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Volume 147B, number 1,2,3 PHYSICS LETTERS 1 November 1984 NEW SQUASHED SOLUTIONS OF d = 11 SUPERGRAVITY Don N. PAGE and C. N. POPE 1 Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA Received 29 May 1984 We construct two homogeneous Einstein metrics on an SO(3) bundle over CP 2, one of which corresponds to the N °1° solution of Castellani and Romans. The new metric is analogous to the squashed Einstein metric on S 7. We then show that there is a second Einstein metric on each N pqr space. The eleven-dimensional supergravity field equa- tions [1,2] admit Freund-Rubin [3] solutions which are products of a four-dimensional anti-de Sitter lorentzian spacetime and an arbitrary compact seven-dimensional positive-definite Ein- stein space M 7 of positive scalar curvature. Many such Einstein spaces have been found, but to date the only example in the literature of two distinct Einstein metrics on seven-dimensional manifolds of the same topology is provided by the seven- sphere. The standard Einstein metric of the round S 7 yields an effective theory in spacetime with N = 8 supersymmetry and SO(8) gauge symmetry [4,5]. However, S 7 also admits a second, squashed, Einstein metric [6] which yields a theory with SO(5) × SU(2) gauge symmetry and N = 1 or N = 0 supersymmetry depending upon the orienta- tion of the S 7 [7,8]. Castellani and Romans [9] examined the coset manifolds N por of the form SU(3) × U(1)/[U(1) × U(1)], where p, q, and r are integers which characterize the embedding of U(1)× U(1) in SU(3) × U(1). They found an Einstein metric for each p, q, and r yielding N = 3 supersymmetry and SU(3) × SU(2) gauge symmetry for N °1°, and N = 1 supersymmetry and SU(3) × U(1) gauge symmetry otherwise. In this letter we give a different construction of metrics on the N °1° space and show that there is in fact a second Einstein 1 Permanent address: Blackett Laboratory, Imperial College, London SW7 2BZ, UK. 0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) metric, which we shall refer to as "squashed", which also yields SU(3) × SU(2) gauge symmetry but N = 1 supersymmetry. Since this result appears to be in conflict with ref. [9], we re-ex- amine the equations therein and show that they do indeed admit a geometrically distinct solution which was overlooked in ref. [9]. We then find that there are in fact two distinct Einstein metrics for each N pqr (with the exception of the spaces corresponding to q = 0). Our construction of the N °1° space is analogous to the description of S 7 as the bundle space of the k = 1 Yang-Mills instanton over S 4. The squashed metrics on S 7 are obtained by scaling the size of the SU(2) fibers by a factor 2~ relative to the S 4 base and give Einstein metrics when h 2 = 1 or h2 = ½ [6]. In general, if a four-dimensional space with metric dg 2 admits a Yang-Mills field liF, i et F t = ~,~ae A e a, where i is an index for the three generators of the SU(2) Lie algebra, then by the generalized inverse Kaluza-Klein mechanism one can write a metric on the seven-dimensional bundle space as ds2=dg2+h2[(~l-A1)2+(~2-A2)2 (1) where ~ is a constant squashing parameter, A t is a Yang-Mills potential which gives F ~ = dA i + l~.tjkAJ A A k, (2) 55

New squashed solutions of d = 11 supergravity

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Page 1: New squashed solutions of d = 11 supergravity

Volume 147B, number 1,2,3 PHYSICS LETTERS 1 November 1984

NEW S Q U A S H E D S O L U T I O N S OF d = 11 SUPERGRAVITY

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

Received 29 May 1984

We construct two homogeneous Einstein metrics on an SO(3) bundle over CP 2, one of which corresponds to the N °1° solution of Castellani and Romans. The new metric is analogous to the squashed Einstein metric on S 7. We then show that there is a second Einstein metric on each N pqr space.

