Yukawa Textures From Heterotic Stability Walls

TL;DR

The paper addresses how stability walls in the Kahler cone of Calabi–Yau compactifications constrain the four-dimensional superpotential in heterotic string theory. By analyzing enhanced anomalous U(1) symmetries on walls and employing holomorphy, the authors show that Yukawa textures established at the wall persist throughout the entire stable chamber, even after the U(1) is Higgsed away. They provide explicit multi-wall and multi-branch examples (SU(3), SU(4), and SO(10) contexts) and demonstrate how stability walls can naturally yield hierarchical Yukawa structures and selective vector-like masses, with implications for realistic model-building. The results offer a framework to pre-screen phenomenologically viable models by incorporating stability-wall data early in the construction process, while noting possible non-perturbative effects that could modify the textures slightly.

Abstract

A holomorphic vector bundle on a Calabi-Yau threefold, X, with h^{1,1}(X)>1 can have regions of its Kahler cone where it is slope-stable, that is, where the four-dimensional theory is N=1 supersymmetric, bounded by "walls of stability". On these walls the bundle becomes poly-stable, decomposing into a direct sum, and the low energy gauge group is enhanced by at least one anomalous U(1) gauge factor. In this paper, we show that these additional symmetries can strongly constrain the superpotential in the stable region, leading to non-trivial textures of Yukawa interactions and restrictions on allowed masses for vector-like pairs of matter multiplets. The Yukawa textures exhibit a hierarchy; large couplings arise on the stability wall and some suppressed interactions "grow back" off the wall, where the extended U(1) symmetries are spontaneously broken. A number of explicit examples are presented involving both one and two stability walls, with different decompositions of the bundle structure group. A three family standard-like model with no vector-like pairs is given as an example of a class of SU(4) bundles that has a naturally heavy third quark/lepton family. Finally, we present the complete set of Yukawa textures that can arise for any holomorphic bundle with one stability wall where the structure group breaks into two factors.

Figures (4)

• Figure 1: The Kähler cone and regions of stability/instability for Calabi-Yau threefold \ref{['eg1cy3']} and the bundle \ref{['eg1bundle 2']}. The stability wall generated by ${\cal O}(-1,3)$ in $V$ occurs on the line with slope $t^{2}/t^{1}=2+\sqrt{7}$.
• Figure 2: The Kähler cone (The set of moduli $t^1,t^2>0$ and $2t^2>t^1$) and the regions of stability/instability for the Calabi-Yau threefold \ref{['twofour']} and the bundle \ref{['doublewalleg']}. At the lower boundary, $V$ decomposes as $V \rightarrow {\cal F}_1 \oplus {\cal K}_1$, where ${\cal F}_1$ is defined in (\ref{['F1']}). At the upper boundary, the poly-stable decomposition is given by $V \rightarrow {\cal F}_2 \oplus {\cal K}_2$, with ${\cal F}_2$ defined by (\ref{['F2']}).
• Figure 3: The Kähler cone ($t^2>0$ and $t^2+3t^1>0$) and the regions of stability/instability for the "downstairs" bundle $\hat{V}=V/(\mathbb{Z}_3\times\mathbb{Z}_3)$ on the quotient manifold $\hat{X}=X/(\mathbb{Z}_3\times\mathbb{Z}_3)$, defined respectively by \ref{['downV']} and \ref{['33q']}. At the line with slope $t^2/t^1=-3-\sqrt{3}$, $\hat{V}$ decomposes as $\hat{V} \rightarrow \hat{{\cal F}_1} \oplus \hat{{\cal F}_2}$ given in \ref{['downV2']} and \ref{['downV3']}.
• Figure 4: The Kähler cone ($t^1,t^2>0$) and the regions of stability/instability for the "upstairs" bundle \ref{['pheno_eg']} on the simply connected Calabi-Yau \ref{['33']}. At the stability wall ($t^2/t^1=1+\sqrt{3}$), $V$ decomposes as $V \rightarrow {\cal F}_1 \oplus {\cal F}_2$ where ${\cal F}_1$ and ${\cal F}_2$ are defined in (\ref{['fdefs']}).