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The potentials of the gauged N = 8 supergravity theories
Nuclear Physics B253 (1985) 675-686 © North-Holland Publishing Company
THE POTENTIALS OF THE GAUGED N = 8 SUPERGRAVITY THEORIES C.M. HULL Department of Mathematics, MIT, Cambridge, MA 02139, USA N.P. WARNER* California Institute of Teehnology, Pasadena, CA 91125, USA
Received 16 October 1984
The potentials of the SO(p, q) gaugings of N = 8 supergravity are investigated for critical points. The SO(7,1) gauging has no GE-invariantcritical points, the SO(6,2) theory has no SU(3) invariant critical points and the SO(5,3) gauging has only one SO(5)-invariant critical point, with positive cosmologicalconstant, SO(5)X SO(3) symmetry and no supersymmetry.
1. Introduction
In [1], a simple method was given for finding all critical points of the scalar potential P(q0) of the N = 8 supergravity [2] with SO(8) gauge symmetry [3] that break the local SO(8) down to a solution with symmetry that is at least some specified subgroup, H, of SO(8). One considers only those scalars qon, which are singlets of H, and seeks critical points of the potential P(CpH) restricted to be a function only of the H-singlets. Schurr's lemma guarantees that any critical point of P(CPH) will be a critical point of the complete potential P(qg) [1]. The problem of finding critical points of the potential P(~0) is then reduced to the simpler one of finding critical points of the potential restricted to the H-singlet sector, P(q0H). In this p a p e r we apply similar techniques to the gauged SO(p, q) supergravities ( p + q = 8) [4, 5]. In ref. [1] the group H was chosen to be SU(3) for the SO(8) model, while in [7] it was taken to be S O ( p ) x S O ( q ) for the S O ( p , q ) and C S O ( p , q) gaugings. Here we will take H to be G 2 for the SO(7,1) gauging, SU(3) for the SO(6, 2) gauging, and SO(5) for SO(5, 3) gauging. Note that the group H is a compact subgroup of SO(p, q). This is necessary for the Schurr's lemma argument to be valid. * Work supported in part by the US Department of Energy under contract no. DE-AC-03-81ER40050 and by the Weingart Foundation. 675
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In the gauged supergravity theories the scalars lie in the coset space E7/SU(8 ) and it is convenient to represent them with a 56 × 56 E 7 matrix, the 56-bein, ~f [2-6]. Indeed, in this paper we adopt the usual SU(8) basis for the Lie algebra, and take [2,3]
[ AtffSjj tl - ¼v~epijk,] ~(= exp _ ¼v~epijkt - a [itk~Atl ] = ( UijIJ ~klIJ
UijKL ) ~lklgL ]
(1.1)
(1.2)
o
The potential, P, for the SO(8) gauged theory is a sixth-order polynomial in the submatrices of eq. (1.2), and is given in ref. [3]. The new potential P', for an S O ( p , q ) gauging may be obtained from P by analytic continuation [4-7]. Define
where X,jkt is the unique S O ( p ) × SO(q) invariant real, self-dual, totally antisymmetric tensor field. (It is unique up to normalization.) Then the potential, P', is given by
P,(V) =
'(t))]
(1.4)
where a and a are real constants, determined by p, q and the normalization of X. Expressions for Xijkl for each S O ( p , q ) gauging may be found in refs. [5,6]. The constant a is taken to be - 1 and a = 1 + p/q. The problem is how to calculate P'(cT). In ref. [8] it was shown how one could obtain explicit and usable expressions for P ( ~ ) , and in ref. [6] it was shown how to modify these expressions to obtain P'(CV). In this work we take a slightly different course since we will consider only those scalar fields that are singlets of some convenient compact subgroup H of SO(p, q). Sect. 2 describes how to calculate the necessary Baker-Hausdorff formulae in order to calculate Y'(t), such that c~' = ~ E - l ( t )
=exp(Y)E-l(t) = exp(Y'(t)), for a particular class of elements, Y, of the Lie algebra of E 7.
(1.5)
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In sect. 3 we use the results of refs. [1, 8,9] to calculate an explicit form for p,(c~-)= P(cV') on the H-singlet subspaces for the SO(p,q) gaugings. Sect. 4 contains a detailed analysis of the G2-invariant sector of the potential for the SO(7,1) gauging, and finally, in sect. 5 we state the results of a similar analysis of the other potentials. 2. Baker-Hausdorff Formulae
The only non-trivial Baker-Hausdorff formulae that we will need are for particular SI(1,1) subgroups of E7(+7). To obtain the subgroup relevant for the SO(7,1) gauging, let X,~kt and X~k t be the unique (up to scaling), completely anti-symmetric self-dual and anti-self-dual tensors which are invariant under SO(7) ÷ and SO(7)-, respectively. Explicitly, define + = {c 1234_+ 8,,k, 5678 ) + x, Tk, \Vijkl
( R 1256
+
R 3478 ]
\ ° i j k l -- ~ijkl ]
+
(t~ 1278 -1- R3456]
\Vijkl
'~ ~ijkl ]
_( Vijkl ,357
,~1368 '--.1- Uijkl ,~ 2457 ]] q- (R1458 ,~ 2367 "~ 2358 q- ~( Uijkl 1467 qL ~ijk, oijkl "~ Oijkl ]'Jc-( ~ijkl )} •
nc
2468
)
-- ~ijkl ,
(2.1)
In addition, let A / b e the purely imaginary, symmetric, traceless matrix which is invariant under SO(7) v. That is, let A / = idiag(1,1,1, 1 , 1 , 1 , - 7 ) ,
(2.2)
Considered as elements of E7(+7), X + and A generate an SU(1,1) subgroup. Indeed, this subgroup is uniquely defined as the subgroup which commutes with the G 2 subgroup of SO(8)c_ E7(+7). Let ~" and A be elements of E7(+7), defined by
0
W= exp aX+_ iflX-
0
A = exp tX +
aX++ iflX- ) 0 '
tX+ ) 0 '
(2.3)
(2.4)
where a, fl and t are real numbers. Observe that we may write [2] coshP
si-~P M]
=
(2.5) P --sinh --~M
( cosh( tX +) A = 1 sinh(tX +)
cosh P ] sinh( tX +) ) cosh(tX +) '
(2.6)
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where M = aX++
P=M~
iflX-,
+ •
(2.7) (2.8)
Let ~'=~A
-1 .
(2.9)
Then ~'~°' is G 2 invariant, being constructed from X +, and so may be written in the form
~' = U
cosh P'
sinh P__~'M ' P
sinh P ' )~, p,
cosh P '
(2.10)
where M' = a'X++ ifl'X-,
P'= M~,
(2.11) (2.12)
for some real numbers, a' and fl'; and when U is an element of SU(8)_c E7(+7 ) generated by A of eq. (2.2). That is,
U=exp
0 ) -7(AtitkBjlq)
0
(2.13)
for some real number 3'. Let S be the submatrix of U: S = exp[ y(A i/'kSjlq)].
