In case the special Lagrangian torus fibrations do not admit any singular fibers, the SYZ picture is particularly nice. McLean’s classic results [126] give us two naturally defined integral affine structures¹¹ 1 An integral affine structure on a manifold is an atlas of charts whose transition maps are all integral affine linear transformations. on the base manifold $B$ of a special Lagrangian torus fibration: the symplectic and complex affine structures, and mirror symmetry can be explained neatly via these structures. More specifically, a normal vector field $v$ to a fiber $L_{b}:=\mu^{-1}(b)$ determines a 1-form $\alpha:=-\iota_{v}\omega\in\Omega^{1}(L_{b};\mathbb{R})$ and an $(n-1)$-form $\beta:=\iota_{v}\textrm{Im }\Omega\in\Omega^{n-1}(L_{b};\mathbb{R})$, where $\omega$ and $\Omega$ are the Kähler form and holomorphic volume form on $X$ respectively. McLean [126] proved that the corresponding deformation is special Lagrangian if and only if both $\alpha$ and $\beta$ are closed. By identifying $TB$ with $H^{1}(L_{b};\mathbb{R})$ using the cohomology class of $\alpha$, we get the symplectic affine structure on $B$, while identifying $TB$ with $H^{n-1}(L_{b};\mathbb{R})$ using the cohomology class of $\beta$ gives us the complex affine structure on $B$. We also have the McLean metric defined by

$$g(v_{1},v_{2}):=-\int_{L_{b}}\iota_{v_{1}}\omega\wedge\iota_{v_{2}}\textrm{Im }\Omega.$$

In his illuminating paper [86], Hitchin explains how these structures are all related through the Legendre transform. If we denote by $x_{1},\ldots,x_{n}$ the local affine coordinates on $B$ with respect to the symplectic affine structure, then locally the McLean metric can be written as the Hessian of a convex function $\phi$ on $B$, i.e. $g\left(\frac{\partial}{\partial x_{i}},\frac{\partial}{\partial x_{j}}\right)=\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}$. Furthermore, setting $\check{x}_{i}:=\partial\phi/\partial x_{i}$ ($i=1,\ldots,n$) gives precisely the local affine coordinates on $B$ with respect to the complex affine structure, and if

$$\check{\phi}:=\sum_{i=1}^{n}\check{x}_{i}x_{i}-\phi(x_{1},\ldots,x_{n})$$

is the Legendre transform of $\phi$, then we have $x_{i}=\partial\check{\phi}/\partial\check{x}_{i}$ and $g\left(\frac{\partial}{\partial\check{x}_{i}},\frac{\partial}{\partial\check{x}_{j}}\right)=\frac{\partial^{2}\check{\phi}}{\partial\check{x}_{i}\partial\check{x}_{j}}$.

If additionally we assume that the fibration $\mu:X\to B$ admits a Lagrangian section, then a theorem of Duistermaat [41] implies that there are global action-angle coordinates so that we can write

$$X=T^{*}B/\Lambda^{\vee},$$

where the lattice $\Lambda^{\vee}\subset T^{*}B$ is locally generated by $dx_{1},\ldots,dx_{n}$, and $\omega$ can be identified with the canonical symplectic form

$$\omega=\sum_{i=1}^{n}dx_{i}\wedge du_{i}$$

on $T^{*}B/\Lambda^{\vee}$. Here $u_{1},\ldots,u_{n}$ are the fiber coordinates on $T^{*}B$.

In this case, the mirror of $X$ is simply given by

$$\check{X}:=TB/\Lambda,$$

where the lattice $\Lambda\subset TB$ is locally generated by $\partial/\partial x_{1},\ldots,\partial/\partial x_{n}$. The quotient $\check{X}$ has a natural complex structure whose holomorphic coordinates are given by $z_{i}:=\exp(x_{i}+\mathbf{i}y_{i})$, where $y_{1},\ldots,y_{n}$ are the fiber coordinates on $TB$ dual to $u_{1},\ldots,u_{n}$. This constructs the mirror of $X$ as a complex manifold with a nowhere vanishing holomorphic volume form

$$\check{\Omega}:=d\log z_{1}\wedge\cdots\wedge d\log z_{n}.$$

Moreover, there is an explicit fiberwise Fourier–type transform, which we call the semi-flat SYZ transform $\mathcal{F}^{\textrm{semi-flat}}$, that carries $\exp\mathbf{i}\omega$ to $\check{\Omega}$; see [22, Section 2] for more details.

Now if we switch to the complex affine structure on $B$, then we get a symplectic structure on $\check{X}$ which is compatible with its complex structure so that the mirror $\check{X}$ becomes a Kähler manifold. Furthermore, if the function $\phi$ above satisfies the real Monge-Ampère equation

$$\textrm{det}\left(\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}\right)=\textrm{constant},$$

then we obtain $T^{n}$-invariant Ricci-flat metrics on both $X$ and its mirror $\check{X}$. The induced metric on $B$ is called a Monge-Ampère metric and $B$ is called a Monge-Ampère manifold. This links mirror symmetry to the study of real Monge-Ampère equations and affine Kähler geometry, where Cheng and Yau had made substantial contributions [31, 32, 33] before even mirror symmetry was discovered. The construction of Monge-Ampère metrics on affine manifolds with singularities has since been an important question in both affine geometry and the study of the SYZ conjecture. The highly nontrivial works of Loftin, Yau and Zaslow [121, 122] constructed such metrics near the “Y” vertex, a typical type of singularity in the 3-dimensional case. But other than this, not much is known.

So the SYZ conjecture indeed paints an appealing picture for mirror symmetry in the semi-flat case; many more details on semi-flat SYZ mirror symmetry were worked out by Leung in [110]. Unfortunately, this nice picture can hold true only at the large complex structure/volume limits where all instanton corrections are suppressed. Away from the limits, special Lagrangian fibrations will have singular fibers and the mirror can no longer be obtained simply by dualizing a fibration.