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On the embedding of d = 4, N = 8 gauged supergravity in d = 11, N = 1 supergravity
Volume 155B, number 1,2
PHYSICS LETTERS
16 May 1985
O N T H E E M B E D D I N G O F d = 4, N ffi 8 G A U G E D S U P E R G R A V I T Y IN dffi 11, N = 1 S U P E R G R A V I T Y Bengt E.W. N I L S S O N lnstttute for Theoretwal Phystcs, Unwerstty of Cahfornta, Santa Barbara, CA 93106, USA The Blackett Laboratory, Irnperlal College, London SW7 2BZ, UK and Instttute for Theorettcal Physzcs, Chalmers Unwerstty of Technology, S-412 96 Goteborg, Sweden
Received 21 December 1984
We derive the Kaluza-Kleln ansatz for the vlelbem of eleven-dlmens~onal supergravlty compactlfied on the round seven-sphere to all orders in the 35 massless scalar fields It is thereby demonstrated how d = 4, N = 8 gauged supergravlty emerges from the d = l l , N = I theory m tins sector The relation between d = 4 and d = l l splnor fields is also gwen exphcltly to all orders in scalar fields
The recent efforts to understand the spontaneous compactification o f d = 1 1 , N = 1 supergravity [1] have unveiled the existence o f infimtely many ground state solutions to the field equations of this theory ,1. Besides the Ricci fiat solutions d = 4 Minkowski spacetime times TT,T 3 × K3 [3,4] or the product o f S 1 with ,he complex hypersurface of degree 5 in CP 4 [4], there are infinitely many solutions of the form (ADS)4 × M 7 where (ADS)4 denotes four-dimensional anta-de Sitter spacetime and M 7 denotes an internal seven-dimensional compact (non-Ricci fiat) Einstein space. For each ground state the corresponding effective field theory on (ADS)4 is obtained by performing a generalized Fourier expansion of the d = 1 1 fields in terms of harmonics on the internal space in question. Clearly, the d = 4 theory so obtained will conrain infinitely many fields and it seems natural to ask if the theory can be consistently truncated to a theory with a finite number of fields, preferably keeping as many massless fields as possible. Here "consistent truncation" refers to a truncation o f the d = 4 field content consistent with the field equations in the following sense: Setting some d = 4 fields to zero in the Fourier decomposed d = I 1 field equations should not lead to unacceptable constraints on the remain,1 For a classification of these solutions see ref [2]. 54
ingd --- 4 fields, see ref. [5] for a more detailed discussion of this point. In this reference it was also pointed out that the compactification on the round seven-sphere with its standard metric [6,7] enjoys unique properties and stands out as being particularly interesting. E.g. the round S 7 is the only non-Ricci flat internal space for which the consistent truncation allows us to retain all the spin-1 gauge fields (corresponding to SO(8)) o f the untruncated theory. (This fact is closely related to the existence of eight Killing spinors on the round S 7 [5] .) In particular, when compactifying on an MPqr space [2] the SU(3) × SU(2) gauge fields cannot be kept in the truncated theory [5]. An attempt to reinstate any of these fields would inevitably lead to the reinstatement of infinitely many massive fields [5]. The problem of consistent truncation may also be addressed in the context of supersymmetry transformation laws [8,9] which, in fact, is the approach we will use in this paper. The similarities between the d = 1 1 supergravity theory compactified on (ADS)4 times the round sevensphere with its SO(8) invariant metric and the d = 4, N = 8 gauged supergravity theory of de Wit and Nicolai [10] were early emphasized [6] and it was conjectured [6,7] that the theory obtained by truncation to the d = 4 massless sector is identical to the 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 155B, number 1,2
PHYSICS LETTERS
16 May 1985
de Wit-Nicolai theory. The Kaluza-Klein ansatz in this case has been known for some time but only to linear order in the d = 4 fields [7,11 ]. Since the appearance of the classificatmn of extrema of the de W~t-Nicolai potential with at least an SU(3) symmetry [ 12], new background solutions of the d = 11 theory have been found and solutions now exist [7, 13-16] which seem to correspond to all but one of the d = 4 extrema. The non-linear nature of these results is a strong indication of the correctness of the conjecture. Another indicatmn in this directmn, which is perhaps less striking but nevertheless will prove very useful (see below), is provaded by the demonstration [9] that the truncated linearized supersymmetry transformation rules coincide