(Constant solution) The simplest solution is where $V^{ij}$ and $W^{p\bar{q}}$ are independent of the base variables and satisfy (1.9). We shall see that many interesting solutions can be thought heuristically as perturbation of the constant solution after introducing some topology. Some important special cases for us are:

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  $N=2,\mathfrak{n}=1$, $V=W=A>0$. The subcase where $\eta$ takes value in $\mathbb{C}$ is relevant for the Taub-NUT metric (cf. Example 1.8), and the subcase where $\eta$ is a periodic variable is relevant for the Ooguri-Vafa metric (cf. Section 1.3). In the periodic case the choice of the connection $\vartheta$ is parametrised by $H^{1}(S^{1}\times\mathbb{R}^{2},S^{1})\simeq S^{1}$.

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  $N=3,\mathfrak{n}=2$, $V^{ij}=a_{ij}$ is symmetric positive definite, and $W=A=\det(a_{ij})$. We write $g_{a}=a_{ij}d\mu_{i}d\mu_{j}+A|d\eta|^{2}$. The subcase where $\eta$ takes value in $\mathbb{C}$ will be relevant for constructing new Taub-NUT type Calabi-Yau metrics on $\mathbb{C}^{3}$, and the subcase where $\eta$ is a periodic variable will be relevant for the positive vertex. In the periodic case the choice of the connection $\vartheta$ is parametrised by $H^{1}(S^{1}\times\mathbb{R}^{3},T^{2})\simeq T^{2}$.
  

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  $N=3,\mathfrak{n}=1$, $W^{p\bar{q}}=a_{p\bar{q}}$ is Hermitian, and $V=A=\det(a_{p\bar{q}})$. We write $g_{a}=\text{Re}(a_{p\bar{q}}d\eta_{p}d\bar{\eta}_{q})+Ad\mu\otimes d\mu$. Here $\eta_{1},\eta_{2}$ are periodic coordinates with period 1. The choice of the connection $\vartheta$ is parametrised by $H^{1}(T^{2}\times\mathbb{R}^{3},S^{1})\simeq T^{2}$. This case will be relevant for the negative vertex. Notice that if we demand that the fibration on $M$ induced by $\mu,\text{Im}(\eta_{1}),\text{Im}(\eta_{2})$ is a special Lagrangian fibration with phase zero, then we would need $a_{1\bar{2}}=a_{2\bar{1}}$, namely $a_{pq}=a_{p\bar{q}}$ is symmetric.