PHYSICAL REVIEW D 96, 026016 (2017)

New Kaluza-Klein instantons and the decay of AdS vacua

Hirosi Ooguri1,2 and Lev Spodyneiko1,3 1Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA 2Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa 277-8583, Japan 3Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia (Received 17 April 2017; published 19 July 2017) We construct a generalization of Witten’s Kaluza-Klein instanton, where a higher-dimensional sphere (rather than a circle as in Witten’s instanton) collapses to zero size and the geometry terminates at a bubble of nothing, in a low effective theory of M theory. We use the solution to exhibit the instability of nonsupersymmetric AdS5 vacua in M theory compactified on positive Kähler-Einstein spaces, providing further evidence for the recent conjecture that any nonsupersymmetric anti–de Sitter supported by fluxes must be unstable.

DOI: 10.1103/PhysRevD.96.026016

I. INTRODUCTION bound [8,9]. When Γ has no fixed point and S5 is Stability is an important criterion for the consistency of sufficiently large, the instability modes are lifted and the Kaluza-Klein vacua. Due to nontrivial of their configuration becomes perturbatively stable. However, in 5 Γ internal spaces, the standard positive energy theorem [1–3] this case, S = is not simply connected, and there is a does not necessarily apply. In fact, it was shown by Witten Witten-type instanton where a homotopically nontrivial 5 [4] that the original Kaluza-Klein theory [5,6] on a product cycle on S =Γ collapses to zero size at a bubble of nothing 5 of the four-dimensional Minkowski and a circle [10]. This eliminates AdS5 × S =Γ as a candidate for a is unstable against a semiclassical decay process, unless stable nonsupersymmetric AdS geometry. protected by boundary conditions on fermions. The instan- In this paper, we present instanton solutions where a ton that mediates the decay is the analytical continuation of higher-dimensional sphere rather than a circle collapses, in a five-dimensional Schwarzschild solution, a low energy effective theory of M theory. Such general- 2 2 izations of Witten’s instanton were attempted earlier, for 2 ¼ dr þ 2 Ω2 þ 1 − R ϕ2 ds 2 r d 3 d ; example in [11], where it was found that fluxes needed to 1 − R r2 r2 cancel the intrinsic on the sphere to prevent it 2 where dΩ3 is the metrics on the unit 3-sphere, and ϕ is the from collapsing. Motivated by the recent conjecture [12] coordinate on the Kaluza-Klein circle. Smoothness of the (see also [13–15]) that any nonsupersymmetric AdS solution at r ¼ R requires ϕ to be periodic with period 2πR, vacuum supported by fluxes must be unstable, we found and the Kaluza-Klein radius at r ¼ ∞ is R.Aswemove examples in nonsupersymmetric setups which avoid the toward small r, the circle collapses and becomes zero size difficulty in the earlier attempt. at r ¼ R, where the geometry terminates. Another analytic We will focus on AdS5 times positive Kähler-Einstein 3 continuation of the polar angle on the 3-sphere turns this spaces, which break [16].AdS5 × CP is into a Lorentzian signature solution, where the “bubble of the only example of this type known to be stable against nothing” expands with velocity that asymptotes to the linearized perturbations [17]. Since its internal speed of light. space is simply connected, we do not expect it to have a Witten’s instanton has also played a role in the stability Witten-type instanton, where an S1 collapses to zero size at of anti–de Sitter (AdS) vacua. There have been several a bubble. On the other hand, CP3 can be realized as S2 proposals for nonsupersymmetric AdS geometries. Among fibration over S4, and it is possible for its S2 fibers to 5 them is AdS5 × S =Γ, where Γ is a discrete subgroup of the collapse. Indeed, we find such an instanton solution with SUð4Þ rotational symmetry of S5 [7]. Supersymmetry is finite . completely broken if Γ does not fit within an SUð3Þ Our solution avoids the difficulty with fluxes encoun- subgroup of the SUð4Þ symmetry. It turns out that, if Γ tered in [11] as follows. The AdS5 geometry in question is has a fixed point on S5 or if the radius of S5 is not large supported by the 4-form flux in M theory, with nonzero 2 4 enough, the perturbative spectrum on AdS5 contains closed components both on the S fibers and on the S base of the that violate the Breitenloner-Freedman internal space. As we move toward the center of AdS5, the

