Four-Flux and Warped Heterotic M-Theory Compactifications

TL;DR

This work advances heterotic M-theory compactifications by solving the Killing-spinor equations on CY_3 × S^1/ℤ_2 with nonzero G-flux beyond the leading order, illuminating how warp factors and four-form flux co-create the internal geometry. It identifies two tractable flux regimes: one reproducing the weakly coupled heterotic string with torsion, and another producing a quadratic Calabi–Yau volume along the orbifold direction, thereby avoiding negative-volume issues and yielding a refined Newton’s constant bound. The analysis shows the four-dimensional cosmological constant vanishes at order $\kappa^{2/3}$, and it clarifies the duality with torsion while accounting for M5-brane contributions and possible general flux beyond warped geometries. These results provide a more complete, supersymmetric-compatible picture of moduli, couplings, and stability in strongly coupled heterotic compactifications, without relying on ad hoc truncations.

Abstract

In the framework of heterotic M-theory compactified on a Calabi-Yau threefold 'times' an interval, the relation between geometry and four-flux is derived {\it beyond first order}. Besides the case with general flux which cannot be described by a warped geometry one is naturally led to consider two special types of four-flux in detail. One choice shows how the M-theory relation between warped geometry and flux reproduces the analogous one of the weakly coupled heterotic string with torsion. The other one leads to a {\it quadratic} dependence of the Calabi-Yau volume with respect to the orbifold direction which avoids the problem with negative volume of the first order approximation. As in the first order analysis we still find that Newton's Constant is bounded from below at just the phenomenologically relevant value. However, the bound does not require an {\it ad hoc} truncation of the orbifold-size any longer. Finally we demonstrate explicitly that to leading order in $κ^{2/3}$ no Cosmological Constant is induced in the four-dimensional low-energy action. This is in accord with what one can expect from supersymmetry.

Figures (3)

• Figure 1: The quadratic dependence of the Calabi-Yau volume on the orbifold direction in the full geometry and its linear approximation to order $\kappa^{2/3}$. If higher order contributions are negligible then the linear approximation is valid for small $x^{11}$. The left boundary corresponds to the "visible" world.
• Figure 2: The figure shows the volume dependence in the presence of an additional M5-brane at $z_{M5}$, where we assume a positive M5-brane source ${\cal S}_{M5}$. In a) the situation for $x_0^{11}<{\tilde{x}}_0^{11}<z_{M5}$ is depicted while b) shows the behaviour for $x_0^{11}>{\tilde{x}}_0^{11}>z_{M5}$. The second boundary at $x^{11}=d$ is not depicted -- it would truncate the solution at the finite distance $d$.
• Figure 3: The dependence of $G_N$ on $d$ is shown in figure a). Qualitatively (modulo an external warp-factor) this can be understood from the corresponding variation of the seven-fold $CY_3\times S^1/\mathbb{Z}_2$ volume $V(\text{7-fold})$ with $d$, figure b). Note that the decrease of $V(\text{7-fold})$ beyond $x_0^{11}$ results from the analytic continuation of the warp-factors into the negative region.