The incompleteness of the Ooguri-Vafa type metric on the negative vertex has a strong analogy with the positive vertex as discussed in detail in Section 3.10. The key point is that asymptotes of the first order corrections $v$ and $w^{p\bar{q}}$ lead naturally to a renormalisation flow equation, which in turn predicts the drifting of coupling constants $a_{p\bar{q}}$ over many log scales.

If we follow the discussion of Section 3.10, but replace the asymptote (3.6) by (4.17), then we find that in the following variables

$$p_{1}=\sqrt{a_{2\bar{2}}},\quad p_{2}=\sqrt{a_{1\bar{1}}},\quad p_{3}=\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}},$$

the renormalisation flow equation for the negative vertex is given as

(4.38)

$$\begin{cases}\frac{d}{d\lambda}p_{1}^{2}=-\frac{1}{2p_{2}}-\frac{1}{2p_{3}},\\ \frac{d}{d\lambda}p_{2}^{2}=-\frac{1}{2p_{1}}-\frac{1}{2p_{3}},\\ \frac{d}{d\lambda}p_{3}^{2}=-\frac{1}{2p_{1}}-\frac{1}{2p_{2}},\\ \frac{d}{d\lambda}\text{Im}(a_{2\bar{1}})=0,\end{cases}$$

where $\lambda$ is the log scale parameter. The rest of this Section is concerned with geometric interpretations.

The renormalisation flow equation implies that

$$\text{Im}(a_{2\bar{1}})=\text{constant}.$$

This is compatible with the fact that $\text{Im}(a_{2\bar{1}})$ is the cohomological invariant determined by integrating the Kähler form on the $T^{2}$-cycle.

More interestingly, the evolution of $p_{1},p_{2},p_{3}$ is formally identical to the renormalisation flow equation (3.16) for the positive vertex. This can be explained in terms of semiflat mirror symmetry (cf. Section 1.1.1) as follows, assuming the homological constraint $\text{Im}(a_{2\bar{1}})=0$, namely $a_{p\bar{q}}$ is a real symmetric matrix.

In general, given a semiflat SYZ fibration, the mirror SYZ fibration is obtained by replacing the torus fibres by their dual tori, interchanging the symplectic moment coordinates on the SYZ base with the complex affine coordinates on the SYZ base, and keeping the same Riemannian metric on the base. We apply this to the constant solution relevant to the positive vertex case (cf. Example 1.6)

$$V^{ij}=a_{ij},\quad W=A=\det(a_{ij}),$$

whose SYZ base is equipped with the Euclidean metric

$$a_{ij}d\mu_{i}\otimes d\mu_{j}+Ady^{2}$$

written in the two symplectic moment coordinates $\mu_{1},\mu_{2}$ and a complex affine coordinate $y=\text{Im}(\eta)$. The SYZ mirror is the constant solution relevant to the negative vertex

$$W^{i\bar{j}}=a_{ij},\quad V=A,$$

whose SYZ base is equipped with the Euclidean metric

$$a_{ij}dy_{i}\otimes dy_{j}+Ad\mu^{2}$$

written in the two complex affine coordinate $y_{1}=\text{Im}(\eta_{1}),y_{2}=\text{Im}(\eta_{2})$ and a symplectic moment coordinate $\mu$. The crucial point is that mirror symmetry means the matrices $a_{ij}$ appearing in both cases are the same.

Now the Ooguri-Vafa type metrics are perturbations of some constant solution at any given log scale, and the coupling constants $a_{ij}$ drift slowly according to the renormalisation flow as the log scale changes. The formal coincidence of the renormalisation flow equations for both the positive vertex and the negative vertex agrees with semiflat mirror symmetry.

Acknowledgement. The author thanks his PhD supervisor Simon Donaldson and co-supervisor Mark Haskins for their inspirations, and Song Sun for discussions.