2470-0010=2017=96(2)=026016(9) 026016-1 © 2017 American Physical Society HIROSI OOGURI and LEV SPODYNEIKO PHYSICAL REVIEW D 96, 026016 (2017) 4-form flux reorients itself. By the time we reach the bubble where ω is the Kähler 2-form of internal space and c is of nothing, the flux has no components in the S2 fiber some constant, which will be related to the AdS radius. direction. Thus, the S2 can collapse at the bubble without With this ansatz, the right-hand side of the second equation violating the flux conservation. in (1) vanishes since F is nonzero only on M6, and the left- 3 AdS5 × CP is only marginally stable with a normal- hand side vanishes by the Kähler integrability condition izable mode at the Breitenlohner-Freedman bound. Thus, on ω. On the other hand, the first equation in (1) gives this vacuum is also in danger of becoming unstable by 2 2 higher derivative corrections to the 11-dimensional super- Rμν ¼ −2c gμν;Rmn ¼ 2c gmn; ð3Þ gravity. It is interesting to point out that in our instanton solution the normalizable mode at the Breitenlohner- where μ ¼ 0; …; 4 are the indices in the noncompact Freedman bound is turned on and is responsible for directions and m ¼ 5; …; 10 are on M6. Therefore, the 2 triggering the collapse of the S fiber. noncompact directions can be chosen to be AdS5, and M6 If there is a bubble of nothing instanton in AdS, it causes must be an Einstein manifold. The configuration breaks an instability that can be detected instantaneously on the supersymmetry [16,19] since there are no nontrivial sol- boundary of AdS [10,18]. This is because any observer in utions to δΨM ¼ 0 for the supersymmetry variation of the AdS can receive signals from any point on a Cauchy gravitini, surface within a finite amount of time, and an observer at the boundary in particular has access to an infinite volume δΨM ¼ DMε ¼ ∇Mε space near the boundary within an infinitesimal amount of 1 þ ðΓ NPQR − 8Γ NPQÞε ð Þ time. Therefore, our new instanton solution in the pertur- 144 MNPQRF NPQFM : 4 batively stable nonsupersymmetric AdS5 configuration offers further evidence for the conjecture of [12]. As we mentioned in the Introduction, the only known The plan of the paper is the following. In Sec. II we 3 3 perturbatively stable case is M6 ¼ CP . This space can be describe the AdS5 × CP solution and introduce our 2 realized as an S fibration, and we look for an instanton instanton ansatz. Boundary conditions on the instanton solution where the fiber collapses. In the next few sections at the bubble and at the infinity are discussed in Sec. III.It 3 we focus on M6 ¼ CP and in Sec. VII we show that an turns out that there are algebraic relations among variables ¼ SUð3Þ in our ansatz, as shown in Sec. IV. These relations reduce instanton solution for M6 Uð1Þ×Uð1Þ can be constructed the problem to a second order ordinary differential equation similarly. on a single function, which we will numerically solve in To make the S2 fibration explicit, we use the following Sec. V. Finiteness of the instanton action is verified in set of coordinates on CP3 [20]: SUð3Þ Sec. VI. In Sec. VII, we discuss AdS5 × Uð1Þ×Uð1Þ.Itisnot pffiffiffiffiffiffiffiffiffi known whether this geometry is stable against linearized e1 ¼ gðrÞdμ; pffiffiffiffiffiffiffiffiffi supergravity perturbation. Regardless, we will show that it ð Þ ¼ g r μΣ ¼ 2 3 4 allows a bubble of nothing solution and is therefore ei 2 sin i−1 for i ; ; ; unstable nonperturbatively. In the final section, we discuss pffiffiffiffiffiffiffiffiffi additional features of our instanton solutions. e5 ¼ hðrÞðdθ − A1 sin ϕ þ A2 cos ϕÞ; pffiffiffiffiffiffiffiffiffi ¼ ð Þ θð ϕ − θð ϕ þ ϕÞþ Þ 3 h r sin d cot A1 cos A2 sin A3 ; II. AdS5 × CP GEOMETRY AND INSTANTON ANSATZ ð5Þ

For any Kähler-Einstein sixfold M6, there exists an where AdS5 × M6 solution [16] to the 11-dimensional super- gravity , Σ ¼ γ α þ γ α β 1 cos d sin sin d ; 1 1 ¼ PQR − PQRS Σ2 ¼ − sin γdα þ cos γ sin αdβ; RMN 3 FM FNPQR 12 gMNF FPQRS ; Σ3 ¼ dγ þ cos αdβ; 1 ∇ FMPQR ¼ − εM1…M8PQRF F : μ 2 M 576 M1M2M3M4 M5M6M7M8 ¼ Σ ð Þ Ai cos 2 i: 6 ð1Þ Such a solution can be found by setting the 4-form field Here the first 4-tetrad corresponds to the base S4 and the strength as last two correspond to the S2 fiber. We multiplied them by the functions gðrÞ, hðrÞ to make their sizes dynamical. We F ¼ cω ∧ ω; ð2Þ take the vierbein on Euclidean AdS space to be

