Construction of N = 8 supergravity theories by dimensional reduction

Nuclear Physics B253 (1985) 541-572 © North-Holland Publishing Company

C O N S T R U C T I O N OF N = 8 S U P E R G R A V I T Y T H E O R I E S DIMENSIONAL REDUCTION

BY

W. BOUCHER

Department of Applied Mathematics and Theoretical Physic.~, Cambridge University, Cambridge, England, UK Received 16 November 1984 in this paper I ask which N = 8 supergravity theories in four dimensions can be obtained by dimensional reduction of the N = 1 supergravity theory in eleven dimensions. Several years ago Scherk and Schwar-z produced a particular class of N = 8 theories by giving a dimensional reduction scheme on the restricted class of coset spaces, G / H , with dim H = 0 (and therefore dim G = 7). 1 generalize their considerations by looking at arbitrary (seven-dimensional) coset spaces. Also, instead of giving a particular ansatz which happens to work, I set about the distinctly more difficult task of determining all ansatzes which produce N = 8 theories. The basic ingredient of my dimensional reduction scheme is the demand that certain symmetries, including supersymmetry, be truncated consistently. I find the surprising result that the only N = 8 theories obtainable within the contexts of my scheme are those theories already written down by Scherk and Schwarz. In particular dim H = 0 and dim G = 7 . Independently of these considerations, 1 prove that any dimensional reduction scheme which consistently truncates supersymmetry must also be consistent with the equations of motion. ! discuss Lorentz-invariant solutions of the theories of Scherk and Schwarz, pointing out that since the ansatz of Scherk and Schwarz consistently truncates supersymmetry, any solution of these theories is also a solution of the N = 1 supergravity theory in eleven dimensions and, hence, in particular that there is a Freund-Rubin-type ansatz for these theories. However I demonstrate that for most gauge groups the ansatz must be trivial which implies that for these theories the cosmological constant of any Lorentz-invariant solution must be zero (classically). Finally, I make some comparisons with work by Manton on dimensional reduction.

1. Introduction

I am interested in constructing N = 8 supergravity theories in four dimensions. The easiest way to do this is by dimensional reduction of N = 1 supergravity in eleven dimensions. C r e m m e r and Julia [1] followed this route to construct the ungauged N = 8 theory, although the largest and hardest part of their paper concerned finding "hidden symmetries" and rewriting the theory in such a way as to reflect these symmetries manifestly. This involved (two kinds of) duality transformations of the fields which were discovered by hard work and insight. This part of the procedure is somewhat contrary to the spirit of dimensional reduction, which leads to N = 8 supergravity theories by a deductive prescription and without having to check explicitly that the theories are supersymmetric. Now, Cremmer and Julia used the "'naive" dimensional reduction scheme, namely, assuming that the fields in the 1 l-dimensional theory were independent of seven of the coordinates, but already in their paper they recognized that other dimensional reduction schemes were 541

542

W. Boucher / N = 8 supergravity theories

possible. Shortly thereafter Schei'k and Schwarz, [2], wrote down a dimensional reduction scheme which produced an N = 8 supergravity theory for any sevendimensional Lie group, G, for which the adjoint representation of G is a subgroup of (the fundamental representation of) SL(7, R), and in these theories G is gauged (although, any factors of U(I) act only trivially). They accomplished this by doing dimensional reduction on coset spaces G / H , with dim H - - 0 and dim G = 7 (i.e. H is a discrete subgroup of G). Of course they had to define what they meant by "doing" dimensional reduction on G / H (see the next section). The dimensional reduction scheme of Scherk and Schwarz reduces to that of Cremmer and Julia (before they make their field redefinitions) when G / H = ( U ( 1 ) ) 7. I do not know how many 7-dimensional Lie groups there are for which the adjoint representation is unimodular, but there certainly are quite a lot, including some which fall into l(or more) parameter families (such as the flat groups of Scherk and Schwarz). Now, the largest possible dimension of a group which can be gauged in an N = 8 theory is 28 since there are only 28 spin-I fields. 7 < 28 and naturally the question arises as to whether one can gauge any other groups, G, with 7 < dim G <~ 28. De Wit and Nicolai [3] answered this in the affirmative by explicitly constructing an N = 8 supergravity theory with a gauged G - - S O ( 8 ) symmetry (the " N = 8 gauged supergravity theory"). They did this by starting from the ungauged theory of Cremmer and Julia, and not by dimensional reduction. The next question is then begged. Can one produce this theory directly by dimensional reduction from N = 1 supergravity in eleven dimensions? Duff and Pope have advocated this point of view, and suggested that dimensional reduction of the N = I theory on S 7 = S O ( 8 ) / 5 0 ( 7 ) would produce the de Wit-Nicolai theory the "easy" way, [4, 5]. They did not explicitly state how one could construct such a dimensional reduction scheme but did a linearized analysis, which I discuss below. Recently, Hull has constructed other N = 8 theories, [6], with gauge groups either SO(p, 8 - p ) , l<~p<~4, or an InSnii-Wigner contraction of SO(8) about SO(p) × SO(8 - p), 1 <~p ~<4. These gauge groups and SO(8) have dimension 28. Even more recently, Pernici, Pilch and van Nieuwenhuizen [7] have produced an SO(5) gauged N = 4 supergravity theory in seven dimensions and by using the procedure of Scherk and Schwarz (which is much more general than the specific application to N = 1 supergravity in eleven dimensions) one sees that there exist gauged SO(5) × G N = 8 supergravity theories in four dimensions, where G is any 3-dimensional Lie group for which the adjoint representation is unimodular. There are six such groups (see sect. 4) and so there are six new N = 8 supergravity theories, with one having a non-trivial 10-dimensional gauge group (when G = ( U ( I ) ) 3) and with the other five having a non-trivial 13dimensional gauge group. Perhaps the most interesting gauge group amongst these is G ' = S O ( 5 ) × S O ( 3 ) . Again one might ask whether one can obtain any of these theories by dimensionally reducing the N = 1 theory in eleven dimensions. In this paper I propose a general dimensional reduction scheme of the N = 1 theory which is partly modeled on that of Scherk and Schwarz, which starts with

543

W. Boucher / N = 8 supergravity theories

an arbitrary 7-dimensional coset space G / H , and in which I am led to an ansatz for the fields in d = I 1 by demanding that certain symmetries are respected in the d = 4 theory, most important of which is N = 8 supersymmetry. I show that this scheme is successful only when dim H = 0, dim G = 7, in which case one can then show that the only successful ansatz is that of Scherk and Schwarz. The N = l supergravity lagrangian in d = I l [8] is* L

4K 2 + (T~4)22 Ke 4,...4, ' F ~ , ...~,F~...~xA~9...,~,, + f e r m i o n

terms,

( 1)

where F~,;~ = 4 ~ t ~ A ~ ~ is the field strength of the "3-index photon", A ~ ; , and /~ is the Ricci scalar. The action is invariant under N = 1 local supersymmetry transformations, given by

(~Q~/t~12 iK~l';' ~l,,~, =

(2)

1 K

= lyre K

+ ,~iF~4~(r~'4~, - 8r~4~a~)e + fermion terms,

= ~eFl~b~l.

(3) (4)

Additionally, the action is invariant under d = 11 general coordinate transformations (parameter sr~), d = 11 local Lorentz transformations (parameter .Q,~) and abelian gauge transformations. For me, dimensional reduction is about building theories with certain desired symmetries in d = 4, from theories in d > 4 such that the d = 4 symmetry algebra is a subalgebra of the symmetries of the higher-dimensional theory. Most bosonic symmetries are reasonably well understood, and the attempt to celebrate dimensional reduction as a "miraculous" method to produce Einstein-Yang-Mills theories in d = 4 from Einstein theory in d > 4 seems over-enthusiastic, since it is easy to build such theories without creating the artifice of higher dimensions. O f course, if d > 4 theories gave natural explanations of observed physical p h e n o m e n a better than d = 4 theories then one would be enthusiastic. But I do not think that anybody would claim that they do. On the other hand, fermionic symmetries are not well * Conventions: my conventions are mostly those of Scherk and Schwarz. In particular, the signature of the metric is (+ . . . . . ). Indices with hats range from 0-10, indices from the latter (beginning) part of the alphabet range from 0-3 (4-10). These are all lower case. Greek (Latin) letters are for world (tangent space) indices. Indices I, J are for Spin(7) spinors. Indices i, j are for ~, the Lie algebra of G. Indices AL B, ,~L /~ are for SL(2, C) spinors. The d = I I /'-matrices, F';', decompose as F ' = y ~ ® l , F ~ = y~®y~, y ~ the d = 4 y-matrices, 7 ° the d = 7 ones, with {y~, yb} = - 2 ~ , 7~ = - i y°)2),27~, ~f~= +1, ~,'" = ~ ) , ' , "y'], etc. g m 2 3 = + l , i take h = c = 1.

544

W. Boucher / N = 8 supergravity theories

understood, at least by me, and it is fruitful to use dimensional reduction to construct theories in d = 4 with extended supersymmetry. From n o w on I restrict myself (except when noted otherwise) to dimensionally reducing N = 1 supergravity in d = I 1 to N -- 8 supergravity in d = 4, although most o f the remarks that follow are valid in more general contexts. Once one has decided which symmetries o f the d = 11 theory are to be preserved one must find an ansatz for the d = 1 i fields in terms o f the d = 4 fields such that these symmetries are truncated consistently. This ansatz may or may not be unique given the dimensional reduction scheme. By consistent truncation o f a symmetry, I mean that if • is a generic d - - 1 1 field and ~l represents its ansatz, then one requires that (~Sq~)I = ~(~]). There are two approaches one can take at this point. Either one can insert the ansatz into the d = 11 action or one can insert the ansatz into the d = 11 field equations. In general the field equations o f the new ( d - - 4 ) action (derived from the original (d = 11) action by inserting the ansatz and doing the integrals over the extra seven coordinates) will not be the same as those derived by inserting the ansatz into the original field equations. The two procedures do not commute. There is some confusion in the literature concerning this point. Inherently, it is healthier to insert the ansatz into the action because it is easier to avoid the "inconsistencies" which can occur when inserting the ansatz into the field equations, such as forcing some fields to be n o n - p r o p a g a t i n g which one wanted to be propagating. For example, in the original d = 5 K a l u z a - K l e i n theory, putting the scalar field to zero in the equations o f motion forces the spin-I field to satisfy F,,~F"~=O [9]. Putting the scalar field to zero in the action causes no such contradiction. However, although it may be inherently better to work with the action rather than with the equations o f motion, there is no a priori reason to do this. Luckily, dimensional reduction schemes which truncate supersymmetry consistently p r o d u c e the same theory whether or not you insert the ansatz into the field equations or into the action (I show this in sect. 3). I will call dimensional reduction schemes which satisfy the latter property ones that are "consistent with the equations o f motion". There are no a priori reasons for preferring such schemes, but fortunately, truncating supersymmetry consistently is enough to guarantee this pleasant property. I take a strictly practical viewpoint concerning extra dimensions. I am only using the higher-dimensional theory in order to construct four-dimensional theories. I consider the extra dimensions to be a mathematical artifice. People who study K a l u z a - K l e i n theory (in contradistinction to dimensional reduction [9]) argue that the extra dimensions are physical, and although 1 am not enthusiastic about this approach, there is nothing inherently wrong with it. However, sometimes these adherents claim that by studying solutions o f the higher-dimensional field equations, they are making a proposal for dimensional reduction, and I think this is where you have to tread carefully. In general, there is no connection between a particular solution o f d > 4 theories and particular theories in d = 4. I f there is such a connection conjectured one should prove it, not merely state it as a desired property.

