Universal isolation in the AdS landscape

We study the universal conditions for quantum nonperturbative stability against bubble nucleation for pertubatively stable AdS vacua based on positive energy theorems. We also compare our analysis with the preexisting ones in the literature carried out within the thin-wall approximation. The aforementioned criterion is then tested in two explicit examples describing massive type IIA string theory compactified on $S^3$ and $S^3\times S^3$, respectively. The AdS landscape of both classes of compactifications is known to consist of a set of isolated points. The main result is that all critical points respecting the Breitenlohner-Freedman (BF) bound also turn out be stable at a nonperturbative level. Finally, we speculate on the possible universal features that may be extracted from the above specific examples.

Figure 1

The three different situations that may occur in presence of a local AdS maximum and a local AdS minimum. Left: Both points obey the positive energy theorem and are hence stable against bubble nucleation. Middle: The maximum turns unstable and decays towards the minimum, which instead stays nonperturbatively stable. Right: Both the maximum and the minimum exhibit nonperturbative instabilities.

Figure 2

The three different situations that may occur in presence of two local AdS minima separated by an unstable maximum. Left: Both points obey the positive energy theorem and are hence stable against bubble nucleation. Middle: The higher minimum turns unstable and decays towards the deeper minimum, which instead stays nonperturbatively stable. Right: Both minima violate the hypothesis of the positive energy theorem and still, the conclusion stays identical to the 2b case.

Figure 3

The nonperturbative stability of massive type IIA on ${\mathrm{AdS}}_7\times S^3$ models summarized. The blue curve shows the profile of the scalar potential (4.2), with a nonsupersymmetric local minimum (left) and a supersymmetric local maximum (right). From both points there starts a globally bounding function $-\frac{3}{5}{f}^2$ ensuring their nonperturbative stability (curves in red and brown, respectively). Note that these branches only cross at one point and respecting crossing rule number 2. This exactly realizes Situation 1a in Fig. 1. Finally, the green curve represents the bounding function defining the static DW (zoomed in on the right).

Figure 4

The nonperturbative stability of massive type IIA on ${\mathrm{AdS}}_4\times S^3\times S^3$ models summarized. The above sheet represents the profile of the scalar potential (4.2) in a particular two-dimensional slice of the scalar manifold, with a supersymmetric local extremum on the left and two additional nonsupersymmetric ones (solutions 3 and 4 in Table 5). From all points there starts a globally bounding function $-3f^2$ ensuring their nonperturbative stability (represented by the paraboloids peaked at each critical point).

Figure 5

The profile of the six real scalars in our STU-model along the flow representing the static DW interpolating between solutions 4 and 1.

Figure 6

The above diagrams represent a plane within the scalar manifold containing Solutions 1 (lower-left), 3 (lower-right) and 4 (upper-middle) of Table 5. Left: The level curves of the scalar potential (4.21). Right: The level curves of the fake superpotential $f$ defining the static DW. In both plots the thick red line denotes the projection of the DW curve connecting solutions 4 to 1. Note that, since such curve is just a projection of the actual extremal trajectory, it needs not be orthogonal to the level curves of $f$.

Figure 7

The potential profile (red), the globally bounding functions $-3f_{\mathrm{SUSY}}^2$ (green) & $-3f_{\cancel{\mathrm{SUSY}}}^2$ (blue), and finally $-3f_{\mathrm{DW}}^2$ (orange). This situation exactly reproduces Situation 1a in Fig. 1. The asymptotic behavior of $f_{\mathrm{SUSY}}^2$ vs $f_{\cancel{\mathrm{SUSY}}}^2$ as $\sigma =\mathrm{Re}(S)\to 0^{+}$; the former diverges exactly as $V$ itself, whereas the latter has a faster slope, i.e. $-\sqrt{3}$, but exactly such that it cancels against $(\partial f{)}^2$ when calculating $V$. Similar arguments correctly predict the asymptotic behavior at other boundaries of ${\mathcal{M}}_{\text{scalar}}$.

Figure 8

Log-plot showing the asymptotic behavior of the potential and the bounding solution starting from the nonsusy solution 4.