026016-2 NEW KALUZA-KLEIN INSTANTONS AND THE DECAY OF … PHYSICAL REVIEW D 96, 026016 (2017)

¼ dr; pffiffiffiffiffiffiffiffiffi ek ¼ fðrÞeˆk for k ¼ 8; 9; 10; 11; ð7Þ

4 where eˆk is any tetrad on the S . The metric in this frame is 1 X3 2 ¼ ð Þ μ2 þ 2μ Σ2 þ ð Þð θ − ϕ þ ϕÞ2 ds g r d 4 sin i h r d A1 sin A2 cos i¼1 2 2 2 2 þ hðrÞsin θðdϕ − cot θðA1 cos ϕ þ A2 sin ϕÞþA3Þ þ dr þ fðrÞdΩ4: ð8Þ

We used the freedom of coordinate redefinition by fixing the coefficient near dr2 to be 1. The next step is to write an ansatz for the 4-form field, utilizing the SUð3Þ-structure of the squashed CP3 given by the 2- form J and 3-form Ω as [21–24]

J ¼ − sin θ cos ϕðe12 þ e34Þ − sin θ sin ϕðe13 þ e42Þ − cos θðe14 þ e23Þþe56; ReΩ ¼ cos θ cos ϕðe126 þ e346Þþcos θ sin ϕðe136 þ e426Þþsin ϕðe125 þ e345Þ − cos ϕðe135 þ e425Þ − sin θðe146 þ e236Þ; ImΩ ¼ − cos θ cos ϕðe125 þ e345Þ − cos θ sin ϕðe135 þ e425Þþsin ϕðe126 þ e346Þ − cos ϕðe136 þ e426Þþsin θðe145 þ e235Þ: ð9Þ

Here e12 ¼ e1 ∧ e2 etc. These forms satisfy latter even after we change the relative sizes of the base S4 and the fiber S2. This is the reason why we use the second 3 structure to build an ansatz for the 4-form field. d6J ¼ W1ImΩ; 2 Left-invariant 2-forms and 3-forms are spanned by J; W2 and ReΩ; ImΩ respectively. Therefore, the most general d6ImΩ ¼ 0; ansatz for the 4-form respecting the symmetries is d6ReΩ ¼ W1J ∧ J þ W2 ∧ J; ð10Þ 56 ¼ ξ1ðrÞJ ∧ J þ ξ2ðrÞJ ∧ e þ dðξ3ðrÞImΩÞ C 3 8;9;10;11 where d6 is an external derivative of P , and W1 and W2 þ dðξ4ðrÞReΩÞþξ5ðrÞe : ð12Þ are torsion classes of the SUð3Þ-structure given by With the ansatz for the metric (8) and the 4-form field 2 gðrÞþhðrÞ strength (12), we are ready to impose the 11-dimensional W1 ¼ pffiffiffiffiffiffiffiffiffi supergravity equations of motion (1). The second equation 3 gðrÞ hðrÞ; in (1), namely the Maxwell equation for the 4-form, can be 2 ð Þ − ð Þ 2 h r g r 56 solved by W2 ¼ pffiffiffiffiffiffiffiffiffi J − 2e : ð11Þ gðrÞ hðrÞ 3 C1 2C1 C2 ξ1ðrÞ¼ ; ξ2ðrÞ¼− þ ; gðrÞ2 gðrÞ2 gðrÞhðrÞ A general manifold with SUð3Þ structure has more terms in pffiffiffi the relations (10), but in the case of our interest (squashed 3 2ξðrÞ ξ3ðrÞ¼0; ξ4ðrÞ¼− ; ξ5ðrÞ¼0; ð13Þ CP3) other torsion classes vanish. gðrÞhðrÞ1=2 Note that this SUð3Þ-structure is different from the usual ξð Þ Fubini-Study Kähler structure of CP3. We use ω to denote where the function r satisfies the differential equation, the Fubini-Study Kähler 2-form to distinguish it from J. 2 0ξ0 4hðξ − 3Þ 2ξ These two SUð3Þ-structures are associated to different ξ00 þ f − 2 − ¼ 0 ð Þ 2 : 14 realizations of CP3 as coset spaces. The first is f g h ð4Þ CP3 ¼ SU , which is a symmetric space and the complex Uð3Þ From nowp on,ffiffiffi we will set the dimensionful constant in (2) structure of SUð4Þ gives the Fubini-Study structure. The to be c ¼ 2. In order for the 4-form (12)pffiffiffito converge to C 3 ¼ Spð2Þ second is P SðUð2Þ×Uð1ÞÞ, which is not manifestly (2) as r → ∞, one must impose C1 ¼ 9 2, C2 ¼ 0 and symmetric but homogeneous. Therefore, we can use the ξð∞Þ¼1. Thus, the 4-form can be expressed as