exactly with the ones of the linearized de Wit-Nicolai theory. This calculation reveals the need for a redefinition of the supersymmetry parameter involvmg scalar and pseudo scalar fields which hints at how to go about generalizing the linear ansatz to all orders. We will take full advantage of this below and derive the ansatz for the d = 11 vlelbein to all orders in the graviton field, SO(8) gauge fields and 35 scalar fields. The full nonlinear redefinition of spmor fields m terms of the 35 scalars is also obtained. We now turn to the derivation of the non-linear ansatze mentioned in the previous paragraph. Due to the lack of space many details of the derivation will have to be left out. These will be published elsewhere [17]. We adopt an approach similar to the one used in refs. [ 18,14] and write the eleven-dimensional vielbein as
where the Fa are the d = 7 Dirac matrices and m is the parameter in the Freund-Rubin ansatz [19,7] :Fuvoa = 3m euvaa and with the rest of the components of FMNPQ set to zero. Da is the round S 7 background covariant derivative. The indices I, J, K .... transform as an 8 s of SO(8) implying that ~/J transform as a 28 of SO(8) (as do the gauge fields Bull ) while the automaticaliy self-dual quantities in (4) and (5) transform in the same way as S HKL under SO(8), i.e. as 35 v [20]. In order to determine the higher order terms in (2) we consider the supersymmetry transformation rule for the elfbem [ t ] , namely
dMA(X,y)
6dMA = --i e r A 6M'
= ( ~ - 1/2(X, y)eua(X )
½BuIJ(x)KblJSba(x,Y)], ~mb(y)sba(x,y)
I
(1) where the d = 11 wond index M and tangent space index A have been decomposed into (/a, m) and (a, a) respectively, with greek indices referring to (ADS)4 with coordinates xU and latin indices to the round S 7 with coordinates y m. ~ma(y)is the siebenbein on the round S 7 so from (1) we see that dma(x,y ) has been designed to have all dependence on the d = 4 scalar fields m Sab(x,y ). A is the determinant of Sab and appears in (1) in such a way as to eliminate scalar
fields in front of the Einstein-Hilbert lagrangian in the d = 4 theory. Furthermore KalJ = rlaJDand by comparison to the previously obtained linearized results [7,9] we see that
1 sHKL, . IJKL 1 o IJKL, Sab = 6ab + 2 [~a,b -- 9 °ab ~c,c ) + "'"
(2)
where terms higher order in the scalar fields S HKL are indicated. Here we use the notation of refs. [7,9], i.e.
(3,4) (Note however the introduction of the comma in (4) and in similar expressxons below.) We will later also need =
(5)
The 7"/I are the eight Killing spinors of the round seven-sphere [7]. They satisfy
D a~?I = ½mPa n I,
(6)
(7)
where the I~A are the d = 11 Dirac matnces while ~m and g denote the d = 11 Rarita-Schwinger field and supersymmetry parameter respectively. We do not glve the corresponding equations for the other fields o f d = 1 1 supergravity since they will not be needed in what follows. We start by imposing the triangular gauge, i.e. we set y) = 0.
(8)
We must then modify the d = 11 supersymmetry transformation rules by the addition of a compensating SO(l, 10)/(SO(I, 3) X SO(7)) local (field-dependent) transformation in order to remain in this 55
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PHYSICS LETTERS
gauge. Unwanted terms are thereby introduced in the transformation rules which, however, can be eliminated by field redefinitions involving ~ and ~m" In order to continue, xnformation about the nonlinear ansatze for the spinonal quantities appearing in (7) must be acquired. This can be done by looking at the linear transformation rules of ref. [9], in particular the redefinition of the supersymmetry parameter. Explicitly, eq. (31) of this reference reads (in our notation)
(9). Also, a factor (-175) -1/2 has been absorbed into el(x), ~bl(x) and xHK(x) to facilitate the comparison with t'*hed = 4 expressions in the de WitNicolal theory. On substituting (1), (11), (12) and (13) into (7) and using the so obtained equation for 8 kma(x,y) in the equation for 6 gua(x,y), one finds by comparison of 6 6 a and the spin-1 transformation rule of the de Wit--Nicolai theory [10] (neglecting pHKL.terms from now on)
~(x,y)
[UIJKL (X ) + oIJKL(x)] r]: L
=
[1 + ~ -aJ K L M :~--A't']a, ~ JKLM ~ J K L M n . a + J'rl-ab,b la)
+ ~ i3,5PSKLM(--3~,c/~pabc)]
eI(x)~I(y), (9)
where the pseudoscalars pIJKL multiply the totally antisymmetric anti-selfdual quantity r//bJ,KL = ~/[IF[ab
.:oKr ] ]-.
(11)
~u(X,y) = A-I/4[Hl(x,y) g/lu(x) + 1 ,),pT5 fflIJK(x, y) X IJK (X)],
(12)
(13)
where u, v and S IJKL are treated as 28 × 28 matrices. Clearly, from (17) we get
(u + V)HKL = exp (--4sIJKL).
