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

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 of the self-duality equation. In this paper we construct such solutions for strongly parabolic $\mathfrak{sl}(2,\mathbb C)$ Higgs fields on a $4$-punctured sphere with parabolic weights $t \sim 0$ using loop groups methods through an implicit function theorem argument. We identify the rescaled limit hyper-Kähler moduli space at $t=0$ to be (the completion of) the nilpotent orbit in $\mathfrak{sl}(2, \mathbb C)$ equipped 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. 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 $\Omega$-values which we believe deserve further investigations.
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https://hal.archives-ouvertes.fr/hal-03689843
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Submitted on : Tuesday, June 7, 2022 - 4:00:42 PM
Last modification on : Wednesday, June 8, 2022 - 3:34:26 AM

### Citation

Lynn Heller, Sebastian Heller, Martin Traizet. Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere. 2022. ⟨hal-03689843⟩

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