A tale of two superpotentials: Stability and Instability in Designer Gravity

TL;DR

The paper addresses stability in asymptotically AdS gravity with tachyonic scalars near the BF bound under boundary conditions $\beta = dW/d\alpha$. It shows that a lower energy bound, proved via a spinor charge, crucially depends on the existence of a $P_-(\phi)$-type superpotential for the scalar potential $V(\phi)$, while a $P_+$-type alone is insufficient and can even lead to divergences in the spinor charge. The authors demonstrate, through analytic proofs and numerical experiments, that two branches $P_\pm$ can exist for a given $V$, and that soliton content correlates with the presence or absence of $P_-$. They also establish global existence of Witten spinors in this context and verify the spinor–energy relation numerically, offering a coherent resolution to earlier apparent paradoxes and clarifying when designer gravity theories are energetically bounded.

Abstract

We investigate the stability of asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound. The boundary conditions in these ``designer gravity'' theories are defined in terms of an arbitrary function W. Previous work had suggested that the energy in designer gravity is bounded below if i) W has a global minimum and ii) the scalar potential admits a superpotential P. More recently, however, certain solutions were found (numerically) to violate the proposed energy bound. We resolve the discrepancy by observing that a given scalar potential can admit two possible branches of the corresponding superpotential, P_{\pm}. When there is a P_- branch, we rigorously prove a lower bound on the energy; the P_+ branch alone is not sufficient. Our numerical investigations i) confirm this picture, ii) confirm other critical aspects of the (complicated) proofs, and iii) suggest that the existence of P_- may in fact be necessary (as well as sufficient) for the energy of a designer gravity theory to be bounded below.

Figures (5)

• Figure 1: Shown here are numerical plots of the derivative $P'(\phi)$ for the example presented in the text. The dashed line corresponds to the $P_+$ solution, which is stationary at the global minimum of the potential, $\phi_{min} = \sqrt{2}$. The solid line corresponds to the $P_-$ solution, whose derivative vanishes at $\phi \approx .27$. Hence, $P_-$ does not meet the global existence criterion.
• Figure 2: The critical potential (solid line) that is on the verge of violating the positive energy theorem for $W \geq 0$ designer gravity boundary conditions. The dashed line shows a potential that violates the positive energy theorem for designer gravity boundary conditions, but satisfies such a theorem for fast fall-off boundary conditions ($\alpha=0$).
• Figure 3: The value of $\alpha$ for the $W=0$ soliton in a range of theories with different values for $A$, the coefficient of the $\phi^8$ term in (\ref{['pot']}). One sees that regular spherical $W=0$ soliton solutions cease to exist precisely when $A \rightarrow A_c^{-}=.283$.
• Figure 4: Numerical results for the example (\ref{['veg']}) confirm the existence of Witten spinors that are regular at the origin and have the correct behavior at large $r$. The solid line is $\textrm{Re}\, R_- = \textrm{Im}\, R_+ = \textrm{Re}\, \tilde{R}_- = \textrm{Im}\,\tilde{R}_+$, which by the arguments in the text asymptotically approach $\frac{\rho_1}{2\sqrt{2}}$, shown here as the dashed line.
• Figure 5: The solid line is $\textrm{Im}\, R_- = \textrm{Re}\, R_+ = -\textrm{Im}\, \tilde{R}_- = -\textrm{Re}\,\tilde{R}_+$, which by the arguments in the text asymptotically approach $-\frac{\rho_2}{2\sqrt{2}}$, shown here as the dashed line.