026016-3 HIROSI OOGURI and LEV SPODYNEIKO PHYSICAL REVIEW D 96, 026016 (2017) pffiffiffi pffiffiffi pffiffiffi 9 2 18 2 56 3 2ξðrÞ F4 ¼ J ∧ J − J ∧ e − d ReΩ : ð15Þ gðrÞ2 gðrÞ2 gðrÞhðrÞ1=2

The next step is to express the first equation in (1), namely the Einstein equations, in our ansatz as ξ02

g00 f0g0 g0h0 g02 ξ02 ξ2 h 2ðξ − 3Þ2 3 − − − − − − − − 2 þ ¼ 0; 2g fg 2gh 2g2 24g2h 12g2h2 3g4 g h00 f0h0 g0h0 ξ02 ξ2 h ðξ − 3Þ2 1 − − − − − þ þ 2 þ ¼ 0; 2h fh gh 24g2h 3g2h2 g2 3g4 h f00 f0g0 f0h0 f02 ξ02 ξ2 ðξ − 3Þ2 3 − − − − þ þ þ 2 þ ¼ 0; 2f fg 2fh 2f2 12g2h 6g2h2 3g4 f 8f0g0 4f0h0 3f02 h02 4g0h0 3g02 ξ02 ξ2 2h ðξ − 3Þ2 12 2 12 þ þ þ þ þ − þ þ þ 2 − − − ¼ 0; fg fh f2 2h2 gh g2 4g2h 2g2h2 g2 g4 g h f 2f0ξ0 4hðξ − 3Þ 2ξ ξ00 þ − 2 − ¼ 0: ð16Þ f g2 h

For our reference below, we added the Maxwell equation in our gauge (8). Therefore, therepffiffi are six linearlypffiffiffiffi inde- for the 4-form (14) in the end. There are four independent pendent modes, and they are e2ð 7−1Þr, e2ð 10−1Þr, e−2r, functions, four Einstein equations and one Maxwell equa- and r · e−2r. Among them, three are normalizable and three tion. Due to the Bianchi identities, only three out of four are non-normalizable. Note that e−2r and r · e−2r are at the Einstein equations are independent. Breitenlohner-Freedman bound. Conformal invariance on As a consistency check, we can easily verify that the −2r C 3 the boundary requires the r · e mode to vanish [26]. This Euclidean version of AdS5 × P , condition also guarantees that the instanton action is finite,

as we will see in Sec.pffiffiVI. For now,p weffiffiffiffi only set the two 2 1 1 ð Þ¼ ð Þ¼ ð Þ¼ ξð Þ¼1 2ð 7−1Þr 2ð 10−1Þr f r sinh r; h r 2 ;gr 2 ; r ; diverging modes, e and e , to vanish at r ¼ ∞. We will keep the r · e−2r mode to be adjustable in ð17Þ the next couple of sections and demand it to vanish in Sec. VI. solves these equations. There is another simple solution, Let us turn our attention to boundary conditions at the 2 4 2 2=3 1 3 5=6pffiffiffi 2 2 2=3 bubble of nothing. In order for the S fiber to shrink to zero ð Þ¼ 2 ð Þ¼ 2 f r 3 3 sinh 2 2 r ;hr 3 ; size, the 4-form flux should not have components on the S ; otherwise the flux conservation would prevent it from 1 4 ξð Þ gðrÞ¼ ξðrÞ¼ ; ð18Þ collapsing. Thus, r in (15) must be chosen in such a way 1=3 2=3 3 4 2 3 that F4 is proportional to the of the base S . For this purpose, it is convenient to rewrite (15) as which is a stretched CP3 solution [25]. One can see that hðrÞ¼2gðrÞ; i.e., CP3 is stretched along its fiber. 4ð3 − ξð ÞÞ 1 2 r 1234 2ξðrÞ 56 pffiffiffi F4 ¼ e − J ∧ e III. BOUNDARY CONDITIONS 3 2 gðrÞ2 gðrÞhðrÞ ξ0ðrÞ In this section, we will study boundary conditions to þ pffiffiffiffiffiffiffiffiffi ReΩ ∧ dr: ð19Þ instanton solutions at the infinity of AdS and at the bubble gðrÞ hðrÞ of nothing. For r → ∞, the solution should approach the vacuum 3 Note that the second and third terms in the right-hand side AdS5 × CP , and we can linearize (16). In this set of pffiffiffiffiffiffiffiffiffi ð Þ ð Þ equations, three are second order differential equations for have h r and h r in the denominators, which should 56 g, h, ξ and one is first order for f (modulo the redundancy vanish at the bubble. However, e andp ReffiffiffiffiffiffiffiffiffiΩ also go to zero by the Bianchi identities). We should also note that there is since they have the factors hðrÞ and hðrÞ respectively. translational invariance in r, which is the residual symmetry Therefore these second and third terms vanish if we set

