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Singular fibers of Hitchin systems

Horn, Johannes

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Abstract

In the recent years, Hitchin systems and Higgs bundle moduli spaces were intensively studied in mathematics and physics. Two major breakthroughs were the formulation of Langlands duality of Hitchin systems and the understanding of the asymptotics of the hyperkähler metric on Higgs bundle moduli space. Both results were considered for the regular locus of the Hitchin system and both results are conjectured to extend to the singular locus. In this work, we make the first steps towards generalizing these theorems to singular Hitchin fibers. To that end, we develop spectral data for a certain class of singular fibers of the symplectic and odd orthogonal Hitchin system. These spectral data consist of abelian coordinates taking value in an abelian torsor and non-abelian coordinates parametrising local deformations of the Higgs bundles at the singularities of the spectral curve. First of all, these semi-abelian spectral data allow us to obtain a global description of singular Hitchin fibers. Moreover, we can construct solutions to the decoupled Hitchin equation on the singular locus of the Hitchin map. These are limits of solutions to the Hitchin equation along rays to the ends of the moduli space playing an important role in the analysis of the asymptotics of the hyperkähler metric. Finally, we can explicitly describe how Langlands duality extends to this class of singular Hitchin fibers. We discover a duality on the abelian part of the spectral data, similar to regular case. Instead, the non-abelian coordinates are symmetric under this Langlands correspondence.

Item Type: Dissertation
Supervisor: Alessandrini, Dr. Daniele
Place of Publication: Heidelberg
Date of thesis defense: 13 October 2020
Date Deposited: 19 Oct 2020 11:04
Date: 2020
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Mathematics
Subjects: 510 Mathematics
Controlled Keywords: Komplexe Geometrie, Algebraische Geometrie
Uncontrolled Keywords: Integrable systems, Moduli theory, Higgs bundle moduli spaces, Hitchin systems
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