SYZ Conjecture and T-Duality - Examples and Constructions

Explicit constructions of SYZ fibrations provide computational laboratories for understanding mirror symmetry. While full SYZ fibrations on Calabi-Yau threefolds remain elusive, partial results and analogous constructions illuminate the general theory.

ExampleTorus Fibered K3 Surfaces

Many K3 surfaces admit elliptic fibrations $\pi: S \to \mathbb{P}^1$ . For special choices, these are SYZ fibrations with:

Base: $B = \mathbb{P}^1$
Fibers: Elliptic curves $E_b$ for $b \in \mathbb{P}^1$
Discriminant: 24 points where fibers degenerate (Kodaira classification)

The mirror K3 $S^\vee$ has dual elliptic fibration with swapped Picard lattice polarizations. The discriminant configurations match, reflecting T-duality symmetry.

Toric Calabi-Yau and Moment Maps

Toric geometryProvides the clearest SYZ examples where fibrations can be constructed explicitly.

DefinitionToric SYZ Fibration

For a toric Calabi-Yau $X_\Sigma$ , the moment map $\mu: X_\Sigma \to P$ to the moment polytope $P$ defines an SYZ fibration:

Base: $B = P$ (the polytope)
Fibers: Lagrangian tori $\mu^{-1}(b)$
Discriminant: Faces of $P$ where fibers degenerate

The mirror $X_{\Sigma^\vee}$ is the toric variety for the dual polytope/fan.

ExampleLocal $\mathbb{P}^2$

For the total space $X = \mathcal{O}_{\mathbb{P}^2}(-3) \to \mathbb{P}^2$ , the moment map gives a fibration over a simplex $P = \{(x,y,z) : x,y,z \geq 0, x+y+z \leq 1\}$ . Fibers over interior points are $(S^1)^3$ , while boundary points have smaller-dimensional tori. The mirror is another toric variety determinedby the dual fan structure.

Gross-Siebert Program

The Gross-Siebert program provides a systematic approach to SYZ via tropical geometry and log structures.

Remark

The Gross-Siebert construction builds mirror pairs from:

A singular affine manifold $B$ with discriminant $\Delta$
A scattering diagram encoding wall-crossing data
Log structures providing algebraic models for degenerating families

This combinatorial/algebraic approach constructs mirrors without requiring the original Calabi-Yau to have a smooth SYZ fibration.

Semiflat Metrics and Quantum Corrections

Near large volume, the SYZ fibration allows explicit metric asymptotics.

DefinitionSemiflat Approximation

The semiflat metric on $X$ ignores quantum corrections: $g_{\text{sf}} = g_B + g_{T^n}$

where $g_B = \partial_i\partial_j \phi \, dt^i \otimes dt^j$ for a convex function $\phi$ on $B$ .

Quantum corrections come from holomorphic discs: $g_X = g_{\text{sf}} + O(e^{-\text{Vol}})$

These corrections are computable via counts of pseudo-holomorphic discs with Lagrangian boundary conditions.

SYZ for Hypersurfaces

Some progress has been made on SYZ for hypersurfaces in toric varieties.

ExampleQuintic Threefold SYZ

The quintic $X_5 \subset \mathbb{P}^4$ is conjectured to admit an SYZ fibration near large complex structure. The base is expected to be: $B \cong S^3$

with generic fibers $T^3$ . The discriminant $\Delta$ is a trivalent graph in $S^3$ .

Constructing this fibration rigorously remains open, though numerical and asymptotic evidence supports its existence.

Wall-Crossing and Stability

SYZ fibrations connect to stability conditions via wall-crossing.

Remark

Crossing the discriminant $\Delta \subset B$ induces wall-crossing in:

Bridgeland stability conditions on $D^b(Coh(X))$
Special Lagrangian moduli in $X$
Fukaya category structure under variation of complex structure

The SYZ picture provides geometric intuition for abstract categorical wall-crossing phenomena.

These constructions demonstrate SYZ in concrete settings, providing evidence for the conjecture and computational tools for mirror symmetry calculations.