(2.14)
S cosh P ' = cosh Pcosh(tX+) - sinh P M sinh(tX+), P
(2.15)
Then
sinh P Mcosh(tX+). . . S sinh P ' M ' = - cosh P sinh(tX+) + -----fi--p,
(2.16)
The negative signs arise because eq. (2.9) contains A - 1. Next observe that 1 2
aij = 6[i6)l - 8 ~iS~l
(2.17)
is simultaneously an eigenvector of X +, X- and W = A [iIkSj]1] with eigenvalues: - 1,
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679
1 and i respectively. (The vector a~j may also be thought of as an element of the Lie algebra of E7(+7), and as such is a generator of G 2. The fact that it is an eigenvector of X e, X - and W follows directly from the observation that G 2 commutes with SU(1,1).) To determine 0/', fl' and y in terms of 0/, fl and t; we consider the action of eqs. (2.15) and (2.16) on aij. We obtain -
e%osh r ' = cosh r cosh t
ei Y 0/' + ifl'
sinh r' =
0/+ ifl sinh r sinh t,
- cosh r sinh t +
r
0/+iflsinhrcosht,
(2.18)
(2.19)
r
rr
w h e r e r 2 = 0/2 --I-~ 2 a n d r '2 = 0/,2 _~_/~t2.
Let e iqa = (0/ "~- ifl)/r,
e i~°'= (0/'+ ifl')/r'.
(2.20)
Since the scalar potential is invariant under SU(8) transformations, we will not need "/. Moreover, one finds that the potential in the sector of interest depends only upon c o s h 2 r and sinh2rcos~. Therefore, we only need determine how these quantities transform under eq. (2.9). By taking the modulus squared of eqs. (2.18) and (2.19) and adding, one obtains cosh 2 r ' = cosh 2 t cosh 2 r - sinh 2 t sinh 2 r cos cp.
(2.21)
Multiplying eq. (2.19) by the complex conjugate of eq. (2.18) one obtains ei~°'sinh2r' = - s i n h 2 t c o s h 2 r + sinh2rcosh2t cos~o + i sinh2r sin~o, (2.22) and in particular sinh 2r 'cos ~' = - sinh 2 t cosh 2r + cosh 2t sinh 2r cos ~o.
(2.23)
Eqs. (2.21) and (2.23) determine the required transformation properties of the Lie algebra under fight multiplication by A - 1. The SU(1,1) subgroup which is relevant for the SO(6, 2) gauging is defined in an analogous manner. It is that SU(1,1) subgroup of E7(+7) which commutes with a certain SU(3) × U(1) × U(1) subgroup of SO(8) c E7(+7 ). Its generators are Xifkl
5678) q- (,gVijkl ~'R1234"t-~ijkl 1256 X 3478 ) q- ( ,R1278 --1-,g 3456 ] = ~,Vijkl '-+ ~.Vijkl ' - - ~ i j k l ] ' -- Vijkl
A ,J= i d i a g ( 1 , 1 , 1 , 1 , 1 , 1 , - 1, - 1).
(2.24) (2.25)
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These generators are stabilized respectively by the three different SO(6)× SO(2) subgroups of SO(8). The common subgroup stabilizing all three is SU(3)× U ( 1 ) × U(1). Once again aij defined in eq. (2.17) is an eigenvector of X -+ and W, and the analysis proceeds exactly as before. Eqs. (2.21) and (2.23) also give the required Baker-Hausdorff formulae. 3. The restricted potentials for the S O ( p , q ) gaugings
To obtain the SO(7,1) potential restricted to the G 2 singlets and the SO(6,2) potential restricted to the SU(3) singlets, we start from the SO(8) potential restricted to the SU(3) singlets. The scalar and pseudoscalar singlets of SU(3) are f [ ~1234` '-..l_ ~ijk I ) + ['R1256"k~ijkl ~ Vijk I ~ 3478) + \ V~1278 yi}~l=6++_[\Vijk 5678 i j k l "1-~34'/6)}, Yi2~l = 6+
{
/"~1357 "1-~ 2468 ]
/ * 1368-1-X 2457 ] + (6i1458 _[_ ,~ 2367 ]
[ X1467 ..I- ,~ 2358 1 }
(~1367--
(,R1468._T_,R2357"~
- - \ ~ i j k l '--Vijkl ] + ~ °ijkl 2_ Vijkl ]
y/3~ =E_+{.T_/R1358i,R2467 ~
2458-
- - ~ i j k l ] + \ ~ijkl '---Wjkl ]
1457
2358
,
(3.1) where e+ = 1, e_ = i, and + gives the scalars and - the pseudoscalars. The parametrization of ref. [1], for the singlet space reduces to q0ijk, = -- { X c o s ( a ) Y l + + ~ksin(c0 Y I - + X'cos cpcos(vq + ~b)r 2+ + ~' sin qocos(~ - ~k)Y 2+~'cos~sin(~+~)Y3++h'simpsin(O-+)Y3-}.
(3.2)
(Note that the sign is chosen to cancel that of eq. (1.1).) The scalar potential is then p = lg2{s,4[(x2 + 3)c 3 + 4x2v3s 3 _ 3v(x 2 _ 1)s 3 + 12xv2cs2 - 6(x - 1) cs 2 + 6(x + 1) c2sv] + 2s '2 [2( c 3 + v3s 3) + 3(x + 1)vs 3 + 6xv2cs 2 - 3(x - 1)cs 2 - 6c] - 12c},
(3.3)
where c = cosh(f~-A.),
s = sinh(~-~ ~),
c'= cosh( X'),
s ' = sinh(~f~-~~'),
O=
COS 0~
x = cos 2cp.
(3.4)
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Note that P is independent of 0 and ~b. The factors of ~ in eq. (3.4) arise through the ¼v~- in eq. (1.1). In order to clear up the ambiguity of the factors of two in matrix products involving q0ijkt, our normalization is taken so that the eigenvalues of ui/J are cosh(l~-~ ?t) and cosh(~/~-~3X) for ?t' = a = 0. Thus the arguments of the hyperbolic functions in eq. (3.4) are twice the fundamental values appearing in cV. To obtain the potential for the G2-invariant scalars ( y l + + y2+ and y1 + y 2 - ) one takes X' = X and cO= a. Then x = 2v 2 - 1, and the potential reduces to P = 2g2( ( 7 v ' - 7v2+ 3)C3S4-t (4v 2 - 7)vSs7+ c5s2+ 7v3c2s5 - 3c3}, "
(3.5)
where c, s and v are given in eq. (3.4). To obtain the SO(7,1) theory one transforms the vielbein ~ according to eq. (1.5), with t = ~i~r [4]. Substituting this into (2.21) and (2.23) we see that the necessary replacements in eq. (3.5) are cosh(~-~X )--, ~-~ [cosh(~-~X )-isinh(~/~-~ X ) c o s a ] ,
sinh(~½ X)cos a --, - ~i [cosh(~-~ X) + isinh(~-~ X)cos ~ l .
