Volume 130B, number 5 PHYSICS LETTERS 27 October 1983

COSMOLOGICAL AND COUPLING CONSTANTS IN K A L U Z A - K L E I N SUPERGRAVITY

M.J. DUFF, C.N. POPE and N.P. WARNER 1

Imperial College o f Science and Technology, The Blackett Laboratory, Prince Consort Road, London SW7 2 BZ, UK

Received 23 July 1983

Weinberg's formula, relating the Yang-MiUs coupling constant, g, to the size of the extra dimensions in a Kaluza- Klein theory, requires modification when gravity is coupled to antisymmetric tensor gauge fields• In particular, in the spon- taneous coml~actification of d = 11 supergravity on the round S 7 of radius m -1 , the Weinberg formula g2 = 64~rGm 2 is changed to g* = 16rrGm 2 owing to the AMN P field. Combined with the cosmological constant relation A = -12m 2 one finds 41rGA = -3g 2, which is precisely the relation in the gauged N = 8 supergravity of de Wit and Nicolai.

Four-dimensional supergravity theories obtained from the N = 1 theory in d = 11 dimensions via the Kaluza-Kle in mechanism have recently received a great deal of at tent ion. In particular, if the eleven-di- mensional ground state is taken to be the product of anti de Sitter space with a (round) seven-sphere the resulting four-dimensional theory has N = 8 supersym- metry (since S 7 admits 8 Killing spinors) and an SO(8) gauge symmetry [since the isometry group of S 7 is SO(8)] , and describes a finite number of massless states together with an infinite tower o f massive states. It is an immediate consequence of supersymmetry that the states fall into N = 8 multiplets, and thus the massless multiplet comprises 1 graviton, 8 gravitini,

• 1 28 vector gauge fields, 56 spin ~ and 70 spin 0 fields (35 scalars +35 pseudoscalars) [1]. An explicit linear- ized calculation of the massless mult iplet confirms these expectat ions [ 1,2] but it should be emphasised that the result was already guaranteed by N = 8 super- symmetry and SO(8) invariance.

An entirely different question is whether the theory can be consistently truncated to include only the mass- less multiplet , and if so whether this truncation coin- cides with the N = 8 gauged SO(8) theory of de Wit and Nicolai [3]. It has been conjectured that this is in- deed the case [1]. Verifying this conjecture would re- quire a complete non-linear analysis of the theory,

x Present address: Lauritsen Laboratory, Californian Institute of Technology, Pasadena, CA 91125, USA.

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which has not yet been carried out. However, a crucial necessary condit ion is that the spacetime cosmological constant A must be related to the SO(8) gauge cou- pling constant g by the same formula

4riGA = - 3 g 2 , (1)

as obtains in the de Wit -Nicola i theory [3], where G is Newton's constant. Weinberg has shown how the coupling constant may be calculated when starting from pure Einstein gravity in higher dimensions [4] ; applied to an n-sphere o f radius m -1 the formula gives

g2 = 8rrGm2(n + 1) , (2)

and so for the seven-sphere solution of d = 11 super- gravity, for which A = - 1 2 m 2 [see (6)], one would appear to obtain 167rGA = - 3 g 2, which disagrees with (1) and would thus disprove the conjecture. (One can check that the normalizations o f g in refs. [3] and [4] agree, and so the discrepancy is not merely due to the use of different conventions). However, the d = 11 supergravity lagrangian contains not only the usual Einstein term but also involves an antisymmetric tensor gauge field FMNPQ and it turns out that this leads to a modification of Weinberg's calculation. When this is taken into account, it changes the relation between A and g to give precisely the de Wit -Nicola i relation (1).

The bosonic field equations o f d = 11 supergravity [5], in the notat ion of ref. [1] are

I ~ M N 1 -- _ SRgMN = I (FMPQRFNPQR 1 2 -- ggMN F ) , (3)

0.031-9163/83/0000--0000[$ 03.00 © 1983 North-Holland

Volume 130B, number 5 PHYSICS LETTERS

7 M F MpQR = -s-L7-6 eMI'"Ms PQR FM1. "'M4 FMs...Ms ,

(4)

w h e r e RMN and/~ denote the d --- 11 Ricci tensor and Ricci scalar. These are solved by taking the F re und - Rubin [6] choice for FMNPQ in the ground state, for which all components vanish except

o

Fuupo = 3meuvao , (5)

where rn is a constant, and we have decomposed d = 11 indices M, N, ... as #, u, ... running over space- time and m, n, ... running over the extra seven dimen- sions. Eq. (5) satisfies (4), and substituting into (3) yields for the ground state metric

o o 2o ( 6 , 7 ) Ruv = -12m2~uv = Aguu , Rmn = 6m gmn ,

which we take to be the product of anti de Sitter space with metric gut, satisfying (6) and the maximally sym- metric seven-sphere solution gmn of (7), with radius m -1 . This yields, via the Kaluza-Klein mechanism, an N = 8 supergravity theory in four dimensions. The mass- less modes are obtained at the linearised level by writing the d = 11 fields as their ground state values plus fluctuations, and thus for the bosons

gMN(X, y ) = ~MN(X, y ) + hMN(X, y ) , (8)