The eleven-dimensional supergravity field equa- tions [1,2] admit Freund-Rubin [3] solutions which are products of a four-dimensional anti-de Sitter lorentzian spacetime and an arbitrary compact seven-dimensional positive-definite Ein- stein space M 7 of positive scalar curvature. Many such Einstein spaces have been found, but to date the only example in the literature of two distinct Einstein metrics on seven-dimensional manifolds of the same topology is provided by the seven- sphere. The standard Einstein metric of the round S 7 yields an effective theory in spacetime with N = 8 supersymmetry and SO(8) gauge symmetry [4,5]. However, S 7 also admits a second, squashed, Einstein metric [6] which yields a theory with SO(5) × SU(2) gauge symmetry and N = 1 or N = 0 supersymmetry depending upon the orienta- tion of the S 7 [7,8].

Castellani and Romans [9] examined the coset manifolds N por of the form SU(3) × U(1)/[U(1) × U(1)], where p, q, and r are integers which characterize the embedding of U(1 )× U(1) in SU(3) × U(1). They found an Einstein metric for each p , q, and r yielding N = 3 supersymmetry and SU(3) × SU(2) gauge symmetry for N °1°, and N = 1 supersymmetry and SU(3) × U(1) gauge symmetry otherwise. In this letter we give a different construction of metrics on the N °1° space and show that there is in fact a second Einstein

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

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

metric, which we shall refer to as "squashed", which also yields SU(3) × SU(2) gauge symmetry but N = 1 supersymmetry. Since this result appears to be in conflict with ref. [9], we re-ex- amine the equations therein and show that they do indeed admit a geometrically distinct solution which was overlooked in ref. [9]. We then find that there are in fact two distinct Einstein metrics for each N pqr (with the exception of the spaces corresponding to q = 0).

Our construction of the N °1° space is analogous to the description of S 7 as the bundle space of the k = 1 Yang-Mil l s instanton over S 4. The squashed metrics on S 7 are obtained by scaling the size of the SU(2) fibers by a factor 2~ relative to the S 4 base and give Einstein metrics when h 2 = 1 or h2 = ½ [6]. In general, if a four-dimensional space with metric dg 2 admits a Yang-Mil ls field

liF, i et F t = ~,~ae A e a, where i is an index for the three generators of the SU(2) Lie algebra, then by the generalized inverse Kaluza-Klein mechanism one can write a metric on the seven-dimensional bundle space as

d s 2 = d g 2 + h 2 [ ( ~ l - A 1 ) 2 + ( ~ 2 - A 2 ) 2

(1)

where ~ is a constant squashing parameter, A t is a Yang-Mi l l s potential which gives

F ~ = d A i + l~. t jkAJ A A k, (2)

55

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Volume 147B, number 1,2,3 PHYSICS LETTERS 1 November 1984

and ~ is a set of left-invariant one-forms on the fibers, which satisfy the SU(2) Lie algebra

1 d~,~ = - - ~ ( i j k ~ j A ~ k "

We define the orthonormal basis e ~= (e", eT), where e ~ is an orthonormal basis for the four- dimensional metric dg 2 with a, fl . . . . = 0,1, 2, 3, and where

(3)

e i = h ( ~ i - A i ) , (4)

with i, j , . . . = 4, 5, 6 = ], 2, 3. The connection one-forms for (1) in this basis are

- - 1 i ~ ½hF2oeO ' to ,~# = to .# + ~ A F~,oe , ~ ,,~ = - t % , =

o~;; = - ~ , , ~ j k ~ - % k A ~ , ( 5 )

where ~,# are the connection one-forms for d s 2 .

