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

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.