ScalingStacks

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.

Aside from its importance in mirror symmetry, the SYZ conjecture is interesting for other diverse fields such as minimal surface theory, Riemannian geometry, Kähler and algebraic geometry.

• •

  Special Lagrangians are minimal submanifolds, and in fact calibrated submanifolds. Currently, there are few methods for producing special Lagrangians in sufficiently large supply of Calabi-Yau manifolds, although there is a series of conjectures initiated by Thomas-Yau [75][74] and further developed in [43][57].

• •

  The behaviour of a family of Einstein metrics depends strongly on whether the volume of geodesic balls satisfies the noncollapsing condition for some uniform constant $\kappa>0$.

  $$\text{Vol}_{g}(B_{g}(r))\geq\kappa r^{\dim_{\mathbb{R}}X},\quad\forall 0<r\leq\text{diam}(X).$$

  A good convergence and regularity theory is available in the non-collapsing case [13]. On the other hand, metric degeneration in the collapsing case is largely terra incognita in Riemannian geometry, and the semiflat metric asymptote is a highly nontrivial emergent feature for a collapsing family of Calabi-Yau metrics.

• •

  A recurring theme of Kähler geometry is the interplay between metric and complex geometry. Käher-Einstein metrics in the non-collapsing case is tied to projective geometry [23]. The large complex structure limit is a very severe kind of polarized degeneration, whose transcendental behaviour (related to exponential and logarithms) is not adequately captured by traditional projective geometry, and instead non-archimedean geometry stands out as a natural framework. One can then ask about the relation between Calabi-Yau metrics and non-archimedean geometry, a problem that turns out to be related to the SYZ conjecture.

An influential early work related to the SYZ conjecture is the gluing description of Gross-Wilson [32] concerning the hyperkähler metric on K3 surfaces $X$ with elliptic fibrations $\pi:X\to\mathbb{CP}^{1}$.

By hyperkähler rotation, the special Lagrangian torus fibres can be viewed as the holomorphic elliptic curves in a different complex structure. In the generic situation, the elliptic fibration has 24 $I_{1}$-type singular fibres, namely the local singularity in the fibration is modelled on $(z_{1},z_{2})\mapsto z_{1}z_{2}$ complex geometrically. They fix Kähler classes $[\omega_{X}]$ on the K3 surface, and $[\omega_{\mathbb{P}^{1}}]$ on $\mathbb{CP}^{1}$, and describe the Calabi-Yau metrics $\omega_{\tau}$ in the Kähler class $\tau[\omega_{X}]+\pi^{*}[\omega_{\mathbb{P}^{1}}]$ for $0<\tau\ll 1$. Some key conceptual features are:

• •

  In the generic region, the metrics $\omega_{\tau}$ are up to exponentially small errors modelled on semiflat metrics, which can be explicitly described via the periods integrals of the elliptic curves. The subset of the K3 surface on which the semiflat metric asymptote breaks down, has length scale $O(\tau^{1/2}|\log\tau|^{1/2})$.

• •

  As $\tau\to 0$, the metrics $\omega_{\tau}$ converge in the Gromov-Hausdorff sense to a singular metric on $\mathbb{CP}^{1}$. The diameter of the K3 surfaces is of constant order $O(1)$. The length scale of generic elliptic curve fibres is $O(\tau^{1/2})$, which shrinks to zero size as $\tau\to 0$.

• •

  In the neighbourhood of the $I_{1}$ type singular fibres, the metrics are modelled on the Ooguri-Vafa metrics, which are explicit $S^{1}$-invariant incomplete hyperkähler metrics constructed by means of the Gibbons-Hawking ansatz.

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  The asymptotic geometry of the Ooguri-Vafa metric matches with the semiflat metric. This is a basic requirement for the gluing construction.

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  Near the nodal singular point of the $I_{1}$-fibre, the Ooguri-Vafa metric contains a small region approximated by the Taub-NUT metric, whose length scale is $O(\tau^{1/2}|\log\tau|^{-1/2})$. These regions concentrate almost the entire $L^{2}$-Riemannian curvature of the K3 surface.

The Gross-Wilson picture represents the best hope on the SYZ conjecture,²² 2 As a caveat sometimes overlooked in the literature, the hyperkähler rotation of the Gross-Wilson setting is not quite a polarized degeneration family, hence does not quite fit into our notion of large complex structure limit. In our perspective, Gross-Wilson is an inspiration, rather than an example of the SYZ conjecture. which includes an almost explicit description of the metric, and a special Lagrangian torus fibration exists globally on $X$. It is partially generalized to higher dimensional hyperkähler manifolds with holomorphic Lagrangian abelian variety fibrations. After hyperkähler rotation, these can be regarded as special Lagrangian fibrations. Tosatti et al. [72] [31] established the semiflat metric asymptote in the generic region, and Gromov-Hausdorff collapse to the base manifold.

For general information on this section, see [42, Chapter 8,9], and the introduction in [54]. In the initial years following the SYZ proposal, there was an overly optimistic belief based on the analogy with the hyperkähler case, and based on topological and complex geometric considerations:

• •

  The special Lagrangian fibration exists globally and is defined by a $C^{\infty}$ map, even though some fibres may be singular.

• •

  The discriminant locus on the base is codimension two. For Calabi-Yau 3-folds, under suitable genericity assumption, the discriminant locus is a trivalent graph, with two types of vertices, known as positive and negative vertices.³³ 3 The names ‘positive/negative vertex’ come from some old fashioned topological models of the torus fibration where the most singular fibres have Euler characteristics $\pm 1$ respectively.

• •

  Along the edges of the trivalent graph, the singularity of the SYZ fibration is transversely modelled on the $I_{1}$ singularity.

