If $\frac{\partial}{\partial\theta_{j}}$ are the Hamiltonian vector fields dual to $\vartheta_{i}$, namely $\vartheta_{i}(\frac{\partial}{\partial\theta_{j}})=\delta_{ij}$, then $d\mu_{i}=-\iota_{\frac{\partial}{\partial\theta_{i}}}\omega$, namely $\mu_{i}$ are the symplectic moment coordinates up to sign. When $N-\mathfrak{n}=1$, namely there is only one $\eta$ coordinate, then $d\eta=\Omega(\frac{\partial}{\partial\theta_{1}},\ldots,\frac{\partial}{\partial\theta_{\mathfrak{n}}},\cdot)$, and accordingly we refer to $\eta$ as the holomorphic moment coordinate. In this situation $M$ admits a fibration

$$M\xrightarrow{(\mu_{1},\ldots,\mu_{\mathfrak{n}},\text{Im}(\eta))}\mathfrak{t}^{*}\times\mathbb{R},$$

where fibres are special Lagrangians with phase zero: the Lagrangian condition follows from $d\mu_{i}=-\iota_{\frac{\partial}{\partial\theta_{i}}}\omega=0$ on fibres, while the special condition is equivalent to $\text{Im}\Omega(\frac{\partial}{\partial\theta_{1}},\ldots\frac{\partial}{\partial\theta_{\mathfrak{n}}},\cdot)=d\text{Im}\eta=0$.