SYZ Conjecture and T-Duality - Core Definitions

The Strominger-Yau-Zaslow (SYZ) conjecture provides a geometric explanation for mirror symmetry through special Lagrangian fibrations and T-duality. This framework connects string theory physics with differential geometry, offering a constructive approach to building mirror pairs.

DefinitionSYZ Conjecture

Let $X$ be a Calabi-Yau $n$ -fold. The SYZ conjecture states that $X$ admits a special Lagrangian torus fibration: $\pi: X \to B$

where $B$ is an $n$ -dimensional base and generic fibers are special Lagrangian $n$ -tori $T^n$ . The mirror manifold $X^\vee$ is constructed as the moduli space of pairs $(L_b, \nabla)$ where:

$L_b = \pi^{-1}(b)$ is a fiber
$\nabla$ is a flat $U(1)$ -connection on $L_b$

The dual fibration $\pi^\vee: X^\vee \to B$ has the same base.

This geometric picture explains mirror symmetry as a fiberwise T-duality transformation. The swap of Hodge numbers arises naturally: $h^{1,1}$ counts fiber deformations (Kähler moduli), while $h^{2,1}$ counts base deformations (complex structure moduli).

DefinitionSpecial Lagrangian Submanifold

A Lagrangian submanifold $L \subset X$ is special Lagrangian if: $\text{Im}(\Omega)|_L = 0, \quad \text{vol}_L = \text{Re}(\Omega)|_L$

where $\Omega$ is the holomorphic volume form and $\text{vol}_L$ is the Riemannian volume form on $L$ .

Equivalently, $L$ is calibrated by $\text{Re}(\Omega)$ and is volume-minimizing in its homology class.

Special Lagrangian submanifolds are minimal submanifolds and satisfy elliptic PDEs (the special Lagrangian equation). Their moduli spaces have expected dimension zero for compact examples in Calabi-Yau threefolds.

DefinitionT-Duality

T-duality is a duality in string theory that exchanges momentum and winding modes on a torus. Mathematically, for a circle fibration $S^1 \to E \to B$ : $H^3(E, \mathbb{Z}) \leftrightarrow H^1(E, \mathbb{Z}) \otimes H^2(B, \mathbb{Z})$

For torus fibrations $T^n \to X \to B$ , T-duality constructs a dual fibration $T^n \to X^\vee \to B$ where fiber and base geometry are exchanged.

The T-duality transformation sends:

Fiber homology classes $\leftrightarrow$ Connection curvatures
Base classes $\leftrightarrow$ Base classes (preserved)
Volume of fiber $\leftrightarrow$ Inverse volume

Remark

The SYZ picture predicts that near large complex structure/large volume limits, Calabi-Yau manifolds develop special Lagrangian fibrations. The discriminant locus $\Delta \subset B$ where fibers become singular encodes the topology and determines instanton corrections in mirror symmetry.

ExampleElliptic Curves

For an elliptic curve $E = \mathbb{C}/\Lambda$ , the SYZ fibration is trivial: $E \to \{pt\}$ with fiber $E$ itself. The dual is $\hat{E} = \mathbb{C}/\Lambda^\vee$ with dual lattice. T-duality exchanges the roles of the two circles in $E \cong S^1 \times S^1$ , swapping Kähler and complex structures.

The SYZ conjecture remains largely conjectural for Calabi-Yau threefolds, though it has been verified in toric cases and near large volume limits. It provides intuition for mirror constructions and explains many mirror phenomena geometrically.