Moduli Spaces - Core Definitions

Moduli spaces parametrize families of geometric objects, encoding how structures can vary while preserving essential properties. For Calabi-Yau manifolds, the moduli spaces of Kähler and complex structures form the geometric foundation for mirror symmetry.

DefinitionModuli Space

A moduli space $\mathcal{M}$ for a class of geometric objects is a space whose points correspond to isomorphism classes of those objects. Local coordinates on $\mathcal{M}$ parametrize continuous families of deformations.

For a moduli problem to be well-posed, we require:

A notion of family (flat morphisms)
An equivalence relation (isomorphism)
Representability or existence of a coarse moduli space

For Calabi-Yau threefolds $X$ , two primary moduli spaces arise naturally from the geometry. The complex structure moduli space $\mathcal{M}_c$ parametrizes complex structures preserving the Calabi-Yau condition, while the Kähler moduli space $\mathcal{M}_K$ parametrizes Kähler forms modulo diffeomorphism.

DefinitionComplex Structure Moduli Space

The complex structure moduli space $\mathcal{M}_c$ of a Calabi-Yau manifold $X$ is the space of complex structures on the underlying smooth manifold that preserve $c_1 = 0$ , modulo diffeomorphisms isotopic to the identity.

The tangent space at a point $[X] \in \mathcal{M}_c$ is: $T_{[X]}\mathcal{M}_c \cong H^1(X, T_X) \cong H^{2,1}(X)$

where the isomorphism follows from Dolbeault cohomology and Serre duality.

The dimension of $\mathcal{M}_c$ equals $h^{2,1}(X)$ near smooth points. Infinitesimal deformations are controlled by the Kodaira-Spencer map, which associates to each tangent vector a class in $H^1(X,T_X)$ .

DefinitionKähler Moduli Space

The Kähler moduli space $\mathcal{M}_K$ parametrizes Kähler classes $[\omega] \in H^{1,1}(X,\mathbb{R})$ with $\omega$ a positive $(1,1)$ -form. The tangent space is: $T_{[\omega]}\mathcal{M}_K \cong H^{1,1}(X,\mathbb{R})$

The dimension of $\mathcal{M}_K$ is $h^{1,1}(X)$ .

The Kähler cone $\mathcal{K}_X \subset H^{1,1}(X,\mathbb{R})$ consists of classes $[\omega]$ representable by Kähler forms. This is an open convex cone whose boundary corresponds to degenerate geometries.

Remark

The total moduli space $\mathcal{M} = \mathcal{M}_c \times \mathcal{M}_K$ has dimension $h^{2,1} + h^{1,1}$ . Mirror symmetry exchanges these two factors: for a mirror pair $(X, X^\vee)$ , the complex structure moduli of $X$ correspond to Kähler moduli of $X^\vee$ and vice versa.

ExampleQuintic Threefold Moduli

For the quintic threefold $X_5$ , we have $h^{2,1} = 101$ and $h^{1,1} = 1$ . Thus $\dim\mathcal{M}_c = 101$ while $\dim\mathcal{M}_K = 1$ . The mirror quintic $X_5^\vee$ has $h^{2,1} = 1$ and $h^{1,1} = 101$ , demonstrating the moduli space exchange.

The geometry of these moduli spaces is highly constrained. They carry natural metric structures (Weil-Petersson and special Kähler metrics) and admit canonical coordinates related to period integrals and Yukawa couplings.