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

STABILITY ANALYSIS OF COMPACTIFICATIONS OF D = 11 SUPERGRAVITY

WITH SU(3) X SU(2) X U(1) SYMMETRY

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

Received 14 May 1984

We show that the M pqr Freund-Rubin compactification of eleven-dimensional supergravity is classically stable if and only if 761 /2 < [p/ql < 1~7 ( 66)1/2.

The eleven-dimensional supergravity field equations [1,2] admit F r e u n d - R u b i n [3] solutions which are products of a four-dimensional anti-de Sitter lorentzian spacetime and an arbitrary compact seven-dimension- al positive-definite Einstein space M 7 . Examples of the latter are the round S 7 [4,5], which yields in space- time a theory with N = 8 supersymmetry; the squashed S 7 [ 5 - 7 ] , w i t h N = 1 o r N = 0 depending upon the orientation; and the MPq r spaces introduced by Witten [8] with Einstein metrics given in ref. [9]. Depending upon the parameters p, q, r, these yield e i the rN = 2 o r N = 0 supersymmetry.

One criterion necessary for a candidate classical ground-state configuration is that it be stable against small perturbations. These perturbations have the in- terpretation of being excitations of physical fields in spacetime, and the requirement of stability translates into the condit ion that there should be no modes grow- ing exponential ly in time. Any solution with a surviv- ing supersymmetry in spacetime is automatically stable [8,10,11]. The general criterion for classical stability has been given in ref. [12], where it was then shown that the squashed S 7 solution even without supersym-

merry is stable. The stability criterion can be expressed as a certain lower bound on the Lichnerowicz opera- tor on M7, but in general it is difficult to determine whether or not this bound is violated for a given M 7 . In this let ter we answer this question for the MPq r spaces and show that they are stable if and only if the ratio p/q lies within a certain range.

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

The MPq r spaces can be described as nontrivial U(1) bundles over CP 2 × S 2 where the integers p, q, r char- acterize the degree of twisting of the U(1) fibers over CP 2 and S 2 [8,9]. We shall follow ref. [13] and give a rather different derivation of the Einstein metrics from that of ref. [9], based upon the Kaluza-Kle in idea of reinterpreting gravity plus a U(1) gauge field as pure gravity in one higher dimension [14,15]. We use an orthonormal basis of one-forms e a, a = 0 . . . . . 6, on M7, with e i, i = 1, . . . , 6, being an orthonormal basis on the base manifold M 6 and e 0 = c ( d r - A ) on the fiber, where c is a constant, r is a coordinate on the fiber with period 47r, andA =Aie i is a potential for the

_ 1 F q e i A e ] = dA. Thus U(1) field strength F - ~ -

ds7 2 = 6ab eae b = c 2 ( d r - A) 2 + ds6 2 , (1)

where ds6 2 =6i/eie ]. Note that neither A nor ds6 2 depends on r. In terms of the six-dimensional connec- tion one-forms 6wi/, the seven-dimensional connection forms are

60ij = 703ij = 6COl/+ff c FijeO , (2)

= 1 eF i j e / ' (3) 6o0i ~ 76o0i -~ -

The seven-dimensional Riemann tensor components are

1 2 RilTc l = 6Rijkl --'~ c ( F i k F l . l - F i l F j k + 2Fi jFkl ) , (4)

_ 1 C2 1 F i / , (5, 6) Roioj - '~ F i kF/k, RijkO ='~ c 6V k

where the covariant derivative 6 ~7 k is defined with the six-dimensional connection 6wij.

