Write the Liouville 1-form as $\lambda$, so $d\lambda=\omega$, and the potential of the immersed Lagrangian $L$ as $f_{L}$, so $df_{L}=\lambda|_{L}$. We consider the potential as part of the brane data, so adding a constant to $f_{L}$ is viewed as a different Lagrangian brane. We shall consider a path of such Lagrangians $L_{t}$, with associated Hamiltonian functions $h_{t}$, so there is a preferred way to parallel transport $f_{L}$, as recalled below.

We observe that by the Kähler condition $\omega\wedge\Omega=0$, so

This means the term $\int_{\cup_{t}L_{t}}\lambda\wedge\text{Im}(e^{-i\hat{\theta}}\Omega)$ is a homological quantity, in the sense that we can replace $\cup_{t}L_{t}$ by any compactly supported $(n+1)$-current $\mathcal{C}$ with $\partial\mathcal{C}=L_{1}-L_{0}$, which would automatically satisfy $[\mathcal{C}-\cup_{t}L_{t}]=0\in H_{n+1}(X)$, since $H_{n+1}(X)=0$ for Stein manifolds. In particular, this explains Solomon’s theorem that his functional is invariant under Hamiltonian deformations of the path of Lagrangians. The advantage of our homological interpretation is to allow more general currents $\mathcal{C}$, which in particular can come from families of holomorphic curves.