The nonzero components of the Riemann tensor are

- - 1 ~ 2 [ 1 7 i K;'i i i i i R, ,#v8 = R,~Ovn - ~'" ~" . . ' # 8 - F,~nFt~v + 2F ,~#F;n) ,

1.)t21g, i K?j __ 1 k R,~ M = 4', *0r" ~ v -'*¢'ijkF20 '

RUi?= lX-2(SikSj t - - 8 i l S j k ) , (6)

plus those obtained by the symmetries of the Riemann tensor, where R,#v8 is the curvature of dg 2 and ~ v is the Yang-Mills covariant deriva- tive, which gives

-~ C u k A . r F 2 0 . (7)

The components of the Ricci tensor are therefore

R~ 0 = ~ # _ 1),2~,i ]u,i 2 ' ' ~t a y L O . t ~

R U = 1-)~21~i K'J 1 - 2 4 , , a a O z aO "~- -~X 3 i j ,

R ,~ = R i, ~ = 1 i - # o e ' o . (8)

We shall now describe the space N °t° as a Yang-MiUs bundle over CP 2. Charap and Duff [10] showed how to obtain a self-dual or an anti-self-dual Yang-Mil ls field on any four-dimen- sional Einstein space by taking A½ or A ~_

56

respectively, where

A ' ± = -T- ~o i l - • - ~c,jk~y k. (9)

We apply this to CP 2 with the orthonormal basis (e.g., ref. [11])

e ° = d# , el = ½ s in#o l ,

e 2 = ½ sin # 02, e 3 = ½ sin # cos# o 3, (10)

where o i is a set of left-invariant one-forms satisfying the SU(2) Lie algebra do 1 = - 0 2 A o 3 etc. The Ricci tensor of (10) is

R,~# = 6 L # . ( 1 1 )

Substituting the connection one-forms calculated f rom (10) in to (9) gives

AI+---- COS/l 01, A 2 = cos~ 0"2,

A 3 = 3(1 + cos2#)o3, (12)

/41_=0, a 2 = 0 , A3_= 3 sin2/z 03 . (13)

A _ has only the U(1) piece A3, which generates a multiple of the (anti-self-dual) Kg.hler form

J = ~ dA3_ = e ° A e 3 - - e 1 A e 2. (14)

The field strength generated by A'~_ is gauge-co- variantly constant and has the form

F i = i ,~ (15) L a o e A e O,

where

i _ _ i _ _ L o y - - L j 0 - S l y , (16) = Z)k - - - - £ i j k "

Substituting (10) and (12) into (1) yields the seven-dimensional metric

ds = d/~ 2 + ¼sin 2/~ (02 + o 2 + cos 2 ~t 02)

+ - cos + ( z 2 - cos o2) 2

+ (17)

In order for (17) to be a metric on a regular manifold, the singularities must be removable by

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Volume 147B, number 1,2,3 PHYSICS LETTERS 1 November 1984

coordinate transformations. Specifically, one has to check that the part orthogonal to ds 2 is regular at the/ t-coordinate endpoints/~ = 0 and/~ = ~r/2. The condition for the compatibility of the coordi- nate transformation needed at /~ = 0, where e 3 = ~ ' ( ~ 3 - °3), with that needed at /~ = ,r /2, where e 3 = X(2~ 3 - ½03), is that the closed line integral of ~3 with ~1 = ~ 2 = 0 be half that of o 3 with 01 = o 2 = 0. Therefore, since the o~ are left-invariant one-forms on SU(2) [11], the 2:~ are left-invariant one-forms on SO(3). Thus (17) is a metric on the manifold of an SO(3) bundle over CP 2. The isometry group is SO(3) x SU(3)/Za for all values of X, or locally simply SU(3) x SU(2).

To obtain Einstein metrics, we substitute (11), (15), and (16) into (8) to find

R ab = diag ( a , a , a , a , f l , f l , fl ) (18)

with

a = 6 - 6 X 2, f l = 4 X 2+½X -2. (19)

The Einstein condition, a = fl, therefore implies

X 2 = ½ or X 2=~o . (20)

As we shall show later, 22 = ½ corresponds to the Einstein metric on N °m found in ref. [9], but 22 = ~ gives a new solution, analogous to the squashed Einstein metric on S 7. Note, however, that here the two solutions have the same isometry group, whereas in the case of S7one solution had SO(8) and the other had SO(5) × SU(2).