(3.6)
Finally, to obtain the SO(7,1) potential one must rescale by e 2~'= ~-;2( 1 - i ) , according to eq. (1.4), with a = - 1 . The result of the substitution and rescaling is P ' = ½ g 2 ( c + vs ) 2 ( ( c + vs ) [ 3 c 2 - 8 c v s + 302s212
-
14(c -
vs ) ( c 2 - 4 c o s +
v2s2) }. (3.7)
To obtain the SO(6, 2) theory, one takes t = + ¼i~r and rescales by e 2at= --i. From eqs. (2.21) and (2.23), this means that one makes the substitutions
c ~ -ivs, vs
--,
-
ic ,
(3.8)
in eq. (3.3). The resulting expression for the SO(6, 2) potential is p , = ½g2{s,4[(c+ v s ) ( 2 x c - ( x -
3)vs) 2 - 3 ( x - 1 ) ( ( x + 1)c + 2vs)]
+ s '2 [2(c + vs)(2c 2 + 2(3x - 1)cvs - ( 3 x - 5)v2s 2) +6((x + 1)c-(x-
3) vs)] + 12vs}.
(3.9)
To obtain the potential for the SO(5, 3) gauging, restricted to the SO(5) singlets, one does not need to use the Baker-Hausdorff formulae.
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There are no SO(5)-invariant pseudoscalars, and there are six SO(5)-invariant scalars. By using the action of the SO(3) in SO(5) × SO(3) _c SO(5, 3), the six scalars may be obtained from 5678 )'~-(~ijk, 1256. ~-~ijkl 3478 )'~-(~,jkl 1278 ..~ ~3456~] fPijkl:~k[(~,12~ 4 "~-~ijk, Vijkl ] ] [(R1234 4(R 1357 -.I~2457]] "~-I"t[~ijkl -- ~ijkl 5678) __\~ijkl -- ,~2468] ~ijkl ] q-('~1368 ~ijkl "}- Vijkl ]] _ [/~1234
R5678~
[ R1458
,~ 2367 ~
/ ,~1467
2358
+P[[oijk, + "-',jkt 1-~=ijk, + Vijk, J--tvijk+ + 8ijk, )].
(3.10)
The action of SO(3) commutes with E(t), and just as the SO(8) potential P is independent of the action of SO(3), so is the SO(5, 3) potential P'. It was shown in ref. [9] that the SO(8) potential when restricted to SO(5) scalar singlets may be written in the form
- 2 ( u v w 3 + uv3w + u3ow) -
(3.11)
Ul)W j
where u = ex / ~ ,
/~ = e "/v~ ,
w = e °/¢~ .
(3.12)
The Lie algebra element, X, generating E ( t ) may be obtained from eq. (3.10) by setting X = / t = p. The analytic continuation is done by transforming:
~-~--+ ~-~- liar,
~-~/~ --+ ~--~/~- ¼iTt,
~--~p--+~-~p- ¼i~r, (3.13)
g 2 ~ g2 e-3~ri/4.
The potential becomes
= g2[(u3V3w + uw3v + o3+)u
15]
-- 2( UVW3 + UV3W + U3VW) -- ~OW "
Thus the analytic continuation results in only one change of sign.
(3.14)
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4. The G2-invariant sector ot the S O ( 7 , 1 ) potential O n e has to be a little cautious in using eq. (3.7) (or for that m a t t e r (3.9)) to find the critical points of the potential. This is because the parametrization of the scalars given in eq. (3.2) can be singular, and therefore a point at which OP/O)~" = 8P/82~ = 8 P / 8 a = O P / 8 ~ = 0 m a y not be a critical point, but a coordinate singularity. Such coordinate singularities have to be treated individually when they arise. For example, h = 0 or )~' = 0 are singular points, and the condition for a critical point at 2~ = 0 ()~' 4: 0) is that all derivatives vanish and 8P/8)~ = 0 independent of a. These singularities arise through the jacobian of the parametrization (3.2) going to zero. It should be noted that this j a c o b i a n is never infinite, and so one is never going to miss a critical point by using eq. (3.2), but one might incorrectly identify a point as being a critical point. The obvious course, therefore, is to ignore this p r o b l e m until one finds a possible critical point, and then check for coordinate singularities. Differentiating eq. (3.7) with respect to a one obtains {(c + us)( c - vs)( c - 3 v s ) [ 3 ( c -
vs) 2 - 2cvs]
- 2 ( 3 c 3 + 5c2os - 17co2s 2 + 5/33s 3) ) s i n a = 0.
(4.1)
T h e r e are two possibilities, either sin a = 0 or the expression in the brackets vanishes. W e consider the two cases separately. 4.1. sina = 0 This implies that a = 0 or a = ~r and thus o = + 1 or v = - 1. However, v appears in P ' always multiplied by s = sinh(f~-)Q, and the case v = - 1 m a y be obtained f r o m v = 1 b y sending h ~ - )~. Thus we only need consider u = + 1. The potential P ' reduces to P'=
-~g2(x7 + l n x 3 - 3 5 x - 1 ) ,
(4.2)
where x = e x / ~ . This has a critical point if, and only if, xs + 6x4 + 5 = O.
( x ' + 1 ) ( x ' + 5) = 0.
(4.3)
This has no roots and x = e x/vff must be real. N o t e that the SO(8) potential, P, when restricted in this manner, is given by P = ~g2{x-7-
14x - 3 -
35x},
(4.4)
a n d the condition for critical points becomes ( x -4 - 1 ) ( x -4 - 5) = 0.
(4.5)
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The two roots of this equation give the symmetric (q) = 0)and SO(7) ÷ critical points [1]. Eq. (4.3) is obviously an analytic continuation of eq. (4.5), but the analytic continuation has moved the critical points out of the physical (i.e., real) scalar sector [6, 7]. 4.2. s i n a ~ 0
The simplest way of dealing with this is to change the independent variables in P ' to
x = c,
y = vs.
(4.6)
This transformation is non-singular since sina 4: O. It is elementary to show that 8 P ' / S x - 8 P ' / S y = 0 reduces to
(x2-y2)(3x2-8xy+
3y2)-(x2-4xy+
y2)-3(x-y)2=O,
(4.7)
where we have divided by (x + y)2. This quantity is non-zero since Icosh( ~/~-~X)I > Icos a sinh( ~--~)t)l.
