Which compactifications of D = 11 supergravity are stable?

Volume 144B, number 5,6 PHYSICS LETTERS 6 September 1984

WHICH COMPACTIFICATIONS OF D = 11 SUPERGRAVITY ARE STABLE?

Don N. PAGE and C.N. POPE 1

Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA

Received 21 May 1984

We complete the stability analysis of all known Freund-Rubin solutions of eleven-dimensional supergravity by deter- mining the necessary and sufficient conditions for Qpqr spaces (U(1) bundles over S 2 x S 2 X S 2) to be stable.

Recently a number of solutions of the field equations of eleven-dimensional supergravity [ 1,2] have been found. One large class consists of F r e u n d - R u b i n [3] compactifications to the product of four-dimensional anti-de Sitter spacetime and a compact seven- dimensional Einstein space M 7 with positive scalar curvature. If such a solution is to be interpreted as a classical ground state of the theory, then it should be stable against small perturbations. In ref. [4] it was shown that the necessary and sufficient conditions for stability of such a solution is that the Lich- nerowicz operator A L acting on symmetric transverse

tracefree tensors hab in M 7 should satisfy

2x L/> { A, (1)

where the Ricci tensor of M 7 satisfies

Rab = Agab. (2)

The question of stability has already been answered for many solutions, and in this letter we complete the task of identifying which of all the known F r e u n d - Rubin solutions are stable and which are not.

For the purpose of discussing stability, it is convenient to divide F r e u n d - R u b i n solutions into three disjoint categories. The first category contains all solutions with one or more unbroken supersymmetries in spacetime. These are all stable, a result which follows from the formal properties of the supersymmetry algebra [ 5 - 7 ] . The complete list of known exam-

i Permanent address: Blackett Laboratory, Imperial College, London SW7 2BZ, UK.

346

ples is as follows: the round S 7 with N = 8 supersymmetry [8,9] ; the squashed S 7 w i t h N = 1 [10] ; the MPq r spaces [5] with Einstein metrics obtained in [11], when p is set equal to q, which have N = 2; tile NPq r spaces with N = 1 (except N 010 which has N = 3) [ 12] ; the Qpqr spaces with p = q = r, which have N = 2 [13] ; the coset space SO(5)/SO(3)max with N = 1 [14] ;and the Stiefel manifold V5, 2 with N = 2 [14]. In ref. [4] it was shown that, with the excep- tion of the round S 7, each solution with supersymmetry has a corresponding solution with no supersymmetry, obtained by reversing the orientation of the M 7. Since the spectrum of the Lichnerowicz operator is insensitive to the orientation, these skew- whiffed spaces, which we also include in the first category, satisfy (1) and hence are also stable [4].

The second category consists of MT'S which are product spaces. In ref. [4] it was shown that these are all unstable to a mode in which one factor grows and the other shrinks at fixed seven-volume. The known homogeneous examples are S 5 × S 2, S 4 X S 3, S 2 X S 2 X S 3, CP 2 X S 3, QpqO, and (SU(3)/SU(2)max) × S 2 [14]. There is also an inhomogeneous Einstein metric on the product of the nontrivial S 2 bundle over S 2 [15] and S 3.

The third category consists of solutions which are not product spaces and are neither supersymmetric nor related to supersymmetric solutions by a reversal of orientation. For this category there are no general arguments which establish whether or not the stability criterion (1) is satisfied. Thus there appears to be no alternative to carrying out a case-by-case analysis.

0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


However, there are in fact just two known families of solutions in this category: the MPq r spaces with general p, q, r [5,l 1], and the Qpqr spaces with gen- eralp, q, r [13]. In ref. [16] we analyzed the stability of the MPq r spaces and found that they were stable if and only if ~761/2 < [P/q] < 1@7 (66) 1/2- In this letter we complete the analysis of all known F r e u n d - Rubin solutions by examining the stability of the Qpqr spaces.

