Asymptotic Geometry of the Moduli Space of Parabolic SL(2,ℂ)-Higgs Bundles

2020/01/10 by Fredrickson, Laura, Mazzeo, Rafe, Swoboda, Jan +1
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.2001.03682

Abstract

Given a generic stable strongly parabolic SL(2,ℂ)-Higgs bundle (E, φ), we describe the family of harmonic metrics h_t for the ray of Higgs bundles (E, t φ) for t≫0 by perturbing from an explicitly constructed family of approximate solutions h_t^app. We then describe the natural hyperKähler metric on M by comparing it to a simpler "semi-flat" hyperKähler metric. We prove that g_L² - g_sf = O(e^-γt) along a generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture. Our results extend to weakly parabolic SL(2,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, γ=4L, where L is the length of the shortest geodesic on the base curve measured in the singular flat metric |det φ|.

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