(18)
The main result of this paper is that in the gauge (17) the y-dependence of the LHS and RHS of eq. (16) can indeed be shown to match for arbitrary x-dependent fields S IJKL . This is possible even though the RHS of (16) corresponds to an infinite series in powers IJKL • of S IJKL ~a[JKL b and S IJKL %b b while the y-depenz dence of the LHS is entirely within the factor 7/aIJ . The crucial Fierz-identity needed to establish this result is
IJ whererltaJb=~lFabrlJand6MN
= ! ( ~ I ,~J _
2~VMVN
(19) J I 8M6N).
[MN.,PQ] _ (1/48m)6ab rleMdNPQDc I r//J. IJ '~a + = t,,s .,MNPQ 'la, b
(20) HIaJK=H[IHJFa HK] ,
(14,15)
and I/lIJK is the F-trace ofHlaJK. Here U(S, P) is an element of SU(8)/SO(7) the form of which, to first order in S IJKL and p I J K L , is of course dictated by 56
= exp - 4 S IJKL
Using (6), we may write (19) as
where
HI=u(s,P)rl I,
(16)
(See ref. [18] for an analogous calculation.) In the symmetric gauge [10] the functions u(x) and v(x) are given by
486 [MNrlPQ]+ = 6rlMNPQ rlI J - rl~I~.NPQrl~aJ, IJ = a,b bc, c
~ m ( X , y ) = A1/a~mb Sb a
X [ItalJK(x,y ) + ~FaI[IIJK(x,y)IxIJK(x),
rIJ.
(10)
It is interesting to note that ~71J, , without antisymGD,KL C metnzation on a, b and c and duality condition on [IJKL] imposed on it, can be expressed in terms of the 7?-symbols (5) and (10). From (9) it seems natural to conclude that the redef'mition of 6 is given by a chiral SU(8)/SO(7) transformation (the second and third terms) together with a multiplicative factor (the first term) which to this order is A -1/4. In fact, the work of ref. [18] shows that this is indeed the correct conclusion. It is also pointed out in ref. [18] that this SU(8) transformation is acting uniformly on all spinor fields. Therefore we write
6(x,y) = A-1/4Hl(x,y)el(x),
= A - 1/2 S - 1ab ~7I U T(S) P b U ( S )
PQ]+ It then follows that in the expressions 6 [MN r.r 6,~_ .. LK,~ X [MN PQ]+ [ v x YZ]+ , riaTU] ,~ 13" 6~Rg ~ TU~ ~a ,etc. the8 scan be removed by using (20) Jteratave. Thus, e.g. the second order (in S IJKL) t e r m on the LHS of (16) can be written +
.1
L
.
j+
•
~(1 ~ ) 2 sIJKL sKLMNrIMaa N
to eq. (26). I.e. if we set A=-{
1 qMNPQ qRSTU [ I .,414NPQ = ~~ ~ t--~ Ua,b
(28,29) in (27) we obtain our main result, namely that (16) (in the gauge (17)) is true to all orders in s I J K L ( x ) if
+ (1/12m)6ab rfffefeQD d]
A - 1/2 S - l a b = exp [-- ½S (1) ~(1)
x
IJ
RS'~aTU
(21)
(u " ~lJ KL ~- o) KL rla = exp r~MNPQ [ ~_~MNPQ
t 2 'la, b
+ (1/12rn)fab rlMN,dPQ Dc] }'r/bJ.
(22)
We now turn to a discussion o f the RHS o f (16), and start by evaluating the factor ~'IIUTFa U'rlJ. At this stage this cannot be done completely since U(S) is not fully known. However, we have seen that U(S) should be an element of SO(8)/SO(7) so we put
e
.
l
-IJKL X
= xp I.~ b
IJKLp , abrlbc, c a),
(23)
where Xab is an as yet unknown function of S IJKL which must satisfy Xab I o = lab" It is very gratifying to note that (modulo aS~eld-dependent SO(7) tangent space rotation) T
J_
7? U Fa Url - e x p [ ( 1 / 1 2 m ) f a b S
IJ rlcd, dDc]rlb ,
(1) (1)
(24) provided we choose 1 c(l)t'.,(1) Xab = ~ab -- z~ ~, ~.,la,b
_
'la,b'
~ a b ~I~ ) + ....
