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$\mathc...
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| Hauptverfasser: | , , , |
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| Dokumenttyp: | Article (Journal) |
| Sprache: | Englisch |
| Veröffentlicht: |
08 May 2022
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| In: |
Journal of the London Mathematical Society
Year: 2022, Jahrgang: 106, Heft: 2, Pages: 590-661 |
| ISSN: | 1469-7750 |
| DOI: | 10.1112/jlms.12581 |
| Online-Zugang: | Verlag, kostenfrei, Volltext: https://doi.org/10.1112/jlms.12581 Verlag, kostenfrei, Volltext: https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12581 
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| Verfasserangaben: | Laura Fredrickson, Rafe Mazzeo, Jan Swoboda, Hartmut Weiss |
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| 100 | 1 | |a Fredrickson, Laura |e VerfasserIn |0 (DE-588)1269734911 |0 (DE-627)1818240149 |4 aut | |
| 245 | 1 | 0 | |a Asymptotic geometry of the moduli space of parabolic SL(2,C)-Higgs bundles |c Laura Fredrickson, Rafe Mazzeo, Jan Swoboda, Hartmut Weiss |
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| 520 | |a Given a generic stable strongly parabolic SL(2,C)-Higgs bundle (E,φ), we describe the family of harmonic metrics ht for the ray of Higgs bundles (E,tφ)t≥ 0 by perturbing from an explicitly constructed family of approximate solutions htapp. We then describe the natural hyperkähler metric on M$\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|. | ||
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