(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.

($\mathbb{C}^{N}$ Harvey-Lawson example) The affine space $\mathbb{C}^{N}$ with the standard Euclidean metric $\omega=\frac{\sqrt{-1}}{2}\sum_{i=0}^{N-1}dz_{i}\wedge d\bar{z}_{i}$ and holomorphic volume form $\Omega=\sqrt{-1}^{N-1}dz_{0}\wedge\ldots dz_{N-1}$ admits a diagonal $T^{N-1}$-action, where the $k$-th circle factor acts by

$$e^{i\theta_{k}}\cdot(z_{0},z_{1},\ldots,z_{N-1})=(e^{-i\theta_{k}}z_{0},z_{1},\ldots,e^{i\theta_{k}}z_{k},z_{k+1},\ldots,z_{N-1}).$$

The corresponding moment coordinates are

$$\mu_{i}=\frac{1}{2}(|z_{i}|^{2}-|z_{0}|^{2}),\quad i=1,2,\ldots N-1,\text{ and }\eta=z_{0}z_{1}\ldots z_{N-1}.$$

This defines a $T^{N-1}$-bundle away from the singular locus $\bigcup\{z_{i}=z_{j}=0\}$. Special cases include (1.1)(1.3). The inverse matrices are

$$(V^{-1})^{ij}=|z_{0}|^{2}+\delta_{ij}|z_{i}|^{2},\quad W^{-1}=|z_{0}z_{1}\ldots z_{N-1}|^{2}\left(\frac{1}{|z_{0}|^{2}}+\ldots+\frac{1}{|z_{N-1}|^{2}}\right),$$

viewed as functions of $\mu_{i}$ and $\eta$. The special Lagrangian fibration described by Remark 1.6 is the well known Harvey-Lawson example.

We notice in particular when $N=3$ that the discriminant locus of the singular $T^{2}$-bundle is given by $\mathfrak{D}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\{0\}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}$ as in (1.2). This is not an accidental feature of the Euclidean metric:

(Taub-NUT) We take $N=2,\mathfrak{n}=1$, and $V=W=\frac{1}{2\sqrt{\mu^{2}+|\eta|^{2}}}+A$, where $A$ is a positive constant. This defines a Calabi-Yau metric whose asymptotic geometry at infinity approaches the constant solution (cf. Example 1.6), with asymptotic circles of length $\frac{2\pi}{\sqrt{A}}$ fibred over a flat 3-dimensional base. Different choices of $A$ define the same metric up to scaling. The first Chern class $c_{1}$ of the $S^{1}$-bundle over $(\mathbb{R}_{\mu}\times\mathbb{C}_{\eta})\setminus\{0\}$ evaluates to $-1$ on any sphere around the origin in $\mathbb{R}_{\mu}\times\mathbb{C}_{\eta}$; equivalently, the 3-current $-\frac{1}{2\pi}dF$ is represented by the origin $0\in\mathbb{R}_{\mu}\times\mathbb{C}_{\eta}$ viewed as a codimension 3 cycle. Written in terms of the delta function,

$$\frac{1}{2\pi}dF=\frac{1}{2\pi}(\frac{\partial^{2}}{\partial\mu^{2}}+4\frac{\partial^{2}}{\partial\eta\partial\bar{\eta}})Vd\mu\wedge d\text{Re}\eta\wedge d\text{Im}{\eta}=-\delta(\mu,\eta,\bar{\eta})d\mu\wedge d\text{Re}\eta\wedge d\text{Im}{\eta}.$$

From the holomorphic perspective, LeBrun [17] observes that the Taub-NUT space is biholomorphic to $\mathbb{C}^{2}$. To see this, recall $\zeta=Vd\mu+\sqrt{-1}\vartheta$ and notice the (1,0) form $\zeta-\frac{\mu}{2\eta\sqrt{\mu^{2}+|\eta|^{2}}}d\eta$ is closed, so locally is the differential of a holomorphic function. The line integrals

$$\log z_{1}=\int\zeta+(\frac{1}{2\eta}-\frac{\mu}{2\eta\sqrt{\mu^{2}+|\eta|^{2}}})d\eta,\quad\log z_{0}=\int-\zeta+(\frac{1}{2\eta}+\frac{\mu}{2\eta\sqrt{\mu^{2}+|\eta|^{2}}})d\eta$$

define holomorphic functions up to $2\pi\sqrt{-1}\mathbb{Z}$ over the regions $\mathbb{R}\times\mathbb{C}\setminus\{\eta=0,\mu\leq 0\}$ and $\mathbb{R}\times\mathbb{C}\setminus\{\eta=0,\mu\geq 0\}$ respectively, so $z_{1}$ and $z_{0}$ are well defined over the respective regions. Since $d\log z_{1}+d\log z_{0}=d\log\eta$ we can normalise $z_{0},z_{1}$ to satisfy the functional equation $z_{1}z_{0}=\eta$, whence $z_{1}$ and $z_{0}$ extend as global holomorphic functions. These coordinates exhibit the biholomorphism to $\mathbb{C}^{2}$. By considering the Hamiltonian vector field acting on $\log z_{1},\log z_{0}$, we idenitfy the $S^{1}$ action as

$$e^{i\theta}\cdot(z_{1},z_{0})=(e^{i\theta}z_{1},e^{-i\theta}z_{0}).$$

The holomorphic volume form is

$$\Omega=-\sqrt{-1}\zeta\wedge d\eta=-\sqrt{-1}d\log z_{1}\wedge d(z_{1}z_{0})=\sqrt{-1}dz_{0}\wedge dz_{1}.$$

Thus $\eta=z_{0}z_{1}$ defines holomorphic fibration of $\mathbb{C}^{2}$ by affine quadrics. The generic quadric fibre is topologically a cylinder, and metrically is also approaching the flat cylindrical metric near spatial infinity. When $\eta=0$, the quadric fibre degenerates into a union of two complex lines with simple normal crossing, where each line looks metrically like a cylinder with one capped end.