Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere
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| Format: | Preprint |
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2022
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| author | Heller, Lynn Heller, Sebastian Traizet, Martin |
| author_facet | Heller, Lynn Heller, Sebastian Traizet, Martin |
| contents | 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-Kähler 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 $λ$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_12106 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere Heller, Lynn Heller, Sebastian Traizet, Martin Differential Geometry Mathematical Physics Algebraic Geometry 53A10, 53C42, 53C43, 14H60, 14H70 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-Kähler 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 $λ$-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. |
| title | Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere |
| topic | Differential Geometry Mathematical Physics Algebraic Geometry 53A10, 53C42, 53C43, 14H60, 14H70 |
| url | https://arxiv.org/abs/2205.12106 |