Bubble instabilities of mIIA on AdS$_4\times S^7$

The authors of this paper study the stability of a non-supersymmetric AdS$_4$ vacuum obtained from compactification of massive type IIA supergravity on $S^6$. There exists a $G_2$ invariant solution that is perturbatively stable and the authors set out to search for non-perturbative decay channels. They identify a bubbling solution that corresponds to such a non-perturbative decay channel via a bubble of nothing.

A previous report identified a few minor points for improvement and the authors addressed these points to the satisfaction of the other referee and myself.

The paper is now very well written and the result are very interesting, in particular, given the recent conjecture that all non-supersymmetric AdS vacua are unstable. The results also generalize to a larger class of solutions for which the internal $S^6$ is replaced by any other nearly Kähler manifold, which will be useful in the future. Therefore, in conclusion I recommend this paper for publication in SciPost.

After the clarifications and modifications on the original manuscript, I recommend the paper for publication in its current form.

A non-perturbative decay channel is proposed for the non-susy and G$_{2}$-invariant AdS$_{4}$ vacuum of massive IIA supergravity on S$^6$. This is further investigated using an effective supergravity description that retains, amongst others, two massive modes with normalised mass $m^2 L^2=20$ not included in the ISO(7) gauged supergravity. The non-perturbative solution describing the decay is obtained numerically and its geometry is qualitatively characterised in a piece-wise manner: and AdS region, a BoN region and a (smeared) D2-brane region. A non-flat but dS$_{3}$ slicing of the external spacetime geometry plays an essential role in each of the three regions, thus selecting very specific boundary conditions for the full solution to exist.

The manuscript contains interesting and timely results regarding the stability of non-susy AdS vacua, which is turning into an active research area in light of the Swampland conjecture. Before the paper can be recommended for publication in SciPost, I would like the authors to address the following comments/questions:

$\bullet$ It is stated at various places in the text that, for the analysed vacuum, a bubble is created and starts to expand, eventually consuming the complete vacuum. How does the existence of an intermediate (and only approximate) BoN-like region in the full solution guarantee this phenomenon? In other words, why having such an intermediate BoN-like region is enough to guarantee that the bubble has enough (physical) time to eat up the analysed vacuum?

$\bullet$ In the ansatz $(3.1)$, two functions $f_{41}(r)$ and $f_{42}(r)$ are introduced to specify $F_{4}$. Later on, when computing the dual $F_{6}=\star F_{4}$ in $(3.4)$, there is an additional function $f_{43}(r)$ appearing. And finally, when computing the Bianchi identity $dF_{4}=H_{3} \wedge F_{2}$ in (3.5), there is yet another function $\zeta(r)$ appearing. Are $f_{43}(r)$ and $\zeta(r)$ introduced somewhere in the text apart from the equations where they suddenly appear?

$\bullet$ Figure 4: I understand that the tuning of parameters in (4.13) is well motivated by the existence of a region with an almost constant $e^{3U-\phi/4}$ function. However, why is this sufficient to identify the resulting region with a BoN-regime? If looking closely at Figure 4, the claimed to be a BoN-like region between $r \sim 1.1$ and $r \sim 1.5$ is somehow not obvious. For example, the blue curve of the full solution and the green curve of the BoN are convex and concave within this window, respectively. Why the physical interpretation of this region as a BoN regime is justified and why the deviation from a truly BoN does not spoil the dynamical eating up process?

$\bullet$ Figure 4: The region of the full solution (blue curve) between $r \sim 0.9$ and $r \sim 1.1$ does not seem to be well approximated by any of the three regimes described above. What does it correspond to?

$\bullet$ There are $4$ coefficients $c_{1,2,3,4}$ compatible with regularity of the boundary conditions about the AdS region, and $2$ constraints in $(4.13)$ required by the intermediate bubble regime. So, naively, one would expect $4-2=2$ free parameters, although one of them could still be set at will by virtue of a further scaling of the radial coordinate. This would yet leave $1$ free parameter to perform a scanning of numerical solutions. Is the solution in $(4.14)$ just an element within a one-parameter family of solutions? If so, why this choice of solution and how do the other solutions look like? Or is this free parameter fixed by other means so the solution is unique?

$\bullet$ When discussing the possible existence of a similar flow for the susy and G$_2$-invariant vacuum the authors write: trying to impose similar conditions as those in $(4.13)$ will completely trivialize the
flow." What do they mean precisely by trivialize the flow"?

$\bullet$ Right afterwards the authors write: without such constraints one obtains solutions displaying an
intermediate regime that is still well approximated by $(2.5)$, but with $\alpha \neq \frac{1}{2}$". What value of $\alpha$ is actually obtained numerically and what is the physical interpretation of the resulting flow? Why, in general, an intermediate regime given by $(2.5)$ with $\alpha \neq \frac{1}{2}$ (thus not coming from Schw$_{5}$) does not represent a potential instability for a given asymptotic AdS solution (irrespective of susy)?