SYZ Conjecture and T-Duality - Key Properties

The SYZ fibration structure imposes strong constraints on Calabi-Yau geometry and explains key features of mirror symmetry. Understanding these properties reveals the geometric mechanism behind mirror correspondence.

Semiclassical Limit and Large Volume

The SYZ picture becomes most transparent in the large volume limit where quantum corrections are suppressed.

DefinitionSemiclassical Approximation

In the semiclassical limit, the Kähler parameter $t \to \infty$ and the SYZ fibration becomes visible. The fiber volumes satisfy: $\text{Vol}(L_b) \sim e^{-t}$

Near this limit, instantons (holomorphic curves) wrapping fibers contribute exponentially suppressed corrections $e^{-t}$ .

This explains why mirror symmetry becomes an equivalence of classical geometries in appropriate limits: quantum corrections on one side (A-model instantons) match classical geometry on the other (B-model periods).

Affine Structures and the Base

The base $B$ of an SYZ fibration inherits a natural affine structure from the fibration.

DefinitionIntegral Affine Structure

The base $B$ carries an integral affine structure: an atlas of charts with transition functions in $\text{Aff}(\mathbb{Z}^n)$ (affine transformations with integer linear part).

This structure arises from identifying fibers $T^n$ with $\mathbb{R}^n/\mathbb{Z}^n$ via action-angle coordinates. The discriminant locus $\Delta \subset B$ where fibers degenerate is a codimension-1 stratified subset.

Near a point in $\Delta$ , the fibration develops singularities (vanishing cycles). The monodromy around $\Delta$ generates wall-crossing phenomena in stability conditions.

Instanton Corrections

Worldsheet instantons wrapping fibers contribute quantum corrections to the mirror map and moduli space metrics.

Remark

Disk instantons are holomorphic discs $D$ with:

Boundary on a special Lagrangian fiber: $\partial D \subset L_b$
Non-trivial homology class: $[D] \neq 0$ in $H_2(X, L_b)$

Their contributions modify the complex structure of $X^\vee$ : $J_{X^\vee} = J_{\text{classical}} + \sum_D e^{-\text{Area}(D)} \cdot(\text{weight})$

This explains instanton corrections in periods and Yukawa couplings.

Mirror Map from SYZ

The SYZ construction provides a geometric derivation of the mirror map connecting moduli spaces.

DefinitionSYZ Mirror Map

The SYZ mirror map identifies coordinates: $t^i_K(X) \leftrightarrow t^j_c(X^\vee)$

where $t_K$ are Kähler parameters and $t_c$ are complex structure parameters. This identification comes from:

Fiber volumes $\leftrightarrow$ Connection holonomies
Base deformations $\leftrightarrow$ Dual base deformations

Including instanton corrections gives the full quantum mirror map.

Singularities and Topology Change

Singular fibers in the SYZ fibration correspond to interesting geometric phenomena.

Remark

Conifold transitions occur when a 3-cycle in a Calabi-Yau threefold shrinks to zero. In SYZ:

A special Lagrangian $S^3$ fiber degenerates
The base $B$ develops a singular point
The mirror manifold undergoes a topology change
The transition exchanges $h^{1,1}$ and $h^{2,1}$

This provides a geometric picture for conifold transitions via SYZ duality.

Metric and Hermitian Structure

The SYZ fibration determines the metric on $X$ and $X^\vee$ through dual geometric data.

DefinitionSYZ Metric

The Calabi-Yau metric on $X$ is locally: $g_X = g_B + g_{T^n}$

where $g_B$ is the base metric (Hessian of a function on $B$ ) and $g_{T^n}$ are fiber metrics varying over $B$ .

On the mirror $X^\vee$ , the roles reverse: $g_{X^\vee} = g_B + g^{-1}_{T^n}$

with fiber metric inverted (T-duality).

This metric prescription, including quantum corrections, produces the Ricci-flat metrics guaranteed by Yau's theorem. The SYZ picture provides an explicit, though highly technical, construction of these metrics near large volume limits.