De Sitter Vacua from Heterotic M-Theory

TL;DR

This work demonstrates metastable de Sitter vacua in strongly coupled heterotic M-theory by balancing open membrane instantons against gaugino condensation on the hidden boundary, which spontaneously breaks supersymmetry via F-terms and yields a positive vacuum energy. It highlights the role of charged matter vevs in stabilizing moduli and explores how non-perturbative effects lead to a robust dS minimum without fine-tuning, while indicating that additional fluxes may stabilize remaining moduli. The analysis connects to weakly coupled heterotic-string stabilization ideas but leverages the non-linear CK1,CK3 background to access large volumes and a controlled expansion regime. The results also show how the SUSY-breaking scale and gravitino mass can be brought into phenomenologically relevant ranges by tuning the hidden sector (gauge group rank and Pfaffians) and the CY data. Finally, the paper outlines how incorporating H-flux could stabilize non-universal moduli, pointing to a broader flux landscape for fully realistic compactifications.

Abstract

It is shown how metastable de Sitter vacua might arise from heterotic M-theory. The balancing of its two non-perturbative effects, open membrane instantons against gaugino condensation on the hidden boundary, which act with opposing forces on the interval length, is used to stabilize the orbifold modulus (dilaton) and other moduli. The non-perturbative effects break supersymmetry spontaneously through F-terms which leads to a positive vacuum energy density. In contrast to the situation for the weakly coupled heterotic string, the charged scalar matter fields receive non-vanishing vacuum expectation values and therefore masses in a phenomenologically relevant regime. It is important that in order to obtain these de Sitter vacua we are not relying on exotic effects or fine-tuning of parameters. Vacua with more realistic supersymmetry breaking scales and gravitino masses are obtained by breaking the hidden $E_8$ gauge group down to groups of smaller rank. Also small values for the open membrane instanton Pfaffian are favored in this respect. Finally we outline how the incorporation of additional flux superpotentials can be used to stabilize the remaining moduli.

Figures (4)

• Figure 1: The dependence of ${\cal{N}}({\cal{L}})$ on ${\cal{L}}$ is plotted for the values $\beta_v=1,d=1,{\cal{V}}_v=300$ which imply ${\cal{G}}_v=0.3$. Notice that the physical range of ${\cal{L}}$ is limited and reaches from zero to ${\cal{L}}_{max}=1/{\cal{G}}_v=3.3$.
• Figure 2: The dependence of the absolute values of the open membrane and gaugino condensation superpotentials, $|W_{OM}|$ (left curve) and $|W_{GC}|$ (right curve) on ${\cal{L}}$. $|W_{OM}|$ decreases with ${\cal{L}}$ while $|W_{GC}|$ increases steeply. For the plot we took the values $\beta_v=d =1, |h|=10^{-7}, {\cal{V}}_v=300$ and hidden gauge group $SO(10)$, i.e. $C_H=8$. The upper bound ${\cal{L}}_{max}=1/{\cal{G}}_v$ on ${\cal{L}}$ lies at 3.7.
• Figure 3: The left picture shows the leading order effective 4-dim. potential in Planck units as a function of the orbifold length ${\cal{L}}$. To give an impression of the global behavior of the potential over the complete interval $[0,{\cal{L}}_{max}]$, we display in the right picture the logarithm of the potential. A de Sitter minimum appears at ${\cal{L}}_0 = 6.9$. Towards its left open membrane instantons dominate the potential, towards its right it's the gaugino condensation. For the assumed values $\beta_v=1, d=10, |h|=10^{-8}, {\cal{V}}_v=800$ and a hidden gauge group $SO(10)$ the upper bound on ${\cal{L}}$ is ${\cal{L}}_{max}=11.0$.
• Figure 4: To check that the minimum lies in a controllable regime we present here ${\cal{V}}$ (upper curve) and ${\cal{V}}_{OM}$ (lower curve) for the same parameter values as in the previous figure. Both are substantially larger than 1 at the minimum's position ${\cal{L}}_0 = 6.9$ to guarantee that the neglected terms or multiply wrapped instantons are adequately suppressed.