Global comparison of pseudo-K\"ahler structures on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component

We compare the closed $2$-form $\omega_f$ constructed by Rungi-Tamburelli and Goldman's symplectic form $\omega_G$ on the $\mathrm{SL}(3,\mathbb R)$-Hitchin component.

In particular, we establish that the semi-pseudo-Kähler structure on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component defined by Rungi-Tamburelli is non-degenerate everywhere and, after aligning the normalization of the Killing form on $\mathfrak{sl}(3,\mathbb{R})$, coincides with the one recently found by Collier-Toulisse-Wentworth.