(4.8)
Similarly, OP'/Ox + OP'/Oy = 0 reduces to
(x + y ) [ 3 x 2 - 8 x y + 3 y 2 ] [ x 2 - 4 x y + y 2] - 4 ( x - y ) ( x 2 - 4 x y +
(4.9)
+2(x+y)2(x-y)=O. Multiplying eq. (4.9) by ( x - y ) using eq. (4.7) one obtains
y 2)
and substituting for ( x 2 - y 2 ) ( 3 x 2 - 8 x y + 3y 2)
xy(x 2 - 4xy + y 2 ) _ 2 ( x
+y)Z(x-y
2
= 0,
(4.10)
or
t 4 - t 3 + 2t 2 - t + 1 + 0 ,
(4.11)
where t = y / x . (Note that x = cosh(vr~--~2t)~ 0.) Eq. (4.11) factorizes into
(t + i)(t - i)(t + to)(t - t~2) = O,
(4.12)
where ¢0 e 2~ri/3. Once again, there are no real roots for this equation, and therefore no critical points. The SO(8) potential has four G2-invariant critical points [1], whereas the SO(7,1) potential has none. The situation is rather reminiscent of the N = 4 theories of Freedman and Schwarz [10], which have potentials without critical points. =
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5. The SO(6, 2) and SO(5, 3) potentials The analysis of the SU(3) singlet sector of the SO(6, 2) potential is rather more complicated than that of the previous section. One has twice as many equations, and several more cases to check. The methods are identical to those given above, but for the final part of the analysis it was necessary to resort to a symbolic manipulation program. Surprisingly, one ultimately obtains a completely factorizable polynomial for the same variable t defined in the previous section. Indeed, the analog of eq. (4.12) is (5t + 2)(2t + 5 ) ( / - 2 + q~-)(t - 2 - (3-)(4 - 7t + 4t2) 2 = 0.
(5.1)
Since t = tanh(~f~-~A)cosa, one must have t real and ttl < 1. The only possible solutions are t = - 2 and t = 2 - v~-. Unfortunately, when substituted back into the other equations defining the critical point, t = - 2 gives a negative value for (s') 2 and t = 2 - ~ gives a value of cos2rp which is bigger than 1. Therefore, there are no SU(3)-invariant critical points. The only solutions of the equations yield complex values for the real parameters of the scalar fields. We see, once again, that the SU(3)-invariant critical points of the SO(8) theory have been analytically continued out of the real scalar section. The analysis of the SO(5)-invariant critical points of the SO(5, 3) potential, eq. (3.14), is almost identical to that given in ref. [7]. The SO(7) ÷ and SO(8) critical points of the SO(8) theory do not remain in the real scalar section of the SO(5, 3) theory. That is, they do not survive the analytic continuation. However, there is a new critical point which comes from the complex section of the SO(8) theory. This is the only critical point of eq. (3.14) and it has symmetry SO(5)× SO(3). It occurs when ~ = # = p, and
P'= -~g2(uS- 1 0 u + 5 u - 3 ) .
(5.2)
The critical points of eq. (5.2) occur at u 4 = 3 and u 4 = - 1 . The latter gives a complex value to u and is not allowed. The other root, u = 3-~/4, is a true critical point with cosmological constants: A = 2.31/4g 2.
(5.3)
The critical point was previously found in ref. [7]. Note that A is positive, and thus all supersymmetry is broken - the de Sitter space solution admits no Killing spinors [7]. It should be noted that it is possible to have supersymmetry in a de Sitter space (A > 0) [11]. However, this implies that the vector kinetic term has the wrong sign [12] and the theory is not unitary. However, in any spontaneous breaking of a non-compact gauging of N = 8 supergravity, the vector kinetic term has the correct (unitary) sign [4, 5]. Thus the critical point described above, must break all supersymmetry.
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F i n a l l y , we have m a d e a very brief e x a m i n a t i o n of the SO(4, 4) potential. T h e r e is o n e k n o w n critical p o i n t at rpuk t = 0, which has a positive c o s m o l o g i c a l c o n s t a n t a n d b r e a k s all s u p e r s y m m e t r y [7]. W e have also l o o k e d for a critical p o i n t which is the a n a l o g o f the S U ( 4 ) - critical p o i n t of the SO(8) gauging. Since SU(4) is n o t a s u b g r o u p o f SO(4) x SO(4), one has to consider some c o n v e n i e n t c o m m o n subgroup. W e c o n s i d e r e d the singlets of a p a r t i c u l a r SU(2) x SU(2) s u b g r o u p of SO(4) x SO(4) in which o n e SU(2) sits in each of the SO(4) factors. O n c e again we f o u n d no new critical point. F r o m our, albeit rather restricted, analysis of the S O ( p , q) potentials, one sees t h a t there is a d e a r t h of critical points. This is rather surprising considering the rich s t r u c t u r e of the SO(8) p o t e n t i a l [1]. T h e p o t e n t i a l s of the S O ( p , q) gaugings seem to b e like the sides of hills, no tops or s a d d l e - p o i n t s within easy sight. C . M . H . was s u p p o r t e d b y an S E R C fellowship. O n e of us (N.P.W.) is grateful to M I T for its h o s p i t a l i t y w h i l e the m a j o r i t y of this w o r k was done. W e are also b o t h grateful to the A s p e n Center for Physics where this w o r k was c o m p l e t e d .
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
N.P. Warner, Phys. Lett. 128B (1983) 169 E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141 B. de Wit and H. Nicolai, Phys. Lett. 108B (1981) 285; Nucl. Phys. B208 (1982) 323 C.M. Hull, Phys. Rev. D30 (1984) 760; Phys. Lett. 142B (1984) 39; The construction of new gaugings of N = 8 supergravity, MIT preprint (1984) C.M. Hull, More gaugings of N = 8 supergravity, MIT preprint (1984), Phys. Lett. B, to appear C.M. Hull and N.P. Warner, Nucl. Phys. B253 (1985) 650 C.M. Hull, Minimal couplings and critical points of the gauged N = 8 supergravity theories, MIT preprint (1984) N.P. Warner, Nucl. Phys. B231 (1984) 250 L.J. Romans, Phys. Lett. 131B (1983) 83 D.Z. Freedman and J. Schwarz, Nucl. Phys. B137 (1978) 333 J. Lukierski and A. Novicki, Supersymmetry in the presence of a positive cosmological constant, Univ. of Wroclaw preprint 609 (1984) K. Pilch, P. van Nieuwenhuizen and M. Sohnius, De Sitter superalgebras and supergravity, Stony Brook preprint ITP-SB-84-46 (1984)
THE POTENTIALS OF THE GAUGED N = 8 SUPERGRAVITY THEORIES C.M. HULL Department of Mathematics, MIT, Cambridge, MA 02139, USA N.P. WARNER* California Institute of Teehnology, Pasadena, CA 91125, USA
Received 16 October 1984
The potentials of the SO(p, q) gaugings of N = 8 supergravity are investigated for critical points. The SO(7,1) gauging has no GE-invariantcritical points, the SO(6,2) theory has no SU(3) invariant critical points and the SO(5,3) gauging has only one SO(5)-invariant critical point, with positive cosmologicalconstant, SO(5)X SO(3) symmetry and no supersymmetry.
