Stability Walls in Heterotic Theories

TL;DR

This work links the mathematical notion of slope stability for heterotic vector bundles to a concrete four-dimensional effective field theory near stability walls. By deriving a U(1) D-term with a Kahler-moduli dependent FI term, the authors show how supersymmetry is preserved in stable regions and broken across stability boundaries, with a decomposable locus V = { cal F} ⊕ { cal K} yielding enhanced E6 × U(1) symmetry. The EFT reproduces the spectrum and moduli matching predicted by bundle stability, and higher-order (ε_S^2) corrections can be absorbed by evaluating moduli on the relevant orbifold plane, leaving the stability picture intact. The framework provides a physical mechanism for slope stability, suggests moduli-stabilization avenues via the perturbative D-term, and generalizes to broader bundle structures and Calabi–Yau geometries, offering guidance for heterotic model-building and beyond.

Abstract

We study the sub-structure of the heterotic Kahler moduli space due to the presence of non-Abelian internal gauge fields from the perspective of the four-dimensional effective theory. Internal gauge fields can be supersymmetric in some regions of the Kahler moduli space but break supersymmetry in others. In the context of the four-dimensional theory, we investigate what happens when the Kahler moduli are changed from the supersymmetric to the non-supersymmetric region. Our results provide a low-energy description of supersymmetry breaking by internal gauge fields as well as a physical picture for the mathematical notion of bundle stability. Specifically, we find that at the transition between the two regions an additional anomalous U(1) symmetry appears under which some of the states in the low-energy theory acquire charges. We compute the associated D-term contribution to the four-dimensional potential which contains a Kahler-moduli dependent Fayet-Iliopoulos term and contributions from the charged states. We show that this D-term correctly reproduces the expected physics. Several mathematical conclusions concerning vector bundle stability are drawn from our arguments. We also discuss possible physical applications of our results to heterotic model building and moduli stabilization.

Figures (4)

• Figure 1: A two-dimensional dual Kähler cone, defined by $s_1 \geq 0$ and $s_2 \geq 0$, where $s_{i}=d_{ijk}t^{j}t^{k}$. Shown are two de-stabilizing sub-sheaves ${\cal F}_1$ and ${\cal F}_2$ with first Chern classes given by $c_1({\cal F}_1)=(-k,m)$ and $c_1({\cal F}_2)=(p,-q)$ for some integers $k,m,p,q$. The bundle $V$ is stable between the lines with slopes $k/m$ and $p/q$.
• Figure 2: The dual Kähler cone and the regions of stability/instability for the monad bundle described in Section \ref{['simpleeg']}. Here $L_1$ and $L_2$ are line bundle sub-sheaves of $V$ and $\wedge^2(V)$ respectively. The boundaries of the dual Kähler cone are denoted by the $s_2$-axis and the line with slope $1/2$.
• Figure 3: The potential in the dual Kähler cone as a functions of the two dual Kähler variables. The potential has been minimized with respect to the $C^{L}$ fields (which are not plotted here). The flat region of the potential is where the bundle is stable. The positive definite potential wall which one encounters upon entering the region where the bundle is unstable can clearly be seen, arising at the line with slope $=1$.
• Figure 4: The D-term potential from Eq. \ref{['dtermex']}, as a function of $s_1$, a dual Kähler modulus, and the absolute value, $|C|$, of a representative singlet matter field $C$. In this plot we have chosen $s_2=4-s_1$ so that we are examining a line in Kähler moduli space perpendicular to the boundary between the supersymmetric and non-supersymmetric regions. The boundary itself is found at $s_1=2$ in this diagram. Since the exact form of the Kähler potential for the matter fields is not known, a simple, canonical form has been chosen for illustrative purposes.

Theorems & Definitions (5)

• Lemma 1
• Lemma 2
• proof
• proof
• Conjecture