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The base metric ds62 is the sum of a standard CP 2 metric [16] ds42 and S 2 metric ds22,

4 6

ds62 = ds42 + ds22 = ~ eiei+ ~ eie i, (7) i =1 i=5

with the orthonormal basis

e I = (-~ A~-I) 1/2 sin p a l , e 2 = (~ A~I ) 1/2 sin p a2,

e 3 = (~ A~-I) 1/2 sin p cos/.t 03, e 4 = (6A~-l) 1/2 d/t,

e 5 = A~ 1/2 dO, e 6 = A~ -1/2 sin 0 de, (8)

where # runs from 0 to rr/2, do 1 = - 02 A 03 etc., and 0 and ¢ are the usual polar coordinates on S 2. The Riemann tensor for ds62 is

6Ri/kl = ~ A4 (4gik 4g]l -- 4gil 4glk + Jik J/l

_ jitJ/k + 2Ji/Jkl) + A2 (2gik 2gfl _ 2gil 2g/k), (9)

where 4g 6 is the metric on CP 2 and 2gt7 is the metric on S 2 , and

J - 2 -~ Jii e l ^ e / = e l ^ e2 +e3^ e4 (10)

is the Kfihler form on CP 2 , satisfying

Ji]~k = -- 4gik , 6 V i J]k =0. (11)

The Ricci tensor is

6Ri /=A 4 4gi/+ A 2 2gi/. (12)

The existence ofnontrivial U(1) bundles corre- sponds to the existence of harmonic two-forms, since these can be taken to represent the second cohomology class. In CP 2 X S 2 the general harmonic two-form is

2 F = -~ mA 4 J + nA 2 e 5 n e 6, (13)

where m and n are constants, with the normalization chosen for later convenience. Locally this can be writ- ten as F = dA, where

A = - m sin2p 0 3 - n cos 0 de. (14)

Thus the metric (1) on the U(1) bundle over CP 2 X S 2 becomes

ds 2 =c2 (d r+ m sin2p o 3 +n cos0 de) 2

+ 6A~-I [dp2 +1 sin2p (Ol 2 + 022 + cos2~t 02)]

The first term of the metric is singular at the north and south poles of S 2, since de is not regular at these points. In order for (15) to be a metric on a nonsingu- lar manifold, it is necessary to be able to make coordi- nate transformations which render the metric regular there. This can be achieved at 0 = 0 by defining for 0 < rr a new coordinate r ' on the U(1) fibers by dr ' = dr +nd¢, and it can be achieved at 0 = 7r by defining for 0 > 0 r" by dr" = dr - n d ¢ . These two redefini- tions must be compatible in the overlap region, e.g.

1 dr" 0 =-~ 7r. Since = d~-' - 2 n d ¢ , a line integral around the equator at fixed r ' changes T" by - 47rn. Since ~- and hence r" have been assigned period 4rr, n must be

a an integer. A similar argument at p= 0 and # =-/rr shows that m must also be an integer. We shall denote the resulting regular manifold by M(m, n), where m and n are arbitrary integers.

When rn and n are relatively prime, the manifold M(m, n) has the topology of the MPq space defined by Witten [8] as (S 5 X S3)/U(1) wi thp = m and q = n. If m and n are not relatively prime and have greatest common divisor r > 1, then it would be consistent for r to be defined modulo 4hr. The fact that r has in- stead been assigned period 4rr by definition means that the space is not simply connected but has funda- mental group Z r. Thus in general M(m, n) corresponds to Witten's Mpqr = M P q / Z r space wi thp = m/r and q = n/r, where r = gcd (m, n), and so in Witten's nota- tion p/q = m/n. However, in the notation of Castellani et al. [9], their p/q - 2 - 3 m/n. This difference apparent- ly arises because the isometry groups of CP 2 and of S 2 are SU(3)/Z 3 and SU(2)/Z 2 rather than simply SU(3) and SU(2). Their r also differs from Witten's, but these differences are merely a matter of retabeling the same set of manifolds, whose covering spaces depend only on m/n.