1. Introduction
In [1], a simple method was given for finding all critical points of the scalar potential P(q0) of the N = 8 supergravity [2] with SO(8) gauge symmetry [3] that break the local SO(8) down to a solution with symmetry that is at least some specified subgroup, H, of SO(8). One considers only those scalars qon, which are singlets of H, and seeks critical points of the potential P(CpH) restricted to be a function only of the H-singlets. Schurr's lemma guarantees that any critical point of P(CPH) will be a critical point of the complete potential P(qg) [1]. The problem of finding critical points of the potential P(~0) is then reduced to the simpler one of finding critical points of the potential restricted to the H-singlet sector, P(q0H). In this p a p e r we apply similar techniques to the gauged SO(p, q) supergravities ( p + q = 8) [4, 5]. In ref. [1] the group H was chosen to be SU(3) for the SO(8) model, while in [7] it was taken to be S O ( p ) x S O ( q ) for the S O ( p , q ) and C S O ( p , q) gaugings. Here we will take H to be G 2 for the SO(7,1) gauging, SU(3) for the SO(6, 2) gauging, and SO(5) for SO(5, 3) gauging. Note that the group H is a compact subgroup of SO(p, q). This is necessary for the Schurr's lemma argument to be valid. * Work supported in part by the US Department of Energy under contract no. DE-AC-03-81ER40050 and by the Weingart Foundation. 675
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In the gauged supergravity theories the scalars lie in the coset space E7/SU(8 ) and it is convenient to represent them with a 56 × 56 E 7 matrix, the 56-bein, ~f [2-6]. Indeed, in this paper we adopt the usual SU(8) basis for the Lie algebra, and take [2,3]
[ AtffSjj tl - ¼v~epijk,] ~(= exp _ ¼v~epijkt - a [itk~Atl ] = ( UijIJ ~klIJ
UijKL ) ~lklgL ]
(1.1)
(1.2)
o
The potential, P, for the SO(8) gauged theory is a sixth-order polynomial in the submatrices of eq. (1.2), and is given in ref. [3]. The new potential P', for an S O ( p , q ) gauging may be obtained from P by analytic continuation [4-7]. Define
where X,jkt is the unique S O ( p ) × SO(q) invariant real, self-dual, totally antisymmetric tensor field. (It is unique up to normalization.) Then the potential, P', is given by
P,(V) =
'(t))]
(1.4)
where a and a are real constants, determined by p, q and the normalization of X. Expressions for Xijkl for each S O ( p , q ) gauging may be found in refs. [5,6]. The constant a is taken to be - 1 and a = 1 + p/q. The problem is how to calculate P'(cT). In ref. [8] it was shown how one could obtain explicit and usable expressions for P ( ~ ) , and in ref. [6] it was shown how to modify these expressions to obtain P'(CV). In this work we take a slightly different course since we will consider only those scalar fields that are singlets of some convenient compact subgroup H of SO(p, q). Sect. 2 describes how to calculate the necessary Baker-Hausdorff formulae in order to calculate Y'(t), such that c~' = ~ E - l ( t )
=exp(Y)E-l(t) = exp(Y'(t)), for a particular class of elements, Y, of the Lie algebra of E 7.
(1.5)
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In sect. 3 we use the results of refs. [1, 8,9] to calculate an explicit form for p,(c~-)= P(cV') on the H-singlet subspaces for the SO(p,q) gaugings. Sect. 4 contains a detailed analysis of the G2-invariant sector of the potential for the SO(7,1) gauging, and finally, in sect. 5 we state the results of a similar analysis of the other potentials. 2. Baker-Hausdorff Formulae
The only non-trivial Baker-Hausdorff formulae that we will need are for particular SI(1,1) subgroups of E7(+7). To obtain the subgroup relevant for the SO(7,1) gauging, let X,~kt and X~k t be the unique (up to scaling), completely anti-symmetric self-dual and anti-self-dual tensors which are invariant under SO(7) ÷ and SO(7)-, respectively. Explicitly, define + = {c 1234_+ 8,,k, 5678 ) + x, Tk, \Vijkl
( R 1256
+
R 3478 ]
\ ° i j k l -- ~ijkl ]
+
(t~ 1278 -1- R3456]
\Vijkl
'~ ~ijkl ]
_( Vijkl ,357
,~1368 '--.1- Uijkl ,~ 2457 ]] q- (R1458 ,~ 2367 "~ 2358 q- ~( Uijkl 1467 qL ~ijk, oijkl "~ Oijkl ]'Jc-( ~ijkl )} •
nc
2468
)
-- ~ijkl ,
(2.1)
In addition, let A / b e the purely imaginary, symmetric, traceless matrix which is invariant under SO(7) v. That is, let A / = idiag(1,1,1, 1 , 1 , 1 , - 7 ) ,
(2.2)
Considered as elements of E7(+7), X + and A generate an SU(1,1) subgroup. Indeed, this subgroup is uniquely defined as the subgroup which commutes with the G 2 subgroup of SO(8)c_ E7(+7). Let ~" and A be elements of E7(+7), defined by
0
W= exp aX+_ iflX-
0
A = exp tX +
aX++ iflX- ) 0 '
tX+ ) 0 '
(2.3)
(2.4)
where a, fl and t are real numbers. Observe that we may write [2] coshP
si-~P M]
=
(2.5) P --sinh --~M
( cosh( tX +) A = 1 sinh(tX +)
cosh P ] sinh( tX +) ) cosh(tX +) '
(2.6)
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where M = aX++
P=M~
iflX-,
+ •
(2.7) (2.8)
Let ~'=~A
-1 .
(2.9)
Then ~'~°' is G 2 invariant, being constructed from X +, and so may be written in the form
~' = U
cosh P'
sinh P__~'M ' P
sinh P ' )~, p,
cosh P '
(2.10)
where M' = a'X++ ifl'X-,
P'= M~,
(2.11) (2.12)
for some real numbers, a' and fl'; and when U is an element of SU(8)_c E7(+7 ) generated by A of eq. (2.2). That is,
U=exp
0 ) -7(AtitkBjlq)
0
(2.13)
for some real number 3'. Let S be the submatrix of U: S = exp[ y(A i/'kSjlq)].
