Interpreting the Ooguri-Vafa symplectic form \`a la Atiyah-Bott

TL;DR

This paper connects the Ooguri-Vafa hyperk"ahler space with Hitchin moduli spaces by interpreting its symplectic form through Atiyah-Bott theory, providing new insights into their geometric structure.

Contribution

It extends the identification of the Ooguri-Vafa space with framed Higgs bundles by relating its symplectic form to a regularized Atiyah-Bott form and links the Ooguri-Vafa metric to Hitchin's $L^2$-metric.

Findings

Identified the Ooguri-Vafa symplectic form with a regularized Atiyah-Bott form.

Proved the analogous statement for semiflat forms.

Connected the Ooguri-Vafa metric to Hitchin's $L^2$-metric on the Hitchin section.

Abstract

Gaiotto, Moore, and Neitzke predicted that the hyperk\"ahler Ooguri-Vafa space $\mathcal{M}^{\rm ov}$ should provide a local model for Hitchin moduli spaces near the discriminant locus. To this end, Tulli identified $\mathcal{M}^{\rm ov}$ 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. Finally, restricting to the Hitchin section, we identify a regularized version of Hitchin's $L^2$-metric with the Ooguri-Vafa metric.