Volume 145B, number 5,6 PHYSICS LETTERS 27 September 1984

INSTABILITIES IN ENGLERT-TYPE SUPERGRAVITY SOLUTIONS

Don N. PAGE and C.N. POPE 1 Department o f Physics, The Pennsylvania State University, University Park, PA 16802, USA

Received 9 June 1984

We show that all eleven-dimensional Englert-type supergravity solutions (in which the four-index field has internal com- ponents) constructed from internal spaces M 7 having two or more Killing spinors, are unstable.

The known solutions of eleven-dimensional super- gravity [ 1,2] of the form of a product of four-dimen- sional anti-de Sitter spacetime with a compact seven- dimensional space M 7 fall into two categories: F r e und - Rubin solutions [3] in which the only nonzero com- ponents of the field strength FABCD lie in spacetime, and Englert-type solutions [4] in which FABCD is also nonzero when the indices lie entirely in the internal space M 7 . Stability analyses have now been performed for all known Freund-Rubin solutions [5 -8 ] . A gen- eral analysis is much more difficult for the Englert- type solutions. De Wit and Nicolai argued [9] that the Englert solution [4] on the round S 7 should be un- stable, and this was confirmed by Biran and Spindel [10], who exhibited specific modes which violate the Breitenlohner-Freedman stability bound [ 11 ]. In this letter we extend this analysis to show that all Englert- type solutions whose internal spaces admit two or more Killing spinors, are unstable.

Englert-type solutions may be obtained from any M 7 whose metric satisfies the Einstein equation

Rab = 6m2gab, (1)

and which admits at least one spinor satisfying the Killing spinor equation

D a - ~ P a 7? =0" (2)

The bosonic field equations o f d = 11 supergravity [1, 2] in the conventions of ref. [12] are

1 Permanent address: Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom.

0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

_ 1 CDE x g F F CDEF RAB - 5 FACDEFB -- ~ AB CDEF , (3)

VAFABCD = -- s76 ~ eBCDEFGHIJKLFEFGHFIJKL . (4)

The Englert-type solution is obtained by setting all of the components Of FABCD equal to zero except [12,13]

Fob.r8 = - 2 m e ~ r 8 , Fabed = --m¢lFabcdr/, (5, 6)

where a,/3, ... run over spacetime and a, b,... run over the internal space M 7. This satisfies (4), and (3) is sat- isfied if the internal space obeys (1) and the anti-de Sitter spacetime has

R ~ = - 10rn2gc¢. (7)

We have made an arbitrary sign choice in (6), since it makes no essential difference in our subsequent calcu- lations.

For the SO(8)-invariant metric on S 7, there are eight Killing spinors r/I satisfying (2), where I = 0 ..... 7. The nonzero Fabed obtained from (6) by making a par- ticular choice for r/, say 77 = 7/0, breaks the SO(8) in- variance down to SO(7). In spacetime this Englert so- lution can be interpreted as a spontaneous symmetry breaking of the SO(8)-invariant Freund-Rubin solu- tion on S 7 in which the 35 of massless pseudoscalars acquires a nonzero expectation value [ 12,13]. Under the SO(7) of the broken phase, the 35 splits as 1 + 7 + 27. The singlet is the pseudoscalar field which ac- quires the nonzero expectation value. The 7 of pseudo- scalars is eaten by the seven gauge bosons of SO(8) which become massive when SO(8) breaks to SO(7) [10]. The stability analysis in ref. [10] showed that the 27 acquires a negative mass-squared which violates

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the Breitenlohner-Freedman stability condition [11]. One might expect that other Englert-type solutions

should exhibit similar instabilities if they have analogues of the 27, i.e. originally massless pseudoscalars over and above the singlet which acquires an expectation value and the Goldstone bosons associated with any breaking of gauge symmetry. This will occur if the numberN of Killing on the M 7 is greater than one. To see this, note that the existence of N Killing spinors implies that there will be ½ N ( N + 1) massless pseudo- scalars in the corresponding Freund-Rubin solution [6], and the isometry group G contains an SO(N) fac- tor, i.e., G = G ' × SO(N) [14]. The Englert-type solu- tion will have invariance group G'X SO(N-1 ) , and hence N - 1 gauge bosons will become massive by eat- i n g N - 1 pseudoscalars. Also taking into account the one pseudoscalar which acquires an expectation value, one will be left with ½ N ( N - 1) pseudoscalars which might be expected to give unstable modes. (Note that N = 8, on the round S 7, is a special case since there are only 35 massless pseudoscalars rather than the 36 given by 1 N ( N + 1).)We now show that Englert-type solutions constructed from M 7,s having two or more Killing spinors, are indeed unstable.

