The lack of examples of special Lagrangian torus fibrations is one main obstacle in implementing the original SYZ proposal for compact Calabi-Yau manifolds (and perhaps this is one of the reasons why Gross and Siebert wanted to develop an algebraic-geometric version). But there are plenty of noncompact examples where one can find explicit special Lagrangian torus fibrations, such as those constructed by Goldstein [61] and Gross [68] in the case of toric Calabi-Yau manifolds. Moreover, open Gromov-Witten invariants which count maps from open Riemann surfaces to the manifold are well-defined in the toric case by the works of Fukaya, Oh, Ohta and Ono [54, 55, 52] (and even in the $S^{1}$-equivariant case by as has been done in the thesis of Liu [120]). So it makes perfect sense to carry out the SYZ proposal directly for toric Calabi-Yau manifolds, without retreating to the tropical world.

This brings us to the realm of local mirror symmetry, which was originally an application of mirror symmetry techniques to Fano surfaces within compact Calabi-Yau 3-folds, and could be derived using physical arguments from mirror symmetry for compact Calabi-Yau manifolds by taking certain limits in the Kähler and complex moduli spaces [98]. Since this mirror symmetry provides numerous interesting examples and predictions, it has been attracting much attention from both physicists and mathematicians [111, 34, 88, 68, 69, 154, 100, 62, 90, 91, 46, 47, 101, 145].

Let $X$ be an $n$-dimensional toric Calabi-Yau manifold, which is necessarily noncompact. To carry out the SYZ construction, we consider a special Lagrangian torus fibration $\mu:X\to B$ constructed by Goldstein and Gross; such a fibration is non-toric, meaning that it is not the usual moment map associated to the Hamiltonian $T^{n}$-action on $X$. The discriminant locus of this SYZ fibration has been analyzed in details by Gross and can be described explicitly. Topologically, the base $B$ is simply an upper half-space in $\mathbb{R}^{n}$, and it admits an integral affine manifold with both singularities and boundary. The pre-image of the boundary $\partial B\subset B$ is a non-toric smooth hypersurface $D\subset X$. The discriminant locus $\Gamma\subset B$ is a real codimension two tropical subvariety sitting inside a hyperplane $H$ which we call the wall in $B$. By definition, the wall(s) inside the base of an SYZ fibration is the loci of Lagrangian torus fibers which bound holomorphic disks with Maslov index zero in $X$. It divides the base into different chambers over which the Lagrangian torus fibers behave differently in a Floer-theoretic sense. In the case of the Gross fibration, the wall $H\subset B$, which is parallel to the boundary hyperplane $\partial B$, divides the base into two chambers.

Now one considers (virtual) counts of holomorphic disks in $X$ bounded by fibers of the SYZ fibration $\mu$ which intersect with the hypersurface $D$ at one point with multiplicity one (i.e. disks with Maslov index two). As a point moves from one chamber to another across the wall, the virtual number of holomophic disks bounded by the corresponding Lagrangian fiber (or genus 0 open Gromov-Witten invariants) jumps, exhibiting a wall-crossing phenomenon. This has been analyzed by Auroux [8, 9] and Chan, Lau and Leung [19] by applying the sophisticated machinery developed by Fukaya, Oh, Ohta and Ono [53]. Notice that there is no scattering phenomenon in this case because there is only one wall. By applying the SYZ dual fibration construction on each chamber in the base, and then gluing the resulting pieces together according to the wall-crossing formulas, we obtain the instanton-corrected or SYZ mirror family $\check{X}$, which is parametrized by the Kähler moduli space of $X$ [19, 3]. The result agrees with the predictions by physical arguments [34, 88].

This SYZ mirror construction is very precise in the sense that it tells us exactly which complex structure on $\check{X}$ is corresponding to any given symplectic structure on $X$ – the defining equation of the mirror $\check{X}$ is an explicit expression written entirely in terms of the Kähler parameters and disk counting invariants of $X$. For example, the SYZ mirror of $X=K_{\mathbb{P}^{2}}$ is given by³³ 3 More precisely, the SYZ mirror of $K_{\mathbb{P}^{2}}$ is the Landau-Ginzburg model $(\check{X},W=u)$; see the next section.

(6.1)

$$\check{X}=\left\{(u,v,z_{1},z_{2})\in\mathbb{C}^{2}\times(\mathbb{C}^{\times})^{2}\mid uv=1+\delta(q)+z_{1}+z_{2}+\frac{q}{z_{1}z_{2}}\right\},$$

where $q$ is the Kähler parameter measuring the symplectic area of a projective line inside the zero section of $K_{\mathbb{P}^{2}}$ over $\mathbb{P}^{2}$, and

(6.2)

$$1+\delta(q)=\sum_{k=0}^{\infty}n_{k}q^{k}$$

is a generating series of genus 0 open Gromov-Witten invariants.

Furthermore, the SYZ construction naturally defines the SYZ map, which is a map from the Kähler moduli space of $X$ to the complex moduli space of $\check{X}$. As conjectured by Gross and Siebert, the SYZ mirror family should be written in canonical coordinates, or put it in another way, the SYZ map should give an inverse to the mirror map. Evidences for this conjecture for toric Calabi-Yau surfaces and 3-folds were given in [19, 107], and Chan, Lau and Tseng [20] proved the conjecture in the case when $X$ is the total space of the canonical line bundle over a compact toric Fano manifold. Recently, by applying orbifold techniques, the conjecture was proved for all toric Calabi-Yau manifolds in [18].

The main challenge in proving these results is the computation of the genus 0 open Gromov-Witten invariants defined by Fukaya, Oh, Ohta and Ono [54]. Since the moduli spaces of holomorphic disks are usually highly obstructed, these invariants are in general very difficult to compute. Currently, there are only very few techniques available (such as open/closed equalities, toric mirror theorems, degeneration techniques, etc). For example, the invariants in (6.2) can be computed:

$\displaystyle n_{k}=1,-2,5,-32,286,-3038,35870,\ldots$

for $k=0,1,2,3,4,5,6,\ldots$, which agrees with period computations in [62].

We should mention that the SYZ construction can be carried out also in the reverse direction [3] (see also [17, Section 5]). For example, starting with the conic bundle (6.1), one can construct an SYZ fibration using similar techniques as in [61, 68]. Although the discriminant locus is of real codimension one in the case, one can construct the SYZ mirror and this gives us back the toric Calabi-Yau 3-fold $K_{\mathbb{P}^{2}}$, as expected.⁴⁴ 4 More precisely, the SYZ mirror of (6.1) is the complement of a smooth hypersurface in $K_{\mathbb{P}^{2}}$.

Nevertheless, outside the toric setting, it is not clear how SYZ constructions can be performed in such an explicit way. One major problem is the well-definedness of open Gromov-Witten invariants. Only in a couple of non-toric cases (see Liu [120] and Solomon [150]) do we have a well-defined theory of open Gromov-Witten invariants.⁵⁵ 5 There are, however, recent works of Fukaya [50, 51] on defining disk counting invariants for compact Calabi-Yau 3-folds.