16 March 2000

Physics Letters B 476 Ž2000. 415–419

D s 7 SU ž2 / gauged supergravity from D s 10 supergravity A.H. Chamseddine, W.A. Sabra Center for AdÕanced Mathematical Sciences (CAMS) and Physics Department, American UniÕersity of Beirut, Lebanon Received 4 January 2000; accepted 31 January 2000 Editor: L. Alvarez-Gaume´

Abstract The theory of SUŽ2. gauged seven-dimensional supergravity is obtained by compactifying ten dimensional N s 1 supergravity on the group manifold SUŽ2.. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction Theories of extended supergravity in various dimensions possess rigid symmetries. A subgroup of these symmetries can be gauged by the vector fields present in the theory. For example, in N s 2 supergravity in five dimensions, one can gauge the UŽ1. subgroup of the SUŽ2. rigid symmetry group of the theory w1x. Gauged supergravity theories exist in higher dimensions in which supersymmetry allows the existence of a cosmological constant. It is well known that a cosmological constant is not allowed in d s 11, d s 10 and d s 9. In d s 7, SUŽ2. and SO Ž5. = SO Ž5. gauged supergravity theories were constructed in w2x and w3x respectively. The theories of gauged supergravity theories in d s 4,5,7 are believed to be related to certain compactifications of d s 10,11 supergravity theories. For instance, the four dimensional gauged N s 8 supergravity w4x is conjectured to be related to the compactification of eleven dimensional supergravity on S 7 w5x. However, this conjecture has not been verified yet and usually it is difficult to find a consistent ansatz for compactification on spheres. Toroidal

compactification of ten dimensional supergravity to four dimensions yields an N s 4 supergravity theory with six vector multiplets w6x. The vector and matter fields obtained in four dimensions are in general linear combinations of the internal components of the ten dimensional metric and antisymmetric tensor. The truncation is then performed by identifying the vector components of the ten-dimensional metric with those of the antisymmetric tensor. Another known compactifications of ten dimensional supergravity are due to Scherk and Schwarz w7x, in which the internal compactified space is taken to be a group manifold. The maximal group manifold allowed is S 3 = S 3 and a compactification of this particular case for the dual formulation of supergravity was performed in w8x. The resulting four dimensional theory is an N s 4 supergravity with a non-compact gauge group containing the factor SUŽ2. = SUŽ2.. In order to obtain the Friedman-Schwarz model w9x, a compactification of the Sherck-Schwarz type is needed. This compactification was performed recently in w10x. The crucial point in this analysis is the specific relation between the components of the metric and antisymmetric tensor along the internal dimensions.

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 1 2 9 - 5

A.H. Chamseddine, W.A. Sabrar Physics Letters B 476 (2000) 415–419

416

This was a long standing problem, the main difficulty was finding the right ansatz for the antisymmetric tensor field. It was suggested by the authors of w11x that a more complicated six dimensional internal manifold is needed in order to obtain the Friedman-Schwarz model. This suggestion was motivated by the general result of Friedman et al. w12x that non-trivial compactifications of ten-dimensional supergravity are inconsistent. This proved not to be the case as a basic assumption made in w12x regarding the dilaton field needs to be violated. In recent years, there has been a renewed interest in gauged supergravity theories. This is mainly due to the recently conjectured duality between supergravity and super Yang Mills w13x. Since this conjecture has been made, anti-de Sitter spaces have received a great deal of interest. The purpose of our work here is to demonstrate that d s 7, N s 2, SUŽ2.-gauged supergravity theory of w2x can be obtained via dimensional reduction of N s 1 ten dimensional and eleven dimensional supergravity theories.

2. Dimensional reduction and D s 7 gauged supergravity In this section we will show explicitly how to obtain the seven dimensional gauged supergravity of w2x as a dimensionally reduced ten dimensional N s 1 supergravity theory. The internal space is taken to be the group manifold SUŽ2.. The bosonic part of N s 1 supergravity action in ten dimensions is S10 s eˆ y Rˆ q E M fˆ E Mfˆ 1 4

Hž

y2 fˆ

q 121 e

1 2

 M 4 s  m s 0, . . . ,3; m s 1, . . . ,6 4 ,  A4 s  a s 0, . . . ,3; a s 1, . . . ,6 4 .