W. Boucher / N = 8 supergravity theories

545

(ADS) 4 × S 7 is by now well-known to be a solution o f the N = ! supergravity theory in d -- I l [9]. It has been speculated that this solution could s o m e h o w be used to construct the de W i t - N i c o l a i theory. The authors concerned have d o n e a linearized analysis in d = I I and one may consider this to be the beginning o f the desired construction, [4, 9-12]. In [10] it was claimed that they had given a " p r o o f that the compactification on S 7 leads to gauged N = 8 supergravity in four dimensions". 1 do not think that they had and I think that indeed they made two important mistakes. Firstly, they did not check that their ansatz truncated supersymmetry consistently; given their ansatz, it did not. (I think that they have now agreed that they were incorrect here, e.g. see [12].) Secondly, even if they had truncated supersymmetry consistently they would only have d o n e so at the linearized level, since they were doing perturbation theory. However, it is not difficult to write down theories with the correct particle spectrum, bosonic symmetries, and linearized supersymmetries o f the de W i t - N i c o l a i theory, but which are not supersymmetric. The first objection was addressed by A w a d a et al. [l 1], who, by making a slightly different ansatz for the supersymmetry parameter, m a n a g e d to show that their linearized ansatz truncated supersymmetry consistently. They did not address the second objection. I think that even their linearized analysis is incomplete in that they made an ansatz for the metric rather than for the vielbein, which is the more fundamental field in supergravity. Given a d = I 1 metric it is always possible to find a c o r r e s p o n d i n g d = I I vielbein, but given a d = I l metric with an ansatz in terms o f d = 4 fields it is not obvious that one can always get a c o r r e s p o n d i n g d = II vielbein with an ansatz in terms o f d = 4 fields which respects the symmetries o f the problem. And even if one can do this, one must check that ~ ' ~ ' ~ truncates consistently, not just ~ , ~ (see sect. 3 for a demonstration that this last point can b e c o m e important). In sect. 2, I review the work of Scherk and Schwarz. In sect. 3, I will give a general scheme for dimensional reduction o f the d = l l theory on coset spaces, G / H , motivated by the work o f Scherk and Schwarz, and show that if supersymmetry truncates consistently, then one is led back to their ansatz. In sect. 4, I consider possible g r o u n d states o f the N = 8 theories provided by Scherk and Schwarz. In sect. 5, I make some final comments.

2. Dimensional reduction h la Scherk and Schwarz In this section I review the dimensional reduction scheme o f Scherk and Schwarz [2], in order to motivate the subsequent development. This scheme is more general than its application to supergravity theories and it is in this more general guise that I will summarize their work, although in the end I will make reference to supergravity. First look at Einstein theory in d = D + E dimensions, which will be reduced to D dimensions (the signature o f the metric is (+ . . . . . ) in both cases). Because o f local Lorentz invariance in D + E dimensions one may choose a triangular para-

546

W. Boucher / N = 8 supergravity theories

meterization of the (inverse) vielbein, ¢',~'~, having the form D

=

E

fo"

1E

Scherk and Schwarz assume that the ( D + E)-dimensional manifold, M D+ r:, is a product space, i.e. M ° ÷ e = M D x M E, with coordinates (x, y), and they choose the following ansatz for the (inverse) vielbein:

V,,,~'(x, y) = w~'(x) Vm"(x) ,

(6)

~",,,'~(x, y) = w~'(x) V,. ~'( x ) A ~' , ( x ) K , " ( y ) ,

(7)

V,"(x, y) = & ',(x) K~'(y)

(8)

( i = I , . . . , E) where w = d e t (4,',), and y is some parameter to be conveniently chosen later (and not to be confused with the "internal" part of a world index). The factor w ~ is known as the "Weyl rescaling" and is simply a conformal transformation which gives a field redefinition of the vielbein in d = D dimensions. This redefinition is not necessary, but it proves convenient to do this in order that the reduced action has the usual normalization for the Einstein term. Note that Scherk and Schwarz insert their ansatz into the action when they do dimensional reduction, but in fact their ansatz is consistent with the equations of motion and they could equally well have chosen to work with these. The K, are a collection of E-vector fields which are linearly independent and which form a Lie algebra, ~3, i.e.

[ K,, KA = f o K k ,

(9)

where f~ are the structure constants of ~3. The assumption that these exist implies that M r is ditteomorphic to G / H , where G is a Lie group whose corresponding Lie algebra is ,~, and where H is a discrete subgroup of G, i.e. dim H = 0. This is why one calls this scheme "dimensional reduction on G / H " . If, for example, G is a compact semisimple group, and if d)i,(x) = 3a,i then in addition the vielbein (8) gives the usual invariant metric on G / H . The reason one wants to have (9) he true is that under the special subgroup of ( D + E )-dimensional general coordinate transformations generated by the collection of

r,°(x,y)=~'(x)K~'(y),

(10)

6V,,'~(x) = 0,

(ll)

one has that i i ) k 6A,.,(x) = -a~,( i (x) +f~kC (x)A~,(x)

&b ',,(x) = f~k~'(X)& ~(X) ,

(12) (13)

SO that under the local gauge transformations (10) generated by ~, V,," is a singlet,

547

W. Boucher / N = 8 supergravity theories

d~'a transforms covariantly under the adjoint representation, and A~, transforms as a guage field. In particular, because 613'~; truncates consistently under the symmetries generated by (10) (there is no y-dependence on the right-hand sides of (I I)-(13)) one will automatically end up with a theory in d = D dimensions which is invariant under the transformations (l 1)-(13) (modulo one small difficulty, see below). I have cheated slightly in deriving (l 1)-(13) (and so did Scherk and Schwarz, who did not seem to notice that they were cheating). Under transformations generated by ( I 0), 6 l~m~' = 0 and therefore 6 V~,~ = 0 (i.e. (11) is true) if and only if 6w = 0. From (13) one finds that 6w=f~k~J(X)W and so 6 w = 0 if and only if .fk k = 0

(14)

(~'J is arbitrary). This is equivalent to saying that the trace of the matrices in the adjoint representation of ~ must be zero, which is equivalent to saying that the adjoint representation of ~ is a subalgebra of (the fundamental representation) of sl(E, R). Therefore if the adjoint representation of ~d is not traceless, one sees that the lagrangian as it stands is not gauge invariant. Because the factor of w* is merely a D-dimensional field redefinition and because one wants the D-dimensional vielbein to be a singlet, one would probably be willing to drop the field redefinition in order not to restrict ~ unnecessarily. However, as I will show below, one will need (14) to be true for other reasons, and so (6) can remain as shown. The other symmetries that the ansatz truncates consistently are D-dimensional general coordinate transformations, generated by ~'~'(x, y ) = ~'~'(x), D-dimensional local Lorentz transformations, generated by £2m"(x, y ) = .Qm"(x), and local SO(E) transformations, generated by Dab(x, y) = .Qo~(x). I have mentioned these for purposes of reference (see the next section). It is straightforward to show (see [2]) that one will get the usual form for the Einstein part of the theory in d = D dimensions if one chooses y - - - l / ( D - 2 ) . Let us see what happens when one drops condition (14), so for now assume it to be false. In this case one finds that the action with the ansatz inserted is not defined because the y-integrals are infinite! To see this, one must use the fact (see [2]) that the only y-dependence in the lagrangian (density) comes from a factor (detK) '.Onehas f d~y(det K ( y ) ) - ' = I d~y'(detK(y'))

'

(15)

for any relabelling of coordinates y ~ y ' (note, there is no prime on the K on the right-hand side of (15)). Because M 7= G / H one may take in particular y ' = g(y) (the coordinate transformation induced by the natural action of G on G / H ) for some (constant) g ~ G. Now for this coordinate transformation, one has [13]

K'~(y') = D / ( g ) K ~ ( y ) Oy''~ c~ya ,

(16)

548

W. Boucher/ N = 8 supergravity theories

where D(g) is (an element of) the adjoint representation of G. Also, dry'= det (~y'/~y)dry, so inserting (16) into (15) one finds that f d r y ( d e t K ( y ) ) - ~ = (det D ( g ) ) - ' f d L y ( d e t

K(y))-'.

(17)

Therefore, if det D(g) ~ l the integral is infinite. But det D ( g ) # ! (for all g ~ G) is equivalent to fkk # 0 (the former is the statement in terms of Lie group theory, the latter in terms of Lie algebra theory). An interesting corollary to this result is that a group, G, for which the adjoint representation is not unimodular, cannot possibly have a discrete subgroup, H, such that G / H is compact, because then the integral (17) would be finite (note, det K is nowhere zero). Another way to see that the integral cannot be finite is to notice that putting aside the integral one still has a factor w-~ multiplying all the terms in the lagrangian and this is not gauge invariant. Now, if one were desperate to have a theory in which (14) is false, then one could just drop the ),-integrals and the factor of w ~ and pretend that they never existed. This is contrary to the spirit of dimensional reduction but leaves you with a G-invariant action. However, this is not a valid thing to do ~f you want to truncate s u p e r s y m m e t r y consistently, which is my ultimate aim, so from now on I impose (14). I have gone on at length about this because Scherk and Schwarz incorrectly justified this and because it is a subtle point. Note, although I am now taking (14) to be true, this does not mean that the integral (17) is necessarily finite. The integrand is an invariant measure on G / H so unless G / H is c o m p a c t the integral is infinite. In contrast to the case where (14) is false, here this does not matter because one can drop the y-integrals: (i) the final action will still be gauge invariant and (ii) s u p e r s y m m e t r y can still be truncated consistently. From the above discussion, (i) is obvious. That (ii) is true is less obvious. U n d e r a s u p e r s y m m e t r y transformation, 6L = V,; (dJ j; ) = V~,(gJ ~' ) + V,,(gJ")

(18)

for some Jd. The first term on the right-hand side of (18) will cause no trouble but the second could because normally the action is invariant only if its integral is zero, but here the ),-integrals are no longer being done. This is not a p r o b l e m within the dimensional reduction scheme of Scherk and Schwarz because when the factor from the volume measure is included the second term becomes ,'~,,((det K) 'K','(y)f'(x))

(19)

for some if(x) (see below) and one can show that this is zero if and only if (14) is true (see [2]). Thus one has the option of whether or not G / H is to be taken to be compact. If one believes that the extra dimensions are physical then obviously one wants G / H to be compact. In this case one sometimes finds interesting consequences, such as mass quantization (see sect. 4).