FMNPQ(X , y ) = FMNPQ(X, y) + fMNPQ(X, y ) , (9)

where xU are spacetime coordinates and ym are coor- dinates on S 7. From ref. [1 ], the correct ansatz for the bosons in the massless multiplet is +x

t I huv(X, y ) = huv(X), (10)

t hun(X, y ) = ~B [IJ] (x)KnlJ(y) , (11)

f hmn(X, y ) = S [HKL] (x )KrnIJ (y )gnKL(y) , (12)

where we have defined h'MN by

, i o ( 1 3 ) hMN = hMN -- ~ gMN hpP ,

and

- ~m em, oohM M , (14) f uvpa - 3

fuvpq = ( 3 /16m)euvpo V°Vqhm m ' (15)

fuvpq = (1 [4m ) euvpo 7 a B o 1JVpKq 1J , (16 )

,1 In order to coincide with the convent ions of refs. [3 ] and 1

[4 ], we introduce a factor o f ~ in eq. (11), and hence also (16), to compensate for the double counting involved in

IJ IJ a c t writing B~ K rather than BuK (~ = 1 . . . . . 28) .

27 October 1983

funpq = ~auptlJKL ] (x)(VtnKpH)Kq] KL , (17)

fmnpq = P tIJKL] ( x ) ( V t m K n l J ) ( V p K q ]KL) . (18)

In these equations h ' (x) is the graviton field, B[IJ](x) Uu are the 28 gauge fields, S [IJKL] (X) are the 35 scalar fields [self-dual in spinor SO(8) indices HKL] and p[IJgL] (x) are the 35 pseudoscalar fields (anti-self- dual in IJKL). KelJ(y) are the 28 Killing vectors of S 7 .

We wish to focus our attention on the gauge fields BuIJ(x), and so for our purposes eqs. (11) and (16) are the important ones. Eq. (11) is part o f the standard Kaluza-Kiein ansatz, yielding a Yang-Mills term in the effective lagrangian in d = 4 dimensions. However, eq. (16) shows that there is a mixing between the anti- symmetric tensor and the vectors, and this is the crucial term which modifies Weinberg's formula [4]. {Biran et al. [2] have also discussed the ansatze for the massless modes, but they did not obtain the com- plete ansatz for the antisymmetric tensor, and in parti- cular they missed the mixing with the vectors of ' (16).}

Substituting these ansatze into (3), we obtain

1-Rg Ruu - 2 uv 1 I JpFKL 1 ,- ,IJ~KLpo.,r- IJt,,'KLm

- o ( Y . u p - ~ , g u ~ ; ~ o 1~ ~ m ~ + ' "

= ( 1 / 1 6 m 2 ) ( F J J o F K L __ g's#~ao"l. Ic;'IJIz, KLao'~S

× (VmKnlJ)(vmKKLn) + ... . (19)

where

_ 1 t-, I Jo KLoMNt~n '~ F u J J = O uBvlS OrB. Is - r~..KLMN . . u v ,~.vj

1_ r, MNKMN (21) [ KIJ, KKL ] = --2 "IJKL

CIJKL MN are the SO(8) structure constants, and we have omitted all terms involving the other massless fields. The terms involving F / J on the left-hand side of (19) are the usual Yang-Mills terms obtained by starting from pure gravity in higher dimensions, while those on the right-hand side are the extra terms coming from (16). As it stands, (16) includes only the terms in Fur 1J linear B I J, but SO(8) gauge invariance implies that (16)must covariantize to involve the correct non- abelian curvature of (20).