We write the perturbations around the Englert- type ground state as

gAB = hAB, ~ FABCD = fABCD" (8)

To first order in the fluctuations, (3) and (4)give [12, 10]

1 f~hiaz ' + V(iaVPhu)p _ ~ V u V , hp °

1 a + V(iaVahv)a - ' i V i a V u h a

= ~ g ia , ( -4me '~ fo l j , r~ - FabCdfabc d

+ 96m2hOp + 48m2haa) - lOm2hiau, (9)

1 ~xhu n 1 o 1 1 +"i V u V hno +'~ VnV°hiao +~ ViaVahna

1 1 x V V n haa +~ Vn Vahiaa - ~ Via V n h ° o - ~ Ia

= _ 2 o4~ 1 ]7 abel 2m2hun,(10) $ m e u fno¢~, +~ n uabc --

1 1 Ahmn + V( m Vahn)a -"£ Vm Vn haa

1 + V(mV°hn) o - -£ V m Vn hPp

= 1 gmn (2me°C~'fa~7, - Fabedfabed

2 abe 2 -- 48m2h°o - 24m2haa) +'~ F(m fn)abe + 2m hmn ,

(11)

e [ r v ~eial)O(r ~wo~,vu.I + V a f avoa)

+ 6m V a ( - h ° o + haa) - 12mVaha

1. ~ a b c d e f g tc ¢ = -- 12 c "abcdJ~.efg, (12)

Vu f unpa + Vrn fmnpa + 2meiaVmr Viahnv

- - 1 eabeclefn e~O~ Fabcd fat~ef , (13)

Via fIanpa + Urn fmnpa _ Vm(Fmnpqhaq)

_ 1 abedenp a~a~, ~ (14) - 7-2 e e rabcd Jotpye,

Viafianpq + Vm fmnpq +~Vml [Fmnpq(hop + haa)]

- Via (Fmnpqhiam) + 4Vm ( Fa [mnPhq] a)

= _ ~ meabcdnpqfabcd

1 a8 eabcdnpq Fabcd e ~ f ~e,y~ • (15)

Since FABCD = 4V [A ABCD] , we also have

V[A fBCDE] = 0. (16)

The operator z~ is the Lichnerowicz operator in eleven dimensions.

Following ref. [10], we now consider a special class of perturbations which gives a small number of the normal modes of the system, including the unstable modes of interest. On a space wi thN Killing spinors satisfying (2), we choose one, which we call B, to gen- erate the background field strength (6). A convenient potential for (6) is given by

A abe _ 1 - -~ -~Pobe n. (17)

We denote the remaining Killing spinors by ~?i, where i = 1 ... . . N - 1. These spinors are chosen to satisfy ~ l = 1, ~ i~ /= ~ i/, ~,?i __. 0. The only nonzero pertur-

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bations we shall consider are

huv(x,y) = Hvv(x), (18)

hab (X,y) = Gi/(x) Yi/b (y) + G(x) Yab (Y) + H(x)gab(Y),

(19)

aabd(X,y ) =¼ oi](x) ri~c(y ) +10(x ) rabc(Y)

+ u (x) A ~b~ (Y), (20)

fo~ya (x ,y) =f(x) e a ~ , (21)

where in (20) we have given the fluctuation in the in- ternal potential, 6Aab c -aabc, which yields

fabcd = 4 V [aabed] , fabed = V c~abcd, (22)

and Aab c (y) is the background internal potential (17). The spacetime coordinates are denoted by x and those on the internal space byy . The spacetime fields G if and q)i/are tracefree,

G ii = 0, dp ii = 0. (23)

The y-dependent harmonics on M 7 are

rab = ri~ , ri~c :~irabcr?], }'abe = Yi~c" (24)

At this s tage(18)-(21) are to be considered as an ansatz for a particular class of solutions of the pertur- bation equations (9) - (16) . We now proceed to show that there are indeed nontrivial solutions of this form.

Inserting the ansatz into the perturbation equations generatesy-dependent terms which all turn out to be linear combinations of the harmonics (24). A useful identity for establishing this result is

~iPabcr~] + "qPabc~78 i/= 6~(iP[ar?~Pbc] 77/), (25)

which can be proved by using Fierz transformations and the orthonormality of the spinors. Equating the spacetime-dependent coefficients of the independent M 7 harmonics (24) yields the following spacetime equations:

* - 1 ) ~ + ~ (26) Ho~ + f = 7 H - ~ ( N u,

AH m, + 2V(uVOHu) p - VuVvH ~ - 7 V u VvH+ 2Om2Huv

_ 8 2 a - S i n guv[4(Ha +f ) + 14H+ ( N - 1) ¢ - 7 u ] ,

(27)

I--1H=g4 m2 [4(H~ + f ) + l lH + L°v(N- 1 ) ¢ - l O u ] ,

(28)

1 ( N - 1) G], (29) []u = 8m 2 [H~ + f - H - - ~

[ ]G = 8m 2 [3G - 4¢], (30)

[3(~ = - 8m2G, (31)

[] G ij = 8m 2 [3G ij - 4~iJ], (32)

[] ¢i/= _ 8m2G q. (33)

& is the Lichnerowicz operator in four dimensions, and [] = Vc, V ~. Eq. (26) comes from (12); (27) from (9); (28), (30), and (32) from (11); and (29), (31), and (33) come from (15). Eqs. (10), (13), (14), and (16) are identically satisfied by the ansatz.