To determine the supersymmetry of the new solution, we seek solutions of the Kilting spinor equation

( D a - ½ m F a ) t l = 0, (21)

where the seven-dimensional metric satisfies the Einstein equation

R a b = 6m2gab . (22)

[We have fixed the scale in (17) so that d s 2 obeys (11), which means that ~2= 1 gives m 2-- ½ in (22), whereas ~2 = 1~ gives m E = 9 . ] The integra-

bility condition for (21) is [7]

1 ( 2 3 ) Cab~ -- ~ c a b ~ r ~ = O.

From (6) and (15), the nonzero components of the Weyl tensor of (17) are

C~t~v , = [(1 - 12X 2 - 20)~4)/20)~ 2 ]

-SasJs~ , + 2J,~sJv8,

- 2 ~ ) e u k L ~ , C,~,L; = - (1 2 k

co , , ; = - [(1 - 2x~) (1 + a o x ~ ) / 2 0 x :] g,~,~a,j

- - 2~ )eijkLafl,

CU~ d = [(1 - 2X2)(1 + 10~2)/10~ 2]

x (~ ,~ , , - 8,,~j,),

plus those given by the symmetries of the Weyl tensor, where J~# are the components of the KMaler form (14). Defining 14 independent Fab combinations by

14o,= }(ro, + },,jkr;.~), Oij: ~(~ij~ l~fi), 14,; = ~(-1",;- }r~ + ½8,jr~- }~,~ro~ ),

(24)

(25)

which generate the exceptional group G 2 [7], we find for the ~2 = ~o squashed Einstein metric that

C01 = 5/-/23 - H 0 1 , Co: = 5H31 - H 0 2 ,

Co3 = _ 10//12 + 14H03, C23 = 5H01 - H23 ,

CaI = 5H02 - / / 3 1 , C12 = _ 10Ho3 + 14/-/12,

Co~ = 4ci jkHjT, , Ci) = 4 H i ; ,

Q : = 4( H i j + c i j kHok ) . (26)

Thus Cab, which generates the holonomy group for (21), can be expressed in terms of the Hab'S, so the holonomy group is G2, and the integrability condition (23) is compatible with at most N = 1 supersymmetry [7]. Substituting the solution of (23) into (21), we find there is indeed one Killing spinor if m is chosen to be the positive root of m 2 = 9 , but no solution of (21) if m is chosen negative. Hence there is N = 1 or N = 0 supersym-

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Volume 147B, number 1,2,3 PHYSICS LETTERS 1 November 1984

metry depending upon the orientation of the space [8,12].

When h 2 = 1, the only nonzero components of the Weyl tensor (24) are given by C~/~rn, which is equal to C~/~vs, the Weyl tensor of the C P 2 metric d s 2, which is anti-self-dual in our conventions:

C a f l y 8 1 - - - - - " ~ . a l h ~ u C t ~ u v s . (27)

Thus in this case the holonomy group is SU(2), which is compatible with at most N = 4 supersym- metry [13]. Substituting the general solution of (23) into (21) yields no solution if rn is chosen to be the positive root of m 2 = ½, but three indepen- dent Killing spinors if m is chosen negative. Hence the orientation for which the h 2 = 1~

Einstein metric yields N = 1 supersymmetry gives N = 0 for h 2 = 1, whereas the orientation giving N = 0 for h 2 = ~ gives N = 3 for h E 1

As in the case of S 7, the existence of two Einstein metrics on N °1° can be interpreted in spacetime in terms of a spontaneous symmetry breaking mechanism in which the scalar field corresponding to the squashing degree of freedom of the internal metric acquires a nonzero expecta- tion value [5,8]. In other words, there are two extrema of the effective potential for the scalar field in spacetime, corresponding to the two Einstein metrics. Unlike the case of S 7 however, here there is no compelling reason to prefer one metric over the other as the natural unbroken ground state. With one choice of orientation of N °1°, one can view the h z = ½ metric as giving the unbroken vacuum state with N = 3 supersymme- try, which then can be broken spontaneously to the h 2 = ~0 metric yielding N = 0 supersymmetry. On the other hand, with the opposite orientation t h e h E = ~0 metric provides a vacuum state with N = 1 supersymmetry, which can be broken to the h E = ½ metric with N = 0. In both cases the gauge group, which is locally SU(3) × SU(2), remains unbroken.