(2.14)
S cosh P ' = cosh Pcosh(tX+) - sinh P M sinh(tX+), P
(2.15)
Then
sinh P Mcosh(tX+). . . S sinh P ' M ' = - cosh P sinh(tX+) + -----fi--p,
(2.16)
The negative signs arise because eq. (2.9) contains A - 1. Next observe that 1 2
aij = 6[i6)l - 8 ~iS~l
(2.17)
is simultaneously an eigenvector of X +, X- and W = A [iIkSj]1] with eigenvalues: - 1,
C.M. Hull, N.P. Warner / Supergraoity
679
1 and i respectively. (The vector a~j may also be thought of as an element of the Lie algebra of E7(+7), and as such is a generator of G 2. The fact that it is an eigenvector of X e, X - and W follows directly from the observation that G 2 commutes with SU(1,1).) To determine 0/', fl' and y in terms of 0/, fl and t; we consider the action of eqs. (2.15) and (2.16) on aij. We obtain -
e%osh r ' = cosh r cosh t
ei Y 0/' + ifl'
sinh r' =
0/+ ifl sinh r sinh t,
- cosh r sinh t +
r
0/+iflsinhrcosht,
(2.18)
(2.19)
r
rr
w h e r e r 2 = 0/2 --I-~ 2 a n d r '2 = 0/,2 _~_/~t2.
Let e iqa = (0/ "~- ifl)/r,
e i~°'= (0/'+ ifl')/r'.
(2.20)
Since the scalar potential is invariant under SU(8) transformations, we will not need "/. Moreover, one finds that the potential in the sector of interest depends only upon c o s h 2 r and sinh2rcos~. Therefore, we only need determine how these quantities transform under eq. (2.9). By taking the modulus squared of eqs. (2.18) and (2.19) and adding, one obtains cosh 2 r ' = cosh 2 t cosh 2 r - sinh 2 t sinh 2 r cos cp.
(2.21)
Multiplying eq. (2.19) by the complex conjugate of eq. (2.18) one obtains ei~°'sinh2r' = - s i n h 2 t c o s h 2 r + sinh2rcosh2t cos~o + i sinh2r sin~o, (2.22) and in particular sinh 2r 'cos ~' = - sinh 2 t cosh 2r + cosh 2t sinh 2r cos ~o.
(2.23)
Eqs. (2.21) and (2.23) determine the required transformation properties of the Lie algebra under fight multiplication by A - 1. The SU(1,1) subgroup which is relevant for the SO(6, 2) gauging is defined in an analogous manner. It is that SU(1,1) subgroup of E7(+7) which commutes with a certain SU(3) × U(1) × U(1) subgroup of SO(8) c E7(+7 ). Its generators are Xifkl
5678) q- (,gVijkl ~'R1234"t-~ijkl 1256 X 3478 ) q- ( ,R1278 --1-,g 3456 ] = ~,Vijkl '-+ ~.Vijkl ' - - ~ i j k l ] ' -- Vijkl
A ,J= i d i a g ( 1 , 1 , 1 , 1 , 1 , 1 , - 1, - 1).
(2.24) (2.25)
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These generators are stabilized respectively by the three different SO(6)× SO(2) subgroups of SO(8). The common subgroup stabilizing all three is SU(3)× U ( 1 ) × U(1). Once again aij defined in eq. (2.17) is an eigenvector of X -+ and W, and the analysis proceeds exactly as before. Eqs. (2.21) and (2.23) also give the required Baker-Hausdorff formulae. 3. The restricted potentials for the S O ( p , q ) gaugings
To obtain the SO(7,1) potential restricted to the G 2 singlets and the SO(6,2) potential restricted to the SU(3) singlets, we start from the SO(8) potential restricted to the SU(3) singlets. The scalar and pseudoscalar singlets of SU(3) are f [ ~1234` '-..l_ ~ijk I ) + ['R1256"k~ijkl ~ Vijk I ~ 3478) + \ V~1278 yi}~l=6++_[\Vijk 5678 i j k l "1-~34'/6)}, Yi2~l = 6+
{
/"~1357 "1-~ 2468 ]
/ * 1368-1-X 2457 ] + (6i1458 _[_ ,~ 2367 ]
[ X1467 ..I- ,~ 2358 1 }
(~1367--
(,R1468._T_,R2357"~
- - \ ~ i j k l '--Vijkl ] + ~ °ijkl 2_ Vijkl ]
y/3~ =E_+{.T_/R1358i,R2467 ~
2458-
- - ~ i j k l ] + \ ~ijkl '---Wjkl ]
1457
2358
,
(3.1) where e+ = 1, e_ = i, and + gives the scalars and - the pseudoscalars. The parametrization of ref. [1], for the singlet space reduces to q0ijk, = -- { X c o s ( a ) Y l + + ~ksin(c0 Y I - + X'cos cpcos(vq + ~b)r 2+ + ~' sin qocos(~ - ~k)Y 2+~'cos~sin(~+~)Y3++h'simpsin(O-+)Y3-}.
(3.2)
(Note that the sign is chosen to cancel that of eq. (1.1).) The scalar potential is then p = lg2{s,4[(x2 + 3)c 3 + 4x2v3s 3 _ 3v(x 2 _ 1)s 3 + 12xv2cs2 - 6(x - 1) cs 2 + 6(x + 1) c2sv] + 2s '2 [2( c 3 + v3s 3) + 3(x + 1)vs 3 + 6xv2cs 2 - 3(x - 1)cs 2 - 6c] - 12c},
(3.3)
where c = cosh(f~-A.),
s = sinh(~-~ ~),
c'= cosh( X'),
s ' = sinh(~f~-~~'),
O=
COS 0~
x = cos 2cp.
(3.4)
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C.M. Hull, N.P. Warner / Supergravity
Note that P is independent of 0 and ~b. The factors of ~ in eq. (3.4) arise through the ¼v~- in eq. (1.1). In order to clear up the ambiguity of the factors of two in matrix products involving q0ijkt, our normalization is taken so that the eigenvalues of ui/J are cosh(l~-~ ?t) and cosh(~/~-~3X) for ?t' = a = 0. Thus the arguments of the hyperbolic functions in eq. (3.4) are twice the fundamental values appearing in cV. To obtain the potential for the G2-invariant scalars ( y l + + y2+ and y1 + y 2 - ) one takes X' = X and cO= a. Then x = 2v 2 - 1, and the potential reduces to P = 2g2( ( 7 v ' - 7v2+ 3)C3S4-t (4v 2 - 7)vSs7+ c5s2+ 7v3c2s5 - 3c3}, "
(3.5)
where c, s and v are given in eq. (3.4). To obtain the SO(7,1) theory one transforms the vielbein ~ according to eq. (1.5), with t = ~i~r [4]. Substituting this into (2.21) and (2.23) we see that the necessary replacements in eq. (3.5) are cosh(~-~X )--, ~-~ [cosh(~-~X )-isinh(~/~-~ X ) c o s a ] ,
sinh(~½ X)cos a --, - ~i [cosh(~-~ X) + isinh(~-~ X)cos ~ l .