026016-4 NEW KALUZA-KLEIN INSTANTONS AND THE DECAY OF … PHYSICAL REVIEW D 96, 026016 (2017) ξ ¼ 0 ξ0 ¼ 0 2 3 2 and , and F4 becomes proportional to the 3 h S 2ðS − 2Þ 1234 − − − volume form e on the base. G G2 12h2G2 3G4 2 ¼ Suppose the S fiber becomes zero size at r r0. This 2 1 2 ð − 3Þ2 0 _ S h h S 2 means we set hðr0Þ¼0. We also require ξðr0Þ¼ξ ðr0Þ¼0 þ G − − − 3G3h G G3 3G5 due to our analysis in the previous paragraph. Combining these boundary conditions with the equations of motion G_ 2 G̈ G_ S_ 2 G_ S_ 2 ¼ −h0ðrÞ2 − þ − − ð22Þ (16), we find 2G2 2G 24G3 2hG 24hG2

2 4 ξðrÞ¼ξ0ðr − r0Þ þ Oððr − r0Þ Þ; and 2 fðrÞ¼f0 þ Oððr − r0Þ Þ; 2 3 2 2 3 2 gðrÞ¼g0 þ Oððr − r0Þ Þ; 4hðS − Þ 2S 2h 2S 2hðS − Þ − 2 − þ S_ 2 þ − þ 2 ð Þ¼ð − Þ2 þ ðð − Þ4Þ ð Þ G2 h G2 3hG2 3G4 h r r r0 O r r0 : 20 2G_ S_ S_ 3 ¼ −h0ðrÞ2 S̈− − ; ð23Þ G 12G2 In order for the geometry to terminate smoothly, we need 2 hðrÞ¼ðr − r0Þ þwith the coefficient 1 in the leading term. This condition turns out to be implied by the Einstein where · ¼ d=dh. Demanding that these equations hold ’ 0 equations. This is in contrast to the case of Witten s independently of h ðrÞ, we obtain four differential equa- 2 instanton, for which an analogue of hðrÞ¼ðr− r0Þ þ tions on GðhÞ and SðhÞ with respect to h. Remarkably, has to be imposed as an additional boundary condition. these four equations can be solved algebraically by impos- Thus, we find that there are three parameters f0, g0 and ing the simple relations, ξ 0 at the bubble. As we will see below, they can be fixedpffiffi by 2ð 7−1Þr demandingpffiffiffiffi the three non-normalizable modes, e , pffiffiffiffiffi 2ð 10−1Þr −2r ξ ¼ 3 − 2 ð3 þ Þ e , and r · e , to vanish at infinity. The location r0 g g h ; 2 pffiffiffiffiffi of the bubble is fixed by demanding fðrÞ= sinh r → 1 1 ¼ 2gðh þ gÞ: ð24Þ for r → ∞.

IV. ALGEBRAIC RELATIONS These algebraic relations are also consistent with the boundary conditions at r ¼ r0 and ∞: setting hðrÞ¼ 2 −1=3 4=3 2 Interestingly, both the equations of motion and the ðr − r0Þ gives gðrÞ¼2 and ξðrÞ¼2 ðr − r0Þ as boundary conditions defined in the last two sections are expected at the bubble, and h ¼ 1=2 at r ¼ ∞ gives g ¼ 3 compatible with two simple algebraic relations between the 1=2 and ξ ¼ 1 as required for AdS5 × CP . We found these ð Þ ð Þ ξð Þ three functions g r , h r , r . In fact, if we set, relations experimentally, and it would be interesting to find their deeper origins. In the following, we will use them to numerically integrate the rest of the equations of motion. gðrÞ¼GðhðrÞÞ; ξðrÞ¼SðhðrÞÞ; ð21Þ V. NUMERICAL SOLUTION With use of the algebraic relations (24), the equations of and substitute them into the equations of motion (16),we motion (16) collapse to the following three equations for find a couple of equations that are independent of fðrÞ: the two functions fðrÞ and gðrÞ,