+ I c(1)c(2)t~(1) .,(2)
~,
Repeating this for the other terms on the LHS of (I 6) we find
U(S)
~"
'la, b
RSTU +(1/12m)6bc4gS, TUDf]rll/. [-{%,c
L~
~(1),~(1)
B = ( I / 1 2 m ) S a b S(1)*'(1) D '~cd, d c'
- 1 i" 4 ~2 sMNPQ sRSTU6MN~)PQ ,~
- "2~-- "
-I
16 May 1985
PHYSICS LETTERS
Volume 155B, number 1,2
(25)
We have here introduced the index notation [lnJnKnLn] = (n). By means of (22) and (24), (16) becomes exp (S(1) [_ _~r/a, 0(1) + (1/12m)6ab~ld),dDe]}rlIbJ
= A - 1 / 2 S - l a b exp [(1/12m)6abS(1)Tled, dLPelrl i-- , IJ, (26) from which we can extract Sab. Here we will just derive A - 1 / 2 S - l a b by applying the following form o f the Campbell-Baker-Hausdorff formula
eA+B = e A - [A,B] [2- [A,[A ,B] ] ] 12- [B,[B,A] ] /12+... e B (27)
~,
,,(1)
,,(2) ~ + ...]
~.~tac,c,lbd, d -- 6ab Ucd, dUce, e j
(30) (the higher order terms are easily obtained from ( 2 7 ) (29)). Thus Sab is an element of GL(7, R)/SO(7). Note that to first order in scalar fields we obtain exactly eq. (2), which is the correct linearized expression of refs. [7,9]. (2) can also be seen to follow from the form of Sab [14] of the particular SO(7) invariant solution o f d = 11 supergravity that corresponds to the SO(7) + invariant extremum [12] of the potential in the de Wit-Nicolai theory. To summarize, we have seen that by utilizing the form of the 56-bein of the de Wit-Nicolai theory in the symmetric SU(8) gauge [10] it is possible to deduce how the scalar fields S 1JKL enter into the d = 11 vlelbein to all orders. In the process of doing so the non-linear form of the compensating SO(8) transformation that must be applied to all d = 11 spinors in order to define their d = 4 counterparts is also obtained. Regarding the pseudo scalars p[HKL]_ we point out that by relaxing the self-duality condition in (19) a further term will appear on the RHS corresponding to the presence of pseudo scalars. We would also like to mention the obvious fact that once the embedding of the de Wit-Nicolai theory in d = 11, N = 1 supergravity is obtained explicitly to all orders the one to one correspondence between extrema of the de Wit-Nicolai potential and solutions o f d = 11 supergravity which are topologically (ADS)4 X S 7 is also established. The extrema discovered in the d = 4 theory can then be used to obtain missing S 7 solutions of the d = 11 theory and vice versa. Finally we note an interesting property of the Fierz equation (19), namely that by combining two products of a 35 v and a 28 of SO(8) only a 28 results although in general [21] 35 v ® 28 = 28 • 35 v • 350 • 567 v. This fact is just another reflexion of the possibility of writing the ansatz entirely in terms of Killing spinors [7,11 ].
57
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We are grateful to M.J. D u f f and C.N. Pope for m a n y useful discussions.
References
[8] [9] [10]
[1] E. Cremmer, B. Julia and J Scherk, Phys Lett 76B (1978) 469. [2] L. Castellam, L.J. Romans and N P. Warner, Nucl. Phys. B241 (1984) 429; M.J. Duff, B.E.W. Nllsson and C N. Pope, Kaluza-Klein Supergravity, Phys. Rep., to be pubhshed. [3] M J. Duff, B E.W. Nllsson and C.N. Pope, Phys. Lett 129B (1983) 39. [4] S -T. Yau, Proc Nat Acad. ScL USA 74 (1977) 1798 [5 ] M.J. Duff, B E.W. Nilsson, C.N. Pole and N.P. Warner, Phys Lett. 149B (1984) 90. [6] M.J. Duff, Nucl. Phys. B219 (1983) 389;in Proc. Marcel Grossman Meeting on General Relatwity (Shanghai, 1982), ed. Hu Ning (Science Press, and North-HoUand, Amsterdam, 1983), M.J. Duff and D.J. Toms, m Unification of the fundamental interactions (II), eds. J. Elhs and S. Ferrara (Plenum, New York, 1982). [7] M.J. Duff and C.N. Pope, m: Supersymmetry and super-
58
[ 11] [12] [13] [14] [15]
[16] [17] [18] [19] [20]
[21]
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gravity 82, eds S. Ferrara, J G. Taylor and P van Nieuwenhuizen (World Scxentlfic, Singapore, 1983) B. de Wit and H Nlcolal, Nucl. Phys B231 (1984) 506 M.A. Awada, B.E.W. Nilsson and C.N Pope, Phys. Rev D29 (1984) 334 B deWlt and H Nlcolai, Phys. Lett. 108B (1982) 285; Nucl. Phys B208 (1982) 323. B. Btran, F. Englert, B. de Wit and H Nlcolai, Phys Lett 124B(1983) 45. N Warner, Phys Lett 128B (1983) 169. F Englert, Phys. Lett. l19B (1982) 339 B. deWit and H. Nlcolal, Phys. Lett. 148B (1984) 60. B. de Wit, H Nicolal and N.P. Warner, The embedding of gauged N = 8 supergravity into d = 11 supergravlty, CERN prepnnt-TH. 4052/84 C.N Pope and N P Warner, Phys Lett. 150B (1985) 352. B.E.W. Nllsson, m preparation. B. deWlt and H Nlcolal, Nucl. Phys B243 (1984) 91. P.G O Freund and M.A. Rubm, Phys Lett. 97B (1980) 233. M.J. Duff, B.E.W. Nllsson and C N. Pope, Phys. Rev. Lett. 50 (1983) 2043, 51 (1983)846 (E); Nucl. Phys. B233 (1984) 433 R Slansky, Phys. Rep 79(1981)1.