(3.6)
Finally, to obtain the SO(7,1) potential one must rescale by e 2~'= ~-;2( 1 - i ) , according to eq. (1.4), with a = - 1 . The result of the substitution and rescaling is P ' = ½ g 2 ( c + vs ) 2 ( ( c + vs ) [ 3 c 2 - 8 c v s + 302s212
-
14(c -
vs ) ( c 2 - 4 c o s +
v2s2) }. (3.7)
To obtain the SO(6, 2) theory, one takes t = + ¼i~r and rescales by e 2at= --i. From eqs. (2.21) and (2.23), this means that one makes the substitutions
c ~ -ivs, vs
--,
-
ic ,
(3.8)
in eq. (3.3). The resulting expression for the SO(6, 2) potential is p , = ½g2{s,4[(c+ v s ) ( 2 x c - ( x -
3)vs) 2 - 3 ( x - 1 ) ( ( x + 1)c + 2vs)]
+ s '2 [2(c + vs)(2c 2 + 2(3x - 1)cvs - ( 3 x - 5)v2s 2) +6((x + 1)c-(x-
3) vs)] + 12vs}.
(3.9)
To obtain the potential for the SO(5, 3) gauging, restricted to the SO(5) singlets, one does not need to use the Baker-Hausdorff formulae.
C.M. Hull, N.P. Warner / Supergravity
682
There are no SO(5)-invariant pseudoscalars, and there are six SO(5)-invariant scalars. By using the action of the SO(3) in SO(5) × SO(3) _c SO(5, 3), the six scalars may be obtained from 5678 )'~-(~ijk, 1256. ~-~ijkl 3478 )'~-(~,jkl 1278 ..~ ~3456~] fPijkl:~k[(~,12~ 4 "~-~ijk, Vijkl ] ] [(R1234 4(R 1357 -.I~2457]] "~-I"t[~ijkl -- ~ijkl 5678) __\~ijkl -- ,~2468] ~ijkl ] q-('~1368 ~ijkl "}- Vijkl ]] _ [/~1234
R5678~
[ R1458
,~ 2367 ~
/ ,~1467
2358
+P[[oijk, + "-',jkt 1-~=ijk, + Vijk, J--tvijk+ + 8ijk, )].
(3.10)
The action of SO(3) commutes with E(t), and just as the SO(8) potential P is independent of the action of SO(3), so is the SO(5, 3) potential P'. It was shown in ref. [9] that the SO(8) potential when restricted to SO(5) scalar singlets may be written in the form
- 2 ( u v w 3 + uv3w + u3ow) -
(3.11)
Ul)W j
where u = ex / ~ ,
/~ = e "/v~ ,
w = e °/¢~ .
(3.12)
The Lie algebra element, X, generating E ( t ) may be obtained from eq. (3.10) by setting X = / t = p. The analytic continuation is done by transforming:
~-~--+ ~-~- liar,
~-~/~ --+ ~--~/~- ¼iTt,
~--~p--+~-~p- ¼i~r, (3.13)
g 2 ~ g2 e-3~ri/4.
The potential becomes
= g2[(u3V3w + uw3v + o3+)u
15]
-- 2( UVW3 + UV3W + U3VW) -- ~OW "
Thus the analytic continuation results in only one change of sign.
(3.14)
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C.M. Hull, N.P. Warner / Supergravity
4. The G2-invariant sector ot the S O ( 7 , 1 ) potential O n e has to be a little cautious in using eq. (3.7) (or for that m a t t e r (3.9)) to find the critical points of the potential. This is because the parametrization of the scalars given in eq. (3.2) can be singular, and therefore a point at which OP/O)~" = 8P/82~ = 8 P / 8 a = O P / 8 ~ = 0 m a y not be a critical point, but a coordinate singularity. Such coordinate singularities have to be treated individually when they arise. For example, h = 0 or )~' = 0 are singular points, and the condition for a critical point at 2~ = 0 ()~' 4: 0) is that all derivatives vanish and 8P/8)~ = 0 independent of a. These singularities arise through the jacobian of the parametrization (3.2) going to zero. It should be noted that this j a c o b i a n is never infinite, and so one is never going to miss a critical point by using eq. (3.2), but one might incorrectly identify a point as being a critical point. The obvious course, therefore, is to ignore this p r o b l e m until one finds a possible critical point, and then check for coordinate singularities. Differentiating eq. (3.7) with respect to a one obtains {(c + us)( c - vs)( c - 3 v s ) [ 3 ( c -
vs) 2 - 2cvs]
- 2 ( 3 c 3 + 5c2os - 17co2s 2 + 5/33s 3) ) s i n a = 0.
(4.1)
T h e r e are two possibilities, either sin a = 0 or the expression in the brackets vanishes. W e consider the two cases separately. 4.1. sina = 0 This implies that a = 0 or a = ~r and thus o = + 1 or v = - 1. However, v appears in P ' always multiplied by s = sinh(f~-)Q, and the case v = - 1 m a y be obtained f r o m v = 1 b y sending h ~ - )~. Thus we only need consider u = + 1. The potential P ' reduces to P'=
-~g2(x7 + l n x 3 - 3 5 x - 1 ) ,
(4.2)
where x = e x / ~ . This has a critical point if, and only if, xs + 6x4 + 5 = O.
( x ' + 1 ) ( x ' + 5) = 0.
(4.3)
This has no roots and x = e x/vff must be real. N o t e that the SO(8) potential, P, when restricted in this manner, is given by P = ~g2{x-7-
14x - 3 -
35x},
(4.4)
a n d the condition for critical points becomes ( x -4 - 1 ) ( x -4 - 5) = 0.
(4.5)
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684
The two roots of this equation give the symmetric (q) = 0)and SO(7) ÷ critical points [1]. Eq. (4.3) is obviously an analytic continuation of eq. (4.5), but the analytic continuation has moved the critical points out of the physical (i.e., real) scalar sector [6, 7]. 4.2. s i n a ~ 0
The simplest way of dealing with this is to change the independent variables in P ' to
x = c,
y = vs.
(4.6)
This transformation is non-singular since sina 4: O. It is elementary to show that 8 P ' / S x - 8 P ' / S y = 0 reduces to
(x2-y2)(3x2-8xy+
3y2)-(x2-4xy+
y2)-3(x-y)2=O,
(4.7)
where we have divided by (x + y)2. This quantity is non-zero since Icosh( ~/~-~X)I > Icos a sinh( ~--~)t)l.
