Dilaton Domain Walls and Dynamical Systems

TL;DR

This work recasts domain-wall solutions in $d$-dimensional gravity with a dilaton and exponential potential $V(\sigma)=\Lambda e^{-{\lambda}\sigma}$ as a 2D autonomous dynamical system, revealing a transcritical bifurcation at $\lambda=\lambda_c$ for $\Lambda<0$. It gives a complete, exact phase-space description for $\lambda=0$, including separatrix walls connecting $adS_d$ to $adS_{d-1}\times\mathbb{R}$ and, in $d=3$, an explicit separatrix solution; for $\lambda>0$ the fixed-point structure remains tractable and flat walls are solvable, with Janus-type trajectories interpreted as marginal bound states of separatrix walls. The flat walls are shown to be supersymmetric for some superpotential $W$ determined by the solution, while non-flat walls with nonzero curvature do not preserve supersymmetry in this single-scalar setup. Overall, the paper provides a unified dynamical-systems framework to classify domain-wall spacetimes and their SUSY properties, with implications for holography and supergravity model-building.

Abstract

Domain wall solutions of $d$-dimensional gravity coupled to a dilaton field $σ$ with an exponential potential $Λe^{-λσ}$ are shown to be governed by an autonomous dynamical system, with a transcritical bifurcation as a function of the parameter $λ$ when $Λ<0$. All phase-plane trajectories are found exactly for $λ=0$, including separatrices corresponding to walls that interpolate between $adS_d$ and $adS_{d-1} \times\bR$, and the exact solution is found for $d=3$. Janus-type solutions are interpreted as marginal bound states of these ``separatrix walls''. All flat domain wall solutions, which are given exactly for any $λ$, are shown to be supersymmetric for some superpotential $W$, determined by the solution.

Figures (6)

• Figure 1: Bifurcation Diagram. The transcritical bifurcation corresponds to an exchange of stable (solid line) and unstable (dashed line) directions between two fixed points.
• Figure 2: (a) The phase plane for $\Lambda<0$. There are four fixed points, connected by separatrices. The solutions corresponding to trajectories along the $v-$axis are foliations of $adS$. (b) The phase plane for $\Lambda>0$. The straight line trajectory along the $v-$axis corresponds to the $dS$ foliation of de Sitter space (\ref{['desitterdesitter']}).
• Figure 3: (a) The phase plane has the same topology as for $\lambda=0$, with four hyperbolic fixed points, but the lateral symmetry is lost. (b) The four fixed points have coalesced to form a pair of non-hyperbolic fixed points.
• Figure 4: (a) The two $k\ne0$ fixed points are now in the $k=-1$ regions. (b) The $k=0$ fixed points have disappeared to $\infty$.
• Figure 5: (a) The phase plane topology is the same as for $\lambda=0$ but the lateral symmetry is lost and there are now two trajectories that are asymptotic to the $v$-axis. (b) For $\lambda>2\alpha$ there are two $k=0$ fixed points, both of which are nodes. All $k=1$ trajectories start at one node and end at the other one.