/

Ž 1.

Our notations are as follows. Ten-dimensional quantities are denoted by hatted symbols. Base space and tangent space indices are denoted by late and early capital Latin letters, respectively. We first comment briefly on the compactification of ten-dimensional supergravity on S 3 = S 3 to four dimensions.

Ž 2.

The general coordinates xˆ M consist of spacetime coordinates x m and internal coordinates z m. The flat Lorentz metric of the tangent space is chosen to be Žq,y, . . . ,y . with the internal dimensions all spacelike. Thus the metric is related to the vielbein by gˆ M N s hˆ A B eˆ A M eˆ B N s ha b eˆ a

M

eˆ b N y d ab eˆ a M eˆ b N ,

Ž 3. and the antisymmetric tensor field strength is HˆM N P s E M BˆN P q E N BˆP M q E P BˆM N .

Ž 4.

The coordinates z m span the internal compact group space, implying that we have the functions Uma Ž z . satisfying the condition

Ž Uy1 . a m Ž Uy1 . b n Ž Em U c n y EnU c m . s

fab c

'2

,

Ž 5.

Here f ab c are the group structure constants and the internal space volume is V s H < Uma < dz. In the maximal case, i.e., SUŽ2. = SUŽ2., each S 3 factor admits invariant 1-form u a s u iadz i , which satisfies du a q 12 e a b c u b n u c s 0 .

Ž 6.

If one chooses Uma ' Uia s y

HˆM N P Hˆ M N P d 4 x d 6 z

' SGˆ q Sfˆ q SHˆ .

For four-dimensional space-time indices, late and early Greek letters denote base space and tangent space indices, respectively. Similarly, the internal base space and tangent space indices are denoted by late and early Latin letters, respectively.

'2 g

u ia ,

Ž 7.

where g is a coupling constant, then the structure constants will be given in terms of the coupling constant by f a b c s g ´ a b c . For the case where the coupling constant of one of the SUŽ2. factors vanishes, the internal space becomes the group manifold SUŽ2. = wUŽ1.x 3. In order to get a consistent truncation, one must set six vector multiplets to zero. This is done by identifying the six gauge fields coming from the components of the metric with six vector

A.H. Chamseddine, W.A. Sabrar Physics Letters B 476 (2000) 415–419

fields from the components of the antisymmetric tensor. In order to do this identification the vectors coming from the antisymmetric tensor must behave like Yang-Mills gauge fields, and here is the main source of difficulty. It is crucial to have the right ansatz for the antisymmetric tensor to get a consistent truncation. We now turn to the compactification of ten-dimensional supergravity theory down to seven dimensions. The three dimensional internal space is taken to be the SUŽ2. group manifold. We shall show that the obtained theory is the SUŽ2. gauged seven dimensional supergravity theory derived in w2x. Following Scherk and Schwarz w7x, we parameterize the Õielbein in the following form 3

eˆMA s



e

10

fˆ

1 y

0

e

2

fˆ

Uma

Ž z.

0

,

Ž 8.

gˆmn s e

5

Bˆmn s Bmn ,

gmn y 2

Ama

ˆ Ana ey f

Hˆmnr s Hmnr ' Em Bnr q En Brm q Er Bmn , a a a '2 Ž Em An y En Am . Um ,

Hˆm m n s 12 f a b c Ama Umb Unc ,

ˆ

gˆ m n s yUmaUna ey f .

1 2'2

f a b c UmaUnb Upc .

LB s e

ž

16

1 y fˆ 5 12

e

Ž 9.

ˆ ˆ n fˆ LG s y 14 R y 18 ey 5 f Fmna F mn a q 103 g mn Em fE

16

8

fˆ

Ž 10 .

where Fmna s Em Ana y En Ama q f a b c Amb Anc .

