The Ooguri-Vafa metric is an incomplete $S^{1}$-invariant hyperKähler metric constructed via the Gibbons-Hawking ansatz (cf. Section 1.2 with $N=2,\mathfrak{n}=1$). In our normalisation conventions, the metric lives on the singular $S^{1}$-bundle $M\to\mathcal{B}\subset\mathbb{R}_{\mu}\times(S^{1}\times\mathbb{R})_{\eta}$ where the complex variable $\eta$ has period 1, and the $S^{1}$-fibre collapses to a point over the origin $(\mu,\eta)=0$. The first Chern class $c_{1}$ of the $S^{1}$-bundle evaluates to $-1$ on a sphere around the origin in $\mathbb{R}_{\mu}\times(S^{1}\times\mathbb{R})_{\eta}$. The composition $M\to\mathcal{B}\xrightarrow{(\mu,\text{Im}\eta)}\mathbb{R}^{2}$ gives a singular $T^{2}$-fibration, and the periodicity condition on $\eta$ amounts to imposing $\int_{T^{2}}\Omega=2\pi$.

Let $A\gg 1$ be a large parameter. The Ooguri-Vafa metric can be thought as a perturbation of the constant solution (cf. Example 1.6) which is encoded by

$$g_{A}=A(d\mu^{2}+|d\eta|^{2}).$$

after incorporating some topology. We denote $|(\mu,\eta)|=\sqrt{\mu^{2}+|\eta|^{2}}$, and set

(1.19)

$$V(\mu,\eta)=W=A+\frac{1}{2|(\mu,\eta)|}+\sum_{n\in\mathbb{Z}\setminus\{0\}}\{\frac{1}{2|(\mu,\eta+n)|}-\frac{1}{2|n|}\}$$

This series is convergent, 1-periodic in the $\eta$ variable, and satisfies the Laplace equation on $\mathcal{B}$ with distributional term which encodes simultaneously the Calabi-Yau condition and the topology:

$$\frac{1}{2\pi}\left(\frac{\partial^{2}}{\partial\mu\partial\mu}+4\frac{\partial^{2}}{\partial\eta\partial\bar{\eta}}\right)Vd\mu\wedge d\text{Re}\eta\wedge d\text{Im}{\eta}=-\delta_{0},$$

where $\delta_{0}$ is the delta measure at the origin in $\mathbb{R}_{\mu}\times S^{1}\times\mathbb{R}$. The metric on $M$

$$g=V(d\mu^{2}+|d\eta|^{2})+V^{-1}\vartheta^{2}$$

is called the Ooguri-Vafa metric. Strictly speaking, the connection $\vartheta$ can be twisted by a flat connection, and this choice is parametrised by $H^{1}(\mathcal{B}\setminus\{0\},S^{1})=H^{1}(\mathcal{B},S^{1})=H^{1}(S^{1}\times\mathbb{R}^{2},S^{1})\simeq S^{1}$ using that a codimension 3 subset in the base does not affect the fundamental group. We sometimes suppress mentioning this choice as it does not affect the geometry significantly. By Remark 1.6 the $\mu,\text{Im}\eta$ coordinates define a special Lagrangian fibration with phase zero on $M$.

The Ooguri-Vafa metric has the important exponential decay property for $\mu^{2}+(\text{Im}\eta)^{2}\geq 1$,

(1.20)

$$|V(\mu,\eta)-A+\gamma_{E}-\log 2+\frac{1}{2}\log(\mu^{2}+|\text{Im}\eta|^{2})|\leq C\exp(-2\pi\sqrt{\mu^{2}+(\text{Im}\eta)^{2}}).$$

where $\gamma_{E}=\lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{k}-\log n$ is the Euler constant. For $\sqrt{\mu^{2}+(\text{Im}\eta)^{2}}\gg 1$, the dependence of $V$ on the periodic $\text{Re}(\eta)$-variable decays exponentially, so up to exponentially small error the Ooguri-Vafa metric is asymptotic to a semiflat metric. This property is the main reason why the Ooguri-Vafa metric is useful for the gluing construction of Gross and Wilson [11]. On the other hand $V$ becomes negative roughly when $\log(\mu^{2}+|\text{Im}\eta|^{2})>2A$, so the metric is only defined on a bounded set and is incomplete.