An $n$-dimensional Calabi-Yau (CY) manifold $(X,\omega,\Omega)$ is a Kähler manifold with a nowhere vanishing holomorphic volume form $\Omega$, satisfying the complex Monge-Ampère (MA) equation

$$\omega^{n}=\text{const}\Omega\wedge\overline{\Omega},$$

(1)

which implies the Ricci flatness of the metric. Furthermore, such manifolds admit parallel spinors, hence are candidates for the target space metrics of supersymmetric type II string theories. Special Lagrangians of phase $\theta$ are $n$-dimensional submanifolds $L$ satisfying

$$\omega|_{L}=0,\quad\text{Im}(e^{-i\theta}\Omega)|_{L}=0.$$

(2)

These are absolute minimizers within their homology classes, due to the calibration inequality

$$\int_{L}\text{Re}(e^{-i\theta}\Omega)\leq\int_{L}dvol_{L}=\text{Vol}(L)$$

(3)

saturated precisely by the special Lagrangians. Physically, these correspond to the support of BPS D-branes.

The Strominger-Yau-Zaslow conjecture [67] in its primitive form asks:

The physical origin of the SYZ conjecture [67] comes largely from mirror symmetry, and a very brief sketch is as follows. From homological mirror symmetry, one expects a compact Calabi-Yau manifold $X$ admits a mirror $X^{\vee}$, such that the category of D-branes on both sides are identified. On the holomorphic side $X^{\vee}$ (‘B-side’), the points $x\in X^{\vee}$ support skyscrapper sheaves $\mathcal{O}_{x}$, which should correspond to certain Lagrangian branes inside the symplectic side $X$ (‘A-side’). The extension groups $\text{Ext}^{*}(\mathcal{O}_{x},\mathcal{O}_{x})\simeq H^{*}(T^{n})$, which suggests the Lagrangian branes are torus objects. For $x\neq y$, the Ext groups between $\mathcal{O}_{x},\mathcal{O}_{y}$ would vanish, which suggests (inconclusively¹¹ 1 The vanishing of Floer cohomology does not imply the vanishing of Floer cochain spaces, and there seems to be no strong argument to rule out intersecting special Lagrangians.) that the tori are disjoint, leading to the speculation of the Lagrangian fibration structure. The assertion about special Lagrangians, is however beyond mere homological mirror symmetry, and comes from the BPS condition on the D-branes. The moduli space of all $\mathcal{O}_{x}$ is the mirror manifold $X^{\vee}$, which should then be identified with the moduli space of BPS branes supported on the special Lagrangian tori. This moduli interpretation gives rise to a zeroth order approximation of Kähler structure on the mirror manifold $X^{\vee}$, subject to the higher order corrections related to the holomorphic discs (‘instanton corrections’), whose effect is supposedly exponentially suppressed except near the singular fibres. Ignoring the subtleties of singular fibres, then the SYZ picture offers a program to reconstruct the mirror, and interpret homological mirror symmetry as a version of Fourier-Mukai transform (‘Mirror symmetry is T-duality’).

Differential geometrically, the main evidence presented in the SYZ paper is the semiflat metrics. Consider the logarithm map $\text{Log}_{t}:(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$ over some open convex subset $U\subset\mathbb{R}^{n}$,

$$\text{Log}_{t}(z_{1},\ldots z_{n})=\frac{1}{\log|t|}(\log|z_{1}|,\ldots\log|z_{n}|).$$

Imposing $T^{n}$ symmetry, then by an elementary Hessian computation, $\phi$ is a smooth strictly convex function downstairs on $U$ if and only if its pullback to $\text{Log}_{t}^{-1}(U)$ is a smooth Kähler potential, and the Calabi-Yau condition

$$(dd^{c}\phi\circ\text{Log}_{t})^{n}=\frac{\text{const}}{|\log|t||^{2n}}\prod d\log z_{i}\wedge d\overline{\log z_{i}}$$

is equivalent to the real Monge-Ampère equation

$$\det(D^{2}\phi)=\text{const}.$$

In this setting, the metric $dd^{c}\phi\circ\text{Log}_{t}$ is called semiflat, because its restriction to the $T^{n}$ fibres are Euclidean, due to $T^{n}$-symmetry. With respect to the Calabi-Yau structure

$$\omega=dd^{c}(\phi\circ\text{Log}_{t}),\quad\Omega=\sqrt{-1}^{n}\prod d\log z_{i},$$

the $T^{n}$ fibres are special Lagrangians of phase zero. The purpose of introducing the normalization parameter $t$, is that as $t\to 0$, the $T^{n}$-fibres shrink down to zero size, and the Calabi-Yau metrics $dd^{c}(\phi\circ\text{Log}_{t})$ converge to the real Monge-Ampère metric on $U\subset\mathbb{R}^{n}$

$$g_{0}=\frac{1}{2\pi}\sum_{i,j}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}dx_{i}dx_{j},\quad\det(D^{2}\phi)=\text{const}.$$

(4)

Now near the large complex structure limit, which is a certain limiting situation for a family of Calabi-Yau metrics, it is expected that the semiflat metrics emerge as an asymptotic description of the degenerating Calabi-Yau metrics, in the generic region of the Calabi-Yau manifolds. In the SYZ picture, the generic region heuristically means away from the singular special Lagrangian fibres. The main point is that in the generic region, the special Lagrangians are just small perturbations of the logarithm maps in local toric charts. As we approach the large complex structure limit, the percentage of the Calabi-Yau volume measure occupied by the generic region should tend to $100\%$.

This recount of this SYZ heuristic reasoning underlines a few precautions:

• •

  The metric predictions are more compelling in the generic region. The non-generic region is subject to instanton correction effects, whose metric significance is much more difficult to analyze.

• •

  The SYZ fibration is likely an emergent behaviour near the large complex structure limit. In particular, the conjecture is concerned with a family of Calabi-Yau manifolds, and features such as semiflat metrics would only appear sufficiently close to the limit.