We now explain how this paper fits into a program to prove the metric version of the SYZ conjecture for Calabi-Yau 3-folds (cf. Conjecture 1.1). This program runs as follows:

1. (1)

   Produce the metric models on the positive and negative vertices.

2. (2)

   The metric structure near the edges in the Gross-Ruan picture are expected to be modelled on a fibration by Ooguri-Vafa metrics. The problem is that Ooguri-Vafa metrics transverse to the edge depend on a moduli parameter which can vary along the edge, possibly governed by an adiabatic equation.

3. (3)

   The SYZ base $B$ as an affine manifold with singularity along a trivalent graph, can be produced from algebraic geometry in some degree of generality [32][16]. The central problem is then to solve the real Monge-Ampère equation with some prescribed singularities along the trivalent graph. This would allow us to produce a semiflat metric which models the generic region of the SYZ fibration.

4. (4)

   One then glues together the metric models in various regions to obtain the global Calabi-Yau metric on the Calabi-Yau 3-fold, similar to Gross and Wilson’s work on K3 surfaces [11]. Some Fourier analysis is needed to prove exponential decay estimates for deviation from the semiflat metric.

5. (5)

   The existence of the SYZ fibration in the generic region is expected to be a straightforward consequence of the gluing construction. To produce the SYZ fibration near the trivalent graph, one needs to produce models for singular SYZ fibrations on the metric models, and set up a Fredholm deformation theory to ensure the SYZ fibration persists when the metric deforms.

The principal contribution of this paper is to carry out Step (1), and our linear analysis is likely to be useful in Step (4). Some informal digressions in this paper go some way towards addressing difficulties in the other Steps:

In Step (3), the singularity of the real Monge-Ampère equation near the trivalent graph in $B$ should match up with the asymptotic behaviour of the metric models around the trivalent graph, in order to enable the gluing construction in Step (4). This requires understanding how the Ooguri-Vafa type metrics on the vertices transition into the generic region of the SYZ fibration. We propose a mechanism called running coupling for this transition to take place over an exponentially long neck region (cf. Section 3.10 and 4.13). Starting from the observation that Ooguri-Vafa type metrics naturally arise in a family parametrised by some positive definite rank 2 matrices referred to as coupling constants, we argue semi-heuristically that these coupling constants drift slowly as the logarithmic scale increases, governed by an ODE called the renormalistion flow equation which can be solved exactly.

The behaviour of the special Lagrangian fibrations is discussed in Corollary 2.29, Corollary 3.35 and Section 4.12. In both the Taub-NUT type $\mathbb{C}^{3}$ case and the positive vertex case, the $T^{2}$-symmetry provides two symplectic moment coordinates and another real coordinate $\text{Im}(\eta)$, which define a map to $\mathbb{R}^{3}$ whose fibres are $T^{2}$-invariant special Lagrangians with phase zero. However, Joyce’s critique suggests the singularity structure of this SYZ fibration is not stable under metric perturbation.

In the negative vertex case (cf. Section 4.12), there is a homological constraint for the SYZ fibration to exist, namely the Hermitian matrix $a_{p\bar{q}}$ needs to be symmetric. When this constraint holds, we outline a speculative description of a $U(1)$-invariant SYZ fibration on the model metric, and explain how it fits with Joyce’s work on $U(1)$-invariant special Lagrangians. The case where this constraint does not hold is possibly relevant for metric degenerations outside the scope of the SYZ conjecture.