LOOP GROUP METHODS FOR THE NON-ABELIAN HODGE CORRESPONDENCE ON A 4-PUNCTURED SPHERE
Résumé
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 of the self-duality equation. In this paper we construct such solutions for strongly parabolic sl(2, C) Higgs fields on a 4-punctured sphere with parabolic weights t ∼ 0 using loop groups methods through an implicit function theorem argument at the trivial connection for t = 0. We identify the rescaled limit hyper-Kähler moduli space at the singular point at t = 0 to be the completion of the nilpotent orbit in sl(2, C) equipped the Eguchi-Hanson metric (modulo a Z2 × Z2 action). Our methods and computations are based on the twistor approach to the self-duality equations using Deligne and Simpson's λconnections interpretation. Due to the implicit function theorem, Taylor expansions of these quantities can be computed at t = 0. By construction they have closed form expressions in terms of Multiple-Polylogarithms and their geometric properties lead to some identities of Ω-values which we believe deserve further investigations.
Domaines
Géométrie différentielle [math.DG]
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