g00 g02 f0g0 1 pffiffiffi − − − − ð1 − 5 2g3=2 þ 12g3Þ¼0; 2g 4g2 fg 6g4 pffiffiffi f00 f02 3f0g0 1 − ð2gÞ3=2 3g02 2g3=2 1 pffiffiffi 3 − − − pffiffiffi þ pffiffiffi þ ð1 − 8 2 3=2 þ 48 3Þþ ¼ 0 2 2 4 g g ; 2f 2f 4fg 1 − 2g3=2 2g 1 − 2g3=2 12g f pffiffiffi pffiffiffi 3 02 6 0 0 1 − ð2 Þ3=2 02 ð9 − 96 2 3=2 þ 192 3Þ 1 1 − 5 2 3=2 12 f þ f g pffiffiffig þ g pgffiffiffi g þ pffiffiffi g − ¼ 0 ð Þ 2 2 4 : 25 f fg 1 − 2g3=2 8g ð1 − 2g3=2Þ2 4g 1 − 2g3=2 f

026016-5 HIROSI OOGURI and LEV SPODYNEIKO PHYSICAL REVIEW D 96, 026016 (2017) Only two of these three equations are independent. Eliminating fðrÞ, one finds the following equation for gðrÞ, pffiffiffi 3 pffiffiffi ð1 − 2 3=2Þ2 6ð4 000 0 − 5 002Þ − ð1 þ 14 2 3=2 − 42 3Þ 4 04 g g g g g 4 g g g g pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 − ð5 − 16 2g3=2 þ 22g3Þg5g02g00 − 2ð1 − 3 2g3=2Þð1 − 2 2g3=2Þð1 − 2g3=2Þ g3g00 pffiffiffi pffiffiffi 2 − ð1 − 2g3=2Þ ð4 − 9 2g3=2 − 12g3Þg2g02 1 pffiffiffi pffiffiffi pffiffiffi − ð1 − 3 2 3=2Þ2ð1 − 2 2 3=2Þ2ð1 − 2 3=2Þ2 ¼ 0 ð Þ 9 g g g : 26

This is a third order differential equation for gðrÞ. Since it We presented the typical behavior of the solution in does not depend on r explicitly, one can lower its order Fig. 1, where we set f0 ¼ 1. The horizontal axis is r − r0, 2 by 1. The numerical integration of this equation shows the where r0 is fixed by demanding fðrÞ= sinh r → 1 as desired behavior; i.e., gðrÞ goes from 2−1=3 to 1=2 as r goes r → ∞. One can see that the solution exhibits the desired from r0 to ∞. behavior: hðrÞ → 1=2, gðrÞ → 1=2, ξðrÞ → 0 and We would like to comment on two important features of f0ðrÞ=fðrÞ → 2 as r → ∞. our numerical solutions: ð Þ (1) The equation for g r is singular at the bubble, and VI. INSTANTON ACTION 000 −1=3 the coefficient of g vanishes with gðr0Þ¼2 . Therefore, instead of numerically integrating the We have shown that there is a family of solutions −1=3 equation with the initial value of gðr0Þ¼2 ,we parametrized by the size f0 of the bubble, which approach 3 computed the first few terms in the Taylor expansion AdS5 × CP at infinity. However, there is one more non- analytically and matched them to a numerical normalizable mode r · e−2r we need to fix. In this section, solution. we show that one can turn off this mode by adjusting f0. (2) In this section, we are not requiring the r · e−2r mode This also makes the instanton action finite. to vanish at r → ∞. Thus, we are left with one free For a general value of the parameter f0, the solution at parameter f0, which is the size of the bubble. infinity behaves as

0.80 0.5

0.75 0.4 0.70 0.3 0.65 0.2 0.60

0.55 0.1

1 2 3 4 5 1 2 3 4 5

1.0

2.0 0.8

1.5 0.6

0.4 1.0

0.2 0.5

1 2 3 4 5 1 2 3 4 5

FIG. 1. Numerical solution to Eq. (26).

026016-6 NEW KALUZA-KLEIN INSTANTONS AND THE DECAY OF … PHYSICAL REVIEW D 96, 026016 (2017)

0.9

0.8 0.010 0.7

0.6 0.005

0.5

0.4 5 10 15 20 25 30

5 10 15 20 25 30

FIG. 2. The graphs of αðrÞ and its derivative. The horizontal axes are r − r0. The values of f0 from top to bottom are 0.63, 0.62, 0.61 respectively.

