Compactifications of N = 1, D = 10 supergravity to 2 and 3 dimensions

Nuclear Physics B261 (1985) 28-40 © North-Holland Pubhshmg Company

C O M P A C T I F I C A T I O N S O F N = 1, D = 10 S U P E R G R A V I T Y T O 2 AND 3 DIMENSIONS E SEZGIN Internat, onal Center for Theorettcal Physics, 34100 Trteste, Italy

Ph SPINDEL Umverstt~ de r Etat, Facultd des Sciences, 7000 Mons, Belgmm

Received 24 May 1985 Utdlzlng Kllhng spmors on spheres we show that N = 1, D = 10 supergravlty admits compactlficatlons to d =2 and 3, which are AdS3 x S7, AdS3×S t ×$6, AdS3x Sl x S3 x S3, AdS2×$8, AdS2xSU(3) In all these solutions the Yang-Mdls field configuration is non-vanishing, and supersymmetry xs broken 1. Introduction C h i r a l N = 1 supergravxty in ten d i m e n s i o n s [1] 1S i n t i m a t e l y c o n n e c t e d with the l o w - e n e r g y limit o f t y p e I s u p e r s t r i n g [2] o r the h e t e r o t l c string [3] theories T h e r e is a m o u n t i n g interest in these theories b e c a u s e t h e y offer a h o p e for h a v i n g a fintte q u a n t u m t h e o r y o f gravity c o u p l e d to a u n i q u e a n o m a l y free Y a n g - M i i l s system [4]. These string theories exist o n l y in ten d i m e n s i o n s , however. Thus, in o r d e r to m a k e c o n t a c t with the f o u r - d i m e n s i o n a l real w o r l d , p r o b a b l y s p o n t a n a o u s c o m p a c tification o f these theories to d = 4 is n e e d e d . This is b y no m e a n s an easy task, since here one is d e a l i n g with e x t e n d e d objects w h o s e field e q u a t i o n s involve f u n c t i o n a l s c o r r e s p o n d i n g to infinitely m a n y c o n v e n t i o n a l fields o f u n b o u n d e d mass a n d spin. M o r e o v e r , at p r e s e n t it a p p e a r s to be &flicult to integrate over all the massive fields so as to b e left with a full a c t i o n for the massless fields alone. In view o f the p r o b l e m s m e n t i o n e d a b o v e , in this p a p e r we c h o o s e to restrict o u r attenUon to the only k n o w n l o c a l l y s u p e r s y m m e t r l c N = 1, d = 10 a c t i o n w h i c h involve the massless fields a l o n e [1], a n d s t u d y s o m e o f its compactlfiCatlons. W e shall n o t resist on m a x i m a l l y s y m m e t r i c f o u r - d i m e n s i o n a l s p a c e - t i m e , since a c c o r d lng to the result o f F r e e d m a n , G i b b o n s a n d W e s t [5] the s i x - d i m e n s i o n a l internal space then m u s t be R a c o flat, e g. T6, K3 ×T2 o r the Y a u - C a l a b l m a n i f o l d s [6]. A l t h o u g h these m a n i f o l d s have no non-trivial isometries, t h e y are p e r f e c t l y accept a b l e s o l u t i o n s in that r e s p e c t since the Y a n g - M i l l s s y m m e t r i e s arise a n y h o w f r o m the Y a n g - M d l s sector o f the t h e o r y a l r e a d y p r e s e n t in d = 10. The o p e n q u e s t i o n is h o w to o b t a i n chiral f e r m i o n s m d = 4, since the Y a n g - M l l l s field is v a n i s h i n g in these solutions. 28