W. Boucher / N = 8 supergravity theories

549

So far I have concentrated on the part of the Scherk and Schwarz dimensional reduction scheme which deals with the pure Einstein action in the higher dimensions. This is very interesting in its own right. More interesting for me is the application to supergravity theories. Now, the vielbein is not the only field in a supergravity theory. One must also make an ansatz for the other fields. Scherk and Schwarz make the ansatz that all the other fields are i n d e p e n d e n t o f y (when one transforms all the " w o r l d " indices of the fields to local Lorentz indices, using the vielbein). This is not a p r i o r i the only thing one could do, as Scherk and Schwarz recognize. But it has two wonderful consequences: (i) the ansatz is consistent with the G gauge transformations so that one ends up with a G-invariant theory (this is obvious); (ii) the ansatz is consistent with supersymmetry (see the next section for a derivation o f this fact the hard way - it is obvious that this is true if you read the paper of Scherk and Schwarz because this ansatz can be seen to imply that the second term in (18) ends up looking the way it is shown in (19)). Although elegant and beautiful, one may w o n d e r whether there are any other dimensional reduction schemes worth considering. This is the subject which I address in the next section.

3. C o n s t r u c t i o n o f N = 8 t h e o r i e s in d = 4

Dimensional reduction is extremely useful in constructing extended supersymmetric field theories, as I have tried to impress u p o n the reader. In this section I state what I mean by the dimensional reduction o f N = 1 supergravity in d -- I 1 and then show that the only N = 8 supergravity theories in d = 4 constructible within this framework are the theories written d o w n by Scherk and Schwarz, [2]. The following is the scheme I will use. 1 will work locally (in the sense o f ditterential analysis and topology) but 1 will often make statements in global terms for the sake o f brevity. Everything should be regarded as only holding locally. M u c h o f what follows is obvious, but I think that stating a complete list o f hypotheses serves some purpose. To construct a 4-dimensional theory from an l l-dimensional one it is natural, and I will assume that one restricts oneself to the class o f manifolds, M ~, which are p r o d u c t spaces, i.e. one has (al)

M II = M 4 x M 7 .

One could think of more general situations, such as M ~ being a fiber bundle over M 4, but the rest o f what I have to say should automatically take this type o f situation into account. Next, there must be an ansatz for the I l-dimensional fields in terms o f the 4-dimensional ones. That is, (a2)

q~(x, y) = q~(qt(x), y ) ,

where x coordinatizes M 4 and y coordinatizes M 7. Here, q~ is a generic d = 11 field,

550

W. Boucher / N = 8 supergravity theories

t/, is a generic d = 4 field, and • is taken to be a local function in ~F (see below, (a3)). I have left indices off • and tp" for the sake o f brevity, I will assume that there are a finite n u m b e r of fields, ~. Specifying • as a function of ~ is specifying a dimensional reduction scheme. There will be a subalgebra of the original (super-) algebra of symmetries which is truncated consistently in the sense that (a3)

~Sq0=

,50.

Here, ~qb is the transformation o f the d = 11 field • which is specified by the relevant d -- 11 symmetry rules, and 8gr is to be uniquely determined by this equation. This is the reason why • should be taken to be a local function o f ~. Biran et al., [ 10], use a non-local redefinition of A,,~p and this would mean that the d = 4 version o f ~oA~,,,, would not be determined from the d = ! I version except implicitly. This is not a serious problem for them because F~,~,a has a local redefinition and if you are working with the equations o f motion and not the action then you may choose to work with F~,~,A instead o f A,~,p. Other than situations such as this, it is ill-advised to have • a non-local function of tp'. The reason for imposing (a3) is that if it holds then the theory obtained by inserting the ansatz (a2) into the original action (and doing the y-integrals) or into the equations o f motion is a theory in d = 4 with those symmetries being symmetries o f the theory automatically. (The only thing one has to worry about is whether or not the theory is well-defined. For example, the y-integrals might not be finite. See the previous section for a discussion o f this point.) Thus, for example, if (a3) can be shown to be consistent for eight (d = 4) supersymmetry transformations then one automatically has an N = 8 lagrangian without having to prove it. This is the beauty of dimensional reduction, at least for me. Note, the aspects o f dimensional reduction in the paper o f C r e m m e r and Julia are relatively uncomplicated and what makes their paper so deep is the fact that they made some duality transformations in order to illuminate the physics, i.e. find hidden symmetries, and these transformations were non-local so they had to determine some of the supersymmetry transformation rules of the redefined fields from first principles. The subalgebra o f the original symmetry algebra which truncates consistently will be assumed to include the following symmetries (cf. the symmetries which truncate consistently in the Scherk and Schwarz case): (bl) d = 4 general coordinate transformations, for which ~'~' = ~r~'(x). (b2) d = 4 local Lorentz transformations, for which 12,." = .Omn(x). (b3) d = 4 local gauge transformations. It is assumed that M 7 is diffeomorphic to G / H for some finite-dimensional Lie groups H and G (with dim G - d i m H = 7) and that the natural action of G on G / H (a s u b g r o u p o f the d = 11 general coordinate transformation group) truncates

W. Boucher / N = 8 supergravity theories

55 I

consistently. In particular it is assumed that sr" = ~" (x) K ~(y)

(20)

is a symmetry that is truncated consistently (this is exactly the ansatz that Scherk and Schwarz give for the case when dim H = 0, cf. (10)). Here i = I, . . . , dim G, and Ki are dim G (left-) invariant vector fields which satisfy (21)

[ gi, Kj] = f k K k ,

where f i g are the structure constants o f ~, the Lie algebra o f G. If G is c o m p a c t and semisimple then with respect to the natural (inverse) metric g~f3 = K T K ~ , the K, are Killing vector fields. More generally, this is true if and only if the Lie algebra has totally antisymmetric structure constants, and it is not true, for example, for the flat groups o f Scherk and Schwarz. If H is a s u b g r o u p o f SO(7) one has the option of a d d i n g a c o m p e n s a t i n g local SO(7) transformation, as Salam and Strathdee do [13], but whether or not this is d o n e will not affect by argument. Note it is n o t necessary to assume that G is compact, nor semisimple. The problems normally encountered when gauging a n o n - c o m p a c t g r o u p do not occur in this context because one can use the scalar fields instead o f the Killing form to contract indices. (b4) d -- 4 global SO(7) transformations, for which /2a b = ~a b (constant). It is natural to assume that the larger symmetry g r o u p generated b y / 2 a b = / / a b ( x ) is truncated consistently, i.e. one would have local SO(7) transformations, but I do not need to have this larger symmetry g r o u p present for my argument to work. (bS) d = 4 local N = 8 supersymmetry transformations. Physicists and mathematicians are still not at ease with the geometry o f supersymmetry and I, in particular, have no intuition as to a " n a t u r a l " ansatz for the d = I l s u p e r s y m m e t r y parameter, e ( x , y), and the major part o f this section will deal with determining an appropriate ansatz for e in terms o f the eight d = 4 supersymmetry parameters, e t ( x ) , I = I , . . . 8. The e I are p r e s u m e d to transform u n d e r the fundamental representation o f Spin(7) (the double cover o f SO(7)). It is only here that assumption (b4) is used. The supersymmetry transformations that are to be truncated consistently are given by

~,(~) =

~ o ( e ) + 8L(o~),

(22)

where 6Q(e) is as given in eqs. (2)-(4) and 8L(w) is a compensating d = 11 Lorentz transformation with parameter tom, ` = -to,~,,, = iKgl',,,~b,~ ,

(23)

which is needed for the reasons given below. By using local Lorentz symmetry in d = I i one may choose the (inverse) vielbein, I~,~'~, to have a triangular parameterization [1,2], i.e.,

o

V~l

"

552

W. Boucher/ N = 8 supergravity theories

Generally, the local Lorentz transformation which puts the vielbein in the above form is not consistent with the symmetries to be truncated consistently, and in particular, one needs to use a compensating d = I l Lorentz transformation for the fermionic symmetry (b5), which is why one uses 6 0 instead of 6Q above. Although the triangular parameterization is not necessarily the most general ansatz that one might consider, it seems natural to use it for dimensional reduction schemes, and a n y o n e who has ever done dimensional reduction with the vielbein has worked with this ansatz. Note that of course one is allowed to use any d = I I symmetry transformation one wants on the fields as long as it does not conflict with the choice of symmetries which are to be truncated consistently. A technical assumption which is motivated by physics is that (c2)

~'r,~'(X, y) = W(X, y) V,,~'(X).