PHYSICS LETTERS
16 May 1985
O N T H E E M B E D D I N G O F d = 4, N ffi 8 G A U G E D S U P E R G R A V I T Y IN dffi 11, N = 1 S U P E R G R A V I T Y Bengt E.W. N I L S S O N lnstttute for Theoretwal Phystcs, Unwerstty of Cahfornta, Santa Barbara, CA 93106, USA The Blackett Laboratory, Irnperlal College, London SW7 2BZ, UK and Instttute for Theorettcal Physzcs, Chalmers Unwerstty of Technology, S-412 96 Goteborg, Sweden
Received 21 December 1984
We derive the Kaluza-Kleln ansatz for the vlelbem of eleven-dlmens~onal supergravlty compactlfied on the round seven-sphere to all orders in the 35 massless scalar fields It is thereby demonstrated how d = 4, N = 8 gauged supergravlty emerges from the d = l l , N = I theory m tins sector The relation between d = 4 and d = l l splnor fields is also gwen exphcltly to all orders in scalar fields
The recent efforts to understand the spontaneous compactification o f d = 1 1 , N = 1 supergravity [1] have unveiled the existence o f infimtely many ground state solutions to the field equations of this theory ,1. Besides the Ricci fiat solutions d = 4 Minkowski spacetime times TT,T 3 × K3 [3,4] or the product o f S 1 with ,he complex hypersurface of degree 5 in CP 4 [4], there are infinitely many solutions of the form (ADS)4 × M 7 where (ADS)4 denotes four-dimensional anta-de Sitter spacetime and M 7 denotes an internal seven-dimensional compact (non-Ricci fiat) Einstein space. For each ground state the corresponding effective field theory on (ADS)4 is obtained by performing a generalized Fourier expansion of the d = 1 1 fields in terms of harmonics on the internal space in question. Clearly, the d = 4 theory so obtained will conrain infinitely many fields and it seems natural to ask if the theory can be consistently truncated to a theory with a finite number of fields, preferably keeping as many massless fields as possible. Here "consistent truncation" refers to a truncation o f the d = 4 field content consistent with the field equations in the following sense: Setting some d = 4 fields to zero in the Fourier decomposed d = I 1 field equations should not lead to unacceptable constraints on the remain,1 For a classification of these solutions see ref [2]. 54
ingd --- 4 fields, see ref. [5] for a more detailed discussion of this point. In this reference it was also pointed out that the compactification on the round seven-sphere with its standard metric [6,7] enjoys unique properties and stands out as being particularly interesting. E.g. the round S 7 is the only non-Ricci flat internal space for which the consistent truncation allows us to retain all the spin-1 gauge fields (corresponding to SO(8)) o f the untruncated theory. (This fact is closely related to the existence of eight Killing spinors on the round S 7 [5] .) In particular, when compactifying on an MPqr space [2] the SU(3) × SU(2) gauge fields cannot be kept in the truncated theory [5]. An attempt to reinstate any of these fields would inevitably lead to the reinstatement of infinitely many massive fields [5]. The problem of consistent truncation may also be addressed in the context of supersymmetry transformation laws [8,9] which, in fact, is the approach we will use in this paper. The similarities between the d = 1 1 supergravity theory compactified on (ADS)4 times the round sevensphere with its SO(8) invariant metric and the d = 4, N = 8 gauged supergravity theory of de Wit and Nicolai [10] were early emphasized [6] and it was conjectured [6,7] that the theory obtained by truncation to the d = 4 massless sector is identical to the 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 155B, number 1,2
PHYSICS LETTERS
16 May 1985
de Wit-Nicolai theory. The Kaluza-Klein ansatz in this case has been known for some time but only to linear order in the d = 4 fields [7,11 ]. Since the appearance of the classificatmn of extrema of the de W~t-Nicolai potential with at least an SU(3) symmetry [ 12], new background solutions of the d = 11 theory have been found and solutions now exist [7, 13-16] which seem to correspond to all but one of the d = 4 extrema. The non-linear nature of these results is a strong indication of the correctness of the conjecture. Another indicatmn in this directmn, which is perhaps less striking but nevertheless will prove very useful (see below), is provaded by the demonstration [9] that the truncated linearized supersymmetry transformation