The Edge Of Supersymmetry: Stability Walls in Heterotic Theory

TL;DR

The paper addresses how non-Abelian gauge bundles in heterotic Calabi–Yau compactifications affect four-dimensional supersymmetry and moduli. It develops a four-dimensional effective field theory that includes an anomalous U(1) D-term arising at the stability wall in Kahler moduli space, and analyzes a concrete two-moduli example where the stability boundary enhances the gauge group to $E_6\times U(1)$ and dictates the spectrum and mass terms. A key result is the explicit D-term $D^{U(1)}=f(t^i)+\frac{3}{2}G_{L\bar M}C^L\bar C^{\bar M}$ with $f(t^i)=\frac{3}{4}\frac{\mu(\mathcal F)}{\mathcal V}$, which vanishes on the wall $\mu(\mathcal F)=0$, while the U(1) vector and Higgs fields acquire masses consistent with the Green–Schwarz mechanism. Away from the wall, the FI term changes sign and cannot be canceled, leading to a positive potential and supersymmetry breaking; this provides a concrete 4D description of SUSY breaking from non-Abelian heterotic bundles and opens avenues for moduli stabilization and phenomenological constraints on bundle choices.

Abstract

We explicitly describe, in the language of four-dimensional N=1 supersymmetric field theory, what happens when the moduli of a heterotic Calabi-Yau compactification change so as to make the internal non-Abelian gauge fields non-supersymmetric. At the edge of the region in Kahler moduli space where supersymmetry can be preserved, an additional anomalous U(1) gauge symmetry appears in the four-dimensional theory. The D-term contribution to the scalar potential associated to this U(1) attempts to force the system back into a supersymmetric configuration and provides a consistent low-energy description of gauge bundle stability.

Figures (2)

• Figure 1: The Kähler moduli space of the Calabi-Yau manifold \ref{['cicy']} in terms of the moduli $t^k=\textnormal{Re}(T^k)$. The allowed set of Kähler moduli (the Kähler cone) is the positive quadrant. The supersymmetric region where \ref{['HYM']} admits a solution is marked in green (light shading), whereas the non-supersymmetric region where it does not is marked in red (dark shaded). The boundary between them is the line $t^{2}=4t^{1}$. The dash-dotted lines parallel to the axes indicate where supergravity breaks down as the Kähler moduli become too small. The additional $U(1)$ vector and Higgs supermultiplets are light compared to the compactification scale between the two dashed lines.
• Figure 2: The $U(1)$ D-term contribution to the scalar potential. The vertical axis is the potential V evaluated at the vacuum expectation values of the $C^{L}$ fields, while the horizontal plane is the Kähler moduli space shown in Figure \ref{['fig1']}. The line $t^{2}=4t^{1}$, where the slope $\mu({\cal F})$ vanishes, indicates the stability wall which separates the supersymmetric and non-supersymmetric regions. Note that for ease of viewing, the axes have been rotated relative to Figure \ref{['fig1']}.