One could easily think o f a more general ansatz. Assumptions (bl) and (b2) imply that one can write ~,',,~'(x, y) = w~(x, y) V,,M(x) , (24) where I is some index which is d = 4 local Lorentz and general coordinate invariant, and where the V,. ~d are functionally independent, and similarly the w ~ are functionally independent, over the vector space o f functions o f x. Because one wants a sensible theory to emerge one only wants a term o f the form w(x, y) V,,~'(x). Other terms would involve fields o f spin less than two and one sees that the lagrangian would then involve terms with higher derivatives (noting (b3)), and one would also find derivatives o f spin-0 or spin-I fields appearing in denominators. This is unphysical. For the same reason, w is taken to be a function only o f y and of the d = 4 scalar fields (and not o f their derivatives, or other fields). Note, there is a possibility that although one ends up with higher derivatives, etc., appearing in the lagrangian, one might be able to make a d = 4 field redefinition at the end o f the day and eliminate this problem. Except in the trivial case when the w i" are functionally dependent over the vector space of functions o f x (which I have excluded above) this does not seem very likely. Finally, I think that assumption (c2) could be considerably weakened or d r o p p e d altogether, but only at the expense o f a considerable increase in complexity of the first part o f the analysis given below. The dimensional reduction scheme set out above is fairly general. One can make objections to it but one has to remember that there has to be some kind o f input, and this particular input seems reasonable, and has an accepted precedent in the literature. One could o f course attempt to study different or even more general situations. The last ingredient in my scheme is without historical precedent but is again not unreasonable. The purely "internal" part o f the d = 1 I general coordinate transformation parameter, sr ' , generates a subalgebra in the obvious way. I assume that (d) the only subalgebra o f d = 11 algebra generated by the sr" which truncates consistently is the subalgebra generated by the gauge transformations given in (b3).

w. Boucher/ N = 8 supergravity theories

553

If the subalgebra which truncates consistently is finite-dimensional (for fixed x) but larger than ~3 then one has that M 7 is ditteomorphic to G ' / H ' where G c G', H c H', H', G' finite-dimensional Lie groups. For example, S 7 = S U ( 4 ) / S U ( 3 ) , but also S 7= SO(8)/SO(7). In doing dimensional reduction that consistently truncates the supersymmetry transformations one may find that although one started by considering M 7 as G / H one may be forced to look upon it as G ' / H ' instead. In this case, in order for assumption (d) to be valid, one should start with G'/H' and not G / H . Actually, I have glossed over one point. I am really assuming that the algebra, ~3, generated by the ~'" which is local in x is finite-dimensional (for fixed x). There may be additional global symmetries, ~3, i.e. symmetries with ~'~ independent of x, but whether there are or not will turn out to be irrelevant. If the (local) subalgebra which truncates consistently is infinite-dimensional (for fixed x), then because the subalgebra generated by ~3 is local (in x), the rest of the subalgebra is also local (in x) and one would have a theory in d = 4 with a finite number of fields but with an infinite number of local symmetries, all but a finite number of which would be non-propagating. This would be something I have never heard of occurring. Before I start analyzing the N = 1 theory in d = 11 I need a trivial lemma. Let Coo(x) = (smooth functions from M 4 to R or C). Let C°~(x, y ) = (smooth functions from M It to [~ or C). For the purposes of the lemma only, I will use indices which do not mean what they mean elsewhere in this paper. One has the following: Lemma. lffT(x, y), I = 1 , . . . n, are n (real or complex) m-vectors such that f i t f i is independent of y, then there exist functions ch(x) and m m-vectors g~(x,y), I = 1. . . . m, such that f ~ = chg ~ and g~*gJ is independent of y. Proof. If n <~ m this is obvious. Suppose k is the first index such t h a t f k is not linearly independent over C°(x, y) from the (f~, a < k). Thus

fk(x, y) = dg"(x, y)fa(x, y)

(25)

for some functions d k" ~ C°~(x, y). Now, by assumption, c ~b = t?b~ = f ~ , f b are functions independent of y such that det (c ab) ~ 0. The functions

eka =_cabdkb = f~* fk

(26)

are also independent of y so by inverting c one finds that the d are y-independent. Thus fk is actually linearly dependent over C~'~(x). Continuing in this way one sees that since only m of the vectors f can be linearly independent over C°°(x, y), only m can be linearly independent over C°°(x). Hence the result follows, with the vectors g chosen such that they contain the f which are linearly independent over C°~(x) and such that the remaining g are chosen to be zero or as convenient. Corollary. If gl(x, y) are m m-vectors such that det ( g 1 ~ g J ) ¢ 0 and such that g1*gJ is independent of y, and if f ( x , y ) is an m-vector such that f * g t is also independent of y, then f = f 1 ( x ) g l .

554

W. Boucher / N = 8 supergravity theories

The key to finding a dimensional reduction scheme consistent with supersymmetry is of course to properly analyze the supersymmetry transformations. Getting the bosonic symmetries to truncate properly is almost a matter of course. However, to analyze the 8~ transformations straight off is not a good idea because (i) they each involve most of the fields and (ii) they involve the supersymmetry parameter e (which a p r i o r i may have to be field dependent if the truncation is to be consistent [l l]). Thus one would have to have ansatzes for all of these, l have only made an ansatz for V,,". The idea is first to analyze the commutator [3~(e~), 6~)(e2)]. Because 8~ truncates consistently, so does its commutator. In short, if ~l represents the ansatz of a generic field qb then (8<)~)1 = 8Q(~l) so that ([6'~, 6~]~)1 = [8'~, 8~](~1). By doing the calculation one finds that (cf. [8]) [ a o'( ~ , ) ,

:
<~b(~)]

•

~/e,-¼o~, ~'e2)

+ ,S, ( f / + ~,(~o,)o~: - ~.(o~2)o~, + [~o,,
+ 6o(e,

)~2

'So(~2)o~,

-

-

,.o')

(27)

+6.t.¢...(A)+a'o(6b(e,)e2-6b(e2)e,).

where ~;' = - i g ~ l ' ~ ' e 2 ,

=~KE,L~

-z,*'r/

A ~ - ~Kell ~ e 2 3

-

-

T/ ~ --

)e2r~,7;;+ ~" to~

3sr~;A~,

, (28)

and to' is defined using e ' = e+~w2" ' y e , - ~ tLo , • ye2 in the same way that to~.2 are defined using e,.2 (see eq. (23)). Also, w . y - - - w m , y " " . In (27) I have allowed for the possibility of eL2 being field dependent. One can verify from (27) that [61, 6~] 1)~~' = 0 which it of course must. Eq. (27) is extremely interesting in that since the commutator truncates consistently, all the terms on the right-hand side must also be symmetries that truncate consistently. Generally speaking, this may lead to the existence of more symmetries being truncated consistently than what one might have first expected. See sect. 5 for a (short) discussion of this. Another interesting consequence of calculating the commutator of two supersymmetry transformations is that, as is well-known, it closes on the bosonic fields, but calculating [61, 6[]~,~ gives the above a n d a term proportional to the ¢,~ equation of motion R~. Thus R~ must truncate consistently. That is, the ~b~ equation of motion derived by varying the action with the d = 4 ansatz inserted must imply the equation of motion derived from the original action. By acting on R,~ with 6~ (which is a symmetry transformation which truncates consistently) one sees that the same must be true for the vielbein and the 3-index photon. Thus any dimensional reduction

W. Boucher / N = 8 supergravity theories

555

scheme consistent with supersymmetry is also consistent with the equations of motion (assuming that the appropriate y integrals are finite). This is remarkable. Note, this result is true independent of any detailed assumptions concerning the dimensional reduction scheme other than that supersymmetry be truncated consistently. It is often stated in the literature (e.g. [9, 14]) that setting the scalar field part of the vielbein to zero is "inconsistent" (they use the word slightly differently from how I do). Well, it depends what you are trying to do. If, as in [9], one is trying to do dimensional reduction by dealing directly with the equations of motion then this is true, but if, as in [14], one is doing dimensional reduction using the original action, then there is no reason to demand that the procedure be "consistent" with the equations of motion (strangely, in [14], after rejecting one scheme because it is inconsistent, he does not check that his ansatz is consistent). The dimensional reduction scheme of Scherk and Schwarz truncates supersymmetry consistently and so it must, by the above argument, also be consistent with the equations of motion. This does not seem to have been realized previously. I have explicitly checked ([15]; details sent upon request) that the pure Einstein part of their action is consistent with the equations of motion (in their scheme, the pure Einstein part must be consistent if the entire action is). This last result is actually independent of dimension (i.e., in the notation of the last section, this is true for arbitrary D and E) and should prove useful for analyzing higher-dimensional Einstein theory independently of whether or not one is interested in supergravity. After I discovered the above result I realized that there is an easy group-theoretic argument why it is true. Manton's dimensional reduction scheme [16], on G / H is also consistent with the equations of motion and, although his ansatz is one of invariance rather than covariance, his group-theoretic explanation of this result can in fact be used here (see sect. 5). Finally, one therefore realizes that the d = 4 ground states that Scherk and Schwarz discovered (see the next section) are also solutions of the d = 1 1 theory and if one is enthusiastic about Kaluza-Klein theory one can then talk about spontaneous compactification. I have not found any use of the above observation in trying to analyze dimensional reduction schemes consistent with supersymmetry. But I think it is truly a fascinating result. ~ " = - i f ~ F " e 2 must be independent of y (this follows from (el) and is needed in proving that [8'~, 8~] ¢'~" = 0; one also wants d = 4 general coordinate transformations to be the only symmetry with a d = 4 vector index on the corresponding parameter, and this implies the same result). This is the key observation of my analysis. The most general ansatz for the d = 11 supersymmetry parameter, e, in terms of the d = 4 ones, e t, is e (x, y) = w-'/2( e i (x) ® #~(x, y) + iyse ~(x) ® IX~(x, y) + iy"e i (x) ® Ix~(x, y ) + y " y s. e ' ( x ) ® t t ~ , , ( x , y ) + ~ y " " e ' ( x ) ®

/.t,.. ' (x, y ) ) ,

(29)

556

w. Boucher / N = 8 supergravity theories

where the Ixo~, etc., are (commuting) 7-spinors. One could add derivative terms such as v p e l ® i x ~ o p, etc. (where V, is the d = 4 covariant derivative) but since e I, Vpe ~, VoV~e I, etc., are all functionally independent one can repeat the argument given below to show that these derivative terms just provide field redefinitions o f e I, and so for the sake o f clarity I shall ignore them. e is a d = I 1 M a j o r a n a spinor, e~ are d = 4 M a j o r a n a spinors, and Ix~, etc., are d - - 7 (pseudo-) M a j o r a n a spinors (I use the terminology o f [17]). I take as d = I1 F-matrices F " = y " ® l and F ~ = y s ® y ~, where y " (y~) are the d = 4 ( d = 7 ) y-matrices. There exists a basis in which the y" are real and antisymmetric, and in which the Ixo~, etc., are real, and I shall work in this basis. It proves convenient to work with 2 - c o m p o n e n t spinors rather than 4 - c o m p o n e n t spinors. With an appropriate choice of basis for the y " eq. (29) can be rewritten as I

e=

1 ,¢~

I -- .I

ra\

/-iA,~-1 ...... \e

tx;Ix

trio"

-IAr:~

)

I

I .~

AA I , ~ - I

mnx

S

1 ~

1

, ...... A -IB~

ea~Ix,~--~O"

) ne

\

I l,

~_)IX,,,,,/

(30)

where ym=

'

Y""=-i\

o

6

Also, ,

Ix =Ix

,

+iIx',

Ix..

,

.

1

=

It is convenient to relabel m as A,4 in the usual way, with o"°6= ½(tr°+ t73), etc., so that (trSB)Aa = 8 As 8 A, etc. Writing out g ~ F " e 2 = w - t V f " g ~ F " e 2 = w - ~ ¢ " ( x ) in full one finds terms involving •- I I -J I~,1 J W etAe2A , WtAe. 2a, and the complex conjugates o f these. Since these are all i n d e p e n d e n t (as functions o f x) it follows that the corresponding coefficients must be independent o f y. The coefficients are e

AS

e

AB

-I¢

-J

Ix

I

IX - - ~ e

"~-IIXltABIXJAB

e

AC

It

/2 /2

J B ( " "F

e

BC

AS-It

IX

dI- IX

IRA("

IX

J

IX

JAB

1

-~e

AB

tx

I~.AB-J

IX

I*AB IX JBA , I

IX-~IX

ItAC

-JS¢~

IX

(31) 1

-~IX

IRA(?

