The non-SUSY $AdS_6$ and $AdS_7$ fixed points are brane-jet unstable

In six- and seven-dimensional gauged supergravity, each scalar potential has one supersymmetric and one non-supersymmetric fixed points. The non-supersymmetric $AdS_7$ fixed point is perturbatively unstable. On the other hand, the non-supersymmetric $AdS_6$ fixed point is known to be perturbatively stable. In this note we examine the newly proposed non-perturbative instability, called brane-jet instabilities of the $AdS_6$ and $AdS_7$ vacua. We find that when they are uplifted to massive type IIA and eleven-dimensional supergravity, respectively, the non-supersymmetric $AdS_6$ and $AdS_7$ vacua are both brane-jet unstable, in fond of the weak gravity conjecture.


Introduction and conclusions
The AdS/CFT correspondence, [1], has provided a framework to study quantum field theories in various dimensions with various amount of supersymmetry through their gravitational duals. When it comes to the non-supersymmetric quantum field theories, even though there are several known perturbatively stable, [2,3,4], non-supersymmetric AdS vacua, due to the limited control over the non-supersymmetry, not much was able to be investigated. 1 Furthermore, recently, as a stronger version of the weak gravity conjecture, [6], a conjecture on non-supersymmetric AdS vacua was suggested: there is no stable non-supersymmetric AdS vacua from string and M-theory, [7]. In support of testing the conjecture, a new non-perturbative decay channel called brane-jet instability was proposed by Bena, Pilch and Warner in [8]. This examines the force acting on the probe branes and when the force is repulsive, the vacua is determined to be unstable. In [8], the authors showed the only known perturbatively stable non-supersymmetric AdS 4 vacuum, [9,10], among the AdS vacua of four-, [11], and five-, [12], dimensional maximal gauged supergravity is, in fact, brane-jet unstable.
The purpose of this note is to examine the brane-jet instability of AdS vacua of six-and seven-dimensional gauged supergravity in [13] and in [14,15,16,17], respectively. In six-and seven-dimensional gauged supergravity, each scalar potential has one supersymmetric and one non-supersymmetric fixed points.
In seven dimensions, minimal gauged supergravity, [14,15], is a subsector of maximal gauged supergravity, [16,17]. As we identify the scalar fields to a scalar field, the maximal theory reduces to the minimal theory. The scalar potentials of the theories have a pair of supersymmetric and non-supersymmetric fixed points. The non-supersymmetric fixed point is known to be perturbatively stable in the minimal theory, [15], but not stable in the maximal theory, [17]. Maximal and minimal theories commonly uplift to eleven-dimensional supergravity, [18,19,20] and [21], but the minimal theory also uplifts to massive type IIA supergravity, [22]. We will examine the brane-jet stability of the AdS 7 fixed points when they are uplifted to eleven-dimensional supergravity.
In F (4) gauged supergravity in six dimensions, [13], there are also a pair of supersymmetric and non-supersymmetric fixed points. The non-supersymmetric AdS 6 fixed point is known to be perturbatively stable, [13]. F (4) gauged supergravity is a consistent truncation of massive type IIA supergravity, [23] and also of type IIB supergravity, [24,25,26]. We will examine the branejet stability of the AdS 6 fixed points when they are uplifted to massive type IIA supergravity.
Indeed we show that when they are uplifted to massive type IIA and eleven-dimensional supergravity, respectively, the non-supersymmetric AdS 6 and AdS 7 fixed points are both branejet unstable in favor of the conjecture on non-supersymmetric vacua in [7].
It would be interesting to consider the alternative uplifts of the AdS 6 and AdS 7 fixed points to type IIB, [24,25,26], and massive type IIA supergravity, [22], respectively. Indeed, the instabilities of AdS 7 solutions in massive type IIA supergravity are already examined with matter couplings in [27,28,29]. It would be interesting to see if a fixed point is uplifted to different theories in higher dimensions and stabilities of the uplifted solutions are different.
In section 2 and 3, we test the brane-jet instabilities of AdS fixed points from six-and seven-dimensional gauged supergravity, respectively. In an appendix, we present the calculation of potentials of the fluxes for supersymmetric flows and show that the probe brane potentials vanish over the whole flows identically.
2 The AdS 6 fixed points