The Qpqr spaces are the coset manifolds SU(2) X SU(2) × SU(2)/U(1) × U(1), where the integers p, q, r characterize the embedding of U(1) X U(1) in SU(2) X SU(2) X SU(2) [13]. They can equiva- lently be regarded as U(1) bundles over S 2 X S 2 × S 2 where p, q, r characterize the degree of twisting of the U(1) fibers over S 2 X S 2 X S 2. We shall follow [17,16] and obtain a seven-dimensional metric on this bundle space by the inverse Kaluza-Klein meth- od of reinterpreting gravity plus a U(1) gauge field as pure gravity in one higher dimension. Thus one can write such a seven-dimensional metric as

d, 2 = c2(dr - A) 2 + ds 2, (3)

where c is a constant, r is a coordinate on the fiber which we assign period 47r, and A is a potential for the U(1) field strength F = dA on the six-dimensional base with metric ds~. In our case the base is S 2 X S 2 X S 2, so we can choose an orthonormal basis e a, a = 0 ..... 6,

e 0 = c(dr - A), e I = A11/2 d01,

e 2 = A11/2 sin 01 dq51 ,

e 4 = A21/2 sin 02 d~2,

e 6 = A31/2 sin 03 d~b3,

e 3 = A21/2d02,

e 5 = A31/2d03 '

(4)

where Oi, ~i are spherical polar coordinates on the ith two-sphere.

The most general U(1) bundle over S 2 X S 2 X S 2 can be represented by the harmonic two-form

F = n l A l e l ^ e 2 +n2A2e3 ^ e 4 +n3A3e5 ^ e 6, (5)

where n l , n2, n 3 are constants. Locally F = dA, where

A = - n 1 cos 01 d~ 1 - n 2 cos 0 2 d~2 - n 3 cos 03 d~b 3.

(6)

With this substituted into (4), the metric is

ds 2 = c2(dr + n 1 cos01 d~ 1 + n2cos02d~ 2

+ n 3 cos 0 3 d~b3)2 + A l l ( d 0 ~ + sin201 d~b 2)

+ A~l(d0 2 + sin202 d~ 2) + A31(d0 ~ + sin203d~2).

(7) In order for (7) to be the metric on a regular mani-

fold, the coordinate singularities at the poles of the spheres must be removable. Since r has been assigned period 4~r, this implies nl , n2, n 3 must all be integers. The resulting manifold Q(nl , n2, n3) is simply con- nected if n l , n2, n 3 are relatively prime. If instead they have greatest common divisor k > 1, then r could have been defined consistently modulo 4rrk. Thus in this case the space has fundamental group Zk, so in general Q(nl , n2, n3) = Q(nl /k , n2/k , n3/k)/ Z k where k = gcd(nl , n2, n3). Note that to characterize these spaces completely, all three integers n l , n2, n 3 are necessary, whereas the MPq r spaces can be completely characterized by just two integers [5,11, 161.

Now to obtain a F reund-Rubin solution, we need to solve (2). Using the formulas of ref. [16] or pro- ceeding directly, one obtains in the orthonormal frame (4) the non-zero Ricci tensor components

2 2 ROo=1C2(n2A2 +n2A22 2 +n3A3) '

2 2 - 2 R l l = R 2 2 = A 1 - ½ c n lA1,

R33 = R 4 4 = A 2 ½ 2 2 - 2 - - C n21~2,

R55 =R66 =A3 ½ 2 2 - 2 - c n3A 3. (8)

Defining cq, c~2, c~ 3 by

c~IA=A l - A , c~2A = A 2 - A , c~3A = A 3 - A , ( 9 )

the Einstein equation (2) implies % , c~2, o~ 3 each lie in the range 0 - 1 and satisfy

~1 + Oe2 + Or3 = 1, (10)

thus defining a planar triangular region in the positive (cq, c~2, c~3) octant. Eq. (2) also implies

c - 2 = ' 5 [(1 + c q ) 2 n 2 + ( 1 +c~2)2n22 +(1 + c~3)2n21 A,

(11)

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Volume 144B, number 5,6

(nl/n2)2 = al (1 + a2)2/a2(1 + a l )2 , (12)

together with the equations obtained from (12) by permuting the subscripts 1,2, and 3. Examining these equations shows one can get any ratio n l:n2:n 3 by an appropriate choice of a l , a2, a 3 . Furthermore, the jacobian of the transformation between the two independent ratios of n's and the two independent a 's is nonsingular. Thus there is exactly one Einstein metric of the form (7) for each set of integers n l , n2, n3, not all zero. These metrics are the same as those in ref. [13] with p, q, r replaced by n l , n2, n 3.

