Hyperkähler metrics on the moduli space of weakly parabolic Higgs bundles

2021/06/30 by Holdt, Maximilian
#34A12 #34M40 #45D05 #53C07 #53C26 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.16017

Abstract

We use the theory of Gaiotto, Moore and Neitzke to construct a set of Darboux coordinates on the moduli space M of weakly parabolic SL(2,ℂ)-Higgs bundles. For generic Higgs bundles (E,RΦ) with R≫ 0 the coordinates are shown to be dominated by a leading term that is given by the coordinates for a corresponding simpler space of limiting configurations and we prove that the deviation from the limiting term is given by a remainder that is exponentially suppressed in R. We then use this result to solve an associated Riemann-Hilbert problem and construct a twistorial hyperkähler metric g_twist on M. Comparing this metric to the simpler semiflat metric g_sf, we show that their difference is g_twist-g_sf=O(e^-μR), where μ is a minimal period of the determinant of the Higgs field.

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