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Non-trivial compactifications of the N = 1, d = 10 supergravity to dimensions greater than four are difficult to obtain as well [7] On the other hand, compactlfications to 2 and 3 dimensions are natural to consider since the theory contains 2-form and 3-form field strengths which can be used as volume forms for d = 2, 3 space-times, respectively [8]. In this paper we shall focus our attention on such solutions with emphasis on allowing non-trivial configurations for the Yang-Mills field Two- and three-dimensional space-times are not just of academic interest One reason for this is that d = 2 (d = 3) field theories with n scalars can be viewed as classical string (membrane) theories in n dimensions. Furthermore, d = 2, 3 field theories constitute by themselves an important and fascinating subject with applications to diverse areas such as statistical physics, solid state and cosmology [9]. Any connection with higher dimensional supergravltIes, in particular with N = 1, d = 10 theory must be of considerable Interest. For instance, as sometimes happens [10], problems which do not seem to have a solution in a given dimension, say 2 or 3, may find one through Kaluza-Klein considerations This paper is organized as follows In subsects 2.1-2.4 we discuss the solutions of the type AdS3 × M 7 where M 7 is either S7, S 6 x S 1 o r S 3 x S 3 x S 1 In all these solutions the Yang-Mllls field is built In terms of (Kalllng) spinors and is non-trivial. In subsects. 2.5, 2.6 the AdS2 × $8 and AdS2 × SU(3) solutions are discussed In sect. 3 the (super)symmetries of the solutions are analyzed In sect 4 we give our conclusions with comments on some of the interesting open problems In the appendix, properties of Killing-like splnors on $3 ×$3 are exhibited We use the signature r / A S = ( - - + + + + + + + + + ) , RlCCl t e n s o r R Q M Q N = O Q F Q N + " (M= 0, 1, . , 9), Clifford algebra {FA,FB} = 2r/aB. The Greek indices refer to AdS2 or AdS3 while the Latin indices refer to the other manifolds involved in the solutions.

2. Compactifications to AdSz and AdS3 We start with the bosonlc field equations of the N = 1, d = 10 supergravity [1] given by

1 RMN -"~

=e2~'GMpoGNm~+2e~'FM o. FN°+-~ OMo'Or,xr--l-~-gMN[-]tr Kor VM (eK~F ~N) + g e'~[A~, F ran ] = - x

e2K~'GNP°F~,

VM (e2K'~GMNP) = 0, 32I-] tr = 2 e2K°GMNpG MNP +

3K

3 eKe'EMIr

'

(1) (2) (3)

F MN ,

(4)

30

E. Sezgm, Ph Spmdel / Compacnficaaons of supergramty

where g is the coupling constant, and GmNe = 30 t mBne] - 3 K Tr ( F t MNAp] -- 2gAt MANApI) =--fMNP -- 3 KXMNV.

In all these equations summation over the Yang-Mills group index is understood For mathemaUcal convenience we take o- to be constant, and wRhout loss of generality to be zero m the background We shall first consider compactifications o n AdS3 × M7 where M7 is a compact seven-dimensional manifold. In this case G~,~o and G ~ p will be taken to be suitable "volume elements" while the Y a n g - M d l s field Fro, will in general be constructed out of Killing spinors. With such an ansatz the field equations, (1)-(4), reduce to algebrmc ones. In the following we choose M7 to be $7, $6 × S~ or $3 × $3 × S~. We now discuss these cases separately

2 1 AdS3xS7 BACKGROUND WITH SO(8) YANG-MILLS FIELD In this solution the only non-vanishing fields are assumed to be

Ol_ I

A~,~ = l - n g

Fm~Tj,

where A, a are real constants, ~ p is the volume element of AdS3 and x 1, . , 8) are lOlling spinors on $7 [11] which satisfy Vpnl :llm7Fp~?t ,

~ '-r/j -----a / .

(6) (I--

(7)

Here m 7 is the reverse radms of $7, and Fp are purely imaginary, antisymmetnc 8 × 8 matrices. Two remarks are m order. Firstly, we have chosen a gauge for the Yang-Mllls field such that the components of the potential are proportional to the 28 IOlhng vectors generating the SO(8) isometry group of $7 Note also that Am~j hes m the defining representation of SO(8) and that the Cartan-Killing metric Ytu~t~:L~is used m the trace of F 2, Tr F F ~ FtU~FtKL]y[1sltrL] = --F[lS]FtKL](~JK~tL )

(8)

The R i c o tensor on AdS3 is given by R~.~ = - 2 m ~ g . ~ ,

(9)

Rm, = 6 m ~ g m , .