Solutions in massive type IIA supergravity
We consider the scalar-gravity action of F (4) gauged supergravity, [13], in the conventions of [23], There are supersymmetric and non-supersymmetric fixed points of the scalar potential at e We employ the uplift formula to massive type IIA supergravity, [30], in [23]. In Einstein frame, the metric, the dilaton, and the four-form flux are non-trivial and are given, respectively, by, [31], 2) where we define The metric and the volume form of the three-sphere are given, respectively, by where σ I are SU (2) left-invariant one-forms, We may introduce explicit SU (2) left-invariant one-forms, Then the metric and the volume form are (2.9)

D4-brane probes
The uplift formula for the six-form flux is given by, [23], (2.10) At the AdS 6 fixed points, it gives and l is the radius of AdS 6 . Thus we obtain that the five-form potential is where we use ∂ r U = 0, ∂ ξ U = 0 at the fixed points. U is so-called geometric scalar potential. 2 We partition the spacetime coordinates, and choose the static gauge, where η a are the worldvolume coordinates. The pull-back of the metric is Now we study the worldvolume action of the D4-branes which is given by a sum of DBI and WZ terms. If the probe branes move slowly, the worldvolume action in Einstein frame is = − ∆ 1/16 X 5/16 sin 5/24 ξ d 5 η e 5A ∆ 15/16 X 5/16 sin 5/24 ξ − 1 2 e 3A ∆ 9/16 X 3/16 sin 1/8 ξG mnẏ mẏn + · · · − √ 2g 3

17)
2 For the supersymmetric flows we can calculate the five-form potential over the whole flow. See appendix A.1. whereC (5) is the pull-back of the five-form potential. 3 Then the worldvolume action reduces to where the kinetic and the potential terms are The final probe brane potential is quite simple. From the probe brane potential, we test the brane-jet instabilities of the supersymmetric and non-supersymmetric AdS 6 fixed points. We set g = 3 √ 2 2 for l = 1. The plots of the brane potential over the hemisphere, 0 ≤ ξ ≤ π, are given in Figure 1. We conclude that the non-supersymmetric AdS 6 fixed point is not stable.
3 The AdS 7 fixed points 3

M5-brane probes
At the AdS 7 fixed points, the seven-form flux is where and l is the radius of AdS 7 . Thus we obtain that the six-form potential is where we use ∂ r U = 0, ∂ α U = 0 at the fixed points. U is so-called geometric scalar potential.  We partition the spacetime coordinates, and choose the static gauge, where η a are the worldvolume coordinates. The pull-back of the metric is Now we study the worldvolume action of the M5-branes which is given by a sum of DBI and WZ terms. If the probe branes move slowly, the worldvolume action is whereC (6) is the pull-back of the six-form potential. 6 Then the worldvolume action reduces to where the kinetic and the potential terms are (3.14) The final probe brane potential is quite simple. From the probe brane potential, we test the brane-jet instabilities of the supersymmetric and non-supersymmetric AdS 7 fixed points. We set g = 2 for l = 1. The plots are given in Figure 2. We conclude that the non-supersymmetric AdS 7 fixed point is not stable.

A Potentials of the fluxes for supersymmetric flows
In the appendix we derive the potentials of the fluxes for supersymmetric flows.

A.1 Flows from AdS 6
We consider the domain wall background, [34], The supersymmetry equations are given by The uplift formula for the six-form flux is given by, [23], For the domain wall solutions, the six-form flux is Then we obtain that the five-form potential is If we employ this five-form potential to compute the probe brane potential, it vanishes identically over the whole flow, V = e 5A (∆ − ∆) = 0 . (A.8)

A.2 Flows from AdS 7
We consider the domain wall background, [37], The uplift formula for the six-form flux is given by, [32], From an analogous calculation of the previous subsection, we obtain that the six-form potential is If we employ this six-form potential to compute the probe brane potential, it vanishes identically over the whole flow, V = e 6A (∆ − ∆) = 0 . (A. 13)