We now give a deeper perspective on the incompleteness of the metric, and a semi-heuristic discussion about how to partially overcome one of the main limitations of the perturbation method: the final metric one constructs is by necessity $C^{0}$-close to the metric ansatz one starts with.

The main insights are as follows. The Ooguri-Vafa type metric is intended as an effective local description below a certain distance scale for the collapsing family of Calabi-Yau metrics on compact manifolds near the large complex structure limit. Our starting assumption is that the metric is a perturbation of a constant solution after incorporating topology. These constant solutions come naturally in a family parametrised by the coupling constants $a_{ij}$, which have a geometric meaning in terms of the size and shape of the generic $T^{2}$-fibres in the local region. The nontrivial topology manifests itself in a distributional equation which dictates the first order corrections $\tilde{\alpha}_{i}$ to the constant solutions, and after Fourier analysis we see the dominant correction terms $\bar{\alpha}_{i}$ depend logarithmically on $\mu_{i},\eta$. The slow growth of $\log$ means it can be treated as a perturbation term in an exponentially long region, but once we attempt to go beyond, the correction will have a perceptible effect on the size and shape of the average $T^{2}$-fibres, which would break down our initial effective description via the original constant solution. This suggests that the coupling constants in the effective description drift slowly as we move up the logarithmic scale, a phenomenon we call running coupling. Morever, the precise formula of these log corrections dictate how these coupling constants change as a function of the logarithmic scale, which we will discuss under the name of renormalisation flow equation. Geometrically, the ansatz metrics naturally come in families, and each time we move up a log scale, we really should glue a different ansatz with slightly changed coupling constants to the previous ansatz. The fact that at very large distance scales the original ansatz should be replaced by another ansatz within the same family, is the deep reason why the ansatz metric is incomplete.

The terminologies are based on the following analogy. According to my rudimentary understanding of high energy physics, Quantum Electrodynamics (QED) is intended as an effective description below a certain energy scale for some more sophisticated theories. The starting assumption of Feynman diagram calculations in QED is that the scattering amplitudes are perturbations of the free field theory, after adding new interaction terms in the Lagrangian. These interaction terms come naturally in a family parametrised by the coupling constants, whose physical meaning is related to the observed charges in low energy experiments. Loop calculations in Feynmann diagrams suggest that the coupling constants depend on the energy scale at which one conducts the experiments, a phenomenon known as running coupling. The equation which governs how the coupling constants change as a function of the cutoff energy scale is known as the renormalisation flow equation.

We now flesh out the ideas in the setting of our Ooguri-Vafa type metrics on the positive vertices. We begin by recalling some main features about the family of ansatz metrics. The construction begins with the choice of parameters $a_{ij}$, and outputs a generalised Gibbons-Hawking metric associated to the data

Here the subscript is to emphasize the dependence on $a_{ij}$. The ambiguity of twisting by a flat connection is not important for the discussions below. The functions $\tilde{\alpha}_{i}$ generically behave like the logarithmic functions $\bar{\alpha}_{i}$. The constants above refer to numbers independent of $\mu_{1},\mu_{2},\eta$ which are up to our choice (cf. Remark 3.3). The significance of this extra freedom is that if we are interested only in the ansatz at one particular logarithmic scale, then we can always adjust the constants to cancel some log factors in $\tilde{\alpha}_{i}$ so that $a_{ij}$ is the average value of $V^{ij}_{a}$ over this log scale. The QFT analogue of these constants are called counterterms. This step is needed to back up the idea that the constant solution defined by $a_{ij}$ really offers an effective description at the given log scale of the metric, suitably away from the discriminant locus $\mathfrak{D}$. This issue did not appear previously, because when $A^{-1/2}\varrho\lesssim 1$ these constants were essentially zero.

The central question is how these effective coupling constants $a_{ij}$ vary as a function of the log scale $\lambda$. Moving up to the next log scale means

Since $a_{ij}$ drifts very slowly, to zeroth order we can treat them as constants. Now $\bar{\alpha}_{i}$ are explicit functions given by formula (3.6), whose values receive a small increment as we move up the log scale:

Since $\tilde{\alpha}_{i}$ are generically almost the same as $\bar{\alpha}_{i}$, this means as we move up a log scale, the average value of $V^{ij}_{a}$ drift by

In our viewpoint, it means the first order change of the coupling constants when we move up a log scale is

We now denote

As we move up a log scale,

We have presented this discussion from a discretized viewpoint, which the author thinks is conceptually simpler. The continuum version is the renormalisation flow equation

The remarkable fact is that this ODE system is exactly solvable.

The rest of the Section offers a heuristic interpretation of the renormalisation flow, whose power is to predict effective metric behvaiour up to a very large distance scale. It is helpful to keep in mind the Gross-Wilson K3 metric [11]. The positive vertex is best understood as part of a global SYZ $T^{3}$-fibration on a Calabi-Yau 3-fold near the large complex structure limit consisting of a finite number of overlapping pieces with simple complex geometric descriptions (cf. review Section 1.1.6). The renormalisation flow breaks down when $\min_{i}p_{i}$ becomes negative, which indicates a metric transition into a different piece in the 3-fold.

In our normalisation convention $\text{Vol}(T^{3})=4\pi^{2}\sim 1$. In the generic region of the 3-fold, it is reasonable to expect the 3 circle factors of the SYZ $T^{3}$-fibres to have comparable length scales, so $\text{diam}(T^{3})\sim 1$. In contrast, our starting point for constructing the Ooguri-Vafa type metric on the positive vertex is that a $T^{2}$ factor inside $T^{3}$ has much smaller diameter compared to $\text{diam}(T^{3}/T^{2})$. In order for the Ooguri-Vafa type metric to smoothly transition into the generic region of the SYZ fibration, we require an exponentially long neck region, modelled by the renormalisation flow.

The renormalisation flow has the curious feature that at smaller distance scales $\sum p_{i}$ becomes larger but $|p_{i}-p_{j}|$ becomes smaller, so the scale invariant ellipticity bound $CA^{1/2}\delta_{ij}\leq a_{ij}\leq CA^{1/2}\delta_{ij}$ works better at smaller distance scales. Suppose this bound holds throughout the renormalisation flow until $A$ decreases to $A\sim 1$ where the metric transitions into the generic region, then in Proposition 3.36 the constants $K_{1},K_{2}=O(1)$. Consequently at smaller distance scales, where $t$ is large, the terms $K_{1},K_{2}$ are neglegible, and the solution of the renormalisation flow is approximated by the special solution

This special solution is invariant under the $S_{3}$-discrete symmetry interchanging the 3 edges $\mathfrak{D}_{i}$. The insight is that the most symmetric configuration of $a_{ij}$ is the attractive fixed point of the renormalisation flow.

We can also use the special solution to approximately count the number of log scales involved in the neck region. At the innermost log scale

and at the outermost log scale

The total number of log scales is roughly $K_{3}\sim\frac{2}{3}(\frac{4A_{\text{max}}}{3})^{3/4}$. The diameter of the neck region is of the order