Hodge Theory and Periods - Examples and Constructions

Computing period integrals explicitly for Calabi-Yau manifolds requires sophisticated techniques from algebraic geometry and special function theory. We examine several important examples that illustrate the general principles.

The Fermat Quintic and its Mirror

The Fermat quintic $X_5 \subset \mathbb{P}^4$ defined by $x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 = 0$ is the simplest Calabi-Yau threefold. Its mirror family is constructed using toric geometry.

ExampleMirror Quintic Family

The mirror quintic is a one-parameter family: $\tilde{X}_\psi: \quad x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 - 5\psi x_0x_1x_2x_3x_4 = 0$

parametrized by $\psi \in \mathbb{C}^*$ . The change of variables $z = \psi^{-5}$ puts this in canonical form. The holomorphic 3-form is: $\Omega_\psi = \text{Res}\left[\frac{dx_0 \wedge dx_1 \wedge dx_2 \wedge dx_3}{dF/dx_4}\right]$

where $F$ is the defining polynomial.

The periods of the mirror quintic satisfy the Picard-Fuchs equation mentioned earlier. Four independent solutions can be expressed as: $\omega_0(z) = \sum_{n=0}^\infty \frac{(5n)!}{(n!)^5} z^n$

This is the fundamental period near $z=0$ (the large complex structure limit).

Hypergeometric Representations

Many Calabi-Yau periods can be expressed in terms of generalized hypergeometric functions. For a one-parameter family, the periods often have the form: $\Pi(z) = \,_nF_{n-1}\left(\begin{array}{c}a_1, \ldots, a_n \\ b_1, \ldots, b_{n-1}\end{array}; z\right)$

DefinitionGeneralized Hypergeometric Function

The generalized hypergeometric function is defined by: $\,_pF_q\left(\begin{array}{c}a_1, \ldots, a_p \\ b_1, \ldots, b_q\end{array}; z\right) = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}$

where $(a)_n = a(a+1)\cdots(a+n-1)$ is the Pochhammer symbol.

The Picard-Fuchs equation is satisfied by such hypergeometric functions when the parameters $a_i, b_j$ are appropriately chosen. The monodromy of the periods corresponds to analytic continuation of the hypergeometric functions.

Toric Period Computations

For Calabi-Yau hypersurfaces in toric varieties, the GKZ (Gel'fand-Kapranov-Zelevinsky) hypergeometric system provides a systematic approach to computing periods.

ExampleGKZ System

For a toric Calabi-Yau with defining data encoded in a matrix $A$ and parameter vector $\beta$ , the periods satisfy: $\left(\prod_{j: A_{ij}=1} \partial_j - \prod_{j: A_{ij}=-1} \partial_j\right)\Pi = 0$ $\sum_j A_{ij} x_j \partial_j \Pi = \beta_i \Pi$

These differential equations determine the periods up to monodromy.

The advantage of the GKZ approach is its systematic nature: given the toric data, one can algorithmically determine the differential equations satisfied by the periods.

K3 Surface Periods

For K3 surfaces, the period domain is the Hermitian symmetric space of type IV. The periods define a map: $\Phi: \mathcal{M}_{\text{K3}} \to \mathcal{D}_{20}$

where $\mathcal{D}_{20}$ is a 20-dimensional period domain.

Remark

The Torelli theorem for K3 surfaces states that the period map is injective with dense image. This means a K3 surface is determined up to isomorphism by its periods, a remarkable rigidity property.

Computing Yukawa Couplings from Periods

The Yukawa coupling can be computed from the period vector $\Pi = (\Pi_0, \Pi_1, \ldots, \Pi_h)$ using: $C_{ijk} = \frac{\partial^3 F}{\partial t^i \partial t^j \partial t^k}$

where $F$ is the prepotential defined by $\Pi_0 F = \sum_{I>0} \Pi_I F_I$ with $F_I = \partial F/\partial t^I$ .

ExampleOne-Parameter Model

For a one-parameter family with coordinate $t$ , the Yukawa coupling is: $C_{ttt} = \frac{d^3 F}{dt^3}$

In the mirror quintic case, this can be computed explicitly from the periods, yielding rational expressions in the coordinate $z$ .

These computations have been essential for verifying mirror symmetry predictions and computing enumerative invariants on Calabi-Yau manifolds through their mirror duals.