper-Kähler moduli space \mathcal Mt at t=0 to be the completion of the nilpotent orbit in \mathfraksl(2, \mathbb C) modulo a \mathbb Z2×\mathbb Z2 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 λ-connections interpretation. By construction we can compute the Taylor expansions of the holomorphic symplectic form \varpit on \mathcal Mt at t=0 which turn out to have closed form expressions in terms of multiple polylogarithms (MPLs). The geometric properties of \mathcal Mt lead to some identities of certain MPLs which we believe deserve further investigations.