In order to obtain a Freund-Rubin solution of the supergravity equations, we must solve the Einstein equations on M 7 . Inserting (9) and (13) into eqs. ( 4 ) - (6) and contracting yields the seven-dimensional Ricci tensor

R00 = c 2 ( 4 m 2 A 2 + I n 2 A 2 ) , (16)

Ri i = A4(1 _ ~ c2m2A4) 4gi] + A2(1 1 c 2 n 2 A 2 ) 2gij" (17)

In order to solve

+ A~ -1 (d02 + sin20 d¢2). (15) Rab =Agab, (18)

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

introduce the parameter x = A4/A2, in terms of which one finds from (16) and (17)

A 4 = [ 4 x / ( l + 2 x ) ] A , A2 = [4/(1 + 2x)]A, (19)

c -2 = ~4(4rn2x 2 + 9n2)/[9(1 + 2x)]}A, (20)

(m/n) 2 = 9(2x - 1)/[4x2(3 - 2x)] . (21)

Thus given m and n, (21) is a cubic equation determin- ingx in terms of the ratio m/n. This equation has ex- actly one real root, and it lies in the range½ ~<x ~<-~.

1 If x - 2 , then m = 0 and M(0,n) = CP 2 X S3/Z n. I f _ 3 x - 2 , then n = 0 and M(m, 0) = (S5/Zm) × S 2. It has

been established [12] that these product spaces are un- stable to a mode in which one factor expands uniform- ly while the other contracts keeping the total volume constant. On the other hand, there are supersymmetric solutions [9] which have x = 1. These will necessarily be stable [8,10,11]. One might therefore expect that there will be a range o f x around x = 1 for which the corresponding M(m n) solutions will be stable. How-

. 1 3 ever, i fx is too close to one of the endpomts2 o r2 , the metric is sufficiently near that o f the correspond-

_ 1 3 ing product metric with x - 7 or x =2 that one might expect the solution to be unstable. We shall now show that this is indeed the case.

In ref. [12] it was shown that the necessary and sufficient criterion for classical stability of a F reund- Rubin solutior~ is that all eigenvalues X of the Lichnerowicz operator A L acting on symmetric trans- verse tracefree tensors hab ,

ALhab = -- [] hab -- 2Raebd hal + 2R(aChb) e = ~thab,

(22)

should satisfy

~ A. (23)

By integrating [ V (ahbc)] 2 over M7, one can readily show

f h ab d V ALhab

= f [ - - 4habRaebd h ed + 4Ahabhab

+ 3 V (ahbC) V(ahbc)] dV. (24)

Since our M 7 is homogeneous, the 27 eigenvalues K of the Riemann tensor, defined by

Racba V~a = ~ Gb (25)

for traceless symmetric eigentensors Vab , are constant. Letting Kma x be the largest such eigenvalue, it follows from (24) that

/> 4A - 4K max- (26)

Thus M 7 gives a stable Freund-Rubin solution if ~ 7

~max " ~ A. (27)

The Riemann tensor for the M(m, n) space is ob- tained by inserting (9) and (13) into eqs. (4), (6). Af- ter using the Einstein equations, and adopting the no- tation that indices s, t, u, v run over 1,2, 3, 4 and w, x , y , z run over 5, 6, the nonzero components o f the Riemann tensor are

_ 1 (2A4 Rosot - ~ - A 2 ) 4gst,

1 (3A2 _ 2A4) Rowox =~ 2gwx ,

1 Rstuv - g A4 (4gsu 4gtv - 4gso 4gtu )

+ ~4 (3A2 - 2 A 4 ) (Jsu Jto -Jso Jtu + 2 Jst Juo),

_ 1 mne2A2A4Jstewx Rstwx = 2Rswtx - - ~

_1 (6A4 _A2) (2gwy 2gxz _2gwz 2gxy), (28) Rwxy z - -ff

plus those implied by the symmetries o f the Riemann tensor. Writing out (25) as a 28 × 28 matrix equation with entries determined from (28), we find the matrix is sufficiently sparse that one can find the eigenvalues by elementary means. The 27 eigenvalues correspond- ing to the tracefree eigentensors Vab are, after using eqs. (19) --(21),