(4.8)
Similarly, OP'/Ox + OP'/Oy = 0 reduces to
(x + y ) [ 3 x 2 - 8 x y + 3 y 2 ] [ x 2 - 4 x y + y 2] - 4 ( x - y ) ( x 2 - 4 x y +
(4.9)
+2(x+y)2(x-y)=O. Multiplying eq. (4.9) by ( x - y ) using eq. (4.7) one obtains
y 2)
and substituting for ( x 2 - y 2 ) ( 3 x 2 - 8 x y + 3y 2)
xy(x 2 - 4xy + y 2 ) _ 2 ( x
+y)Z(x-y
2
= 0,
(4.10)
or
t 4 - t 3 + 2t 2 - t + 1 + 0 ,
(4.11)
where t = y / x . (Note that x = cosh(vr~--~2t)~ 0.) Eq. (4.11) factorizes into
(t + i)(t - i)(t + to)(t - t~2) = O,
(4.12)
where ¢0 e 2~ri/3. Once again, there are no real roots for this equation, and therefore no critical points. The SO(8) potential has four G2-invariant critical points [1], whereas the SO(7,1) potential has none. The situation is rather reminiscent of the N = 4 theories of Freedman and Schwarz [10], which have potentials without critical points. =
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685
5. The SO(6, 2) and SO(5, 3) potentials The analysis of the SU(3) singlet sector of the SO(6, 2) potential is rather more complicated than that of the previous section. One has twice as many equations, and several more cases to check. The methods are identical to those given above, but for the final part of the analysis it was necessary to resort to a symbolic manipulation program. Surprisingly, one ultimately obtains a completely factorizable polynomial for the same variable t defined in the previous section. Indeed, the analog of eq. (4.12) is (5t + 2)(2t + 5 ) ( / - 2 + q~-)(t - 2 - (3-)(4 - 7t + 4t2) 2 = 0.
(5.1)
Since t = tanh(~f~-~A)cosa, one must have t real and ttl < 1. The only possible solutions are t = - 2 and t = 2 - v~-. Unfortunately, when substituted back into the other equations defining the critical point, t = - 2 gives a negative value for (s') 2 and t = 2 - ~ gives a value of cos2rp which is bigger than 1. Therefore, there are no SU(3)-invariant critical points. The only solutions of the equations yield complex values for the real parameters of the scalar fields. We see, once again, that the SU(3)-invariant critical points of the SO(8) theory have been analytically continued out of the real scalar section. The analysis of the SO(5)-invariant critical points of the SO(5, 3) potential, eq. (3.14), is almost identical to that given in ref. [7]. The SO(7) ÷ and SO(8) critical points of the SO(8) theory do not remain in the real scalar section of the SO(5, 3) theory. That is, they do not survive the analytic continuation. However, there is a new critical point which comes from the complex section of the SO(8) theory. This is the only critical point of eq. (3.14) and it has symmetry SO(5)× SO(3). It occurs when ~ = # = p, and
P'= -~g2(uS- 1 0 u + 5 u - 3 ) .
(5.2)
The critical points of eq. (5.2) occur at u 4 = 3 and u 4 = - 1 . The latter gives a complex value to u and is not allowed. The other root, u = 3-~/4, is a true critical point with cosmological constants: A = 2.31/4g 2.
(5.3)
The critical point was previously found in ref. [7]. Note that A is positive, and thus all supersymmetry is broken - the de Sitter space solution admits no Killing spinors [7]. It should be noted that it is possible to have supersymmetry in a de Sitter space (A > 0) [11]. However, this implies that the vector kinetic term has the wrong sign [12] and the theory is not unitary. However, in any spontaneous breaking of a non-compact gauging of N = 8 supergravity, the vector kinetic term has the correct (unitary) sign [4, 5]. Thus the critical point described above, must break all supersymmetry.
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F i n a l l y , we have m a d e a very brief e x a m i n a t i o n of the SO(4, 4) potential. T h e r e is o n e k n o w n critical p o i n t at rpuk t = 0, which has a positive c o s m o l o g i c a l c o n s t a n t a n d b r e a k s all s u p e r s y m m e t r y [7]. W e have also l o o k e d for a critical p o i n t which is the a n a l o g o f the S U ( 4 ) - critical p o i n t of the SO(8) gauging. Since SU(4) is n o t a s u b g r o u p o f SO(4) x SO(4), one has to consider some c o n v e n i e n t c o m m o n subgroup. W e c o n s i d e r e d the singlets of a p a r t i c u l a r SU(2) x SU(2) s u b g r o u p of SO(4) x SO(4) in which o n e SU(2) sits in each of the SO(4) factors. O n c e again we f o u n d no new critical point. F r o m our, albeit rather restricted, analysis of the S O ( p , q) potentials, one sees t h a t there is a d e a r t h of critical points. This is rather surprising considering the rich s t r u c t u r e of the SO(8) p o t e n t i a l [1]. T h e p o t e n t i a l s of the S O ( p , q) gaugings seem to b e like the sides of hills, no tops or s a d d l e - p o i n t s within easy sight. C . M . H . was s u p p o r t e d b y an S E R C fellowship. O n e of us (N.P.W.) is grateful to M I T for its h o s p i t a l i t y w h i l e the m a j o r i t y of this w o r k was done. W e are also b o t h grateful to the A s p e n Center for Physics where this w o r k was c o m p l e t e d .
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
N.P. Warner, Phys. Lett. 128B (1983) 169 E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141 B. de Wit and H. Nicolai, Phys. Lett. 108B (1981) 285; Nucl. Phys. B208 (1982) 323 C.M. Hull, Phys. Rev. D30 (1984) 760; Phys. Lett. 142B (1984) 39; The construction of new gaugings of N = 8 supergravity, MIT preprint (1984) C.M. Hull, More gaugings of N = 8 supergravity, MIT preprint (1984), Phys. Lett. B, to appear C.M. Hull and N.P. Warner, Nucl. Phys. B253 (1985) 650 C.M. Hull, Minimal couplings and critical points of the gauged N = 8 supergravity theories, MIT preprint (1984) N.P. Warner, Nucl. Phys. B231 (1984) 250 L.J. Romans, Phys. Lett. 131B (1983) 83 D.Z. Freedman and J. Schwarz, Nucl. Phys. B137 (1978) 333 J. Lukierski and A. Novicki, Supersymmetry in the presence of a positive cosmological constant, Univ. of Wroclaw preprint 609 (1984) K. Pilch, P. van Nieuwenhuizen and M. Sohnius, De Sitter superalgebras and supergravity, Stony Brook preprint ITP-SB-84-46 (1984)