JBC

IX

(32)

(and their c o m p l e x conjugates) where IX~R = ~e t ABIXA,aSS I . is the self-dual part o f i x ~ , , IX1,as = (IX,As),, and IX,*AB = ( i x , s a ) * . It can be shown ([15]; details sent u p o n request) that the y - i n d e p e n d e n c e o f (31) and (32) implies that by a redefinition o f e ~ one may choose Ix~a and IX~s to be zero. That is, one may assume e = w

\:,A®121},

(33)

557

W. Boucher / N = 8 supergravity theories

or equivalently, (34)

e = w - l / 2 ( e ' ®tx~ + i y s e J ® I x ~ ) ,

and the y - i n d e p e n d e n c e of f,l'~'e2 implies that c U =/.t tt/.t j z

IT

J--

IT

Jx--

.z

=

(~IJ

IT

J

IT

J\

= ~ z o /Zo-r/.t5 /~5/-rzt/Zo p - s - / z 5 /Xol

(35) is i n d e p e n d e n t o f y. c is a matrix with real non-negative eigenvalues and by Spin(7) invariance it in fact has no zero eigenvalues (for all field configurations except on a set of measure zero). By a redefinition of e I one m a y take c " = 8 " , which I do. That is, /z I t tt J

.

(36)

Note, if one is willing to accept eq. (33) from the beginning as the ansatz for e then one can d r o p a s s u m p t i o n (c2) because it is then implied by the fact that flF'~e2 is i n d e p e n d e n t of y (assuming of course that the d = 4 vielbein a p p e a r s s o m e w h e r e in the sum (24)). Next, analyze ~'~ = - i f , F ' ~ e 2 . By a s s u m p t i o n (d), ~'~ = ~ i ( x ) K ?(y). Note, no global symmetries (which are not also local ones) can occur in ~'~ because it is h o m o g e n e o u s in f~(x), e J ( x ) . Using (33) and (36) one has that --

f,F'~e2 = e l f -,,, = W

m

A

a

e2 V,, + e l i "~e2Va ~~ -

(tEl

.~ - I A - J

IA-IA_-IA

/

el e2A(#

t~

A

"I-E I

E 2

I't

IA, ^ ~ )VAA"~EI

IA

E 2

a-J',

J ,-1, E2A([.I,

" a

3' # )V~ ),

3,

a

J)f,~

(37)

a

where VAa ~ V,, . Each term in (37) is i n d e p e n d e n t of the rest. Thus one must have f',,° = w V , , " ( x ) A ~ ( x ) K T ( y )

(38)

for some (real) functions A~,, and ( ~ l*3"aI,LJ ) Vac" = wglJi(x) K ~(y)

(39)

for some (possibly c o m p l e x ) functions gU,. /j~,3~/zJ = _/2J,3o/x~ so gU~= _ g J , . Eq. (38) determines the ansatz for V,, ~ exactly. If one had bothered to check, A c¢ gauge invariance would already have told you that V,, has to include a term of the form (38) but whereas gauge invariance in principle allows other terms, supersymmetry says that V',," must have exactly the above form. By using (36), eq. (39) implies that

3alj J = wg KJi(x)l~ KK ~(y) ~/ a = wc~agKJi(x)ft K, a

ct

(40)

where ~b~--- V,, K i . Having (partially) analyzed the [8'1, 8~] c o m m u t a t o r it is time to turn to the s u p e r s y m m e t r y t r a n s f o r m a t i o n of the fields. First, study 8~ 5',~'~. Using eqs. (2) and

W. Boucher / N = 8 supergravity theories

558

(22) one finds that

6bf',." = iKgr"6,.f'J',

(41)

^ ~. 6 o' V ,^" " = ~KeF ", Or. V. ^ . +. .I K. e F 6,. Vo ^ , , + o" c e- F , . 6 ° V.

(42)

! I.t t N o w ~',,~' = w V , , ~ ' ( x ) by assumption (c2), so 6Q(,',, ~'= w 6 ' o V , , ~ ' + V,, 6 o w . w, being a function o f the scalar fields, is such that the 6 ' 0 w are determined by their supersymmetry transformations (by assumption (a3)). One can write

•

-

6'oV,,," = l K e l

-n

6inV,

t.t

- Vm"6 o' ln w.

(43)

The left-hand side o f (43) is independent o f y so the right-hand side must also be. 6,. is a sum o f terms o f the form 4'aaB ® ;L'~ ~aSa ® ~ ] ,

(44)

and thus gF"6,,, is a sum o f terms of the form (45)

e IBd/AABli t ~ . + gIBI/L~AB/X t* h .

N o w e t is arbitrary and so it is not possible to choose an ansatz for ~,, so that (45) B R is proportional to 8a6 a and both terms in (43) must be y - i n d e p e n d e n t separately• In particular, 6 ~ l n w is y-independent, so that In w = f ( x ) + g ( y ) . But by gauge invariance, one must then have g ( y ) = constant and so w is just a function o f x and merely provides a d = 4 field redefinition. From now on in this section, for the sake o f clarity, I set w = 1 (although it is easy to drag it through all the calculations if this is desired). Note that the people who have done the linearized analysis o f d = 11 supergravity a r o u n d the ( A d S ) 4 x S 7 solution, [9, 11, 12], have had a w which was y-dependent. This is because they were using the metric, and not the vielbein, in their ansatzes. It is important to stress that the vielbein is the fundamental field, not the metric. The equation corresponding to (43) for the metric is 6'0g "~ = 2 i K g F ~ " 6 ") - 2g"~6'~ In w,

(46)

and in this case a term in 6 ~ o f the form F " 6 would create the possibility o f having w y - d e p e n d e n t , if it were indeed correct to consider only the metric. G o i n g back to (43) (with w = 1) one sees, by using (36) or the corollary to the lemma, that t/,,. must be o f the form I

6,. = 6 m ( x ) ® t Z o

I

•

I

+ z'/s6,.(x)®tx5

/

(47)

for some (Majorana) spinors 6 ~ ( x ) . N o w 6'0(/m ~ = 6'0( V , , " ( x ) A , . ( x ) ) K i ' '~ and the right-hand side of (42) must have the same y - d e p e n d e n c e as this. The first two terms on the right-hand side are already o f this form, by eqs. (38), (39) and (47). Thus

559

W. Boucher / N = 8 supergravity theories

the last term on the right-hand side must also be of this form and so ~ba = tk"(~b't(x)® ~o~ + iT5~" ( x ) ® / z s ~)

(48)

for some ( M a j o r a n a ) spinors &it(x). a ot V^, , cl does not seem to be useful to analyze at this point. Not enough is known A about V," yet. So turn to the s u p e r s y m m e t r y t r a n s f o r m a t i o n of #',1. From (3), (22), and (47) one finds that

(''

~O

~lra = /

')

~',

7lar:zx

-I

\ OQ~rn ~ p . I

^

= - - Vm

~

I

I

e-raT

np

-- I

n

a

etOmnp~r~Y "YSY etOmna

K _~ l

ah

I •

aY e~mab--3617~ t

np ah

•

~17,~ 7

~

I •

-

np

a

r'.

T57 ernpqa

I

•

n

e r n p a b - ~ r Y m 757abCeFnabc

I . abcd r q- i-dZll~lm~ El'abcd

--6IT

npq

-- l

a

I •

l'}lnpqEFmnpq n

~/s'y eF, nnp,,-r, zT 7

tab

eF,,,,,at,

(49)

! • abcel'mabc. ,~ T~lT5),

In (49) I have put all the fermion fields to zero on the right-hand side for the sake of brevity. This does not change any of the results given below. e involves both I and 7s terms. Thus the terms on the right-hand side of (49) which involve (i) 0 or 4, (ii) I or 3, or (iii) 2 7P-matrices must separately have the same y - d e p e n d e n c e as the left-hand side of (49) (this is easiest to see in 2-spinor notation). There are five terms which have l or 3 7P-matrices: I

n

a

I

I •

~7 757 eWmna±~ZT,~ .}_

I • ~|~/mg

abed

np

7

r EF'abcd

ah

r~

er.pab

I. npq ~2 - - ig/~// e, mnpq,

(50)

or equivalently,

3\__g,a®¢,

rh.,,F--6 \e,a®,),at,12,,J f,,,.,,t,

,

(51)

where *F,.,ah =½e,.,PqFpqah and F,,,pq = e,.,pqF, y" flips the u p p e r and lower 2c o m p o n e n t spinors of the 4 - c o m p o n e n t spinor. Let M be the matrix defined by (/~1) ( a is a d = 7 spinor index in this line only). Note M t M = 1 (cf. (36)). Eq. (51) and the above remarks imply that I

a

l~mnab

-~M~=.F

.t- ~ | T

lVl~mnFabcd

-61~' = /~?.,.(X) t. ~ . ~""m°ab .~

(52)

560

W. Boucher/ N = 8 supergravity theories

for some (matrix of) functions f m . ( x ) . Multiplying by M T on the left leads to a y-independent quantity. By explicit calculation, using (38) with w = 1, one finds that I

i

w,..a = -~F,..~bai, m

n

(53)

i

i

where F~. = V. V~ F,.. is the Yang-Mills field strength corresponding to A.. In (52) there are thus two terms proportional to rl,.. and three terms antisymmetric in ran. These both lead to y-independent quantities. In particular, lilT

all

--

I

IIT

~tw 3' P,qto,,,.,,-rT~M 2/

abl.*

I.

T

tVl F , . . . b - ~ i M

ab

3, M F , . . . b = f , . . ( x )

(54)

for some (matrix of) functions f , . . ( x ) . Similarly (putting fermion fields to zero) t ,I 1 , ..,, { 8o¢,,,®,.,. '/ 8oq, o ,-°,\SbF,,A ® ~z, ) =

1

.

--

tr

I

m

b

K I

bc

--

1

.

+~3' eo~ab~±iazt3" I.

-36t3"

mnp

I • --~|3"m3"a

b

r'~

mnpq

r~

y53,~er~.pq

--|.

3"~ er~.pb t ~ t 3 " bcd

I

•

mn

eF~bcd + i~t3"a

bt"

r -~

Ys3,~ er~,s,.

bode

t-

eFhca~

li3ranPeFam.p - - 6 3I, ,n. 3"5"111 be ,~ Farnnb I"

m

bc

r'.

1 •

-~t3, 3, e r ~ h ~ - - ~ t 3 , s y

bcd

r.

er~h~a.

