Effective Actions for Heterotic M-Theory

TL;DR

The paper develops a four-dimensional effective description of heterotic M-theory using a moduli space approximation, first for two boundary branes and then including a bulk brane. The authors derive a simple gravity-plus-scalars action in the no-bulk-brane case, $S_{\mathrm{mod}} = 6 \int_{4d} [-\dot{a}^2 + a^2 (\dot{\psi}^2 + \dot{\chi}^2)]$, and show how the colliding-brane (LMT) scaling solution emerges as a moduli-space trajectory with a hard-wall boundary at $\chi=0$, corresponding to a bounce of the negative-tension brane. Including a bulk brane yields a richer moduli space with an extra modulus $Y$; in the large harmonic function limit the action reduces to a gravity-plus-two-scalar system with exponential couplings to $Y$, while a symmetric case with two negative-tension boundaries reproduces a two-distance moduli description. These results provide a framework to study cosmological dynamics and perturbations in heterotic M-theory by incorporating inter-brane forces via effective potentials and to examine how higher-dimensional structures constrain four-dimensional trajectories.

Abstract

We discuss the moduli space approximation for heterotic M-theory, both for the minimal case of two boundary branes only, and when a bulk brane is included. The resulting effective actions may be used to describe the cosmological dynamics in the regime where the branes are moving slowly, away from singularities. We make use of the recently derived colliding branes solution to determine the global structure of moduli space, finding a boundary at which the trajectories undergo a hard wall reflection. This has important consequences for the allowed moduli space trajectories, and for the behaviour of cosmological perturbations in the model.

Figures (4)

• Figure 1: A Kruskal plot of the colliding branes solution described in LMT. In suitable coordinates, the bulk geometry is static and contains a timelike naked singularity (denoted by thick black lines) corresponding to a zero of the bulk warp factor. The dashed lines indicate representative orbits of the bulk Killing vector field. The boundary branes then move through this bulk geometry according to the Israel junction conditions. The trajectory of the positive-tension brane is shown in red and that of the negative-tension brane in green. The collision of the branes, as well as the two bounces of the negative-tension brane off the naked singularity, are shown at a magnified scale in the inset. The regularisation of these bounces is discussed in detail in LMT.
• Figure 2: The trajectory of the scaling solution as seen in the $\psi$ - $\chi$ plane. $\chi = 0$ corresponds to the scale factor on the negative-tension brane shrinking to zero, and thus the $\chi = 0$ plane represents a boundary to moduli space, at which the scaling solution trajectory is reflected. The brane collision occurs as $\psi$, $\chi \rightarrow - \infty$. Also shown are the directions of increasing logarithm of the distance between the boundary branes ($\ln{d}$) and increasing logarithm of the Calabi-Yau volume at the location of the positive-tension brane ($\phi_+$). Away from the $\chi=0$ boundary, $e^{\phi_+}$ is approximately equal to the volume of the Calabi-Yau at the location of the negative-tension brane and thus also approximately equal to the average Calabi-Yau volume.
• Figure 3: An alternative viewpoint on the colliding branes scaling solution: this plot shows how the volume $V_{\pm}$ of the Calabi-Yau manifold at the location of the boundary branes at $y=\pm 1$ depends on the inter-brane distance $d$. The upper curve represents $V_+$ while the lower curve represents $V_-.$
• Figure 4: The harmonic function in the presence of a bulk brane at $y=Y$.