This Section is an informal discussion concerning special Lagrangian 3-tori $L\simeq T^{3}$ on the negative vertex $M^{-}_{\nu}$.

The generic special Lagrangian 3-tori $L$ are expected to be isotopic to the $T^{3}$ lying over the 2-tori in the 5-dimensional base defined by

(4.33)

$$(y_{1},y_{2},\mu)=\text{const}.$$

The Lagrangian requirement then imposes a homological constraint in the light of Proposition 4.34:

(4.34)

$$\int_{T^{2}}\omega_{-}=\text{Im}(a_{2\bar{1}})=0,$$

or equivalently $(a_{p\bar{q}})$ is real symmetric. The interpretation is that the Ooguri-Vafa type metrics we constructed on the negative vertex can be the metric model for the SYZ fibration only if the homological constraint is satisfied; when this fails, they may still be the local model for other types of degenerating 3-fold Calabi-Yau metrics which do not admit a global special Lagrangian 3-torus fibration.

From now on in this Section we assume the homological constraint, and proceed to speculate on the geometric features of the special Lagrangian 3-tori $L$, without attempting to prove existence results.

First, notice that outside a tubular neighbourhood of the singular locus $S$, the metric $g_{-}$ is a perturbation of the flat model $g_{\text{flat}}$ (cf. Example 1.6), namely the generalised Gibbons-Hawking construction applied to the constant solution

$$V=A,\quad W^{p\bar{q}}=a_{p\bar{q}}.$$

On the flat model it is elementary to check that the map to $\mathbb{R}^{3}$ defined by $(y_{1},y_{2},\mu)$ have special Lagrangian fibres, which are flat 3-tori invariant under the $S^{1}$-action. In other words, to crudest approximation the map

(4.35)

$$M^{-}_{\nu}\xrightarrow{(y_{1},y_{2},\mu)}\mathbb{R}^{3}$$

is an approximate special Lagrangian fibration. Most of these $T^{3}$-fibres stay far away from the curvature radius along $S$, so it is likely that in the generic region these $T^{3}$ can be perturbed into a genuine special Lagrangian fibration with respect to the Calabi-Yau structure $(g_{-},\omega_{-},J,\Omega)$, while maintaining the $S^{1}$-invariance.

Near $S$ the features of the special Lagrangians $L$ have strong resonance with Joyce’s work [14] (cf. Section 1.1.5). It is natural to expect $L$ to be $S^{1}$-invariant. Around $P\in S$, the Calabi-Yau structure is transversely modelled on $g_{\text{NUT}}$ (cf. Section 4.5), so the $S^{1}$-reduction of the special Lagrangian condition

$$\omega_{-}|_{L}=0,\quad\text{Im}(\Omega_{-})|_{L}=0$$

approximately reads:

(4.36)

$$\begin{cases}\mu=\text{constant},\\ \{(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})d\xi_{1}\wedge d\bar{\xi}_{1}+Ad\xi_{2}\wedge d\bar{\xi}_{2}\}|_{L/S^{1}}=0,\\ \text{Im}(d\xi_{1}\wedge d\xi_{2})|_{L/S^{1}}=0.\end{cases}$$

To render the analogy with Joyce [14] more transparent, we introduce real variables $\tilde{x}_{1},\tilde{x}_{2},\tilde{u}_{1},\tilde{u}_{2}$ such that $\xi_{1}=\tilde{x}_{1}-\sqrt{-1}\tilde{u}_{1},\xi_{2}=\tilde{x}_{2}+\sqrt{-1}\tilde{u}_{2}.$ Representing $L$ locally by

$$\tilde{u}_{1}=\tilde{u}_{1}(\tilde{x}_{1},\tilde{x}_{2}),\quad\tilde{u}_{2}=\tilde{u}_{2}(\tilde{x}_{1},\tilde{x}_{2}),\quad\mu=\text{constant},$$

then (4.36) takes the form of the nonlinear Cauchy-Riemann equation

(4.37)

$$\frac{\partial\tilde{u}_{1}}{\partial\tilde{x}_{1}}=\frac{\partial\tilde{u}_{2}}{\partial\tilde{x}_{2}},\quad\frac{\partial\tilde{u}_{1}}{\partial\tilde{x}_{2}}=-(1+\frac{1}{2A\sqrt{\mu^{2}+\tilde{x}_{1}^{2}+\tilde{u}_{1}^{2}}})\frac{\partial\tilde{u}_{1}}{\partial\tilde{x}_{2}},$$

which is very similar to the key equations in [14]. Morever $L$ should asymptotically match up with fibres of (4.35) at far distance from $S$, described by the affine condition (4.33).

We can use this information to speculate on the nature of singularities in line with Joyce [14]. For $\mu\neq 0$ the equations are nonsingular, so the special Lagrangians $L$ will be smooth. When $\mu=0$ the $T^{3}$-fibres of (4.35) intersect $S$ if and only if $(y_{1},y_{2})$ lies in the amoeba $\text{Im}(S)$, and we expect a perturbation of such fibres to produce special Lagrangians $L$ with singularities. In the subcase where $(y_{1},y_{2})$ lies in the interior of the amoeba, there are two transverse intersection points $L\cap S$, at which we expect to create a pair of special Lagrangian $T^{2}$-cone singularities. At the boundary of the amoeba these two intersection points merge together, and the singularities disappear outside the amoeba.

The fine details of a special Lagrangian $L$ near a transverse intersection point with $S$ is conjecturally modelled by an entire solution to the nonlinear Cauchy-Riemann equation (4.37) over $\mathbb{R}^{2}_{\tilde{x}_{1},\tilde{x}_{2}}$, which has a local special Lagrangian $T^{2}$-cone singularity at the origin and is asymptotic to (4.33) at infinity. $L$ is then obtained by gluing this local picture to the corresponding fibres of (4.35) away from $S$.

A salient feature of Joyce [14] is that the special Lagrangian fibration can fail to be defined by smooth maps (cf. Section 1.1.5). This is compatible with this Chapter. The key point is that the absence of an a priori smooth topology forces us to work with tensors of low regularity (cf. Section 4.5), and the smooth structure along $S$ only emerges a posteriori after solving the Monge-Ampère equation. Thus one neither expects to produce a model special Lagrangian fibration defined by smooth maps, nor expects smoothness properties to persist after perturbation inside function spaces of low regularity.