1 2 −2r the CP base. The last fact is best understood from the coset gðrÞ¼ þðaðf0Þþbðf0ÞrÞe þ: ð27Þ 2 point of view. One can choose the SUð2Þ subgroup in ð3Þ ð3Þ¼SUð3Þ ð2Þ Numerically, bðf0Þ vanishes at f0 ¼ 0.6203025… To show SU and decompose SU SUð2Þ × SU . The for- 5 that bðf0Þ can be set exactly equal to zero by adjusting f0, mer term in the product is homogeneous space S and the 1 2r 3 3 5 we present Fig. 2 for αðrÞ¼ðgðrÞ − 2Þe . Since αðrÞ ∼ latter is S . Therefore, SUð3Þ is an S fibration over S . The aðf0Þþbðf0Þr for r → ∞, we see that bðf0Þ changes its two Uð1Þ subgroups in the denominator of the coset ð3Þ sign near 0.62. Therefore, there must be f0 near 0.62 such SU Uð1Þ×Uð1Þ turn each sphere into a complex projective space that bðf0Þ¼0. 2 ↪ SUð3Þ → C 2 The action for 11-dimensional supergravity takes the form resulting in S Uð1Þ×Uð1Þ P fibration. C 3 SUð3Þ Z It worth mentioning that both P and Uð1Þ×Uð1Þ are pffiffiffiffiffiffiffiffiffiffiffi 1 1 4 3 ¼ − MNPQ þ ð Þ twistor spaces of S and CP [22]. This fact seems to be the S det G 4 R 48 FMNPQF ; 28 M11 main reason of the similarity between the collapsing solutions of the models. Choosing the vielbein as [20,27] where we have ignored the Chern-Simons terms and pffiffiffiffiffiffiffiffiffiffiffi fermions, which are irrelevant to our discussion. Using e1 ¼ 2gðrÞdμ; the supergravity equations (1), the 4-form kinetic term is rffiffiffiffiffiffiffiffiffi related to the Einstein term, F FMNPQ ¼ 36R. ð Þ MNPQ ¼ g r μΣ Therefore, the instanton action reduces to e2 2 sin 1; Z rffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffi gðrÞ ¼ − ð Þ e3 ¼ sin μΣ2; S 2 det GR: 29 2 M11 rffiffiffiffiffiffiffiffiffi gðrÞ It is straightforward to see that, with the r · e−2r mode ¼ μ μΣ e4 2 sin cos 3; removed, the instanton action is finite and positive after pffiffiffiffiffiffiffiffiffi subtracting the value for the vacuum AdS. e5 ¼ hðrÞðdθ − A1 sin ϕ þ A2 cos ϕÞ; 3 pffiffiffiffiffiffiffiffiffi We conclude that AdS5 × CP is unstable due to the e6 ¼ hðrÞ sin θðdϕ − cot θðA1 cos ϕ þ A2 sin ϕÞþA3Þ; finite action instanton. ð30Þ SUð3Þ VII. INSTABILITY OF AdS5 × Uð1Þ×Uð1Þ with SUð3Þ In this section, we will show that the AdS5 × Uð1Þ×Uð1Þ model has an instanton that mediates its decay. It is not Σ1 ¼ cos γdα þ sin γdβ; known whether this solution is perturbatively stable or not. Σ ¼ − γ α þ γ β In either case, the existence of the bubble of nothing 2 sin d cos d ; solution shown here means that it is unstable. Σ3 ¼ dγ þ cos αdβ; To construct the solution, let us first review the geometry A1 ¼ μΣ1; SUð3Þ Fð1 2 3Þ cos of Uð1Þ×Uð1Þ. It can be viewed as a flag manifold ; ; or 2 A2 ¼ cos μΣ2; a twistor space over CP . It admits Kähler structure with an Einstein metric and therefore is a solution of the super- 1 2 A3 ¼ ð1 þ cos μÞΣ3; ð31Þ gravity equations of motion. It is also an S2 fibration over 2

