Interpreting the Ooguri-Vafa symplectic form à la Atiyah-Bott

Gaiotto, Moore, and Neitzke predicted that the hyperkähler Ooguri-Vafa space ${\mathcal{M}}^{\mathrm{o}\mathrm{v}}$ should provide a local model for Hitchin moduli spaces near the discriminant locus. To this end, Tulli identified ${\mathcal{M}}^{\mathrm{o}\mathrm{v}}$ 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.