- ~ (6x - 1) A 2 2 times,

_ 1 (3 - 2x) A 2 2 times,

1 - ~ (2x - 1) A 2 4 times,

- ~4 (9 - 2x) A 2 6 times,

- ~ (lOx - 9 ) A 2 3 times,

3 - ~ [(3 - 2x) (2x - 1)] 1/2 A2 4 times, (29)

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

3 + ~ [(3 - 2x) (2x - 1)] 1/2 A2

I [2x ( 2 5 - 4 8 x + 3 2 x 2 ) l / Z ] A 2 -

1 [2x + (25 - 48x + 32x2) 1/2 ] A 2

4 times,

1 time,

1 t ime.(29cont 'd)

Although it is not relevant for stability, we note for completeness that the 28th eigenvalue, corresponding to the pure trace mode, is¼ (1 + 2 x ) A 2 a = A.

Using (19) and the fact that~ ~<x ~<~, one can easily check that for all permitted x, the last eigenval- ue listed is the largest one,

max = ~ (1 + 2 x ) - 1 [2x + (25 - 48x + 32x2) 1/2 ] A,

(30) and all the other eigenvalues are less than ~ A. The con- dition (27) is satisfied by (30) if

9 39 - ~ < x ~ < ~ . ( 3 1 ) 14

From (21), this implies that the M(m, n) solution is definitely stable against arbitrary small perturbations if

2 61/2 < Im/nl < .~ (66) 1/2 (32) 18

In the notation of Castellani et al. [9], where their 2 m/n, this condition is p/q ='~

61/2 <[P/q[< ~x7 (66) 1/2, (33)

or, approximately, 0.6350529 < IP/q[< 1.1804158. As one might expect, the supersymmetric solution, which hasp/q = 1 [9], lies within this range. However, there are also infinitely many other solutions which are stable despite having no supersymmetry.

We have so far shown that stability is guaranteed if condition (31) is satisfied. We shall now demonstrate that i fx lies outside the range (31), then the solution is unstable. The proof proceeds by explicitly exhibiting the mode which violates the stability criterion (23). Its components in the orthonormal frame e a are con- stants, given by

ha b e: diag (a, /3, /3, /3,13, 7, 7),

a = 1 6 ( x - l ) ,

= 3 + (25 - 48x + 32x2) 1/2 ,

? = 2 - 8x - 2(25 - 48x + 32x2) 1/2.

(34)

(35)

(36)

(37)

This symmetric tensor, which is obviously tracefree, is

the eigentensor o f the Riemann tensor corresponding to the maximum eigenvalue (30), as can readily be checked from (28). Since the hi/part is clearly covari- antly constant in ds62, it is a straightforward matter using (2) and (3) to verify that hab given by (34) is a transverse Killing tensor:

Vahab = O, V(ahbc ) = 0. (38, 39)

By acting on (39) with V a and using (38), one obtains in an Einstein background

ALhab = -- 4Racbd hed + 4Ahab. (40)

Thus hab given by (34) is an eigenmode of AL, and from (30) it follows that this minimum eigenvalue of the Lichnerowicz operator is

~min = 4 A - 4 K m a x

= 2 ( 1 + 2 x ) ~1 [ 2 + 2 x - ( 2 5 - 4 8 x + 3 2 x 2 ) 1/2] A.

(41)

Therefore this mode actually saturates the bound (26), so the M(m, n) solutions are unstable if rain does not satisfy (32). Combining this demonstration with our previous result, we conclude that the M(m, n) solutions are classically stable if and only if [mini lies within the interval (32), i.e. if and only if the p/q of Castellani et al. [9] satisfies (33).

C.N. Pope acknowledges useful discussions with B.E.W. Nilsson. This work was supported in part by NSF grant PHY-8316811 and by an Alfred P. Sloan Fellowship to D.N. Page.

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