(55)

Doing an analysis on the terms in (55) with two yP-matrices similar to the one done above for (49), one finds that M*yh~MF~.b~ + 4M,3,hMF~.h. = ~b,~'i f,..(x)

(56)

for some (matrix of) functions f~,.(x). Now M ' 3 , h M is antihermitian while M*3"b*"M is hermitian (with my conventions 3'~ is antihermitian), so one must have that each term separately is of the form given on the right-hand side of (56), and in particular. M *yb MF,,,,,ab = ¢.,fm.(x) i

(57)

for some (matrix of) functions f ~ . ( x ) , and so MT3,ab MFm.,a, = M T 3,'~M M t3"hMFm.ab T

a

i

= ¢b,,~M Y M f m . ( X ) .

(58)

Note, (57) implies that f~.. is hermitian (and tracefree). Inserting (58) in (54) gives 1

"F

a

i

4 - _ 1 . tci

z M 3' M 6 . , ( - F , . . - 3

j,..-~/f',..)=f,..(x).

(59)

W. Boucher / N = 8 supergravity theories

561

One can write

f ~ . ( x ) = A ( x ) F m' . ( x ) + B(x) F~. +" ..

(60)

for some hermitian matrices A, B. Both the term multiplying F~,. and that multiplying , i F... in (60) must separately be y-independent. The term multiplying F~,. is I T nM ya Mth~(-l-½B-~ia),

(61)

and that multiplying *F~,. is

¼Mry"~b.,(~A - }iB) .

(62)

Multiplying (61) by - 2 i and adding it to (62) implies that MVy"M&,,(I +~iA)

(63)

is y-independent. Since A' = A, det ( ! + ½iA) ~ 0 and by inverting this matrix (which is y-independent) one finds that

Mry~Mcb., = f ( x )

(64)

is y-independent. (64) implies that - / " M ~ , =/~]tf.

(65)

which in turn implies that

yby"Mrb.~dabj =

"),bl~fi~bbj

=

Mfjf,

(66)

and so MJ~ ) f ) = yby"M~b~bbj) = ~ l ( b y a ) MqS~,qSb~

= Mep~,~oj.

(67)

Thus h 0 -= -~b," ~b,j = - ~ Ji)(x)

(68)

is independent of y. The rank of K," is 7 since G acts transitively on MTand so the rank of ~b~ = K~ ~,a is also 7. The lemma and its corollary now imply that

ebb(x, y) -- 4)~(x)g)b"(x, y)

(69)

for some functions 4~b(x), and with 4,bc4,ca independent of y. Thus

K'~(y) = g~(x)(CS~b~'~").

(70)

For fixed x this says that only seven of the K~ are linearly independent (over ~). Thus dim H = 0 and dim G = 7. In particular i = 1 , . . . , 7. In the same manner as was used to show that one can choose M * M = 1, one can redefine ~,~ so that d,b"(b,.~= - 6 ~ , , i.e. ((bb")c 0(7). (Remember; my signature for

562

W. Boucher / N = 8 supergravity theories

M 7 is negative.) The lagrangian is invariant under d = 11 local 0 ( 7 ) transformations and by doing such a transformation one ends up with a theory equivalent to the original one, i.e. it is possible to take ~bah = 6~. This is assuming that this transformation does not affect the symmetries ( b l ) - ( b 5 ) , which one will eventually see below. Then ~.Zaa

i

(~

= ~,,(x)K, (y),

(71)

where ~b~a~ = ,5]. The ansatz for the vielbein is now complete. ^ (v,~(x) V'~'~ = 0

V,. ," ( x ) A ~, ( x ) K , ( .y ) ~ qb~a(X)K ~ (y) ],

(72)

with dim G = 7. This is the ansatz of Scherk and Schwarz for the vielbein (cf. (6)-(8) with w-- !). It is consistent with the symmetries (bl)-(b4). Eq. (64) now implies that

MTT"M= f ( x ) ,

(73)

so that MTM

1

--

• nr 7! e'~'"7~vt "Ya I " " " T ~ M

I

= 7--'.co,. .... f " , f ' ~ 2 . . , f"6f°~

(74)

is y - i n d e p e n d e n t . Thus M ( x , y ) = A(x, y ) U ( x ) , A T A = 1, A real, U t U = I, and f a = UT AT.yaAU = UTybUDff~(A),

(75)

where D = (Db a) is the adjoint representation of 0(7). (75) implies that Db"(A) = ~ tr ((Jf~U*yb)

(76)

is y - i n d e p e n d e n t , i.e. A = A ( x ) . Thus M = M ( x ) . Therefore by redefining ~ . A~- . - ) ~ . A'~ = M n e ~ , and letting h i be the 7-spinors with c o m p o n e n t s h ~. = 6~, one has (dropping primes) e = e t ( x ) ® h t.

(77)

By going back and studying (49) and (55) (which are unaffected by the previous 0 ( 7 ) t r a n s f o r m a t i o n since that will just add fermion terms to the equations, and these do not change the argument) and using a technique similar to that used above, one finds in quick succession that F,,,pq = Fm,vq(x),

(78)

w. Boucher / N = 8 supergravity theories

563

F,,,;~ = F , , , p a ( x ) ,

(79)

Fmnab= F,,,~b(x) ,

(80)

F,,abc = F,,,~bc(x),

(81)

Fabcd=

(82)

F~bcd ( X ) ,

and of course ~b,. = ~ ( x ) ® A ' ,

(83)

~b~ = ~b~(x)®A'

(84)

Eqs. (72) and (78)-(84) are exactly the ansatz of Scherk and Schwarz (with w = 1). This is consistent with all the symmetries ( b l ) - ( b 5 ) . Remember, however, for the action to be consistent one must have det D ( G ) = 1, D ( G ) the adjoint representation. of G, or equivalently, f k = 0, f j k the structure constants of qd.

4. Ground states of N --8 supergravity theories

The theories of Scherk and Schwarz provide us with a large class of N = 8 supergravity theories. But do any of these theories have ground states? In this section I will first analyze Lorentz-invariant ground states in d = 4 of the Scherk and Schwarz theories which have been dimensionally reduced from N = 1 supergravity in d -- 11. Because I am only looking for Lorentz-invariant ground states I will only have to examine the scalar potential for stationary points to find such solutions (modulo one small point, see below). This may seem like a trivial task until one remembers that there are 70 scalar fields and thus one is trying to find stationary points of a function of 70 variables, which in this case is fortunately not too complex. One should mention that one has a big advantage over people who study the scalar potential in the N = 8 de Wit-Nicolai theory [18, 19], because for this theory, just to write down the scalar potential in closed form in a "unitary gauge" is difficult. Despite the relative lack of complexity, I have not had time to do an exhaustive survey, so these results are preliminary and incomplete. The bosonic part of the N = 1 supergravity lagrangian is given in eq. (1). When one reduces the theory to d = 4 the part of the lagrangian which is important for finding Lorentz-invariant ground states is straightforward to determine using the recipe of Scherk and Schwarz [2], and the result is i L'-

I

4K 2 R - i - - ~ r 2 wfjkfm~(h~,h' '

J"hk" +26mt~Jthk.)

K'#),po" - a ~I, ~~A,--31~ • ~,~,~r-- ~ w F , j k t F Okl

- CW l e~'"('F~,~p,,eiiktr~'pFijklAm, p

(85)

W. Boucher / N = 8 supergravity theories

564

(i,j,...--

1,...,7)

where F~,,,~ = 4at,A,p~, ] ,

(86)

F,jk, = 6 f ~ A a , j m ,

(87)

and c is a constant whose exact value need not concern us (but which is 3K/(144)2). I have included the Einstein term for reasons o f comparison. Recall also that w = (det hiJ)l/2 > 0. The Aok are 35 of the scalars coming from the purely "internal" part, A , ~ , o f the 3-index photon, A,~;~; (the other 7 come from A~,,,, and do not contribute to the scalar potential). One is looking for stationary points o f L'. The h o and the A o k a r e constants but F,~p. satisfies F~,.t,,. = ae~,,,,,

(88)

for some constant a. (88) is the analogue in d = 4 o f the F r e u n d - R u b i n ansatz [20], in d = 11, although, again, because any solution in d = 4 o f the dimensionally reduced theory o f Scherk and Schwarz is also a solution of the original theory in d = 11, one m a y view (88) merely as the usual F r e u n d - R u b i n ansatz. Historically, at the same time that Freund and Rubin were using (88) in d = I1, Aurilia et al. [21 ], were using it in d = 4 to introduce an extra parameter, a, into the C r e m m e r - J u l i a theory. They noticed that this induced a positive "cosmological constant" in the lagrangian (it is, in fact, not a cosmological constant but rather a scalar potential because o f the factor o f w -3 in front of F,,,p.,F "~p" in (85)). Although the scalar potential has no stationary points they w o n d e r e d whether when the scalar potential was introduced in the de Wit-Nicolai theory (yet to be completely constructed when Aurilia et al. wrote their article), which induces a negative cosmological constant, one might be able to find stationary points o f the total potential for which the sum o f the two cosmological constants cancel. Although the F,..,,~.F '*''~ term occurs in the (naively) dimensionally reduced C r e m m e r - J u l i a theory, it is not clear to me that in the final version o f t h e C r e m m e r - J u l i a theory (after all the duality transformations have been made) that such a term can still exist ( C r e m m e r and Julia just quietly drop it). Thus, since the de Wit-Nicolai theory is obtained directly from the C r e m m e r - J u l i a one, I do not know if the c o m m e n t s o f Aurilia et al. are relevant for the de Wit-Nicolai theory. However their c o m m e n t s are certainly relevant here. 1 will show below that unfortunately the positive cosmological constant induced by (88) for a # 0 is more than compensated by the negative cosmological constant coming from the normal scalar potential. However I will also prove the slightly more fortunate result that for many choices of the 7-dimensional Lie algebra, ~, the only stationary points of the total (effective) scalar potential, V~, are when a = 0, so the F r e u n d - R u b i n ansatz becomes trivial, and in this case V~f~= 0. Therefore, there is no induced cosmological constant and the g r o u n d state is Minkowski space. This is g o o d news. This result may actually be true for all choices of ~ but 1 have not m a n a g e d to prove it in this generality yet.

W. Boucher / N = 8 supergravity theories

565

Let V = l l K 2 f j k f t ~ , ( h i , h J m h k" +2tSmS~hk").