(10)

and on $7 by

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Consider first eq. (2). Using the closure property of the Killing spinors 8

n,~'=- E a%m'.,=~,

(11)

l,J=l

the Yang-Mills field strength is found to be

F~.~ =2a (m7-~)~lrmnnj.

(12)

g

The Chern-Simons 3-forms XMNP vanishes m this background since A~, = 0, and Fm,p is traceless. Accordingly eq. (3) is automatically satisfied. Substituting (12) into the Yang-Mills equation (2) one finds the condition

a(a - mT)(2a - m7) = 0.

(13)

To satisfy eq. (4) on the background considered we must choose the non-trivial root tr =½m7.

(14)

Wxth this value of a, the Einstein equation reduces to g2 m---~= 4 ,

m]=A 2,

(15)

while the scalar field equation gives A2 = ~21 -m 2 7.

(16)

Thus, the solution depends on one parameter

22

AdS 3 xS 7 BACKGROUND

W I T H SO(7) Y A N G - M I L L S

FIELD

In this background we break the SO(8) invariance of the previous solution by picking up one of the eight Killing spinors, say *78~ ~/, and by expressing the Yang-Mills field in terms o f the rest as 7 ×7 real antisymmetric matrices, ' i ag fl'r,,,~?.,, Amj=

(t,j = 1 .....

7),

(17)

which correspond to the Killing vectors of the spin(7) lsometry subgroup of $7. The Yang-Mills field strength now reads t

Ot

Fm.j = 2--[(m7 - a)~7'r,,,,,ns + ot~'rt,,m~r,,lnj]. g

(18)

The Chern-Simons 3-form X~.,.p is still vamshing because A~, = 0, while X,~.p is no

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longer vanishing and is given by

X,n.p =

0/

2

4~ g--5(m7 -

a)~Ymnpn.

(19)

Although X,,,.p is gauge d e p e n d e n t , its curl is gauge m v a r l a n t and is given by 2 O/ 4 m 7 ~ - ~ ( m 7 - ot )

O[qXmnpI -~-~F[qmF.pI =

CqI'qrnnpn

(20)

Using eqs. (18) and (19), from the Y a n g - M i l l s equation (2) we deduce the algebraic relation

6a(cl-mT)(3a-2m7)

1+4

=0

(21)

The two non-trivial cases c o r r e s p o n d to case 1:

a

=

m7 ,

case 2:

a

----2 m 7 .

(22a) (22b)

In case 1, X,..v and consequently F[,..FpqI vanishes although F,.. is non-vanishing. In this case, eq. (3) is automatically satisfied, while the Einstem and scalar field equations require that*

£

m27=4,

m 2 = A 2,

A 2 = ~ m 2.

(23)

Notice the similarity with the previous solution for which we have no explanation In case 2, Xm.p as well as its curl are non-vanishing. The Einstein equatmns, in this case implies m32=A 2 ,

4 m72= 128 m,,_.7q. 256 m 6 2~- g2 81 g4

(24)

m2 3 g--T= 1-6"

(25)

From the last equation we obtain

Accordingly the value o f A 2 is obtained from eq (4) to be Az = ~ r n 2

23

AdS3 x S l x S 6 B A C K G R O U N D

WITH

SO(7) YANG-MILLS

(26)

FIELD

The possibility of having a flat factor in this solution is due to the fact that we assume or to be zero, and thus all the trace terms on the right-hand side of the * In d e n y i n g t h e s e r e l a t i o n s t o a v o i d Flerz r e a r r a n g e m e n t s w e q u o t e t w o u s e f u l i d e n t i t i e s o n 7d i m e n s i o n a l m a m f o l d s Fk'tl~lFk = ~ -- "t/~ a n d ~7'~, = ~ - r/~

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Einstein equation vanish. In this background, the non-vamshing objects are assumed to be

Avp-~'Al"lt~vp,

Or_ I

A , , ~ = t - ~ l FmrlJ,

(m=4,..,9)

g

R~v = - 2 m 2 g , v,

Rmn = 5m2gmn,

(27)

where i are 8 killing spinors on $6 whose radius is m6 ~. Note that Am½ lies m the 8-dimensional representation of SO(7). Computation of the parameters of this solution proceeds p r e o s e l y as m subsect. 2.1. In particular, the Chern-Slmons 3-form is vanishing. The field equations (1)-(4) lead to the non-tnwal solution Ot

:lm6,

m~-4,c,

mE = A2 = ! ~ m 2 .

