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.

As a hyperKähler metric, the Ooguri-Vafa metric admits a 2-sphere of compatible integrable complex structures. There is one distinguished complex structure giving rise to a holomorphic elliptic fibration, which is described in detail in [11]. Here we wish to focus on another distinguished complex structure where $\eta$ is holomorphic and $\mu$ is the symplectic moment map, which is more natural for special Lagrangian fibrations. The author is not aware of explicit references for the content of this Section.

Our main goal is to identify the complex structure on $M$ explicitly, which requires us to construct holomorphic functions on $M$. We start with the type $(1,0)$ form $\zeta=Vd\mu+\sqrt{-1}\vartheta$ and recall formula (1.14). To turn $\zeta$ into a holomorphic differential, we need to subtract a function times $d\eta$, whose differential cancels out $d\zeta$. Inspired by the Taub-NUT example 1.8, and taking care of periodicity requirement, we introduce the functions

$$\begin{cases}\beta_{+}=\frac{\pi\sqrt{-1}}{2}-\sqrt{-1}\theta_{\infty}+\lim_{k\to\infty}\sum_{n=-k}^{n=k}\{\frac{1}{2(\eta+n)}-\frac{\mu}{2(\eta+n)\sqrt{\mu^{2}+|\eta+n|^{2}}}\}\\ \beta_{-}=\frac{\pi\sqrt{-1}}{2}+\sqrt{-1}\theta_{\infty}+\lim_{k\to\infty}\sum_{n=-k}^{n=k}\{\frac{1}{2(\eta+n)}+\frac{\mu}{2(\eta+n)\sqrt{\mu^{2}+|\eta+n|^{2}}}\}\end{cases}$$

These series are convergent and 1-periodic in $\eta$, such that the forms $\zeta^{\prime}=\zeta+\beta_{+}d\eta$, $\zeta^{\prime\prime}=-\zeta+\beta_{-}d\eta$ are closed. The real number $\theta_{\infty}$ is chosen to cancel the asymptotic holonomy of the $S^{1}$-connection $\vartheta$ along the $\text{Re}(\eta)$-circle as $\text{Im}\eta\to\infty$. We have

$$\begin{split}\zeta^{\prime}+\zeta^{\prime\prime}&=\pi\sqrt{-1}d\eta+\lim_{k\to\infty}\sum_{n=-k}^{k}\frac{1}{\eta+n}d\eta\\ &=\pi\sqrt{-1}d\eta+d\log(\eta\prod_{n=1}^{\infty}(1-\frac{\eta^{2}}{n^{2}}))\\ &=\pi\sqrt{-1}d\eta+d\log(\frac{\sin(\pi\eta)}{\pi})=d\log(1-e^{2\pi i\eta}),\end{split}$$

where we made use of Euler’s factorisation identity of $\frac{\sin(\pi\eta)}{\pi\eta}$. The line integrals $\int\zeta^{\prime}$ and $\int\zeta^{\prime\prime}$ are locally holomorphic functions on $M$, defined over the complement of $\{\mu\leq 0,\eta=0\}$ and $\{\mu\geq 0,\eta=0\}$ inside $\mathcal{B}$. Their $T^{2}$-periods lie in $2\pi\sqrt{-1}\mathbb{Z}$: the periods along the $S^{1}$-fibre over $\mathcal{B}$ is $\int_{S^{1}}\sqrt{-1}\vartheta=2\pi\sqrt{-1}$, while the periods along the $S^{1}$-cycle in $M$ lifting $S^{1}\subset\mathbb{R}\times S^{1}\times\mathbb{R}$ can be evaluated by their asymptotic value as $\text{Im}(\eta)\to+\infty$; in particular if we twist the connection $\vartheta$ by a flat connection, then we can always use the choice of $\theta_{\infty}$ to cancel that twist. Thus we can define the holomorphic functions without multivalue issues

$$z_{1}=\exp(\int\zeta^{\prime}),\quad z_{2}=\exp(\int\zeta^{\prime\prime}).$$

We are free to choose the multiplicative constants on $z_{1},z_{2}$ to satisfy the functional equation $z_{1}z_{2}=1-e^{2\pi i\eta}$, from which we see that $z_{1},z_{2}$ extend to global holomorphic functions on $M$, with zero locus $\{\mu\leq 0,\eta=0\}$ and $\{\mu\geq 0,\eta=0\}$ inside $\mathcal{B}$ respectively. Setting $z_{3}=\exp(2\pi\sqrt{-1}\eta)$, we obtain a holomorphic map

$$M\to\{z_{1}z_{2}=1-z_{3}\}\subset\mathbb{C}^{2}_{z_{1},z_{2}}\times\mathbb{C}^{*}_{z_{3}},$$

which is easily seen to be an open embedding.

By looking at the action of the Hamiltonian vector field $\frac{\partial}{\partial\theta}$, we can identify the circle action as

$$e^{i\theta}\cdot(z_{1},z_{2},z_{3})=(e^{i\theta}z_{1},e^{-i\theta}z_{2},z_{3}).$$

By construction the holomorphic volume form $\Omega$ satisfies $\iota_{\frac{\partial}{\partial\theta}}\Omega=d\eta$, which implies $\Omega=-\frac{1}{2\pi}\frac{1}{z_{3}}dz_{1}\wedge dz_{2},$ or equivalently

$$\Omega\wedge d((z_{1}z_{2})^{-1}(1-z_{3})-1)=\frac{1}{2\pi}d\log z_{1}\wedge d\log z_{2}\wedge d\log z_{3}.$$

The reader is advised to compare this discussion to Section 1.1.6.

