We plan to construct an approximate Calabi-Yau metric using the generalised Gibbons-Hawking ansatz, on a singular $T^{2}$-bundle $M$ over (the complement of a compact subset of) the real 4-dimensional base $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}$, whose discriminant locus is

(2.1)

$$\begin{split}\mathfrak{D}&=\mathfrak{D}_{1}\cup\mathfrak{D}_{2}\cup\mathfrak{D}_{3}\cup\{0\}=\{\mu_{1}=0,\mu_{2}>0\}\cup\{\mu_{2}=0,\mu_{1}>0\}\cup\{\mu_{1}=\mu_{2}<0\}\cup\{0\}\\ &\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\{0\}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}\simeq\mathbb{R}^{4}.\end{split}$$

The topological situation is the same as in the Harvey-Lawson Example 1.7 in complex dimension 3. Our primary concern is that this metric should be approximately Calabi-Yau near spatial infinity, while on a compact set this approximation is allowed to fail.

The basic heuristic idea is to perturb the constant solution (cf. Example 1.6) in a way which incorporates the topology. The information in the constant solution is contained in the base metric

$$g_{a}=a_{ij}d\mu_{i}\otimes d\mu_{j}+A|d\eta|^{2},$$

where $(a_{ij})$ is a real symmetric positive definite $2\times 2$ matrix with inverse matrix $(a^{ij})$, and $A=\det a$. The matrix $a^{ij}$ describes the asymptotic metric on the $T^{2}$-fibres. The associated volume measure is

$$d\text{Vol}_{a}=A^{3/2}d\mu_{1}\wedge d\mu_{2}\wedge d\text{Re}\eta\wedge d\text{Im}\eta.$$

Now in terms of the local potential $\Phi$ the Calabi-Yau condition (1.9) reads

$$\det(\frac{\partial^{2}\Phi}{\partial\mu_{i}\partial\mu_{j}})=-4\frac{\partial^{2}\Phi}{\partial\eta\partial\bar{\eta}},$$

whose linearised equation at the constant solution is the Laplace equation

$$\Delta_{a}\phi=a^{ij}\frac{\partial^{2}\phi}{\partial\mu_{i}\partial\mu_{j}}+4A^{-1}\frac{\partial^{2}\phi}{\partial\eta\partial\bar{\eta}}=0.$$

Here $\Delta_{a}$ is unsurprisingly the Laplacian of $g_{a}$. This suggests that at least away from the discriminant locus, the first order correction to $V^{ij}$ and $W$ from the constant solution

(2.2)

$$v^{ij}=\frac{\partial^{2}\phi}{\partial\mu_{i}\partial\mu_{j}},\quad w=-4\frac{\partial^{2}\phi}{\partial\eta\partial\bar{\eta}}$$

ought to be given by $\Delta_{a}$-harmonic functions,

(2.3)

$$\Delta_{a}v^{ij}=0,\quad\Delta_{a}w=0,\quad Aa^{ij}v^{ij}=w.$$

To incorporate the topology we need to recall the distributional equation (1.16) on $V^{ij}$ and $W$. Since $v^{ij}$ and $w$ are linearisations, it makes sense to require the equation on currents

(2.4)

$$\frac{\sqrt{-1}}{4\pi}\left(\frac{\partial^{2}w}{\partial\mu_{i}\partial\mu_{j}}+4\frac{\partial^{2}v^{ij}}{\partial\eta\partial\bar{\eta}}\right)d\mu_{i}\wedge d\eta\wedge d\bar{\eta}\otimes e_{j}=\mathfrak{D}_{1}\otimes e_{1}-\mathfrak{D}_{2}\otimes e_{2}+\mathfrak{D}_{3}\otimes(e_{2}-e_{1}).$$

The task is to find a compatible solution to (2.2)(2.3)(2.4). Notice that $v^{ij}$ and $w$ are global quantities while $\phi$ is only local. We view $v^{ij}$ and $w$ as the unknown functions in this system of equations, and the existence of a local $\phi$ solving (2.2) is equivalent to some integrability conditions on $v^{ij}$ and $w$ away from $\mathfrak{D}$,

(2.5)

$$\frac{\partial v^{11}}{\partial\mu_{2}}=\frac{\partial v^{12}}{\partial\mu_{1}},\quad\frac{\partial v^{22}}{\partial\mu_{1}}=\frac{\partial v^{21}}{\partial\mu_{2}},$$

and

$$\frac{\partial^{2}w}{\partial\mu_{i}\partial\mu_{j}}=-4\frac{\partial^{2}v^{ij}}{\partial\eta\partial\bar{\eta}}.$$

We now move on to a more formal construction.

