Asymptotic Geometry of the Moduli Space of Parabolic $SL(2,\mathbb{C})$-Higgs Bundles

Abstract

Given a generic stable strongly parabolic $SL(2,\mathbb{C})$-Higgs bundle $(\mathcal{E},\phi )$, we describe the family of harmonic metrics $h_t$ for the ray of Higgs bundles $(\mathcal{E},t\phi )$ for $t\gg 0$ by perturbing from an explicitly constructed family of approximate solutions $h_t^{\mathrm{app}}$. We then describe the natural hyperKähler metric on $\mathcal{M}$ by comparing it to a simpler "semi-flat" hyperKähler metric. We prove that $g_{L^2}-g_{\mathrm{sf}}=O({\mathrm{e}}^{-\gamma t})$ along a generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture. Our results extend to weakly parabolic $SL(2,\mathbb{C})$-Higgs bundles as well. In the case of the four-puncture sphere, we describe the moduli space and metric more explicitly. In this case, we prove that the hyperkähler metric is ALG and show that the rate of exponential decay is the conjectured optimal one, $\gamma =4L$, where $L$ is the length of the shortest geodesic on the base curve measured in the singular flat metric $|\det \phi |$.

75 pages, revised version

References in corpus (2)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• From $S^1$-fixed points to $\mathcal{W}$-algebra representations

Cited by in corpus (4)

• Moduli spaces of generalized hyperpolygons
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• Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study
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