Calabi-Yau Metrics for Quotients and Complete Intersections

TL;DR

This work extends numerical Calabi–Yau metric computations to free quotients and complete intersections by combining Donaldson’s algorithm with invariant theory. It shows how to construct Ricci-flat metrics on quintic quotients, four-generation models, and Schoen complete intersections using invariant polynomials and equivariant line bundles, with explicit primary/secondary invariants and Molien/Hironaka decompositions guiding the basis selection. The results demonstrate rapid convergence of balanced metrics across diverse moduli, quantify dependence on Kahler and complex structure moduli, and illustrate substantial efficiency gains when exploiting symmetries such as $\mathbb{Z}_5\times\mathbb{Z}_5$ and $\mathbb{Z}_3\times\mathbb{Z}_3$ in computing metrics on quotients. The methods enable practical numerical access to physically relevant data for heterotic string compactifications, including metrics on multiply-connected Calabi–Yau spaces relevant to Wilson lines and generation reduction. The work thus provides a robust framework for computing geometric data essential to deriving Yukawa couplings and gauge dynamics from explicit Calabi–Yau geometries.

Abstract

We extend previous computations of Calabi-Yau metrics on projective hypersurfaces to free quotients, complete intersections, and free quotients of complete intersections. In particular, we construct these metrics on generic quintics, four-generation quotients of the quintic, Schoen Calabi-Yau complete intersections and the quotient of a Schoen manifold with Z_3 x Z_3 fundamental group that was previously used to construct a heterotic standard model. Various numerical investigations into the dependence of Donaldson's algorithm on the integration scheme, as well as on the Kahler and complex structure moduli, are also performed.

Figures (9)

• Figure 1: The error measure $\sigma_k$ for the metric on the Fermat quintic, computed with the two different point generation algorithms described in \ref{['sec:integrating']}. In each case we iterated the T-operator $10$ times, numerically integrating over $N_p= 00000.$ points. Then we evaluated $\sigma_k$ using $0000.$ different test points. The error bars are the numerical errors in the $\sigma_k$ integral.
• Figure 2: The error measure $\sigma_k$ for the balanced metric on the Fermat quintic as a function of $k$, computed by numerical integration with different numbers of points $N_p$. In each case, we iterated the T-operator $10$ times and evaluated $\sigma_k$ on $000.$ different test points. Note that we use a logarithmic scale for the $\sigma_k$ axis.
• Figure 3: The error measure $\sigma_k$ for the balanced metric on the Fermat quintic as a function of $N_k^2 =$ number of entries in $h^{\alpha{\bar{\beta}}}\in \mathop{\mathrm{Mat}}\nolimits_{N_k\times N_k}$. In other words, evaluating the T-operator requires $N_k^2$ scalar integrals. In each case, we iterated the T-operator $10$ times and finally evaluated $\sigma_k$ using $000.$ different test points. We use a logarithmic scale for both axes.
• Figure 4: The error measure $\sigma_k$ as a function of $k$ for five random quintics, as well as for the Fermat quintic. The random quintics are the sum over the $126$ quintic monomials in $5$ homogeneous variables with coefficients random on the unit disk. We use a logarithmic scale for $\sigma_k$.
• Figure 5: The error measure $\sigma_{5\ell}(Q_F)$ on the non-simply connected threefold $Q_F={\widetilde{Q}_F}/(\mathbb{Z}_5\times\mathbb{Z}_5)$. For each $\ell\in\mathbb{Z}_>$ we iterated the T-operator $10$ times, numerically integrating using $N_p= 000000.$ points. Then we evaluated $\sigma_{5\ell}(Q_F)$ using $0000.$ different test points. Note that all three plots show the same data, but with different combinations of linear and logarithmic axes.

Theorems & Definitions (6)

• Definition 1
• Theorem 1: Donaldson MR1916953
• Theorem 2: Donaldson, DonaldsonNumerical
• Theorem 3: Molien
• Theorem 4: Hironaka decomposition
• Theorem 5: Shifman, Zelditch