Now we define the Kähler ansatz $(g^{(1)},\omega^{(1)},J,\Omega)$ via the generalised Gibbons-Hawking construction, using the functions

$$V^{ij}_{(1)}=a_{ij}+v^{ij},\quad W_{(1)}=A+w.$$

Here the script $(1)$ signifies first order approximation. The complex structure and the holomorphic volume form are not scripted because they turn out to agree with the standard structures on $\mathbb{C}^{3}$ and will not be corrected in a later stage. Notice that the positive definite condition on $V^{ij}$ is implied by $\alpha_{i}\geq 0,i=1,2,3$, which can be checked from the explicit formula. A grain of salt is that there is no a priori guarantee that the metric is smooth over the discriminant locus, a problem we shall take up in Section 2.3.

Let us examine the approximation to the Calabi-Yau condition. This is measured by the volume form error function

(2.10)

$$E^{(1)}=\frac{W_{(1)}}{\det(V^{ij}_{(1)})}-1=\frac{A+w}{A+Aa^{ij}v^{ij}+\det(v^{ij})}-1=-\frac{\det(v^{ij})}{A+w+\det(v^{ij})},$$

where $\det(v^{ij})=\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{3}+\alpha_{2}\alpha_{3}$ and $w=a_{22}\alpha_{1}+a_{11}\alpha_{2}+(a_{11}+a_{22}+2a_{12})\alpha_{3}$. We denote $|\vec{\mu}|_{a}=\sqrt{a_{ij}\mu_{i}\mu_{j}+A|\eta|^{2}}$. Near spatial infinity $E^{(1)}=O(\frac{1}{A^{1/2}|\mu|_{a}^{2}})$ sufficiently far away from the discriminant locus, and $E^{(1)}=O(\frac{1}{A^{1/4}|\mu|_{a}})$ near the discriminant locus. However in standard analytic packages [12] which construct Calabi-Yau metrics from asymptotic approximate solutions, it is essential to have faster than quadratic volume error decay rate, which is not satisfied by our ansatz, so this error must first be corrected. This issue will be explained more amply in Section 2.7.

Now we comment on the symmetry of the ansatz. Apart from the $T^{2}$-symmetry from the construction, there is an additional $U(1)$-symmetry for the Kähler metric commuting with the $T^{2}$-action:

$$\mu_{i}\mapsto\mu_{i},\quad\eta\mapsto e^{i\theta}\eta.$$

However, the holomorphic volume form will be rotated by a phase angle under this action. This may be compared to the Taub-NUT metric, which has an $SO(3)$-symmetry acting on the base. Our ansatz has less continuous symmetry because the base contains a distinguished trivalent graph, which is a new higher dimensional phenomenon. Another analogy to draw from this comparison is that when the Taub-NUT metric glues into the Ooguri-Vafa metric, these additional symmetries are broken, and the same phenomenon shall happen when we construct the Ooguri-Vafa type metrics on the positive vertex. This is because the Ooguri-Vafa type metrics involve an extra periodicity condition on $\eta$ which is not compatible with rotation; an alternative viewpoint is that the special Lagrangian fibration selects out a preferred phase angle.

In some special cases there can be some discrete symmetries from permuting the 3 edges of the trivalent graph. The most symmetric situation is where

$$a_{ij}d\mu_{i}d\mu_{j}\propto(d\mu_{1})^{2}+(d\mu_{2})^{2}+(d(\mu_{1}-\mu_{2}))^{2},$$

or equivalently

$$\begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix}\propto\begin{bmatrix}2&-1\\ -1&2\end{bmatrix}$$

This choice of parameters has a special significance in the theory.

Morever, the family of ansatzs have a scaling symmetry which will be fundamental when we construct the Ooguri-Vafa type metrics later. This symmetry is prescribed by (1.17). In our concrete construction, this means replacing

$$a_{ij}\mapsto\Lambda a_{ij},\quad A\mapsto\Lambda^{2}A.$$

The region near $(\mu_{1},\mu_{2},\eta)$ in the $(a_{ij})$-ansatz correspond to the region near the point $(\Lambda^{-1}\mu_{1},\Lambda^{-1}\mu_{2},\Lambda^{-1.5}\eta)$ in the $(\Lambda a_{ij})$-ansatz. For example, a useful scaling-invariant quantity is $A^{1/4}|\vec{\mu}|_{a}$ :

$$(\Lambda^{2}A)^{1/4}\sqrt{(\Lambda a_{ij})(\Lambda^{-1}\mu_{i})(\Lambda^{-1}\mu_{j})+(\Lambda^{2}A)|\Lambda^{-1.5}\eta|^{2}}=A^{1/4}|\vec{\mu}|_{a}.$$

It defines the approximation scale $A^{1/4}|\vec{\mu}|_{a}\gtrsim 1$, namely the region where the ansatz is approximately Calabi-Yau. Another scaling-invariant quantity is $A^{1/4}\ell$ where $\ell=A^{-1/4}+\text{dist}_{g_{a}}(\cdot,\mathfrak{D})$. The scaling symmetry enables us to easily extract information about the $(\Lambda a_{ij})$-ansatz by analysing the $(a_{ij})$-ansatz, which is very useful for analytical questions.