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 $\int_{T^{2}}\Omega=2\pi$.

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  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.

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  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 $\mathbb{R}^{3}$.

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  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 $\mathbb{R}_{\mu}\times S^{1}\times\mathbb{R}$.

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  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.

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  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.