The non-SUSY Ad S 6 and Ad S 7 fixed points are brane-jet unstable
TThe non-SUSY
AdS and AdS fixed pointsare brane-jet unstable Minwoo Suh
Department of Physics, Kyungpook National University, Daegu 41566, Korea [email protected]
Abstract
In six- and seven-dimensional gauged supergravity, each scalar potential has one super-symmetric and one non-supersymmetric fixed points. The non-supersymmetric
AdS fixedpoint is perturbatively unstable. On the other hand, the non-supersymmetric AdS fixedpoint is known to be perturbatively stable. In this note we examine the newly proposednon-perturbative instability, called brane-jet instabilities of the AdS and AdS vacua. Wefind that when they are uplifted to massive type IIA and eleven-dimensional supergravity,respectively, the non-supersymmetric AdS and AdS vacua are both brane-jet unstable,in fond of the weak gravity conjecture. April, 2020 a r X i v : . [ h e p - t h ] A p r ontents AdS fixed points 2 AdS fixed points 5 A Potentials of the fluxes for supersymmetric flows 8
A.1 Flows from
AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8A.2 Flows from AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The AdS/CFT correspondence, [1], has provided a framework to study quantum field theoriesin 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 severalknown perturbatively stable, [2, 3, 4], non-supersymmetric
AdS vacua, due to the limited controlover the non-supersymmetry, not much was able to be investigated. Furthermore, recently, asa 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 theprobe branes and when the force is repulsive, the vacua is determined to be unstable. In [8], theauthors showed the only known perturbatively stable non-supersymmetric AdS 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- andseven-dimensional gauged supergravity in [13] and in [14, 15, 16, 17], respectively. In six- andseven-dimensional gauged supergravity, each scalar potential has one supersymmetric and onenon-supersymmetric fixed points. There is a recent search of curious 5 d non-supersymmetric CFTs in [5].
1n seven dimensions, minimal gauged supergravity, [14, 15], is a subsector of maximal gaugedsupergravity, [16, 17]. As we identify the scalar fields to a scalar field, the maximal theory reducesto the minimal theory. The scalar potentials of the theories have a pair of supersymmetric andnon-supersymmetric fixed points. The non-supersymmetric fixed point is known to be pertur-batively stable in the minimal theory, [15], but not stable in the maximal theory, [17]. Maximaland 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 examinethe brane-jet stability of the
AdS fixed points when they are uplifted to eleven-dimensionalsupergravity.In F (4) gauged supergravity in six dimensions, [13], there are also a pair of supersymmetricand non-supersymmetric fixed points. The non-supersymmetric AdS fixed point is known to beperturbatively stable, [13]. F (4) gauged supergravity is a consistent truncation of massive typeIIA supergravity, [23] and also of type IIB supergravity, [24, 25, 26]. We will examine the brane-jet stability of the AdS 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-dimensionalsupergravity, respectively, the non-supersymmetric AdS and AdS fixed points are both brane-jet unstable in favor of the conjecture on non-supersymmetric vacua in [7].It would be interesting to consider the alternative uplifts of the AdS and AdS fixed pointsto type IIB, [24, 25, 26], and massive type IIA supergravity, [22], respectively. Indeed, theinstabilities of AdS solutions in massive type IIA supergravity are already examined with mattercouplings in [27, 28, 29]. It would be interesting to see if a fixed point is uplifted to differenttheories 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- andseven-dimensional gauged supergravity, respectively. In an appendix, we present the calculationof potentials of the fluxes for supersymmetric flows and show that the probe brane potentialsvanish over the whole flows identically.