To examine the stability of these solutions, we need to find the lowest eigenvalue X = ~'min of the Lichnerowicz operator AL, defined by

AL hab = _ [] hab -- 2RacbdhCd + 2R(aChb) c = Xhab ,

(13) acting on symmetric transverse traceless tensors hab. such tensors obey the identity

h ab AL ha b d V

= f [--4habRacbdhCd + 4Ahabhab

+ 3V(ahbC)V(ahbc)] dV (14)

on an Einstein space satisfying (2). If M 7 is homogeneous, the 27 eigenvalues K of the Riemann tensor, defined by

Raebd V ed = t¢ Vab, (15)

for symmetric traceless eigentensors Vab, are independent of position. It follows from (14) that if Kma x is the maximum eigenvalue of (15),

~kmi n ~ 4A - 4tCma x, (16)

with equality if the corresponding eigenmode hab satisfies (15) with K = tCma x and is a Killing tensor, obeying

7(ahbe ) = 0. (17)

(If the space is not homogeneous, (16) still holds if tCma x is now taken to be the maximum value at- tained by any of the eigenvalues of (15) anywhere in the space.)

In order to find tCmax, we need the components of the Riemann tensor. We put a subscript 1, 2, or 3

PHYSICS LETTERS 6 September 1984

on indices a, b .... , to indicate that they run over 1 and 2, 3 and 4, or 5 and 6 respectively. Using the formulas of ref. [16] or calculating directly, one obtains from (4), after using the Einstein equations, the nonzero Riemann tensor components

Roal 0bl = ~alA6alb 1 '

Ra, b,c, dt = (1 -- I ~ ' ) A ( ~ " ", ~1 ~ 6h~1 ~,d - 6axd, fb ,c ) ,

Ra~blc2d2 = 2Ratc2bld2

- ½c2nln2A1A2ealbleC2d2

= --sgn(n 1 n2) (0t I ot2)l/ZAeaabl ec2d2 , (18)

together with those obtained by permuting the subscripts 1,2, and 3 and those implied by the symme- tries of the Riemann tensor. The e's are the Levi- Civita symbols on the three S2's, e12 = - e 2 l = e34

= -e43 = e56 = -e65 = 1. The 27 eigenvalues K of (15) are as follows:

First,

- l a l A , - ( 1 - I ( ~ I ) A ,

-~(ot2a3)l /2A, +~(~2a3)l/2A, (19)

each of which occurs twice. There are a further 16 eigenvalues obtained by cyclically permuting the subscripts 1,2, and 3 in (19). The remaining three eigenvalues are given by inserting into

K = (1 - ¼7)A (20)

the three (real) roots o f

,),3 _ 6,,/2 + 2 0 ( a l a 2 + o~2a 3 + a3al),),

- 56c~1c~2a 3 = 0. (21)

One can use (10) and the fact that a l , a2, a 3 are all nonnegative to show that none of the first 24 eigenvalues, defined by (19) and permutations, is larger than ~ A. On the other hand, the lowest root of the cubic (21) is ')'min ~< ~, which gives K/> ~ A. Thus this eigenvalue is nCma x-

The three eigentensors of (15) corresponding to the eigenvalues given by (20) and (21) have the form

Vab = diag(130' 131'/31' ~32' /32' /33' /33)' (22)

where

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30 + 2(31 +/32 + 33 ) = O, (23)

4a 2 6a 2 - ' / 4a 2 /32 = 0 . (24)

4a 3 4~ 3 6a 3 - 3, 133

The overall scale is still arbitrary as a function of position on M7, but if the 3's are chosen to be constant, one can show that the eigentensors Vab are transverse traceless Killing tensors and hence eigenmodes of A L with X = 4A - 4K [16]. Thus the stability criterion (1) implies that the Q(n l , n2, n3) spaces are stable

7 if and only i f~ma x <~ gA or

(25) '/min • 5"

In order to see what part of the planar triangular region defined by (10) corresponds to stability, it is convenient to choose ~1 and ~2 - c~3 to be the independent variables. Then (21) implies that the stability criterion (25) is satisfied if and only if both

~< ~ 1 ~< -~ (26)

and

(14oq - 3)(1 - 2 a l ) ( 3 - 4 a l )

(°L2 - °~3)2 ~< 4(28e¢ 1 - 5) (27)

are satisfied. Although it is not manifest, these in- equalities are completely symmetric in a l , a2, and % , as will presently be shown.

