SYZ Conjecture and T-Duality - Applications

The SYZ framework provides powerful tools for computing mirror symmetry predictions and understanding dualities in string theory. These applications demonstrate the practical utility of the SYZ picture beyond its conceptual elegance.

Mirror Construction

SYZ provides a geometric algorithm for constructing mirror manifolds.

TheoremSYZ Mirror Construction Algorithm

Given a Calabi-Yau $n$ -fold $X$ with SYZ fibration $\pi: X \to B$ :

Identify the integral affine base $B$ with discriminant $\Delta$
Construct the dual fibration moduli space: $X^\vee = \{(b, [\nabla]) : b \in B \setminus \Delta, \, \nabla \text{ flat on } \pi^{-1}(b)\} / \sim$
Add quantum corrections from holomorphic discs
Compactify to obtain smooth mirror $X^\vee$

This produces the mirror with swapped Hodge numbers and dual moduli spaces.

This algorithm has been successfully implemented for toric varieties and K3 surfaces, producing known mirrors and verifying HMS predictions.

Computing Instanton Corrections

SYZ allows systematic computation of quantum corrections.

Remark

Disc instantons contribute to the superpotential on $X^\vee$ : $W = \sum_D n_D e^{-\int_D \omega + i\int_{\partial D} \theta}$

where:

$D$ is a holomorphic disc with boundary on fiber $L_b$
$n_D$ counts discs (with multiplicity from obstruction theory)
$\theta$ is the connection holonomy

These corrections modify the complex structure of $X^\vee$ , explaining B-model prepotential from A-model geometry.

String Dualities

SYZ connects various string theory dualities.

TheoremType IIA/IIB Exchange

Compactifying Type IIA string theory on a Calabi-Yau $X$ with SYZ fibration is dual to Type IIB on the mirror $X^\vee$ : $\text{IIA}(X) \leftrightarrow \text{IIB}(X^\vee)$

This duality exchanges:

D2-branes wrapping fibers $\leftrightarrow$ D1-branes (strings)
D4-branes wrapping base cycles $\leftrightarrow$ D3-branes
RR fluxes $\leftrightarrow$ NS-NS fluxes

The SYZ fibration provides the geometric mechanism for this exchange.

Tropical Geometry Applications

SYZ connects to tropical geometry, providing combinatorial methods.

Remark

The tropicalization of a Calabi-Yau $X$ with SYZ fibration produces the affine base $B$ with integral structure. Tropical curves in $B$ correspond to holomorphic curves in $X$ : $\text{Trop}: \text{Curves}(X) \to \text{Tropical Curves}(B)$

This allows counting curves combinatorially using tropical intersection theory, then lifting to genuine Gromov-Witten invariants.

Homological Mirror Symmetry from SYZ

The SYZ picture explains HMS categorically.

TheoremSYZ Explanation of HMS

The SYZ fibration induces HMS as follows:

Objects on B-side: Coherent sheaves $\mathcal{F} \in D^b(Coh(X))$
Objects on A-side: Lagrangian sections $s: B \to X^\vee$ of dual fibration

The correspondence: $\mathcal{F} \leftrightarrow L_{\mathcal{F}} := \{(b, \nabla_b) : \nabla_b \text{ determined by } \mathcal{F}|_{L_b}\}$

associates sheaves to Lagrangian sections, explaining the categorical equivalence geometrically.

Moduli Space Geometry

SYZ illuminates the special geometry of moduli spaces.

ExampleSpecial Coordinates from SYZ

The SYZ construction yields natural coordinates on moduli spaces:

Kähler moduli: Volumes of fiber homology classes
Complex moduli: Deformations of affine structure on $B$

These coordinates are special coordinates where the prepotential and metric take simple forms, explaining special Kähler geometry from SYZ.

Wall-Crossing and Stability

SYZ explains wall-crossing phenomena geometrically.

Remark

Crossing the discriminant $\Delta \subset B$ causes:

Fiber topology changes: Vanishing/appearing cycles
Stability wall-crossing: Bridgeland stability changes
BPS spectrum jumps: New/old BPS states

The SYZ picture makes these abstract categorical phenomena geometric and computable.

Arithmetic Applications

For Calabi-Yau varieties over number fields, SYZ provides arithmetic insights.

TheoremSYZ and Rational Points

For $X$ defined over $\mathbb{Q}$ , the SYZ fibration structure constrains rational points. The discriminant locus encodes arithmetic information about reduction types, and T-duality relates arithmetic on $X$ to that on $X^\vee$ .

These applications show SYZ as a unifying framework connecting geometry, physics, and arithmetic through the lens of special Lagrangian fibrations.