In the summer of 1996, Strominger, Yau and Zaslow [153] made a ground-breaking proposal which gave the first geometric explanation for mirror symmetry:

In a nutshell, this is saying that the mysterious mirror phenomenon is simply a Fourier transform! This remarkable and far-reaching conjecture not only provides a beautiful geometric explanation to mirror symmetry, but also suggests that a mirror partner of any given Calabi-Yau manifold $X$ can be constructed by fiberwise dualizing a special Lagrangian torus fibration on $X$ (or $T$-duality). It immediately attracted much attention from both mathematicians and physicists, and has lead to a flourishing of research work aiming at either solving the conjecture or applying it to understand the geometry underlying mirror symmetry.

Before going on, let us go through briefly the heuristic arguments behind the SYZ conjecture. First of all, a key feature in string theory is the existence of Dirichlet branes, or D-branes. Physical arguments suggest that D-branes in the B-model (or simply B-branes) are coherent sheaves over complex subvarieties while D-branes in the A-model (or A-branes) are special Lagrangian submanifolds equipped with flat connections. As mirror symmetry predicts an isomorphism between the A-model of $X$ and the B-model of $\check{X}$, the moduli space of an A-brane on $X$ should be identified with the moduli space of the mirror B-brane on $\check{X}$.

Now, points on $\check{X}$ can certainly be regarded as B-branes. And as $\check{X}$ itself is the moduli space these B-branes, it should be identified with the moduli space of certain A-branes $(L,\nabla)$ on $X$, where $L\subset X$ is a special Lagrangian submanifold and $\nabla$ is a flat $U(1)$-connection on $L$. Also, since $\check{X}$ is swept by its points, $X$ should be swept by these special Lagrangian submanifolds $L$ as well. By McLean’s theorem [126], the moduli space of a special Lagrangian submanifold $L\subset X$ is unobstructed and modelled on $H^{1}(L;\mathbb{R})$, while the moduli space of flat $U(1)$-connections (modulo gauge) on $L$ is given by $H^{1}(L;\mathbb{R})/H^{1}(L;\mathbb{Z})$. Therefore, in order to match the dimensions, we should have $\textrm{dim }H^{1}(L;\mathbb{R})=\textrm{dim}_{\mathbb{C}}\check{X}=n$. Hence $X$ should admit a special Lagrangian torus fibration

$$\mu:X\to B.$$

Moreover, the manifold $\check{X}$ itself can be regarded as a B-brane whose moduli space is a singleton and it intersects each point in $\check{X}$ once, so the corresponding A-brane should give a special Lagrangian section $\sigma$ to $\mu$ with $H^{1}(\sigma;\mathbb{R})=0$. In particular, the base $B$ should have first Betti number $b_{1}=0$.

Applying the same argument to $\check{X}$ yields a special Lagrangian torus fibration with section

$$\check{\mu}:\check{X}\to\check{B}.$$

Now for a torus fiber $L_{b}:=\mu^{-1}(b)\subset X$, its dual $L_{b}^{\vee}$ can be viewed as the moduli space of flat $U(1)$-connections on $L$ which, under mirror symmetry, correspond to points in $\check{X}$. This shows that $L_{b}^{\vee}$ is a submanifold in $\check{X}$. With more elaborated arguments, one can see that $L_{b}^{\vee}$ can in fact be identified with a special Lagrangian torus fiber of $\check{\mu}$, and hence deduce that $\mu$ and $\check{\mu}$ are fibrations over the same base which are fiberwise dual to each other.

Notice that we have a transform carrying special Lagrangian torus fibers in $X$ (A-branes) to points in $\check{X}$ (B-branes). This is an instance of a fiberwise Fourier(-Mukai)–type transform. More generally, there should exist geometric Fourier transforms mapping symplectic-geometric data on $X$ to complex-geometric data on $\check{X}$. We call these SYZ transforms. In the original SYZ paper [153], it was inferred that the behavior of the Ricci-flat metrics on the mirror $\check{X}$ should differ from the semi-flat Calabi-Yau metrics, constructed earlier by Greene, Shapere, Vafa and Yau in an important paper [64], by contributions from instanton corrections. As we shall see, a key step in the investigation of mirror symmetry is to understand these corrections, which should come from higher Fourier modes of the SYZ transforms.