_

(28)

2 4 AdS3xS t ×$3 xS3 WITH SO(6) YANG-MILLS FIELD

In this subsection we shall use the following notation. Greek indices refer to ADS3, Latin radices p, q, r = 1, 2, 3 are used for one $3, and primed Latin indices refer to the other $3. We assume that the non-vanishing components of f~Ne are proportional to the 3-volume forms on AdS3 and on the two 3-spheres, A ~ --- ; ~ ,

f , qr = ~'7~r,

f,'q'r' = ~''7~'q','

(29)

It lS easy to show that (29) constitutes a two-parameter solution characterized by R ~ = -2A2g,~,

Rpq=-2~,2gpo,

Rp,q,=-2v':gp,q,,

(30a)

with A2 - v 2 - p'2 = 0.

(30b)

We emphaszze that unlike all the solutions m Kaluza-Klein supergravities known to the authors, here there are two different (and non-vanishing) mass scales in the theory. The limit in which p ' = 0 possesses N = 4 supersymmetry was found in ref. [12]. In this solution, one $3 is replaced by Ta. Note that one can also replace T3 × S1 by any four-dimensional euclidean Ricci flat manifold, e g. K3. We now consider a non-trivial extension of the solution dzscussed above. Guided by the results ofref. [ 12], we choose the non-vanishing components of the Yang-Mills field to be o/

Ap5 = Tg n,qr, 'rq'n,,

O/

t

- - _ _ ~

,~IFq'r'~

Ap'I - 2g ,tp'q'r',t

,tJ.

(31)

The spinors involved m this equation are given by the 8 hnearly independent soluUons of 1 VpTq l --- ~ a "rlpqrFq r 17I ,

--

1

r~q" r '

Vp,r/~-~a,~p,q,,,l

1

77 ,

(32)

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34

where due to the lntegrability conditions [a[ and [a'l are the radii of the 3-spheres, and I = 1, . , 8 (see appendix). The solutions of (32) are normalized as fff~Ts= 8/. It should be noted that 7?t are not Killing spmors on $3 × $3. Actually there are no Killing spinors on $3 x $3; however, m a suitable representation rff can be built out of the product of Killing spinors on each 3-sphere. From (31) It follows that

y\a_

,\

i

Using this result, one finds that the Chern-Slmons 3-form is given by

X , , , p - c ~ 2 ( 2ct -1)~7,,,~,

X,.','p'-c~'2(2a'-l)rlm','p'.

(34)

We note that if a = a -1 The Yang-Mills field is pure gauge although Xm.v is non-vanishing. This can be understood if one recalls that X,.., is a gauge-dependent object. Having established eq. (34), it is convenient to replace the ansatz (29) by

G~,~o = ;t ~ , ~ ,

Gpq, = z'rlpqr ,

Gp, q,r, = v ' ~Tv'q'r' ,

(35)

which automatically solve eq (3). Considering the non-trivial case of a ~ a -l, the Yang-Mills equaUon gives

1)0

(36)

and simdady for the primed parameters. We next turn to the Einstein equation. It yields the relations 1 = a---i

+ 16--7

,

1 + 16-~T],

m2=h 2,

(37)

where we have used v = ( a - a-l). The scalar field equation, on the other hand, gives the condition A2= v 2 ( l + 1 2 ~ ) +

v'2(1+ 12~22) •

(38)

Obviously, this solution can be parametrized in terms of two variables, say v and v':

g2 a = 2 g2_ 16=,~,

1 jg _+16 2 a-

\ g 2 _ 1--~v2],

idem or' and a '-1 ,

(39)

whale the value of A2 is determined from (38) and of m 2 from (37). Note the true singularity appearing when v 2= ~rg 2. We now turn to a brief description of the solutions of the type AdS2 × Ms, where Ms is a compact dimensional manifold. We shall discuss only two cases. Ms = Ss

or SU(3).

