Moduli Spaces - Key Properties

The moduli spaces of Calabi-Yau manifolds possess rich geometric structures that govern deformations and encode physical information. Understanding these properties is essential for applications to mirror symmetry and string theory.

Special Kähler Geometry

The complex structure moduli space $\mathcal{M}_c$ carries a natural Kähler metric called the Weil-Petersson metric. This metric has special properties making $\mathcal{M}_c$ a special Kähler manifold.

DefinitionSpecial Kähler Manifold

A special Kähler manifold is a Kähler manifold $(\mathcal{M}, g, J)$ equipped with a flat symplectic connection $\nabla$ and a holomorphic prepotential $F$ such that the Kähler potential is: $K = -\log i(\bar{X}^I F_I - X^I \bar{F}_I)$

where $X^I$ are homogeneous coordinates and $F_I = \partial F/\partial X^I$ .

For Calabi-Yau threefolds, the special Kähler structure on $\mathcal{M}_c$ arises from the variation of Hodge structure. The flat coordinates are related to period integrals, and the prepotential $F$ determines the Yukawa coupling via: $C_{ijk} = \partial_i\partial_j\partial_k F$

Remark

The Weil-Petersson metric is incomplete: geodesics can reach the boundary of moduli space in finite time. The boundary points correspond to singular Calabi-Yau manifolds, such as those with conifold singularities.

Quantum Corrections and the Kähler Moduli

While the complex structure moduli space has a classical special Kähler structure from Hodge theory, the Kähler moduli space receives quantum corrections from worldsheet instantons.

The quantum-corrected Kähler potential on $\mathcal{M}_K$ is: $K_{\text{quantum}} = K_{\text{classical}} + \sum_{d>0} n_d e^{-d t}$

where $n_d$ are genus-zero Gromov-Witten invariants counting degree $d$ rational curves, and $t$ is the complexified Kähler parameter.

DefinitionQuantum Cohomology

The quantum cohomology ring $QH^*(X)$ deforms the classical cup product by including contributions from curves: $\alpha \star_q \beta = \alpha \cup \beta + \sum_{d>0} q^d \langle\alpha, \beta, \cdots\rangle_d$

where $\langle\cdots\rangle_d$ are Gromov-Witten invariants.

Singularities and Boundary Structure

The moduli space $\mathcal{M}_c$ is generally singular. Singularities arise from:

Enhanced symmetries: Points where automorphism groups become larger
Conifold points: Where the manifold develops ordinary double points
Large complex structure limit: Asymptotic regions in moduli space

Near a conifold point, the local structure is described by the equation $xy = t$ for local coordinates. The monodromy around $t=0$ has a single Jordan block, characteristic of conifold degenerations.

ExampleQuintic Conifold Locus

The quintic threefold $X_5$ has a conifold divisor in its 101-dimensional complex structure moduli space. Points on this divisor correspond to quintics with ordinary double points. There are ${5 \choose 2} \cdot 2^4 = 10 \cdot 16 = 160$ such points in general position.

Mirror Symmetry and Moduli Exchange

Mirror symmetry predicts a correspondence: $\mathcal{M}_c(X) \leftrightarrow \mathcal{M}_K(X^\vee), \quad \mathcal{M}_K(X) \leftrightarrow \mathcal{M}_c(X^\vee)$

The classical geometry of $\mathcal{M}_c(X)$ (periods, Yukawa couplings) equals the quantum geometry of $\mathcal{M}_K(X^\vee)$ (quantum cohomology, Gromov-Witten invariants). This allows computation of difficult quantum invariants using classical Hodge theory on the mirror.