rules coincide exactly with the ones of the linearized de Wit-Nicolai theory. This calculation reveals the need for a redefinition of the supersymmetry parameter involvmg scalar and pseudo scalar fields which hints at how to go about generalizing the linear ansatz to all orders. We will take full advantage of this below and derive the ansatz for the d = 11 vlelbein to all orders in the graviton field, SO(8) gauge fields and 35 scalar fields. The full nonlinear redefinition of spmor fields m terms of the 35 scalars is also obtained. We now turn to the derivation of the non-linear ansatze mentioned in the previous paragraph. Due to the lack of space many details of the derivation will have to be left out. These will be published elsewhere [17]. We adopt an approach similar to the one used in refs. [ 18,14] and write the eleven-dimensional vielbein as
where the Fa are the d = 7 Dirac matrices and m is the parameter in the Freund-Rubin ansatz [19,7] :Fuvoa = 3m euvaa and with the rest of the components of FMNPQ set to zero. Da is the round S 7 background covariant derivative. The indices I, J, K .... transform as an 8 s of SO(8) implying that ~/J transform as a 28 of SO(8) (as do the gauge fields Bull ) while the automaticaliy self-dual quantities in (4) and (5) transform in the same way as S HKL under SO(8), i.e. as 35 v [20]. In order to determine the higher order terms in (2) we consider the supersymmetry transformation rule for the elfbem [ t ] , namely
dMA(X,y)
6dMA = --i e r A 6M'
= ( ~ - 1/2(X, y)eua(X )
½BuIJ(x)KblJSba(x,Y)], ~mb(y)sba(x,y)
I
(1) where the d = 11 wond index M and tangent space index A have been decomposed into (/a, m) and (a, a) respectively, with greek indices referring to (ADS)4 with coordinates xU and latin indices to the round S 7 with coordinates y m. ~ma(y)is the siebenbein on the round S 7 so from (1) we see that dma(x,y ) has been designed to have all dependence on the d = 4 scalar fields m Sab(x,y ). A is the determinant of Sab and appears in (1) in such a way as to eliminate scalar
fields in front of the Einstein-Hilbert lagrangian in the d = 4 theory. Furthermore KalJ = rlaJDand by comparison to the previously obtained linearized results [7,9] we see that
1 sHKL, . IJKL 1 o IJKL, Sab = 6ab + 2 [~a,b -- 9 °ab ~c,c ) + "'"
(2)
where terms higher order in the scalar fields S HKL are indicated. Here we use the notation of refs. [7,9], i.e.
(3,4) (Note however the introduction of the comma in (4) and in similar expressxons below.) We will later also need =
(5)
The 7"/I are the eight Killing spinors of the round seven-sphere [7]. They satisfy
D a~?I = ½mPa n I,
(6)
(7)
where the I~A are the d = 11 Dirac matnces while ~m and g denote the d = 11 Rarita-Schwinger field and supersymmetry parameter respectively. We do not glve the corresponding equations for the other fields o f d = 1 1 supergravity since they will not be needed in what follows. We start by imposing the triangular gauge, i.e. we set y) = 0.
(8)
We must then modify the d = 11 supersymmetry transformation rules by the addition of a compensating SO(l, 10)/(SO(I, 3) X SO(7)) local (field-dependent) transformation in order to remain in this 55
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gauge. Unwanted terms are thereby introduced in the transformation rules which, however, can be eliminated by field redefinitions involving ~ and ~m" In order to continue, xnformation about the nonlinear ansatze for the spinonal quantities appearing in (7) must be acquired. This can be done by looking at the linear transformation rules of ref. [9], in particular the redefinition of the supersymmetry parameter. Explicitly, eq. (31) of this reference reads (in our notation)
(9). Also, a factor (-175) -1/2 has been absorbed into el(x), ~bl(x) and xHK(x) to facilitate the comparison with t'*hed = 4 expressions in the de WitNicolal theory. On substituting (1), (11), (12) and (13) into (7) and using the so obtained equation for 8 kma(x,y) in the equation for 6 gua(x,y), one finds by comparison of 6 6 a and the spin-1 transformation rule of the de Wit--Nicolai theory [10] (neglecting pHKL.terms from now on)
~(x,y)
[UIJKL (X ) + oIJKL(x)] r]: L
=
[1 + ~ -aJ K L M :~--A't']a, ~ JKLM ~ J K L M n . a + J'rl-ab,b la)
+ ~ i3,5PSKLM(--3~,c/~pabc)]
eI(x)~I(y), (9)
where the pseudoscalars pIJKL multiply the totally antisymmetric anti-selfdual quantity r//bJ,KL = ~/[IF[ab
.:oKr ] ]-.