(89)

Inserting (88) in (85) one finds that L'-

l

4K:

R - w V + ½a2w-3-~WFvktF'Jk~ + 2 4 a c w - l e°ktm"PFoktAm,p

(90)

(Although in (90) it looks like the third term on the right-hand side is inducing a negative cosmological constant it is in fact inducing a positive one as can be seen by calculating the stress-energy tensor from (85) - one cannot calculate it from (90)). Note, (90) is p o l y n o m i a l in Aij k and although it is not quite p o l y n o m i a l in h 'j, it is still r e a s o n a b l y simple c o m p a r e d to the de Wit-Nicolai potential [3, 18, 19]. The part of (90) that involves Aqk is a quadratic form, Q, in Auk, where Q=2swFijkiFijkt - z4~, a c w -l e i j k l m n p rrok,Am, -p.

(91)

Stationary points of a quadratic form are trivial to calculate. In particular, at a stationary point, Q = 0. If Q is non-degenerate then the only stationary point is the trivial one, A0k = 0. On the other hand, Q is m a x i m a l l y degenerate if fjk = 0, and in this case the Auk can be chosen to be arbitrary constants. If Q is not maximally degenerate then for general choice of b = a w - 2 it will have a non-zero rank, p, p <~ 35. H o w e v e r for certain choices of b (which is a constant determined by the rest of the variational p r o b l e m ) the rank of Q will dip below p. Because b a p p e a r s in a simple way in w- t Q one sees that there are no more than p choices of b where this happens. Exact details d e p e n d on what .~ is. Below I will show that one is often forced to have b = a = 0 and in this case the only stationary points of Q are when F, ik~ = 0 (see (91)), which does not necessarily imply that Aok = 0 (again, it d e p e n d s on what ,~ is). The equation one gets by d e m a n d i n g that L' be stationary with respect to h 0 is ~V

( ½ V + 3 a 2 w 4+~Fk,,..Fk"")hO

--=

8ho

(92)

(where I have used Q = 0) but from (89) one sees that 8V h~j 8ho

-V,

(93)

so combining (92) and (93) one finds that V

=

3 2 " - ~1 4 /~4a w " + AFuk~F 'Jar) ~ O,

8V tSh,i

}Vh 'j .

(94) (95)

From (91) (with Q = 0) one sees that the equality occurs in (94) if and only if a = 0.

566

W. Boucher / N = 8 supergravity theories

From (85) one may calculate the stress-energy tensor and hence the cosmological constant and one finds that A = -K2(~a2w 3 + 2~WF,jk,FiJk~) <~O,

(96)

again with equality if and only i f a = 0. Thus if the F r e u n d - R u b i n ansatz is non-trivial (i.e., a < 0 ) the induced cosmological constant is negative. Note, however, that if hij is a solution o f (98) then so is Ah~j for all A > 0 . When h ~ A h , V - - , A - ~ h so that one can make the cosmological constant as small as one wants, if it is negative. 1 remind the reader concerning some points in Lie algebra theory [22, 23]. Let r be a Lie algebra, r is called solvable if r ~") = 0 for some n, where r(~)=[r ~-~), r (~- ~)] and r ~°J= r. ~ has a unique maximal solvable ideal, called the radical o f ~, which I will denote by r. s = ~3/r is again a Lie algebra and it is semisimple (an algebra is semisimple if its radical is trivial). Note, this does not mean that ~3 = s@r. There are seven (non-trivial semisimple Lie algebras o f dimension less than or equal to seven, and these fall into two classes: (i) d i m s = 3 ; s = s o ( 3 ) , so(2, 1). (ii) dim s = 6 ; s = s o ( 3 ) e s o ( 3 ) , s o ( 3 ) ® s o ( 2 , 1), so(2, I)E)so(2, 1), so(3, 1), so(2, 2). Suppose dim s = 6. Let T be the generator o f t and 7", (a = 1 , . . . , 6) the generators ofs. Then [T,, T] = c , T for some constants c,, since r is an ideal. The Jacobi identity for To, Th and T then implies that co = 0 (one has to use the fact that s is semisimple). Therefore ~ = s(~r. This implies in particular that [ ~d, ~] ~ ~. Similarly, if s = 0 then ".'g= r is solvable and again [~3, ~ ] # ~3. On the other hand, if dim s = 3 then it is possible to have [~, ~ ] = ~3. In general, if T~ (a = 1,2,3) are the generators o f s and Ta (a = 1, 2 , 3 , 4 ) are the generators o f r, then T~ =

K,

'

La

Ma

"

J is a 3 ×3 representation o f s, while K is a 4 x 4 representation of s. M is a 4 ×4 e M h and KoL~ - LeJ~ = f,~a L~. representation o f r. Also, 2 M IaLt, i =.f'c,' Lc~ [ K,, M ~ ] = f ~ c, J, K, L and M are all real matrices. For example, ifs -- so(2, 1) then one may choose J ( K ) to be the usual irreducible 3 x3 (4 x4) representation o f the algebra (these are both real). If in addition one takes M = 0 and sets (L~)a,. = (K,.)bc, then it is not difficult to see that this defines an algebra for which [ ~3, ~] = (~. The only non-trivial real 4 x 4 representation o f so(3) is the sum o f the trivial representation and the irreducible 3 x3 one and I am not sure ifthat precludes the possibility that[Y, ~3] = ~,. There are o f course m a n y examples when dim s = 3 and [,% ~3] ¢ ~. In particular, if ~3= s ® r (an important class o f examples) then this is true. I will not be able to say much about the case when [~, ~] = '~. From now on I assume that [~, ~3]~ ~ .

(98)

w. Boucher / N = 8 supergrauity theories

567

I summarize the remarks made above. (98) is always true when dim s = 0, 6 but is sometimes false when dim s = 3. 1 will now show that the only solutions of (95) (assuming (98)) are when V - - 0 and hence a = 0. Roughly speaking, the semisimple part of the algebra wants the cosmological constant to be negative but the solvable part of the algebra wants it to be zero. Solvability wins. Split the generators, T,, of ~d into T = T~ and Ta (a -- 2 . . . . ,7), where T~ ~[~d, ~3], or equivalently, flj = 0. From (89) one finds 16K2 8hkk ~V .= --2fj,(f~r + fj:,'h,'h Jj') h k'h k ' l ' + f~f~j,h "'h ~' ,

(99)

and thus 16K2 8V = -2f~t(f~r +fS, rh,,h j/) h I Ih Jr

8h~z

-- - 2 tr ( T,( Tj + h -I T~h ) )h t'h u = - 2 tr ( S ( S + h--~S'rh)),

(100)

where S - - Tkh ~k. There is a remarkable identity, due to Scherk and Schwarz (who use it for a different purpose than I will), which says that tr ( S ( S + h-~STh))>>-O ,

101)

S+ h-LSrh =0.

102)

with equality if and only if

Thus the right-hand side of (100) is non-positive. But (94) and (95) imply that the left-hand side of (100) is non-negative. Therefore one has a contradiction unless V = 0, which is the result I promised. This reduces the problem to that studied by Scherk and Schwarz and means that the scalar fields coming from the 3-index photon enter the variational problem for the potential only trivially. Note, this may be true even if [ ~, ~d] = ~, but the proof would presumably be more complicated. Scherk and Schwarz, working with the number of extra dimensions, E, being arbitrary instead of 7, found some algebras for which the corresponding scalar potentials have stationary points with h o --c50. Because the induced metric on the "'internal" manifold is flat, they called any such algebra a flat algebra (more precisely, they called the corresponding groups, flat groups). One class of flat algebras which they found are defined by

[T,, T~]= -M~Tb, b [To, T b ] = 0 (a = 2 , . . . ,

(103)

E), where M is a traceless matrix (which implies that fkk =0). Scherk

568

w. Boucher/ N = 8 supergravity theories

and Schwarz restricted their attention to matrices, M, which were antisymmetric. The reason they did this is because they were looking for stationary points with h o -- ~,,. For this choice o f h~j it is not difficult to show that (99) is zero if and only if t r ( M ( M + M t ) ) = O which implies that M + M r - - 0 (cf. (101) and (102)). Moreover, for antisymmetric M one finds more generally that h~ = )t~, hab = A , ~ At, A2 > 0, are stationary points. At this moment, I do not have anything more to add to the discussion about stationary points. However, I would like to point out that for the fiat algebras defined in (103), if one wants the corresponding coset space, G / H , to be c o m p a c t then one finds the interesting prediction that the masses o f the theory are quantized. More precisely, the masses must be (integer) multiples o f a fundamental unit o f mass, or equivalently (since there are only a finite n u m b e r of particles in the theory) the masses must be rational multiples of each other*. There is a unique simply connected group, G, whose Lie algebra is one o f the flat algebras (103). The adjoint representation o f G is given by [ 13] D(L,. ) =

(~

-YCM~ (eY,M)~},

(104)

where L.,. is the element o f the group whose coordinates are y~. From (104) one finds that the g r o u p multiplication law is given by

(y,, ya) ×(z ~, z ~)~ (y~ + z ~, y~ + zh(e y'M)~,) .

(105)

From (105) one sees that any discrete s u b g r o u p o f G must be represented by (y~= n'k'; n~E 2r), with k ~ satisfying e k : ~ c Z. This leads to a coset space which is compact. For M antisymmetric, with non-trivial eigenvalues ,-ibm, m -- 1 , . . . , [½E], this translates into the condition that e k'b,', ~ Z for all m. This implies that there must be a (real) n u m b e r c such that b,, = N,,c, for some integers N,, (with greatest c o m m o n factor 1), and such that ek'"~ 2v. A n o t h e r way to see this is to notice that for G, the c o r r e s p o n d i n g matrix, K, o f generators is given by

e ), M , and because K must be well-defined on G / H one again finds the same condition. For the g r o u n d states discovered by Scherk and Schwarz the masses o f all the particles in the theory (calculated from the quadratic fluctuations in the usual way) are simple linear functions o f the b,,,. Hence the result concerning mass quantization. Therefore people who believe that the extra dimensions are physical (and hence compact) have a prediction that can be tested experimentally. Unfortunately, (i) any real n u m b e r can be approximated by rational numbers so unless by some " This has also been realized by P.K. Townsend and G.W. Gibbons independently of each other and of me, and presumably by other people, although I am unaware of any statement in the literature.

