G-Fluxes and Non-Perturbative Stabilisation of Heterotic M-Theory

TL;DR

This work analyzes moduli stabilization in heterotic M-theory with a parallel M5-brane, focusing on non-perturbative open membrane instantons (OM) that connect the M5 to the boundaries. The authors derive the four-dimensional N=1 supergravity potential including subleading terms and show that the leading OM contribution vanishes at the minimum, yielding a positive vacuum energy and stabilizing the M5 at the interval midpoint, with the CY volume running due to G-flux. Crucially, stabilisation requires a flux equality between the visible boundary and the M5 (r_v = r_M5), which fixes the relation between the orbifold length $R$ and the CY volume $V$ (in particular $R = V_1/r_v$) and yields a controlled perturbative regime (ε, ε_R < 1). They also compute the SUSY-breaking scale and gravitino mass in this vacuum, finding an exponential suppression of the vacuum energy and a SUSY-breaking scale that, while potentially TeV-scale with extreme parameter choices, generally favors larger fluxes to maintain perturbativity. The results illustrate a viable, calculable open-membrane vacuum in which non-perturbative OM effects stabilise moduli and generate a positive vacuum energy, with the phenomenology tightly tied to the G-flux configuration on the boundaries and the M5.

Abstract

We examine heterotic M-theory compactified on a Calabi-Yau manifold with an additional parallel M5 brane. The dominant non-perturbative effect stems from open membrane instantons connecting the M5 with the boundaries. We derive the four-dimensional low-energy supergravity potential for this situation including subleading contributions as it turns out that the leading term vanishes after minimisation. At the minimum of the potential the M5 gets stabilised at the middle of the orbifold interval while the vacuum energy is shown to be manifestly positive. Moreover, induced by the non-trivial running of the Calabi-Yau volume along the orbifold which is driven by the G-fluxes, we find that the orbifold-length and the Calabi-Yau volume modulus are stabilised at values which are related by the G-flux of the visible boundary. Finally we determine the supersymmetry-breaking scale and the gravitino mass for this open membrane vacuum.

Figures (6)

• Figure 1: The CY volume dependence on the orbifold coordinate $x^{11}$ in the eleven-dimensional picture which is implied by the stabilised moduli and G-flux values found within the four-dimensional effective description.
• Figure 2: The CY volume behaviour which is found beyond leading order under the assumptions that $x=1/2$ and $r_v = r_{M5}$ remain true in the full theory. Over the first half of the orbifold-interval the volume varies quadratically and stays constant over the second half. The zero volume interval gets lifted to a positive value $V_1/4$.
• Figure 3: The logarithm of the OM potential, $\ln((\kappa_4)^4U_{OM})$, is depicted as a function of the orbifold modulus $R$ for parameters $|h|=\beta=1$, $V_1=3000$, $r_{OM}=200$, $d=30$. It exhibits a minimum at $R=11$. At $R=15$ the average CY volume $V$ vanishes thus leading to the steep increase there. The reason for this is that the CY volume becomes negative to the right of the minimum and one can strictly trust the potential only up to its minimum. The possibility of a saddle point at $R=11$ is however excluded since the potential exhibits a positive second derivative there.
• Figure 4: The figure shows that $\epsilon$ stays smaller than one in the $(V_1,r_v)$ parameter region given by $525 \le V_1\le 5000$, $80\le r_v\le 250$ and $d = 30$.
• Figure 5: The figure shows that also $\epsilon_R$ stays smaller than one in the same $(V_1,r_v)$ parameter region as considered in the previous figure.