SYZ Conjecture and T-Duality - Main Theorem

While the full SYZ conjecture remains open for generic Calabi-Yau threefolds, important theorems establish the framework and verify key cases. These results provide mathematical rigor for the SYZ picture and connect it to mirror symmetry.

TheoremMcLean's Theorem

Let $L \subset X$ be a compact special Lagrangian submanifold in a Calabi-Yau $n$ -fold. The moduli space of special Lagrangian deformations of $L$ is smooth near $L$ with tangent space: $T_L\mathcal{M}_{\text{SLag}} \cong H^1(L, \mathbb{R})$

The obstruction space is $H^2(L, \mathbb{R})$ , which vanishes for $n=3$ when $L$ is a 3-torus.

This theorem ensures that special Lagrangian $T^3$ fibers deform smoothly, supporting the existence of SYZ fibrations. For Calabi-Yau threefolds, the unobstructed deformation theory is crucial for constructing families of special Lagrangians.

TheoremHitchin-Joyce Existence for SYZ Fibers

Given an integral affine manifold $B$ with singularities satisfying certain conditions, and a compatible scattering diagram, there exists a family of special Lagrangian tori fibering a Calabi-Yau manifold over a dense subset of $B$ .

The construction uses:

Asymptotic analysis near large volume limits
Gluing special Lagrangian tori using implicit function theorem
Quantum corrections from holomorphic disc counting

This result, developed by Gross-Siebert and refined by others, provides existence of SYZ-type fibrations in limiting regimes, though the fibers may have singularities.

TheoremSYZ for Toric Varieties

Every toric Calabi-Yau manifold $X_\Sigma$ admits a special Lagrangian torus fibration given by the moment map: $\mu: X_\Sigma \to P$

where $P$ is the moment polytope. The mirror $X_{\Sigma^\vee}$ is the toric variety for the polar dual polytope $P^\vee$ , and T-duality establishes: $D^b(Coh(X_\Sigma)) \simeq \mathcal{F}uk(X_{\Sigma^\vee})$

This theorem provides a large class of examples where SYZ is fully realized and HMS can be verified explicitly using toric methods.

TheoremT-Duality for Circle Bundles (Bunke-Schick)

For principal $S^1$ -bundles $E \to B$ with connection, T-duality produces a dual bundle $\hat{E} \to B$ satisfying: $H^*(E, \mathbb{Z}) \cong H^*(hat{E}, \mathbb{Z})$

with degrees shifted. The T-dual exchanges:

Curvature $\omega \in H^2(B)$ $\leftrightarrow$ Euler class $e(\hat{E})$
Fiber integral $\int_{S^1}$ $\leftrightarrow$ Cup product with $e$

For K3 surfaces with elliptic fibrations, this rigorously establishes T-duality as a topological correspondence.

TheoremSemiclassical SYZ

In the large volume limit $\text{Vol}(X) \to \infty$ , a Calabi-Yau threefold $X$ develops an approximate SYZ fibration $\pi: X \to B$ where:

Most of $X$ fibers over an integral affine base $B$
Fibers are approximately special Lagrangian tori
Corrections are exponentially small: $O(e^{-\text{Vol}})$

The mirror $X^\vee$ is constructed via T-duality with quantum corrections from holomorphic discs.

This asymptotic result provides rigorous justification for SYZ near large volume and explains how mirror symmetry emerges in this limit.

ExampleApplications to Counting Problems

Using SYZ, enumerative invariants on $X$ can be computed from geometry of $X^\vee$ : $\text{GW}(X) = \int_{X^\vee} \Omega \wedge \text{(cycles)}$

The SYZ fibration explains this correspondence geometrically: curves in $X$ wrap fibers multiple times, and their count equals intersection theory on $X^\vee$ .

These theorems establish SYZ on firm mathematical ground in specific contexts, providing evidence for the full conjecture and computational tools for mirror symmetry.