I t is of interest to determine what happens to the masses of the originally massless gravitinos as one goes f rom either of the supersymmetric phases to the corresponding broken phase. We follow the eigenspinors of the Dirac operator which are equal to the Killing spinors at h 2 = ½ and h 2=

58

respectively, as was done in ref. [8] for S 7. We first scale the metric (17) by the constant conformal factor

~22 = (28h2m2)- l (1 + 16h 2 - 8h4), (28)

so that the Ricci scalar becomes R = 42m 2 for all values of h. In the natural spin frame, the components of the relevant eigenspinors are simply equal to those of the corresponding Killing spinors on the Einstein metrics, and so it is straightforward to substitute them into the Dirac equation. We find that the triplet of eigenspinors which are the three Killing spinors at h 2 = ½ has Dirac eigenvalue

K3 = 71/2(5 + 4h2)m/2(1 + 16h 2 - 8h4) 1/2 (29)

and that the single eigenspinor which is the Killing spinor at h 2 = ~ has

it; 1 = -71 /23(1 + 4h2)m/2(1 + 16h 2 - 8h4) 1/2.

(30)

Thus at h 2 -~- 1, 11;3 = 7m/2 and x 1 = - 9 m / 2 ; at h 2 = i!6o, r3 = 9rn/2 and t¢ 1 -m" - - 7m/2. Hence in each of the two supersymmetry-breaking scenarios, the originally massless gravitinos acquire (mass) 2 = m 2 [14,12]. In each supersymmetric phase, the corresponding + 9rn/2 Dirac eigenvalue yields massless spin- ½ fields [14,12]. In the N = 3 phase this gives the single spin- ½ member of the massless graviton multiplet; in the N = 1 phase it gives the triplet of spin-1 partners of the SU(2) gauge bosons.

Our solution with h 2 = ½ is equivalent to the Einstein metric on N °1° found in ref. [9]. To see this, note that since our SO(3) bundle over CP 2 is a special case of the SU(3)/U(1) cosets considered in ref. [9], the topology should be contained there. In general, the N pqr spaces have the smaller isometry group SU(3)× U(1), except for N °1°, which, like the SO(3) bundle over CP 2, has SU(3) × SU(2). Furthermore, the Weyl tensor (24) with h 2 = ½ agrees precisely with that of the N 010 solution given in ref. [9], and in each case the same N = 3 supersymmetry is found.

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Volume 147B, number 1,2,3 PHYSICS LETTERS 1 November 1984

W e now turn to the question of why our ?~2 = solut ion was not found in ref. [9]. There, Einstein metr ics on N pqr spaces were parametr ized by a cons tan t c, related to p, q, and r, and lying in the range - 1 < c < 1. It was argued that all inequiva- lent geometr ies are exhausted by the restricted range - 1 < c < - 2 /q3- . However , this turns out to be incorrect , and in fact our ?~2 = ~ solution cor responds to setting c = + 1.