The massless graviton mode in (27), described by t 7 H'uv =Huv +~ guvH, (34)

decouples from the other fields [10]. Thus for the re- maining analysis we set H~v = 0, and then (26) and (28) imply that (27) is satisfied identically.

Using (26) to eliminate H~ +f , (28) - (31) become four coupled equations in four fields. Acting on a nor- mal mode, the laplacian, V], can be replaced by the eigenvalue 3,. Solving for X, we obtain the four eigen- values

X = - 8m 2, 20m 2, 32m 2, 60m 2 , (35)

The eigenvalues 20m 2 and 60m 2 were obtained in the singlet sector of the Englert solution on the round S 7 in ref. [ 10]. The corresponding eigenmodes both have G = ~ = 0. For X = 20m 2,u = - 6 H a n d f = 0; forX = 60m 2,u = ~ H and f = los "Z- H. The eigenvalues - 8 m 2 and 32m 2 correspond to modes both havingH = 0

_ 1 ( N - 1) q~, and f = 0. For (and hence Huv = 0), u --~ =-8m2 ,G=¢ ,whereas fo r X = 3 2 m 2 , G = - 4 ~ .

These two modes in the singlet sector were absent in the special case of the Englert solution on the round S 7 analyzed in ref. [10], since fo rN = 8 they have u = ~b, and hence (20) gives aab c =1 dp~iPabcr?l, which is zero because the 35 of harmonics ~II'abcr~J on S 7 is tracefree.

IfN~> 3, eqs. (32) and (33) give½ ( N - 2 ) (N+ 1) normal modes with X = - 8 m 2 and an equal number with X = 32m 2 . These have G ij = ¢i] and G i/= - 4 ~ i/

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respectively. In the c a s e N = 8, these are the 27 modes found in ref. [10] for the round S 7 solution.

In the anti-de Sitter metric satisfying (7), the Bre i ten lohner-Freedman stability criterion [ 11 ] is

X ~> -- ~ m 2. (36)

Hence the modes with X = - 8 m 2 are unstable. There- fore, an Englert-type solution constructed from an in- ternal space M 7 havingN Killing spinors is unstable if N / > 2, since there are ~ N ( N - 1) unstable modes of the form ( 1 8 ) - ( 2 1 ) (except f o r N = 8, in which the singlet is absent, leaving 27 unstable modes).

Aside from the Englert solution [4] on the round S 7 , which was proved to be unstable in ref. [10], the known spaces M 7 with N >~ 2 are M PP r with N = 2 [15], N 010 w i t h N = 3 [16], QPPP w i t h N = 2 [17], and the Stiefel manifold V 5,2 with N = 2 [ 18]. These can all be used to construct Englert-type solutions, which by the results of this paper are all unstable.

There are also spaces M 7 with only one Killing spinor. The known examples are the squashed S 7 [19, 20] , the Einstein metrics on NPq r given in ref. [16] (except N 010 discussed above), the new "squashed" Einstein metrics on NPq r given in ref. [2 t ] , and the coset space SO(5)/SO(3)max [18]. The unstable modes exhibited in this paper do not occur for Englert-type solutions constructed from such spaces h a v i n g N - 1 Killing spinor. It is therefore an open question whether such solutions are stable. However, we shall now give an argument which suggests that at least some of these solutions are unstable.

The basic idea of our argument is that each of the two inequivalent Einstein metrics on the NPq r spaces given respectively in refs. [ 16] and [21 ] depends local- ly only on the ratio p/q and is a continuous function of this ratio (which itself can of course only take ra- tional values). As we have already observed, the N 010 solution of ref. [16] which has three Killing spinors will give rise to an Englert-type solution which is un- stable. Since this solution has p/q = 0, then, locally at least, a solution with small but nonzero p/q will ad- mit almost Killing spinors, i.e., spinors which satisfy (2) up to terms o f orderp/q. If such spinors are used in place of~7 i in (24), the resulting perturbations (19) and (20) will reduce the value of the effective poten- tial in spacetime almost as much as the unstable nor- mal modes do whenp/q = 0. (Even if the almost Killing spinors do not exist globally, they can presum-

336

ably be used to construct a perturbat ion nearly every- where on M 7 which then can be smoothly extended without changing the value of the effective potential significantly.) Since the unstable normal modes on N 010 lie below the Bre i ten lohner-Freedman bound

1 (36) by an amount ~ m 2, then for sufficiently small p/q the corresponding perturbations on NPq r should

also be unstable. Thus in this case at least it would seem that there are Englert-type solutions constructed from spaces havingN = 1 Killing spinor which are un- stable, so in generalN = 1 apparently does not guarantee stability.

This work was supported in part by NSF Grant PHY-8316811 and by an Alfred P. Sloan Research Fel- lowship to D.N.P.

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