Fmna

F

mn a

g2 y 16

8

e5

fˆ

/

,

Ž 15 .

X Hmnr s Hmnr y vmnr ,

vmnr s y6 Ž Awam En Ara x q 13 f a b c Ama Anb Arc . .

Ž 16 .

In deriving equation Ž 15 . care must be taken to include the off-diagonal components of the metric. This can be effectively done by first defining HˆA B C s e AM e BN eCP HˆM N P and then writing 9

ˆ

X Hˆa bg s ey 10 f eam ebn egr Hmnr ,

1

ˆ

Hˆa bg s ey 10 f eam ebn Fmna ,

Ž 11 .

3

Hˆabg s 0 ,

When using this formula, we rescale the gauge fields.

X Hmnr H X mnr

where

8

e5 ,

Ž 14 .

The reduction of SBˆ gives the following

y e

Using Eq. Ž38. of w7x 1 , for the reduction of SGˆ , one obtains the reduced Lagrangian, which reads

3g2

Ž 13 .

where we also require that

,

ˆ

1

1

Hˆmn m s y

1 y fˆ 5 8

gˆm m s y'2 Uma Ama ey f ,

q

Bˆm n s B˜m n ,

Ama Uma ,

'2

where Bmn is a function of x and B˜m n is a function of the internal coordinates z only. The field strengths are given by

8

fˆ

1

Bˆm m s y

Ž 12 .

Hˆm n p s

where we have set k s 1 and rescaled the gauge fields by 1r '2 . Here U depends on the internal coordinates Ž7,8,9. and fˆ s fˆ Ž x . . Our ansatz in terms of the metric components is thus given by 3

For the antisymmetric tensor our ansatz is

Hˆm n p s Em B˜n p q En B˜p m q Ep B˜m n ,

1

'2 ey 2 fˆ Amm Ž z . dma

ema Ž x .

417

Hˆab c s e

2

fˆ

g 2'2

eab c .

The appearance of the Chern-Simons term vmnr in X Hmnr is a crucial test of the consistency of the ansatz.

A.H. Chamseddine, W.A. Sabrar Physics Letters B 476 (2000) 415–419

418

The reduction of the scalar part of ten dimensional supergravity, Sfˆ gives the following contribution to the seven dimensional Lagrangian e ˆ mfˆ . LS s Em fE Ž 17 . 2 Therefore, combining LG , LB and LS , the seven dimensional theory is described by the Lagrangian

ž

8

1 y fˆ 5 4

1 4

L7 s e y R y e

q e

X

Hmnr H

F

X mnr

mn a

4 5

q g

g2 q 8

8

e

5

fˆ

mn

/

ˆ n fˆ Em fE Ž 18 .

4

48'2

e mnrsk lh Fmnrs Fk l i j Ah j i q a 2sy2 ,

Ž 19 . 1

where s s e '5 f . The Lagrangian in Ž 18 . can be seen to be identical to Ž 19 . , after multiplying equation Ž 18 . by an overall factor of 2 and under the following identifications y

fˆ s

'5 4

5

fˆ

f,

Am i s yAma

Ž 20 .

a

Žt .

j i.

Fmnrs F mnrs

/

e mnrsk lh Fmnrs vk lh .

Integrating the last term by parts and using the identity

one obtains the Chern-Simons term 7

Hd x e

mnrsk lh

Fmna Frsa Ak lh

and this is seen to agree with the Lagrangian in Ž 19 . after integrating by parts 2 . We note in passing that the seven dimensional Lagrangian can also be obtained by compactifying eleven dimensional supergravity and then truncating. This can be seen by embedding the ten-dimensional supergravity into eleven-dimensional supergravity after truncating half the degrees of freedom. One has the following identifications for the eleven dimensional fields: 3 1 ˆ e MA s ey 6 f e MA

ˆ ,

gs2a , j

96

16

e

144'2

24'2

Fmn i j F mn i j y 12 Em fE mf

i q

y

1

ey1 L s y 12 R y 481 sy4 Fmnrs F mnrs

s2

ž

e

E w mvnrs x s y 32 Fmna Frsa ,

The N s 2 SUŽ2. gauged d s 7 supergravity which was constructed in w2x is given by

y

H

d7x

1 Fmna

16

1 y fˆ 5 12

two form Bmn . On the other hand, now the Hmnr appears quadratically and linearly in the Lagrangian, the Gaussian integration of Hmnr can be carried out resulting in the terms

e 11 M s0 ,

4 11 e11 ˙ se

fˆ

,

A e11 ˙ s0 ,

Ž 21 .

