Physical Yukawa Couplings in Heterotic String Compactifications

Abstract

One of the challenges of heterotic compactification on a Calabi-Yau threefold is to determine the physical $(27{)}^3$ Yukawa couplings of the resulting four-dimensional $\mathcal{N}=1$ theory. In general, the calculation necessitates knowledge of the Ricci-flat metric. However, in the standard embedding, which references the tangent bundle, we can compute normalized Yukawa couplings from the Weil-Petersson metric on the moduli space of complex structure deformations of the Calabi-Yau manifold. In various examples (the Fermat quintic, the intersection of two cubics in ${\mathbb{P}}^5$, and the Tian-Yau manifold), we calculate the normalized Yukawa couplings for $(2,1)$-forms using the Weil-Petersson metric obtained from the Kodaira-Spencer map. In cases where $h^{1,1}=1$, this is compared to a complementary calculation based on performing period integrals. A third expression for the normalized Yukawa couplings is obtained from a machine learned approximate Ricci-flat metric making use of explicit harmonic representatives. The excellent agreement between the different approaches opens the door to precision string phenomenology.

33 pages, 11 figures, 2 tables, 3 lemmas, 1 theorem. v2: Minor edits