(11)
~u(X,y) = A-I/4[Hl(x,y) g/lu(x) + 1 ,),pT5 fflIJK(x, y) X IJK (X)],
(12)
(13)
where u, v and S IJKL are treated as 28 × 28 matrices. Clearly, from (17) we get
(u + V)HKL = exp (--4sIJKL).
(18)
The main result of this paper is that in the gauge (17) the y-dependence of the LHS and RHS of eq. (16) can indeed be shown to match for arbitrary x-dependent fields S IJKL . This is possible even though the RHS of (16) corresponds to an infinite series in powers IJKL • of S IJKL ~a[JKL b and S IJKL %b b while the y-depenz dence of the LHS is entirely within the factor 7/aIJ . The crucial Fierz-identity needed to establish this result is
IJ whererltaJb=~lFabrlJand6MN
= ! ( ~ I ,~J _
2~VMVN
(19) J I 8M6N).
[MN.,PQ] _ (1/48m)6ab rleMdNPQDc I r//J. IJ '~a + = t,,s .,MNPQ 'la, b
(20) HIaJK=H[IHJFa HK] ,
(14,15)
and I/lIJK is the F-trace ofHlaJK. Here U(S, P) is an element of SU(8)/SO(7) the form of which, to first order in S IJKL and p I J K L , is of course dictated by 56
= exp - 4 S IJKL
Using (6), we may write (19) as
where
HI=u(s,P)rl I,
(16)
(See ref. [18] for an analogous calculation.) In the symmetric gauge [10] the functions u(x) and v(x) are given by
486 [MNrlPQ]+ = 6rlMNPQ rlI J - rl~I~.NPQrl~aJ, IJ = a,b bc, c
~ m ( X , y ) = A1/a~mb Sb a
X [ItalJK(x,y ) + ~FaI[IIJK(x,y)IxIJK(x),
rIJ.
(10)
It is interesting to note that ~71J, , without antisymGD,KL C metnzation on a, b and c and duality condition on [IJKL] imposed on it, can be expressed in terms of the 7?-symbols (5) and (10). From (9) it seems natural to conclude that the redef'mition of 6 is given by a chiral SU(8)/SO(7) transformation (the second and third terms) together with a multiplicative factor (the first term) which to this order is A -1/4. In fact, the work of ref. [18] shows that this is indeed the correct conclusion. It is also pointed out in ref. [18] that this SU(8) transformation is acting uniformly on all spinor fields. Therefore we write
6(x,y) = A-1/4Hl(x,y)el(x),
= A - 1/2 S - 1ab ~7I U T(S) P b U ( S )
PQ]+ It then follows that in the expressions 6 [MN r.r 6,~_ .. LK,~ X [MN PQ]+ [ v x YZ]+ , riaTU] ,~ 13" 6~Rg ~ TU~ ~a ,etc. the8 scan be removed by using (20) Jteratave. Thus, e.g. the second order (in S IJKL) t e r m on the LHS of (16) can be written +
.1
L
.
j+
•
~(1 ~ ) 2 sIJKL sKLMNrIMaa N
to eq. (26). I.e. if we set A=-{
1 qMNPQ qRSTU [ I .,414NPQ = ~~ ~ t--~ Ua,b
(28,29) in (27) we obtain our main result, namely that (16) (in the gauge (17)) is true to all orders in s I J K L ( x ) if
+ (1/12m)6ab rfffefeQD d]
A - 1/2 S - l a b = exp [-- ½S (1) ~(1)
x
IJ
RS'~aTU
(21)
(u " ~lJ KL ~- o) KL rla = exp r~MNPQ [ ~_~MNPQ
t 2 'la, b
+ (1/12rn)fab rlMN,dPQ Dc] }'r/bJ.
(22)
We now turn to a discussion o f the RHS o f (16), and start by evaluating the factor ~'IIUTFa U'rlJ. At this stage this cannot be done completely since U(S) is not fully known. However, we have seen that U(S) should be an element of SO(8)/SO(7) so we put
e
.
l
-IJKL X
= xp I.~ b
IJKLp , abrlbc, c a),
(23)
where Xab is an as yet unknown function of S IJKL which must satisfy Xab I o = lab" It is very gratifying to note that (modulo aS~eld-dependent SO(7) tangent space rotation) T
J_
7? U Fa Url - e x p [ ( 1 / 1 2 m ) f a b S
IJ rlcd, dDc]rlb ,
(1) (1)
(24) provided we choose 1 c(l)t'.,(1) Xab = ~ab -- z~ ~, ~.,la,b
_
'la,b'
~ a b ~I~ ) + ....