E Sezgm,Ph Spmdel/ Compact~cattonsofsupergravlty

35

25 AdS2xS s SOLUTION In order to build this solution we shall again make use of Kilhng spinors, but now on $8. Let us consider the 16 linearly independent solutions of

Vp~7'=½msFp~',

p=l,...,8,

I=1,

. ,16,

(40)

normalized as #~r/1 = 6~. Here rn 8 represents the inverse radms of $8 and the Dirac matrices used are 16 x 16, purely imaginary and antxsymmetric As an ansatz we assume f M N P -~" 0 ,

ApIj =, g ~ tFp~Tj.

(41)

Accordingly, the Yang-Mills field is given by (see subsect 2.1) 1

O[

F,,. j = 2 - ( m 8 - a)#IF,,,~7_, g

(42)

and the associated C h e r n - S l m o n s 3-form vanishes. As a consequence the GMup matter field vanishes and eq. (3) become empty The Yang-Mills equation (2) gives us the condition a ( a - m8)(24 - ms) = 0,

(43)

whose non-trivial solution is obtained from the root" a = ½ms

(44)

Moreover, we see from the scalar field equation that we need in the sum of the squared fields a negative contribution. Previously, this was obtained from some of the GMNp field components. Now we pick a U(1) direction in the Yang-Mills gauge algebra, and consider for it the ansatz F~,~ = Ar/~,

(45)

where A is a constant and ~7~,~the volume 2-form on AdS2 Obviously, the corresponding field equation is trivially satisfied, while the value of A is obtained from the scalar field equation 4

A2= 112~-~

(46)

Finally, the Einstein equation (1) fixes the value of the radn m81 of $8 and rn~-1 of AdS2 as follows.

m~=¼g 2,

m ~ = 2 8 m 2.

(47)

36

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2 6 AdS2×SU(3) SOLUTION

Here we can make use of the non-vanishing 3-de Rham cohomology space of the SU(3) group manifold in order to let GmN. have non-vanishing components on SU(3) proportional to the SU(3) structure constants. Moreover, as we assume the Yang-Mills field to be identically zero on SU(3), we also have to introduce, as before, a U(1) Yang-Mills field on AdS> Then the scalar equation will relate the field strengths of both. Finally the Einstein equation gives the value of the radius of AdS2 and fixes the scale of the Killing metric used on SU(3)

3. (Super)symmetries of the solutions Consider first the bosonic symmetries of the solutions. In subsect. 2.1, as pointed out earlier, the components of the Yang-Mllls configuration are nothing but the 28 IOlling vectors of the SO(8) lsometry group of $7. Under these isometry transformations the Killing vectors ~:~ themselves transform as ~e,,~KL = f,S.KLMNI~MN '

(48)

where f • are the SO(8) structure constants. Since the right-hand side of eq. (48) is not proportional to an infimteslmal gauge transformation, the SO(8) symmetry is broken. Thus, we conclude that the (bosomc) symmetry of the solution is the centralizer of SO(8) in the Yang-Mills group G (of the N = 1, d = 10 theory) times the obvious SO(2, 2) symmetry of AdS3 Similar reasoning leads to the conclusion that also m the cases of subsects. 2 2-2 5 the bosonic symmetries of the solutions are that of the anti-de Sitter space-time U(1) symmetries associated with the flat factors m the solution, and the centrahzer of the group H m which the Yang-Mills configuration lies, m the Yang-Mills group G. In subsect. 2.6, the Yang-Mflls field is vanishing on SU(3); therefore the bosomc invariance group of the soluUon is SO(I, 2) x SU(3)L X SU(3)r. To investigate the supersymmetry of our solutions we must study the supersymmetry transformaUon rules for the fermlomc fields in the theory. They are given by [ I ] 1 8X = 2x/2 eK~FmNFmNe' (49) 8a, = ( x ~ F r a O M t r + I - ~ e K ~ F m N O G M N Q ) e , 8tb~ = V Me - ~ s ( F ~ F ~ a + 2FPORFM) e.

