Moduli Spaces - Examples and Constructions

Explicit constructions of moduli spaces reveal their structure and provide computational tools. Different Calabi-Yau families exhibit varying levels of complexity in their moduli space geometry.

One-Parameter Families

The simplest non-trivial case involves one complex structure parameter. These families are tractable both theoretically and computationally.

ExampleMirror Quintic Family

The mirror quintic family is: $X_\psi: x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 - 5\psi x_0x_1x_2x_3x_4 = 0$

The complex structure moduli space is $\mathcal{M}_c \cong \mathbb{P}^1 \setminus \{0,1,\infty, \text{and } 3125\text{th roots of unity}\}$ .

The coordinate $z = \psi^{-5}$ gives a canonical parameter. The Picard-Fuchs operator is: $\mathcal{L} = \theta^4 - 5z(5\theta+1)(5\theta+2)(5\theta+3)(5\theta+4)$

where $\theta = z\partial_z$ .

The large complex structure limit corresponds to $z \to 0$ , where the periods have logarithmic expansions. The conifold points occur at the 3125th roots of unity in the $z$ -plane.

Toric Descriptions

Toric geometry provides systematic constructions of Calabi-Yau moduli spaces. The toric data (fan or polytope) encodes the ambient toric variety, while the GKZ system describes the period integrals.

DefinitionSecondary Fan

For a Calabi-Yau hypersurface in a toric variety, the secondary fan parametrizes different triangulations of the Newton polytope. Each chamber corresponds to a different phase of the Kähler moduli space, related by flops or extremal transitions.

The Kähler cone $\mathcal{K}_X$ has walls corresponding to divisors with negative intersection. Crossing these walls corresponds to birational transformations (flops) connecting different Calabi-Yau birational models.

Multi-Parameter Families

For Calabi-Yau manifolds with $h^{2,1} > 1$ , the moduli space becomes higher-dimensional. Complete intersection Calabi-Yau (CICY) manifolds often have multi-parameter moduli spaces.

ExampleBi-Cubic Calabi-Yau

The bi-cubic in $\mathbb{P}^2 \times \mathbb{P}^2$ defined by two $(3,3)$ forms has $h^{1,1} = 2$ and $h^{2,1} = 19$ . The complex structure moduli space has dimension 19, parametrized by coefficients in the defining polynomials modulo automorphisms.

K3 Surface Moduli

K3 surfaces provide an important special case where the moduli space structure is completely understood.

DefinitionK3 Moduli Space

The moduli space of K3 surfaces is non-compact and has dimension 20 (for complex structures) or 19 (for polarized K3s). It is a quotient: $\mathcal{M}_{\text{K3}} = \Gamma \backslash \mathcal{D}_{20}$

where $\mathcal{D}_{20}$ is a period domain (type IV Hermitian symmetric domain) and $\Gamma$ is a discrete group.

The Torelli theorem for K3 surfaces states that the period map $\mathcal{M}_{\text{K3}} \to \mathcal{D}_{20}/\Gamma$ is an isomorphism onto its image, which is dense in the period domain.

Compactifications and Boundary Divisors

Understanding the compactification of moduli spaces requires analyzing boundary components.

Remark

The Deligne-Mumford compactification adds boundary divisors corresponding to stable degenerations. For Calabi-Yau threefolds, the boundary consists of:

Type I: Manifolds with ordinary double points (conifolds)
Type II: Manifolds with more severe singularities
Type III: Limiting configurations with multiple components

Each boundary stratum has its own moduli space of lower dimension.

These constructions provide concrete realizations of abstract moduli spaces and enable explicit calculations in mirror symmetry.