Date of Award

Spring 1-1-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Neitzke, Andrew

Abstract

In this thesis we study the hyperkähler structure of a particular moduli space of Higgs bundles near the discriminant locus of the Hitchin fibration, which in this case consists of quadratic differentials with a double zero. Gaiotto, Moore, and Neitzke predicted that the hyperkähler Ooguri-Vafa space $\mathcal{M}^\textrm{ov}$ should provide a local model for Hitchin moduli spaces near the discriminant locus. To this end,Tulli identified $\mathcal{M}^\textrm{ov}$ with a certain space of framed Higgs bundles with an irregular singularity. We extend this result by identifying the Ooguri-Vafa holomorphic symplectic form with a regularized version of the Atiyah-Bott form on the associated space of framed connections. We also prove the analogous statement for the corresponding semiflat forms, which are more explicit and play a role in the asymptotic description of the hyerperkähler structure. Finally, restricting to the Hitchin section, we identify a regularized version of Hitchin's $L^2$-metric with the Ooguri-Vafa metric.

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