(50) (51)

From 8)( = 0 it follows that F MNPOFMNFpqe = 1F~rvF MNe .

In all the solutions except that of subsect. 2 2, F ^ F = 0 , but F . F # 0

(52) m the

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37

background. Therefore, in these cases all supersymmetries are broken In subsect. 2.2, although both F ^ F and F . F are non-vanishing in the background, it is easy to show that eq. (52) is incompatible with 8A =0. This is essentially due to the appearance of the factor A = ±x/~(2m7) -1 in an eigenvalue equation. We conclude that all the supersymmetries are broken in the backgrounds discussed in sect. 2

4. Conclusions In this paper, we have shown that N = 1, d = 10 Yang-Mllls coupled supergravlty admits several compactificatlons to 2 and 3 dimensions (the cases of subsects. 2.1-2 6). In most of these solutions the Yang-Mills field is constructed in terms of Kllhng splnors on the internal space In two of the solutions (subsects 2 2 and 2 4) the Chern-Slmons 3-form, and m one solution (subsect. 2.2) its curl are nonvanishing. In two solutions (subsects 2.3, 2.4) the S~ factor can be replaced by a real hne to extend the solutions to include four-dimensional spacetime homeomorphic to R 4 In subsects. 2.1-2.5 the isometrles of the internal supace are broken (except the trivial U(1) associated with $1), while in subsect. 2 6 all isometrics survive In all the cases supersymmetry is completely broken Some of the open problems are. (i) to find new compactifications which preserve some of the supersymmetrles and to make contact with the known supergravlty theories &rectly constructed in two [13] and three dimensions [14] (n) To investigate the stability of the solutions presented in this paper, especially the case of subsect. 2.4 in which there exist a singular value of the solution parameters (iu) It would be interesting to find compactificatlons in which the internal space admits complex (or more general) structures which might be relevant in connection with problem (1). The authors are indebted to Professor Abdus Salam for illuminating discussions One of the authors (Ph. S) would like to thank him for hospitality at the International Center for Theoretical Physics in Trieste where part of this work was carried out.

Appendix Here we discuss in some detail eq (32) of subsect 2.4. Consider a 3-dimensional euclidean manifold with volume 3-form ~Tpq.On such a space we analyze the spinorlal equation Vp~7 = A~Tpq,Fqr~7, where the

(A 1)

Fp matrices satisfy the Clifford algebra {/'q, Fq} =

2gpq

but are not necessarily of minimal &mension 2.

(A.2)

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The first m t e g r a b d l t y c o n d i t i o n o f eq (A 1) is In rrs ~l~pqrsl r / = 8A2/'pqr/,

( A 3)

a n d as a c o n s e q u e n c e , we o b t a i n

1RpsFS,1 = 16X2Fp~.

(A.4)

M o r e o v e r , as in e u c h d e a n space the o n l y s o l u t i o n o f an e q u a t i o n hke r l ¢ 0 is xp = 0, we d e d u c e that

Rps : 32A2gp,.

xpFPrl

--

0 for

(A.5)

A c c o r d i n g l y s o l u t i o n o f eq (A.1) exists o n l y on the 3-sphere* a n d m terms o f its r a d i u s a we o b t a i n A 2=

1

(A6)

16a 2"

So the m t e g r a b d l t y c o n d i t i o n ( A 3) b e c o m e s an identity a n d as a c o n s e q u e n c e s t a n d a r d t h e o r e m s assure the c o m p l e t e l n t e g r a b d l t y o f eq. ( A 1) This result can be u n d e r s t o o d in the f o l l o w i n g w a y Let us c o n s i d e r the Pauli m a t r i c e s o-,. A n y set o f matrices o b e y i n g eq. (A.2) can be r e p r e s e n t e d as

)

(A.7)