It is useful to introduce rectangular coordinates on the planar region (10) with origin located at the center of the triangle,

x = 6o~ 1 -- 2, y = 2X/'3(~ 2 -- oe3). (28)

Then from (21) the contours of fixed 3, are

y2 = (4 -- 6 ' / - x ) [ (4 -- 67 + x) (14 - 3' /) - 7x 2] 3(14 - 15-i + 7x)

(29)

If we go to polar coordinates so that x = r cos 0, y = r sin 0, this becomes

7r 3 cos30 - 3(14 -- 157)r 2 + (14 - 3 ' / ) (4 -- 6'/) 2

= 0, (30)

making the threefold rotational symmetry manifest. For 3' < ~, the smallest positive root r(O) maps out a closed curve around the origin as 0 is varied. When 7 is just slightly below ~ 3, the curve is very nearly a circle of radius 4 - 67. As ' / i s reduced further below -~, the circles expand and distort to become like rounded triangles. The stable region thus corresponds to points on or within the closed curve (30) for ' /

2 "

14r 3 cos 30 - 39r 2 + 25 = 0. (31)

As 0 varies, r(O) oscillates between -~ and 1. The size of this region can be compared with that of the triangle of allowed a 's defined by (10), whose vertices lie on the circle of radius r = 4.

In principle (31) determine the range of ratios of

n l , n2, and n 3 for which stable solutions occur. How- ever, i f n l , n2, and n 3 are specified, it is difficult to solve (12) and its permutat ions for a l , a2, and a 3 to see whether they lie within the stable region bound- ed by (31), i.e., whether they satisfy (26) and (27). Hence it is instructive to look on an axis of symmetry, where the cubic equation (21) can be solved without resorting to the Cardan formula. Thus if we set a 2 = a3, the three roots o f (21) are

7 0 = I - a l ,

T+ = } [5 + 0¢ 1 -+ (25 -- 46~ 1 + 57a2)l /21. (32)

Thus for ~ ~< c~ 1 ~< 1, ' / r a i n = ' / 0 , whereas for 0 ~< cq 3, '/rain = 3'-" At cq = -~, there is a double root,

' / m i n = ' / 0 = 7 _ = ~. In fact, at this point one has % = oe 2 = ol3, which corresponds to the supersymmetric solution with n 1 = n 2 = n 3 [13]. Since (16) and (20) imply that Xmin = A'/min, the lowest eigenvalue of the Lichnerowicz operator in this case is -~A with degeneracy 2. The corresponding eigenmodes give two massless scalars in spacetime [4] ; these are the two superpartners of the massless pseudovectors as- sociated with the two harmonic two-forms on Q(nl , n2, n3). It is markworthy that these modes have/3 o = 0 in (22) and thus correspond to perturbations in which the length of the U(1) fiber is unchanged. The same was true for the single A = ~A mode in the supersymmetric MPq r spaces [ 16].

On the axis a 2 = % we can express the range of stability (26) directly in terms of n2/n 1 = n3/nl:

21/2 < ]n2/nll <~ ~ (66) 1/2, (33)

349


or, approximately, 0.84852814 < In 2/n 11 < 1.1804158. Interestingly enough, the upper limit in (33) is precisely the same as the upper limit for LP/ql in the stability criterion for M Pq r spaces [ 16]. Further-

more, if')'min had been 7_ (instead of 70) for o~ 1 > 1 as well as for a 1 ~< ~ on the axis a 2 = a3, the lower limit for In2/nll would have been ~7 61/2, precisely that for [P/ql in ref. [16]. The stronger actual lower limit of (33) results from the existence of the mode

with X = 70 A, which has/30 =/31 = 0 and/32 = -/33 in (22).

This concludes the classical stability analysis of all known Freund - R u b i n solutions of eleven-dimen-

sional supergravity. In ref. [ 14] homogeneous spaces

of dimension seven were examined, and it was con-

cluded that there are no further homogeneous Ein- stein metrics. There may well be other seven-dimensional Einstein metrics, but since they would there- fore have to be inhomogeneous, they are likely to be

more difficult to find.

This work was supported in part by NSF Grant PHY-8316811 and by an Alfred P. Sloan Research

Fellowship to D.N.P.

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Which compactifications of D = 11 supergravity are stable?