W. Boucher / N = 8 supergravity theories

569

accident the integers N,, happen to be small, it is unlikely that one would ever be able to detect anything, and (ii) the theories are a joke phenomenologically. Still one might claim that there is a principle i n v o l v e d . . . I have also searched for Lorentz-invariant solutions of the six new gauged N = 8 supergravity theories obtained by dimensional reduction of the recently discovered N = 4 SO(5) gauged supergravity theory in d = 7 dimensions [7]. I was unable to find any such solutions ([15]; details sent upon request). For completeness, I state that the six gauge groups are given by SO(5) x G, where G is a Lie group corresponding to one of the following Lie algebras, ~3: (i) ~ = 0, abelian. (ii) ~3= so(3). [ x y ] -- z, [ y z ] = x, [ z x ] = y. (iii) ~3= so(2, !). [ x y ] - - - z , [ y z ] = x , [ z x ] = y . (iv) ~3 = e(2). [ x y ] = O, [ y z ] = x, [ z x ] = y. (v) ~d = e( 1, 1). [ x y ] = O, [ y z ] = x, [ z x ] = - y . (vi) ~3 = Heisenberg algebra. [ x y ] = z, [ y z ] = 0, [ z x ] = O. For a physicist, the most interesting gauge group is SO(5) x SO(3). If, as seems likely, these theories have no (Lorentz-invariant) ground states, one might become discouraged. Well, sooner or later physicists are going to have to become accustomed to this type of situation. It occurs over an over again in supergravity theories.

5. Comments

A few brief comments are in order. Of all the symmetries, ( b l ) - ( b 5 ) , that I have assumed truncated consistently, I do not think that anybody will argue with (bl) and (b2). It is hard to imagine ending up with a d = 4 general coordinate and local Lorentz-invariant theory if one does not assume that these symmetries explicitly truncate consistently. Global SO(7) invariance (b4) is also a fairly mild symmetry to have truncated consistently. The only place I have used this is when assuming that the eight d = 4 supersymmetry parameters, e~, transform under the fundamental representation of Spin(7). One could think of more general situations. In particular, one might want to construct N < 8 supergravity theories in d = 4 by means of dimensional reduction. I have no helpful suggestions along these lines, but would encourage somebody to look into the matter. I make no apologies for assuming that local gauge invariance (b3) is a symmetry that is consistently truncated, nor do I think that assumption (d) is unreasonable. I am, after all, trying to build g a u g e d supergravity theories. Again, however, one could look at other (or more general) situations. Finally, one arrives at supersymmetry itself, (b5). I want to build N = 8 supergravity theories so one would at first sight arrive at the conclusion that certainly ~

570

W. Boucher / N = 8 supergravity theories

should truncate consistently. However, in some sense this assumption is the weakest link in the chain o f argument. One reason why bosonic symmetries such as ( b l ) - ( b 4 ) are so well understood is that we have a geometric picture which accompanies the algebraic formulae. This means that one has a natural choice for the ansatz o f the parameters o f the bosonic symmetries. The same is not true for supersymmetry. Most o f sect. 3 was spent finding an ansatz for e. This was done assuming that ~5~ was the fermionic symmetry which truncated consistently. The reason ! have chosen ~5~ as in (22) is because this is the natural and minimal prescription from an algebraic point o f view. But ~5~ is not the only fermionic symmetry one could consider. One should be able to consider a combination o f c5o and any bosonic symmetry o f the d = I I theory, as long as it respects the form (cl) o f the vielbein. For example, one ~j.(gFr~@bl). I do not might consider theories which consistently truncate ,5o" - ~ + know if this will lead to new or even sensible results. One way to find a " n a t u r a l " ansatz for e is to look to superspace. The on-shell N = l, d = l l superspace supergravity formalism has been written d o w n [24]. I am not suggesting that one should try to do the entire dimensional reduction process in superspace (although in theory one could) but merely that one should explore the formalism and hopefully gain some insight into the fermionic symmetry involved in the problem. One may find for general manifolds G / H that ~5~ is not the symmetry that one should be considering. This is all very speculative. As far as I know, not even people who work on 7-spheres have thought o f this possibility yet. I mentioned in sect. 3 that d e m a n d i n g that a fermionic symmetry truncates consistently can lead to more bosonic symmetries being truncated consistently than one at first demanded. Eq. (27) makes this obvious. The c o m m u t a t o r [80(e~ ), 8~(e2)] contains tSGCT, cSL and tS,be,a,. All o f these contain field-dependent parameters so the corresponding symmetries are local in x. This is what makes dimensional reduction which is consistent with supersymmetry so restrictive a process. If one does not choose the correct y - d e p e n d e n c e then the c o m m u t a t o r will force you into having a bosonic symmetry group which is too large, for example, an infinitedimensional (for fixed x) spin-I gauge group. Finally, M a n t o n [16], has given a version of dimensional reduction on coset spaces, G / H . I will assume below that l am considering the dimensional reduction down to d = 4 o f N = l supergravity in d = I l, although much o f what I will say is valid in more general contexts. When dim H > 0, it is clear from M a n t o n ' s paper that he will never p r o d u c e an N = 8 supergravity lagrangian in d = 4, and indeed he states this implicitly. However when dim H = 0, he claims that he will end up with an N = 8 supergravity lagrangian by using his dimensional reduction scheme (he only makes this claim explicitly for G = SU(2) x SU(2) x U ( I ) , SU(2) x U(l)4, or U(I) 7, the latter case producing the usual C r e m m e r - J u l i a theory). I now show that indeed this is true and that although at first sight there is a fundamental difference between his ansatz and that o f Scherk and Schwarz, in the final analysis the two ansatzes are e x a c t l y t h e s a m e .

W. Boucher/ N = 8 supergravity theories

571

Manton has an ansatz for the fields in the theory which is one of invariance rather than covariance, and one might think that this would make his ansatz and that of Scherk and Schwarz incompatible. To illustrate what happens I will look specifically at the case of a vector field, v a. If H is not a normal subgroup of G then one needs to modify slightly the remarks made below. For the sake of clarity I will indeed assume that H is a normal subgroup of G and since in this case G / H is a group, I will also assume that H = 0. Scherk and Schwarz have the ansatz

v"(x,y) = v " ( x ) ;

v'~(x, y ) = K'~(y)v'(x).

(107)

v"(x, y) = K ~ ( y ) D j ( L;')v'(x) ,

(108)

Manton has the ansatz

v"(x,y)

= v"(x);

where D is the adjoint representation of G and Ly is the element in G whose coordinates are y. Manton in fact allows an extra factor to occur on the right-hand side of the second term in (108) if there happens to be another group action associated with v but this is irrelevant for the problem at hand. Using eq. (16) one finds that under y - y' = g(y) (g ~ G constant) Scherk and Schwarz have

,, Oy''~ v"(x, y ) = ~yO K ~ ( y ) D / ( g ) v ' ( x ) ,

(109)

while Manton has 0

ta

v"(x, y') = 0-~ K f ( y ) D / ( L y t ) V ' ( x ) ,

(1 lO)

so that Scherk and Schwarz have

v°(x) = D ] ( g ) v i ( x ) ,

(111)

v°(x) = ta(x).

(112)

while Manton has

This would seem to make all the difference in the world. It does not. Recall that any group G has two actions of G on itself, one on the left and one on the right. For elements which are not in the center of G, these actions are distinct. These actions both induce a collection of vector fields on G, which separately form Lie algebras (isomorphic to ~) which commute with each other. I have the convention that the K~(y) are the left-invariant vector fields. Then D, IJ(Ly)Kj(y) are the right-invariant vector fields! Thus Manton's ansatz, (108), is equivalent to that of Scherk and Schwarz (107), but uses the other set of invariant vector fields. Both ansatzes are covariant under one action of G and invariant under the other. I prefer the formulation of Scherk and Schwarz to that of Manton (although to be fair, Manton is looking at a more general situation). When Manton calculates the commutator of two supersymmetry transformations he will discover that with his ansatz it will indeed be the right-invariant vector fields that appear on the right-hand side in the ~ C T term (el. (27)). Thus in eq. (20) he needs to include the right-invariant

572

W. Boucher / N = 8 supergravity theories

vector fields (cf. the comments made after assumption (d) in sect. 3). Observe that the symmetry o f invariance is global whereas the symmetry o f covariance is local, so Manton should not include the left-invariant vector fields in eq. (20) and Scherk and Schwarz should not include the right-invariant vector fields (in addition, these only act trivially on the fields in the theory). Note, Manton uses the invariance with respect to one o f the group actions (rather than the covariance with respect to the other group action) to give an easy group theoretic argument that his ansatz is consistent with the equations of motion, generalizing an argument due to Coleman [25]. Therefore the ansatz of Scherk and Schwarz is also consistent with the equations of motion, as I mentioned in sect. 3. In conclusion, the basic point I want to emphasize is that although truncating supersymmetry consistently is not a trivial matter, it is something which c a n be studied successfully. In this paper, I have introduced several new ideas on how to deal with the practicalities of demanding that supersymmetry be truncated consistently, and even if readers do not like by particular ansatz, at least they should see the possibility of carrying through the procedure with another ansatz. Certainly, one would expect the key to any such attempt to be the analysis of the commutator of two supersymmetry transformations.

References [I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [l l] [12] [13] [14] [15] [ 16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141 J. Scherk and J. Schwarz, Nucl. Phys. BI53 (1979) 61 B. de Wit and H. Nicolai, Nucl. Phys. B208 (1982) 323 M.J. Duff, in Supergravity '81, ed. S. Ferrara and J.G. Taylor (CUP, 1982) M.J. Duff, seminar at DAMTP, Michaelmas term, 1981 C.M. Hull, Phys. Rev. D30 (1984) 760 M. Pernici, K. Pilch and P. van Nieuwenhuizen, Phys. Left. 143B (1984) 103 E. Cremmer, B. Julia and J. Scherk, Phys. Lett. 76B (1978) 409 M.J. Duff, in Supergravity "82, ed. S. Ferrara, J.G. Taylor and P. van Nieuwenhuizen (World Scientific, Singapore, 1983) B. Biran, F. Englert, B. de Wit and H. Nicolai, Phys. Lett. 124B (1983) 45 M.A. Awada, B.E.W. Nilsson and C.N. Pope, Phys. Rev. D29 (1984) 334 B. de Wit and H. Nicolai, Nucl. Phys. B243 (1984) 91 A. Salam and J. Strathdee, Ann. Phys. 141 (1982) 316 M. Awada, Phys. Lett. 127B (1983) 415 W. Boucher, PhD thesis, Cambridge University, Cambridge, England (1984) N.S. Manton, Generalized dimensional reduction of supergravity, Santa Barbara preprint, NSF-1TP83-04 T. Kugo and P. Townsend, Nucl. Phys. B221 (1983) 357 N.P. Warner, Phys. Lett. 128B (1983) 169 B. de Wit and H. Nicolai, Nucl. Phys. B231 (1984) 506 P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B (1980) 233 A. Aurilia, H. Nicolai and P.K. Townsend, Nucl. Phys. BI76 (1980) 509 N. Jacobson, Lie algebras (Wiley, New York, 1962) J.E. Humphreys, Introduction to Lie algebras and representation theory (Springer, Berlin, 1972) L. Brink and P. Howe, Phys. Lett. 91 (1980) 384 S. Coleman, in New phenomena in subnuclear physics, ed. A. Zichigi (Plenum, New York, 1977)