+ I c(1)c(2)t~(1) .,(2)
~,
Repeating this for the other terms on the LHS of (I 6) we find
U(S)
~"
'la, b
RSTU +(1/12m)6bc4gS, TUDf]rll/. [-{%,c
L~
~(1),~(1)
B = ( I / 1 2 m ) S a b S(1)*'(1) D '~cd, d c'
- 1 i" 4 ~2 sMNPQ sRSTU6MN~)PQ ,~
- "2~-- "
-I
16 May 1985
PHYSICS LETTERS
Volume 155B, number 1,2
(25)
We have here introduced the index notation [lnJnKnLn] = (n). By means of (22) and (24), (16) becomes exp (S(1) [_ _~r/a, 0(1) + (1/12m)6ab~ld),dDe]}rlIbJ
= A - 1 / 2 S - l a b exp [(1/12m)6abS(1)Tled, dLPelrl i-- , IJ, (26) from which we can extract Sab. Here we will just derive A - 1 / 2 S - l a b by applying the following form o f the Campbell-Baker-Hausdorff formula
eA+B = e A - [A,B] [2- [A,[A ,B] ] ] 12- [B,[B,A] ] /12+... e B (27)
~,
,,(1)
,,(2) ~ + ...]
~.~tac,c,lbd, d -- 6ab Ucd, dUce, e j
(30) (the higher order terms are easily obtained from ( 2 7 ) (29)). Thus Sab is an element of GL(7, R)/SO(7). Note that to first order in scalar fields we obtain exactly eq. (2), which is the correct linearized expression of refs. [7,9]. (2) can also be seen to follow from the form of Sab [14] of the particular SO(7) invariant solution o f d = 11 supergravity that corresponds to the SO(7) + invariant extremum [12] of the potential in the de Wit-Nicolai theory. To summarize, we have seen that by utilizing the form of the 56-bein of the de Wit-Nicolai theory in the symmetric SU(8) gauge [10] it is possible to deduce how the scalar fields S 1JKL enter into the d = 11 vlelbein to all orders. In the process of doing so the non-linear form of the compensating SO(8) transformation that must be applied to all d = 11 spinors in order to define their d = 4 counterparts is also obtained. Regarding the pseudo scalars p[HKL]_ we point out that by relaxing the self-duality condition in (19) a further term will appear on the RHS corresponding to the presence of pseudo scalars. We would also like to mention the obvious fact that once the embedding of the de Wit-Nicolai theory in d = 11, N = 1 supergravity is obtained explicitly to all orders the one to one correspondence between extrema of the de Wit-Nicolai potential and solutions o f d = 11 supergravity which are topologically (ADS)4 X S 7 is also established. The extrema discovered in the d = 4 theory can then be used to obtain missing S 7 solutions of the d = 11 theory and vice versa. Finally we note an interesting property of the Fierz equation (19), namely that by combining two products of a 35 v and a 28 of SO(8) only a 28 results although in general [21] 35 v ® 28 = 28 • 35 v • 350 • 567 v. This fact is just another reflexion of the possibility of writing the ansatz entirely in terms of Killing spinors [7,11 ].
57
Volume 155B, number 1,2
PHYSICS LETTERS
We are grateful to M.J. D u f f and C.N. Pope for m a n y useful discussions.
References
[8] [9] [10]
[1] E. Cremmer, B. Julia and J Scherk, Phys Lett 76B (1978) 469. [2] L. Castellam, L.J. Romans and N P. Warner, Nucl. Phys. B241 (1984) 429; M.J. Duff, B.E.W. Nllsson and C N. Pope, Kaluza-Klein Supergravity, Phys. Rep., to be pubhshed. [3] M J. Duff, B E.W. Nllsson and C.N. Pope, Phys. Lett 129B (1983) 39. [4] S -T. Yau, Proc Nat Acad. ScL USA 74 (1977) 1798 [5 ] M.J. Duff, B E.W. Nilsson, C.N. Pole and N.P. Warner, Phys Lett. 149B (1984) 90. [6] M.J. Duff, Nucl. Phys. B219 (1983) 389;in Proc. Marcel Grossman Meeting on General Relatwity (Shanghai, 1982), ed. Hu Ning (Science Press, and North-HoUand, Amsterdam, 1983), M.J. Duff and D.J. Toms, m Unification of the fundamental interactions (II), eds. J. Elhs and S. Ferrara (Plenum, New York, 1982). [7] M.J. Duff and C.N. Pope, m: Supersymmetry and super-
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16 May 1985
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