AdS fixed points We consider the scalar-gravity action of F (4) gauged supergravity, [13], in the conventions of[23], e − L = R − ∂ µ φ∂ µ φ − g (cid:18) e √ φ − e √ φ − e − √ φ (cid:19) . (2.1)There are supersymmetric and non-supersymmetric fixed points of the scalar potential at e − √ φ = 1 and e − √ φ = 1 / / , respectively. 2e employ the uplift formula to massive type IIA supergravity, [30], in [23]. In Einsteinframe, the metric, the dilaton, and the four-form flux are non-trivial and are given, respectively,by, [31], ds = X / sin / ξ (cid:18) ∆ / ds + 2 g ∆ / X dξ + 12 g cos ξ ∆ / X ds S (cid:19) , (2.2) e Φ = ∆ / X / sin / ξ , (2.3) F (4) = − √ U sin / ξ cos ξg ∆ dξ ∧ vol S − √ / ξ cos ξg ∆ X dX ∧ vol S , (2.4)where we define X = e − φ √ , ∆ = X cos ξ + X − sin ξ ,U = X − sin ξ − X cos ξ + 4 X − cos ξ − X − . (2.5)The metric and the volume form of the three-sphere are given, respectively, by ds S = (cid:88) I =1 (cid:0) σ I (cid:1) ,vol S = σ ∧ σ ∧ σ , (2.6)where σ I are SU (2) left-invariant one-forms, dσ I = − (cid:15) IJK σ J ∧ σ K . (2.7)We may introduce explicit SU (2) left-invariant one-forms, σ = − sin α cos α dα + sin α dα ,σ = sin α sin α dα + cos α dα ,σ = cos α dα + dα . (2.8)Then the metric and the volume form are ds S = dα + dα + dα + 2 cos α dα dα ,vol S = sin α dα dα dα . (2.9)3 .2 D4-brane probes The uplift formula for the six-form flux is given by, [23], F (6) = e Φ / ∗ F (4) = − √ g U vol + 4 √ g sin ξ cos ξX ∗ dX ∧ dξ + · · · . (2.10)At the AdS fixed points, it gives F (6) = − √ g U e A dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dr , (2.11)where A = rl , U = U ( ξ ) , X = constant , (2.12)and l is the radius of AdS . Thus we obtain that the five-form potential is C (5) = √ g l U e A dx ∧ dx ∧ dx ∧ dx ∧ dx , (2.13)where we use ∂ r U = 0, ∂ ξ U = 0 at the fixed points. U is so-called geometric scalar potential. We partition the spacetime coordinates, x a = { x , x , x , x , x } , y m = { r, ξ, α , α , α } , (2.14)and choose the static gauge, x = t = η , x a = η a , y m = y m ( t ) , (2.15)where η a are the worldvolume coordinates. The pull-back of the metric is˜ G ab = G µν ∂x µ ∂η a ∂x ν ∂η b . (2.16)Now we study the worldvolume action of the D4-branes which is given by a sum of DBI andWZ terms. If the probe branes move slowly, the worldvolume action in Einstein frame is S = − e Φ / (cid:90) d η (cid:113) − det( ˜ G ) − (cid:90) ˜ C (5) = − ∆ / X / sin / ξ (cid:90) d η (cid:18) e A ∆ / X / sin / ξ − e A ∆ / X / sin / ξG mn ˙ y m ˙ y n + · · · (cid:19) − (cid:90) √ g l e A U dx ∧ dx ∧ dx ∧ dx ∧ dx , (2.17) For the supersymmetric flows we can calculate the five-form potential over the whole flow. See appendix A.1. - - ξ - - ⅇ - A V - - - ξ ⅇ - A V Figure 1:
The probe brane potentials of the supersymmetric and non-supersymmetric fixed pointsat X = 1 and X = 1 / / , respectively. where ˜ C (5) is the pull-back of the five-form potential. Then the worldvolume action reduces to S = (cid:90) d η ( K − V ) , (2.18)where the kinetic and the potential terms are K = 12 e A ∆ / X / sin / ξ G mn ˙ y m ˙ y n + · · · ,V = e A (cid:32) ∆ + √ g l U (cid:33) . (2.19)The final probe brane potential is quite simple. From the probe brane potential, we test thebrane-jet instabilities of the supersymmetric and non-supersymmetric AdS fixed points. We set g = √ for l = 1. The plots of the brane potential over the hemisphere, 0 ≤ ξ ≤ π , are givenin Figure 1. We conclude that the non-supersymmetric AdS fixed point is not stable. AdS fixed points We consider the minimal scalar-gravity action of seven-dimensional gauged supergravity, [14, 15]and [16, 17], in the conventions of [32], e − L = R − ∂ µ λ∂ µ λ + g (cid:18) X + 4 X − − X − (cid:19) , (3.1) The sign of the ˜ C (5) term is determined by the orientation of our solution. X = e λ . There are supersymmetric and non-supersymmetric fixed points of the scalarpotential at X = 1 and X = 1 / / , respectively.We employ the uplift formula to eleven-dimensional supergravity, [33], in [32]. The metricand the seven-form flux are given by, ds = ∆ / ds + 1 g ∆ − / (cid:32) X − dµ + (cid:88) i =1 X − i (cid:0) dµ i + µ i dφ i (cid:1)(cid:33) , (3.2) F (7) = U vol + 12 g (cid:88) α =0 X − α ∗ dX α ∧ d ( µ α ) , (3.3)where vol and ∗ are volume form and Hodge dual on ds . We define X = X = X = e λ , X = ( X X ) − , ∆ = (cid:88) α =0 X α µ α ,U = 2 g (cid:88) α =0 (cid:0) X α µ α − ∆ X α (cid:1) + g ∆ X , (cid:88) α =0 µ α . (3.4)We introduce explicit coordinates, µ = cos α , µ = sin α cos β , µ = sin α cos β . (3.5) At the