The Ooguri-Vafa metric comes with an intrinsic parameter $A$, and admits different geometric behaviours at different scales, which can be formalised in terms of blow up limits. Recall by periodicity we may assume $\text{Re}(\eta)$ lies in some interval $[0,1]$. The periodicity condition we chose amounts to the normalisation that $\int_{T^{2}}\Omega=2\pi$.

• •

  When $\sqrt{\mu^{2}+|\eta|^{2}}\lesssim\frac{1}{A}$, the leading order behaviour is $V\sim A+\frac{1}{2\sqrt{\mu^{2}+|\eta|^{2}}}$, and the metric is modelled on the Taub-NUT metric with parameter $A$. The length of the circle fibres $\sim\frac{2\pi}{\sqrt{A}}$. After scaling up the metric by a factor $A$ and taking the limit $A\to\infty$, the pointed Gromov-Hausdorff limit based at the origin is the standard Taub-NUT metric with parameter 1. Most Riemannian curvature is concentrated in this region.

• •

  When $\frac{1}{A}\ll\sqrt{\mu^{2}+|\eta|^{2}}\ll 1$, the leading order behaviour is the constant solution $V\sim A$, and the metric is locally modelled on a flat circle bundle over a flat base $\mathbb{R}^{3}$. A suitable blow up limit space is flat $\mathbb{R}^{3}$.

• •

  When $\sqrt{\mu^{2}+|\eta|^{2}}\sim 1$, the leading order behaviour is still $V\sim A$, but the periodicity condition is now visible. The metric is locally modelled on a flat circle bundle over a flat base $\mathbb{R}_{\mu}\times S^{1}\times\mathbb{R}$. The length of the circle factor of the base is approximately $\sqrt{A}$. If we scale down the metric by a factor $\frac{1}{A}$ and take the limit $A\to\infty$, the pointed Gromov-Hausdorff limit based at the origin is the flat $\mathbb{R}_{\mu}\times S^{1}\times\mathbb{R}$.

• •

  When $1\ll\sqrt{\mu^{2}+|\eta|^{2}}\ll\exp(A)$, the metric becomes almost semiflat up to exponentially small errors, and the leading order behaviour is

  $$V\sim A+\log 2-\gamma_{E}-\frac{1}{2}\log(\mu^{2}+|\text{Im}(\eta)|^{2}).$$

  We remark that the log function grows very slowly. Thus within an exponentially long neck region, the constant solution $V\sim A$ is a good approximation. If we view $A$ as related to the average length of circles, then we can think of $V$ as approximated by a family of constant solutions whose parameter slowly drifts down as we move up the logarithmic scale. The author finds it attractive to call this phenomenon running coupling.

• •

  When $\sqrt{\mu^{2}+|\eta|^{2}}\sim\exp(A)$ the incompleteness of the metric is manifested. This is best understood by viewing the Ooguri-Vafa metric as an effective local description of the hyperKähler metric on a family of collapsing K3 surfaces, and incompleteness is an indication that there is a scale beyond which this description must break down. On the other hand, if we are looking at smaller distance scales, then the Ooguri-Vafa metric becomes better approximations of the K3 metric. In particular, the K3 hyperKähler structure involves 60 parameters while the Ooguri-Vafa metric only involves one scaling parameter $A$ and a gauge parameter in $S^{1}$, but the metric at smaller distance scales are not sensitive to many extra parameters as long as the K3 surfaces are sufficiently collapsed. This phenomenon may be called effective uniqueness or local universality, which is an essential aspect of Gross and Wilson’s gluing construction [11]. The analogy with K. Wilson’s philosophy of effective quantum field theory will be further explained in Section 3.10.

The analysis of the blow up limits reveals the cause d’etre of the Ooguri-Vafa metric. Recall the Taub-NUT metrics arise in a 1-parameter family, which are related by the scaling symmetry. The Ooguri-Vafa metric is obtained conceptually by gluing the Taub-NUT metric to the constant solution. The periodicity condition, which is a kind of integral lattice structure, breaks down the scaling symmetry, and results in an intrinsic gluing parameter $A$. Another major effect of the periodicity condition is the exponential decay of higher Fourier modes, which works via spectral theory, and results in the semiflat asymptotic picture.

A notable feature in the Ooguri-Vafa metric is the appearance of the Green’s function $V$. This is because we are perturbing from the constant solution, and the first order correction to the flat ambient solution natually invovles harmonic functions at least away from the singular locus. The precise nature of the singularity of $V$ is dictated by the topology, or more precisely the Chern class, via the distributional equation.

These principles are sufficient to lead to the discovery of the Ooguri-Vafa metric. While the exact linearity of the equation governing the Gibbons-Hawking ansatz in complex dimension 2 is a fortunate simplifying feature, it does not appear essential in our discussions above. A core idea in this paper is that essentially the same principles dictate how to generalise the Ooguri-Vafa metric to dimension 3.