026016-7 HIROSI OOGURI and LEV SPODYNEIKO PHYSICAL REVIEW D 96, 026016 (2017) we see that all the formulas and results become exactly the Blanco-Pillardo et al. [28] considered the Einstein gravity C 3 SUð3Þ ð3Þ in six dimensions coupled to SUð2Þ Yang-Mills gauge field same as in the P case. Namely, Uð1Þ×Uð1Þ has the SU - and an adjoint Higgs field with a specific potential to break structure defined by (9) and has torsion classes (11). Since ð1Þ the SUð3Þ-structure is the same, the ansatz for the flux will the gauge to U and found a smooth solution of have the same solution (13). Finally and most importantly, this type. Brown and Dahlen [29] considered the Einstein- the equations of motion of supergravity take exactly the Maxwell system without the Higgs and added a domain same form (16). The last fact makes all the results of the wall as a source. It would be interesting to realize such previous sections applicable to this case. The only thing solutions in a low energy effective theory of M/string which is different is the expression of the vielbein in terms theory in a controlled approximation. C 1 C 2 of the coordinates. According to [17],AdS5 × P × P and AdS5 × 1 1 1 One might wonder why the equations are exactly the CP × CP × CP are not stable perturbatively. Thus, we same. It follows from the fact that we constructed the ansatz do not need instantons for these geometries to be consistent which respects all the symmetries of the model with with the conjecture of [12]. In fact, our ansatz is not squashed fiber. The bases of the compact manifold in both applicable to them since the configurations are too restric- cases are Einstein manifolds and therefore their contribu- tive for the fluxes to slip off. In our solution, it is important tion to the Einstein equations will enter in a similar manner. that CP3 has the nontrivial fibration structure and not a Besides, the SUð3Þ-structure is rooted in the twistor origin direct product since this allows our nontrivial solution to of both spaces and it is used to build the ansatz for the flux. the 4-form equations. A similar argument applies to the Because the spaces have the same origin, the ansatz gives Freud-Rubin-type compactification (when the flux is pro- the same result in both cases. portional to the volume form of the AdS) and the compactifications where the flux is proportional to the VIII. DISCUSSION volume form of the compact space (unless one can turn on another lower dimensional form). We would like to end this paper by explaining how our Finally, we want to mention that it is possible that our solution evades the issue raised in [11]. Suppose we try to solution is related to resolved M-theory solu- collapse a d-dimensional sphere in the internal space ¼ tions with G2-holonomy [30]. They are Ricci-flat seven- supported by a flux, at the bubble located at r r0.The dimensional manifolds which have conic structure and its flux gives a contribution to the Einstein equations propor- d metrics reads tional to 1=hðrÞ , where hðrÞ is the square radius of the 2 sphere, while the contribution from curvature is proportional 2 1 2 1 2 1 2 r 2 2 2 ¼ þ 1 − j j þ ð Þ 1 ð − Þ ð Þ ∼ ð − Þ ds 1 dr r 4 dAu dsM4 ; 33 to = r r0 . Taking into account that h r r r0 for 1 − 4 4 r 2 the smoothness, it was argued in [11] that the only possible r 4 2 2 way to make these two terms of the same order is to set where M4 is either S or CP and jdAuj is the metrics on the d ¼ 1, i.e., a circle. However, this does not apply to our case S2 fiber which are exactly the same as in the present paper ð3Þ since the amount of flux on the sphere can vary. for CP3 and SU respectively. Moreover, the S2 fiber It is instructive to see it explicitly in the Einstein Uð1Þ×Uð1Þ collapses at finite r ¼ 1, while the M4 radius stays finite. equation (16) for hðrÞ, Unfortunately, the radii grow linearly at infinity. It may be 00 0 0 0 0 02 3 2 2 h f h g h ξ h ðξ − 2Þ 1 ξ possible to construct a desirable solution by multiplying this − − − − þ þ þ − ¼ 0: 4 2h fh gh 24g2h g2 3g4 h 3g2h2 geometry with the flat R andbyaddingsomefluxalongthe conifold in order to change the behavior at infinity from the ð32Þ linear growth to the constant. An idea along this line may One can see that the curvature contribution (first term) is of allow us to generalize our solutions further. 2 order 1=ðr − r0Þ , while the flux (last term) is of order ACKNOWLEDGMENTS ξðrÞ2=hðrÞ2 [flux for the S4 is proportional to the fourth power of gðrÞ as it should be]. This is consistent with the The authors would like to thank G. Horowitz, S. Gukov, estimate of [11] for d ¼ 2. However, in our solution, these I. Klebanov, and H. Reall for discussions. This research is two terms can balance each other since ξðrÞ → 0 when supported in part by the U.S. Department of Energy Grant 2 hðrÞ → 0. In this way, the flux evaporates from the S fiber, No. DE-SC0011632. H. O. is also supported in part by the and the Einstein equations can be satisfied. Simons Investigator Award, by the World Premier Another possibility to deal with flux conservation is to International Research Center Initiative, Ministry of introduce a domain wall at the bubble to absorb the flux. Education, Culture, Sports, Science and Technology This idea was used in [10] to collapse the supersymmetry (MEXT), Japan, by the Japan Society for the Promotion 1 5 breaking S in the AdS5 × S =Γ geometry of [7]. More of Science (JSPS) Grant-in-Aid for Scientific Research recently, instanton solutions with S2 collapsing have No. C-26400240, and by the JSPS Grant-in-Aid for been constructed in some models in six dimensions. Scientific Research on Innovative Areas No. 15H05895.

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