AdS fixed points, the seven-form flux is F (7) = U e A dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dr , (3.6)where A = rl , U = U ( α ) , X = constant , (3.7)and l is the radius of AdS . Thus we obtain that the six-form potential is C (6) = l U e A dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dx , (3.8)where we use ∂ r U = 0, ∂ α U = 0 at the fixed points. U is so-called geometric scalar potential. For the scalar fields in [32], X = e − √ ϕ − √ ϕ and X = e √ ϕ − √ ϕ , we set ϕ = 0 and ϕ = − √ λ . For the supersymmetric flows we can calculate the six-form potential over the whole flow. See appendix A.2. .5 1.0 1.5 2.0 2.5 3.0 α - - ⅇ - A V α ⅇ - A V Figure 2:
The probe brane potentials of the supersymmetric and non-supersymmetric fixed pointsat X = 1 and X = 1 / / , respectively. We partition the spacetime coordinates, x a = { x , x , x , x , x , x } , y m = { r, α, β, φ , φ } , (3.9)and choose the static gauge, x = t = η , x a = η a , y m = y m ( t ) , (3.10)where η a are the worldvolume coordinates. The pull-back of the metric is˜ G ab = G µν ∂x µ ∂η a ∂x ν ∂η b . (3.11)Now we study the worldvolume action of the M5-branes which is given by a sum of DBI andWZ terms. If the probe branes move slowly, the worldvolume action is S = − (cid:90) d η (cid:113) − det( ˜ G ) − (cid:90) ˜ C (6) = − (cid:90) d η (cid:18) e A ∆ − e A ∆ / G mn ˙ y m ˙ y n + · · · (cid:19) − (cid:90) l U e A dx ∧ dx ∧ dx ∧ dx ∧ d ∧ dx . (3.12)where ˜ C (6) is the pull-back of the six-form potential. Then the worldvolume action reduces to S = (cid:90) d η ( K − V ) , (3.13) The sign of the ˜ C (6) term is determined by the orientation of our solution. K = 12 e A ∆ / G mn ˙ y m ˙ y n + · · · ,V = e A (cid:18) ∆ + l U (cid:19) . (3.14)The final probe brane potential is quite simple. From the probe brane potential, we test thebrane-jet instabilities of the supersymmetric and non-supersymmetric AdS fixed points. We set g = 2 for l = 1. The plots are given in Figure 2. We conclude that the non-supersymmetric AdS fixed point is not stable. Acknowledgements
We are grateful to Nakwoo Kim for helpful discussions and to Krzysztof Pilch for reading adraft. This research was supported by the National Research Foundation of Korea under thegrant NRF-2019R1I1A1A01060811.
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 We consider the domain wall background, [34], ds = e A ds , + dr . (A.1)The supersymmetry equations are given by φ (cid:48) = g (cid:16) e − φ √ − e φ √ (cid:17) ,A (cid:48) = g √ (cid:18) e − φ √ + 13 e φ √ (cid:19) . (A.2)The uplift formula for the six-form flux is given by, [23], F (6) = e Φ / ∗ F (4) = − √ g U vol + 4 √ g sin ξ cos ξX ∗ dX ∧ dξ . (A.3)For the domain wall solutions, the six-form flux is F (6) = ω r dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dr + ω ξ dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dξ , (A.4)8here ω r = − √ g e A U ,ω ξ = 4 √ g e A X (cid:48) sin ξ cos ξX . (A.5)Employing the supersymmetry equations, (A.2), they satisfy an relation, ∂ω ξ ∂r = ∂ω r ∂ξ . (A.6)Then we obtain that the five-form potential is C (5) = − e A ∆ dx ∧ dx ∧ dx ∧ dx ∧ dx . (A.7)If we employ this five-form potential to compute the probe brane potential, it vanishes identicallyover the whole flow, V = e A (∆ − ∆) = 0 . (A.8) A.2 Flows from
AdS We consider the domain wall background, [37], ds = e A ds , + dr . (A.9)The supersymmetry equations are given by λ (cid:48) = 25 e − λ − e λ ,A (cid:48) = 15 e − λ + 45 e λ . (A.10)The uplift formula for the six-form flux is given by, [32], C (6) = l U e A dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dx . (A.11)From an analogous calculation of the previous subsection, we obtain that the six-form potentialis C (6) = − e A ∆ dx ∧ dx ∧ dx ∧ dx ∧ dx ∧ dx . (A.12)If we employ this six-form potential to compute the probe brane potential, it vanishes identicallyover the whole flow, V = e A (∆ − ∆) = 0 . (A.13) This is an analogous calculation of (3.13), (3.14), (3.28), (3.29) from [35] and (8) from [36]. eferences [1] J. M. Maldacena, The large N limit of superconformal field theories and supergravity,
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