Mot iva t ed by this discovery, we ask whether any of the o ther N pqr spaces admits a second, inequivalent , Einstein metric. F r o m eqs. (3.3) and (3.4) of ref. [9] one can show

x = 3 p / q = - ( 6 + 5 c ) d / ( 2 + 3 c ) ,

d = ÷ ( 1 - - C 2 ) 1/2. (31)

The N pqr spaces are topologically equivalent to those ob ta ined by permut ing 3p + q, - 3p + q, and 2q [9], which is equivalent to the fractional l inear t r ans format ions of x generated by

x ~ - x , x ~ ( x - 3 ) / ( x + l ) . (32)

These t r ans format ions permute the six intervals [ - oo, - 3], [ - 3, - 1], [ - 1, 0], [0,1], [1, 3] and [3, oo] of x and allow one to t ransform any x into the interval [0,1]. For such an x, (31) has exactly two real solut ions for c, one satisfying - 1 < c < - 2 / q 3 - with d = + ( 1 - c 2 ) 1/2 and the other sat isfying 2 / ~ - _< c < 1 with d = - (1 - c 2 ) 1/2.

These each vary monotonical ly with x, going f rom Icl = 1 at x = 0 to {c{ = 2 /¢5 - at x = 1. To show that these give two distinct Einstein metrics, we calculate the square of the Weyl tensor of the Einstein metr ics in ref. [9] and normalize by A 2

( R a b = A ga b) to obta in the dimensionless curva- ture invar iant

I =- A-2CabcdCabcd = - ~ [ 1 - 4 c 2 ( 1 + c ) / ( 2 + c)3]. (33)

Clear ly if two Einstein metrics have different values of I , they are geometrically different. N o w one can easily see that for - 1 _< c _ - 2/~/5-, I

lies in the range 32 /3 _> I_> 8, whereas for 2 / ~ < c _< 1, one finds 8 >_ I >__ 608/81. Thus for {cl > 2 / ¢ 3 - or, equivalently, 0 _< x < 1, there are indeed two inequivalent Einstein metrics on each N pqr space. Conversely, by considering the action of the pe rmuta t ions generated by (32), we can p rove that if two N pqr Einstein spaces give the same value of I , they are geometrically identical and have the same topology up to identifications (i.e., their universal covering spaces are the same). In part icular , this means there is only one Einstein metr ic if {c[ = 2/~/5-, i.e. x = 1.

T o summar ize the argument above, all possible inequivalent N pqr spaces are represented by those with 0 _< 3p _< q. They each admit two inequivalent Einstein metrics, with the exception of 3p = q, which has only one.

Final ly we note that the arguments in ref. [9] tha t showed that each N pqr metric there (except N m°) had one Killing spinor generalize to show tha t the new squashed Einstein spaces also each have one Kill ing spinor. However , as in the case of N m° which we discussed above, the Killing spinor on the squashed metric obeys (21) with the oppos i te sign of m to that on the original metric. Thus in each case there are two inequivalent spon taneous supersymmet ry breakings f rom N = 1 to N = 0 .

C.N.P. acknowledges useful discussions with C.J. I sham. This work was suppor ted in par t by N S F G r a n t PHY-8316811 and by an Alfred P. Sloan Fel lowship to D.N.P.

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[2] E. Cremmer and B. Julia, Nucl Phys. B159 (1979) 141. [3] P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B (1980)

233. [4] M.J. Duff, in: Supergravity 81, eds. S. Ferrara and J.G.

Taylor (Cambridge University Press, London, 1982). [51 M.J. Duff and C.N. Pope, in: Supersymmetry and

supergravity 82, eds. S. Ferrara, J. G. Taylor and P. van Nieuwenhuizen (World Scientific, Singapore, 1983).

[6] G. Jensen, Duke Math. J. 42 (1975) 397. [7] M.A. Awada, M.J. Duff and C.N. Pope, Phys. Rev. Lett.

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[8] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Rev. Lett. 50 (1983) 2043; 51 (1983) 846 (E).

[9] L. CasteUani and L.J. Romans, Nucl. Phys. B238 (1984) 683.

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(1978) 239.

[12] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Lett. 139B (1984) 154.

[13] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Lett. 129B (1983) 39.

[14] R. D'Auria and P. Fr~, Torino preprint ITP-SB-83-57 (1983).

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