• •

  The local region near the positive vertex is complex geometrically a large open subset inside

  $$\{z_{0}z_{1}z_{2}=1-z_{3}\}\subset\mathbb{C}^{3}\times\mathbb{C}^{*}_{z_{3}},\quad\Omega\propto\frac{1}{z_{3}}dz_{0}\wedge dz_{1}\wedge dz_{2},$$

  while the negative vertex region is modelled on a large open subset inside

  $$\{z_{3}z_{4}=1-z_{1}-z_{2}\}\subset(\mathbb{C}^{*})^{2}_{z_{1},z_{2}}\times\mathbb{C}^{2}_{z_{3},z_{4}},\quad\Omega\propto\frac{1}{z_{1}z_{2}}dz_{2}\wedge dz_{3}\wedge dz_{4}.$$

The naïvete was challenged by Joyce [41], based on his observations concerning special Lagrangian singularities, which are markedly different from those visible in holomorphic fibrations. A global special Lagrangian fibration on the compact Calabi-Yau manifolds near the large complex structure, should it exist at all, is expected to have a much more subtle structure:

• •

  The special Lagrangian fibration is typically not defined by $C^{\infty}$ maps, but are at best piecewise smooth.

• •

  The discriminant locus on the base is typically not codimension two, but the trivalent graph is expected to be thickened to a codimension one ‘ribbon’. The amount of thickening probably tends to zero in the large complex structure limit.

• •

  The $I_{1}\times\mathbb{R}$ local singularity model does not lead to a Fredholm deformation theory for the singular special Lagrangian fibres, so should be replaced by some other singularity models.

Stepping aside from the substantial difficulties of the special Lagrangian local singularities, another major difficulty is to understand the Calabi-Yau metrics near the large complex structure limit, in complex dimension three. The optimistic expectations are inspired by the Gross-Wilson picture in complex dimension two. Hypothetically, the Calabi-Yau 3-fold is Gromov-Hausdorff close to a real 3-dimensional manifold, whose topology is believed to be the 3-sphere⁴⁴ 4 This expectation comes from the topology of the essential skeleton, see section 3.3 below., containing a trivialent graph, such that

• •

  Away from the trivalent graph, the Calabi-Yau metric is semiflat up to exponentially small errors.

• •

  Transverse to the edges in the trivalent graph, the metric is modelled on the Ooguri-Vafa metric appearing in Gross and Wilson’s picture.

• •

  Near the positive and negative vertices of the trivalent graph, the local metric is modelled on some generalization of the Ooguri-Vafa metrics.

It was recently realized that there exist almost canonical constructions of Ooguri-Vafa type metrics in complex dimension three, with the predicted topology and complex structure of the positive and negative vertices, constructed from a (nonlinear) generalized Gibbons-Hawking ansatz [54]. The asymptotic geometry of these Ooguri-Vafa type metrics matches with semiflat metrics in the generic region. The positive vertex metric contains a local region modelled on a generalized Taub-NUT type metric on $\mathbb{C}^{3}$, analogous to the way the Ooguri-Vafa metric contains a region modelled on the Taub-NUT metric.

The following questions are widely open:

Due to the analytic difficulty of the SYZ conjecture, especially the nongeneric regions with large Riemannian curvature, the literature has developed many interpretations of the conjectures, with somewhat diverging goals.

• •

  (Soft versions) For applications to homological mirror symmetry, one is primarily interested in constructing Lagrangian fibrations without requiring $\text{Im}(\Omega)|_{L}=0$, and therefrom build a mirror manifold $X^{\vee}$ and prove the categorical predictions $D^{b}Coh(X^{\vee})\simeq D^{\pi}Fuk(X,\omega)$. This viewpoint separates the symplectic and holomorphic data of the Calabi-Yau manifold, and has a topological/algebraic flavour.

  002H

  Remark 1. The special Lagrangian condition usually left out of homological mirror symmetry discussions, is supposedly related to Bridgeland stability conditions, which is a popular categorical interpretation of the BPS condition on D-branes.

• •

  (Strong metric version) On compact Calabi-Yau manifolds near the large complex structure limit, find a global special Lagrangian torus fibration.

• •

  (Weak metric version) Prove for a suitable class of compact Calabi-Yau manifolds near the large complex structure limit that a special Lagrangian $T^{n}$-fibration exists on a large subset with at least $99\%$ of the Calabi-Yau measure. More precisely, the percentage converges to $100\%$ in the limit.

The metric versions are highly sensitive to the Calabi-Yau metric, which is a much more rigid structure compared to the soft versions. They conform to the PDE spirit of the SYZ paper, while the soft versions are closer to the mirror symmetry motivations of the SYZ conjecture.

The strong metric version is perhaps the most faithful to the original intention of the SYZ paper. Its main evidence comes from the special case of hyperkähler metrics, mentioned in section 2.3. Other peripheral evidence comes from the construction of Lagrangian fibrations in many examples, ignoring the $\text{Im}(\Omega)|_{L}=0$ condition [58]. There are however many subtleties besetting this strong version, mentioned in section 2.4, making the conjecture very formidable, and by comparison the supporting evidence seems inadequate. Notably, the special Lagrangian singularities are not sufficiently understood, and we are not aware of any argument that definitively rules out special Lagrangians intersecting each other in the non-generic regions with large Riemannian curvature. As food for future thought, a somewhat weakened version bypassing these possible objections is

The weak metric version, on the other hand, concerns only the generic region, which is more accessible than the strong version. It conforms to the more cautious expectation, that the special Lagrangian fibration is only a limiting phenomenon. The bulk of the survey will focus on this weak version.