After integration over S 7, the terms on both sides of (19) give standard SO (8) Yang-Mills stress tensors. This follows from

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Volume 130B, number 5 PHYSICS LETTERS 27 October 1983

(KmIJKKLm ) = c2(¢sIK6JL _ 6IL6JK) , (22)

where ( f ) d e n o t e s the S 7 average o f f ( y ) ; i.e. V -1 f ~"gf d7y, where V = f Vtff dYy is the volume of S 7 , and e is a constant which depends upon the norma- lization of the Killing vectors which, following ref. [4], will be chosen to give the canonical Einstein equa- tion for a Yang-Mills field. In the case of the terms on the right-hand side of (19), we also need to perform an integration by parts to give

( (V m K n l J ) ( v m K KLn ) ) = _ ( K m l J I-] K KLm )

= 6 m 2 ( K m l J K KLm ) , (23)

the final expression following from -[-]Km 1J = t~mnKn IJ and (7). Defining the canonical Y a n g - Mills stress tensor by

l tl:;,IJ01 ~, IJ 1 !JJri,IJ,oo~ (24) T u v = - ~ " u --vo - g~uv* p o - - J '

we therefore obtain

1 1 2 _ 3 2 - - - ~ C T . v ( 2 5 ) Ruv - ~Rguv ~c Tuv

and so the Einstein tensor is 2c 2 Tuv, of which ~c 2 Tuv is the standard contr ibut ion from reducing/~ from d

• . 3 2 = 11 to d = 4 dimensions and ~c Tuv is the contribu- t ion from the antisymmetric tensor. Thus in Weinberg's calculation [4] the Yang-Mil ls term in the four-dimen- sional lagrangian should be increased by a factor of 4, and so we must choose c 2 = 4rrG rather than c 2 = 167rG in order to obtain the canonical Einstein equa- tion

1 Ruv - ~Rg , v = 87rGTuv . (26)

Using (2), appropriately rescaled by this factor of 4, or by direct calculation of the structure constants as de- freed by (21), we obtain the relation (,1) be tweeng 2 and A.

Thus the SO(8) gauge coupling constant for the round seven-sphere solution o f d = 11 supergravity is the same as that of the de Wit -Nicola i theory, an ex- act result following only from knowledge of the lin- earized theory together with gauge invariance o f the non-linear theory. Note, however, that although this result is consistent with the conjectured equivalence discussed earlier, a p roof of the complete equivalence of the massless sector of the Ka luza-Kle in theory and the de Wi t -Nico la i theory is still lacking. We emphasise of course, that any such equivalence would in any case disappear once the massive states are included [7].

Note also that the relation (1)obta ins in the N-super- symmetric vacua o f all known gauged N-extended super- gravities and may indeed be a model-independent con- sequence of the de Sitter supersymmetry.

It has also been conjectured [1 ] that non-symmetric extrema of the de Wit -Nicola i effective potential are described by other solutions of the d = 11 theory which deviate from the maximally symmetric geometry but which still have the same S 7 topology. (Not all d = 11 solutions with S 7 topology can correspond to de Wit -Nicola i extrema, however, [7]). A possible equi- valence of this kind [1,2,8] is provided by the SO(7) invariant extremum of ref. [9] and the parallelised S 7 solution of Englert [10], where in both cases only the massless pseudoscalars acquire non-zero vacuum expec- tation values. Once again, however, doubt has been cast on the equality of the cosmological constants [11 ]. Yet until a calculation along the lines described in this paper has been carried out, nothing can be said.

Finally, we emphasise that the modification to Weinberg's formula will need to be considered in all Ka luza-Kle in theories with antisymmetric tensor gauge fields of any rank and not merely in d = 11 supergravity.

[1 ] M.J. Duff and C.N. Pope, in: Supergravity 82, eds. S. Ferrara, J.G. Taylor and P. van Nieuwenhuizen (World Scientific, Singapore, 1983); M.J. Duff, in: Supergravity 81, eds. S. Ferrara and J.G. Taylor (Cambridge U.P., Cambridge); M.J. Duff and D.J. Toms, in: Unification of the funda- mental interactions II, eds. J. Ellis and S. Ferrara (Plenum, New York, 1982); M.J. Duff, Nucl. Phys. B219 (1983) 389; and to be pub- fished in the Proceedings of the Marcel Grossman Meeting (Shanghai, August 1982).

[2] B. Biran, F. Englert, B. de Wit and H. Nicolai, Phys. Lett. 124B (1983) 45.

[3] B. de Wit and H. Nicolai, Phys. Lett. 108B (1982) 285; Nucl. Phys. B208 (1982) 323.

[4] S. Weinberg, Phys. Lett. 125B (1983) 265. [5 ] E. Cremmer, B. Julia and J. Scherk, Phys. Lett. 76B

(1978) 409. [6] P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B (1980)

233. [7] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Rev.

Lett. 50 (1983) 2043. [8] L. CasteUani and N.P. Warner, The relationship between

dimensional reduction and gauged N = 8 supergravity, in preparation.

[9] N.P. Warner, Phys. Lett. 126 B. (1983) 169. [10] F. Englert, Phys. Lett. l19B (1982) 339. [11 ] B. de Wit and H. Nicolai, The paraUelizing S 7 torsion in

gauged N = 8 supergravity, NIKHEF preprint H/83-8.

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