A M N 11 ˙ s BˆM N ,

Ž 22 .

This identification is important in lifting special solutions from seven to ten and eleven dimensions. We now comment on previous work that appeared in the literature. First in the work of Duff, Townsend and van Niewenhuizen w11x a compactification of ten dimensional supergravity on S 3 was given, but this

The Lagrangian Ž 19 . contains a three-form Amnr instead of the two-form Bmn appearing in Ž 18 . . These forms are, however, related by a duality transformation. To see this we add to the seven dimensional compactified action in Ž 18 . the term

AM NP s 0 .

y 361 d 7 x e mnrsk lh Hmnr Es Ak lh

H

and assume that Hmnr is not the field strength of Bmn but an independent field. The equation of motion of Aklh then implies that Hmnr is the field strength of a

2 The different signs for the kinetic terms as well as the i factor appearing with the epsilon tensor in Ž 19 . are due to the choice of the metric Ž qqqqqqq. in w2x.

A.H. Chamseddine, W.A. Sabrar Physics Letters B 476 (2000) 415–419

breaks all supersymmetries. More recently, Lu and Pope w14x gave an ansatz for the reduction and truncation of eleven dimensional supergravity to seven dimensions. Their ansatz was shown to be consistent by checking that the equations of motion for eleven dimensional supergravity are satisfied. When, however, the ansatz was substituted into the eleven-dimensional action, the Lagrangian Ž Eq. Ž 19. . was not obtained. In this work we have not given the ansatz for the reduction of the fermionic parts. This should be straightforward but tedious. The fact that bosonic part of the supergravity Lagrangian fixes completely the fermionic parts as well, implies that the fermionic terms have also to agree.

Acknowledgements W. Sabra would like to thank N.N. Khuri and his group for hospitality at Rockefeller University where most of this work was done.

419

References w1x E. Cremmer, in; Supergravity and superspace, eds. S. Hawking and M. Rocek, Cambridge University Press Ž1981. w2x P.K. Townsend and P. Van Nieuwenhuizen, Phys. Lett. B 125 Ž1983. 41. w3x M. Pernici, K. Pilch and P. van Nieuwenhuizen, Phys. Lett. B 143 Ž1984. 103. w4x B. de Wit and H. Nicolai, Nucl. Phys. B 208 Ž1982. 323. w5x M. J. Duff, B. E. Nilsson and C. Pope, Phys. Rep. 130 Ž1986. 1. w6x A.H. Chamseddine, Nucl. Phys. B 185 Ž1981. 403. w7x J. Scherk, J.H. Schwarz, Nucl. Phys. B 153 Ž1979. 61. w8x A.H. Chamseddine, Phys. Rev. D 24 Ž1981. 3065. w9x D.Z. Freedman, J.H. Schwarz, Nucl. Phys. B 137 333 Ž1978.. w10x A.H. Chamseddine and M.S. Volkov, Phys. Rev. Lett. 79 Ž1997. 3343; Phys. Rev. D 57 Ž1998. 6242. w11x M.J. Duff, P.K.Townsend and P. van Niewenhuizen, Phys. Lett. B 122 Ž1983. 232. w12x D.Z. Freedman, G.W. Gibbons and P.C. West, Phys. Lett. B 124 Ž1984. 491. w13x J. Maldacena, Adv. Theor. Math. Phys. 2 Ž1998. 231; S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Phys. Lett. 428B Ž1998. 105; E. Witten, Adv. Theor. Math. Phys. 2 Ž1998. 253. w14x H. Lu and C. N. Pope, hep-thr9906168.