In this r e p r e s e n t a t i o n the s o l u t i o n o f eq (A 1) Is e x p r e s s e d m terms o f 2 - c o m p o n e n t IOihng s p m o r s e j ( I = 1, 2), o b e y i n g the w e l l - k n o w n e q u a t i o n I + t.OjktOrjk e±I -~ ± 2___Z ~ ' c~lS± a errs±

( A 8)

as a direct s u m

r/=

( A 9)

F i n a l l y c o n s i d e r the f o l l o w i n g e q u a t i o n s on $3 × $3 V p ~ = A ~ p q r F qrg~ ,

V p , ~ = A t ~p'q'r I~q'r' 7]

* Of course the hmmng case A = 0 gwes R3, R2 × St, R x T2 o r

T3

(A ~0)

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A c o n v e n i e n t r e p r e s e n t a t i o n o f the 8 x 8 F matrices c o n s i d e r e d in these e q u a t i o n s is given by

Fp, = T 5 ® % ,

rp = yp®~2,

(A 11)

where

0

I n this r e p r e s e n t a t i o n , eq (A.10) d e c o m p o s e s into 3 Killing s p i n o r e q u a t i o n s The s o l u t i o n can be written as

where e+ a n d e_ are K i l l i n g spinors o n o n e $3,

1 IA[ = ~ a '

Vpe:~ = +2A,o-pe±,

(A.14)

while e' is a Killing s p i n o r defined o n the other $3

Vp,E'= 2A'ltrp,e'

IA'[ = - "

1

4a'

(A.15) "

Each of these K l l h n g e q u a t i o n s admits 2 i n d e p e n d e n t solutions By c o n s i d e r i n g all possible c o m b i n a t i o n s o f t h e m we o b t a i n the 8 I n d e p e n d e n t s o l u t i o n s ~7~ o f eq (A.10) Note that by a similar type o f a r g u m e n t we can easily see that there are no Killing spinors o n $3 ×$3. However, the vectors built out of the 77 1 a s =

Cp, = ¢l 'Fp,Tb

(A 16)

are K l l h n g vectors o n $3 x $3.

References [1] A H Chamseddlne, Nucl Phys B185 (1981) 403, E Bergshoeff, M de Roo, B de Wltt and P van Nleuwenhmzen, Nucl Phys B195 (1982) 97, (3 F Chaphne and N S Manton, Phys Lett 120B (1983) 105 [2] J H Schwarz, Phys Reports 89 (1982) 223, M B Green, Surveys m high-energy physics 3 (1983) 127, L Bnnk, Superstrsngs, CERN prepnnt TH4006 (1984) [3] D J Gross, J Harvey, E Martmec and R Rohm, Princeton preprmt 0984) [4] M B Green and J H Schwarz, Caltech preprmt CALT-68-1182 (1984) [5] D Z Freedman, G W Gibbons and PC West, Phys Lett 124B (1983) 491 [6] E Calabl, m Algebraic geometry and topology a symposmm m honor of S Lefschetz (Princeton Umverslty Press, 1957) p 78, S T Yau, Proc Natl Acad So 74 (1977)1798, P Candelas, G T Horowltz, A Strommger and E Witten, Caltech preprmt, (1984)

40 [7] [8] [9] [10] [11]

E Sezgm, Ph Spmdel / Compactlficatlons ofsupergrawty

M J Duff, P K Townsend and P van Nleuwenhmzen, Phys Lett 122B (1983) 232 P G O Freund and M A Rubm, Phys Lett 97B (1980) 233 R Jacklw, MIT prepnnt, CTP 1204, (1984) M Pernlcl, K Pflch and P van Nleuwenhmzen, Phys Lett 143B (1984) 103 M J Duff and C N Pope, m Supergravlty 81, ed S Ferrara and J G Taylor (Cambridge Umverslty Press, 1982) [12] J Kowalskl-Gllkman, Phys Lett 139B (1984)159 [13] S Deser and B Zummo, Phys Lett 65B (1976) 369, L Bnnk, P Dl Wecchla and P Howe, Phys Lett 65B (1976) 471 [14] N Marcus and J H Schwarz, Nucl Phys B228 (1983)145