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Asymptotic geometry of the moduli space of parabolic SL(2,C)-Higgs bundles

Given a generic stable strongly parabolic SL(2,C)-Higgs bundle (E,φ), we describe the family of harmonic metrics ht for the ray of Higgs bundles (E,tφ)t≥ 0 by perturbing from an explicitly constructed family of approximate solutions htapp. We then describe the natural hyperkähler metric on M$\mathc...

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Bibliographic Details
Main Authors: Fredrickson, Laura (Author) , Mazzeo, Rafe (Author) , Swoboda, Jan (Author) , Weiß, Hartmut (Author)
Format: Article (Journal)
Language:English
Published: 08 May 2022
In: Journal of the London Mathematical Society
Year: 2022, Volume: 106, Issue: 2, Pages: 590-661
ISSN:1469-7750
DOI:10.1112/jlms.12581
Online Access:Verlag, kostenfrei, Volltext: https://doi.org/10.1112/jlms.12581
Verlag, kostenfrei, Volltext: https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12581
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Author Notes:Laura Fredrickson, Rafe Mazzeo, Jan Swoboda, Hartmut Weiss
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Summary:Given a generic stable strongly parabolic SL(2,C)-Higgs bundle (E,φ), we describe the family of harmonic metrics ht for the ray of Higgs bundles (E,tφ)t≥ 0 by perturbing from an explicitly constructed family of approximate solutions htapp. We then describe the natural hyperkähler metric on M$\mathcal M$ by comparing it to a simpler ‘semiflat’ hyperkähler metric. We prove that gL2−gsf=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. A centerpiece of this paper is our explicit description of the moduli space and its L2 metric for the case of the four-punctured sphere. We prove that the hyperkähler manifold in this case is a gravitational instanton of type ALG and that its rate of exponential decay to the semiflat metric 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 |detφ|.
Item Description:Gesehen am 07.10.2022
Physical Description:Online Resource
ISSN:1469-7750
DOI:10.1112/jlms.12581

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