Nuclear Physics B185 (1981) 403-415 ~) North-Holland Publishing Company
N=4
S U P E R G R A V I T Y C O U P L E D TO N = 4 AND HIDDEN SYMMETRIES
MATTER
Ali H. C H A M S E D D I N E 1
CERN, Geneva, Switzerland Received 25 September 1980 (Revised 22 January 1981) We present the N = 1 supergravity in 10 dimensions obtained by truncating the reduced N = 1 supergravity from 11 dimensions. This is further reduced to 4 dimensions to give SU(4) supergravity coupled to six SO(4) vector multiplets. As the reduction is from 10 dimensions, the theory is expected to have the symmetry SL(6R)gtobat x SO(6hocab but we give a theoretical argument that this can be extended to SO(6, 6) x SU(1, 1)~lobal and SO(6) × SO(6) × U(1)local.
1. Introduction The study of the dual spinor model and interacting closed strings revealed the existence of N = 1 supergravity in 10 dimensions and predicted the necessary fields [1]. This knowledge was then used in constructing the pure SU(4) supergravity [2]. F r o m a different angle, after the establishment of N = 1 supergravity in 11 dimensions [3], it was argued by Scherk that this can be truncated to give the 10-dimensional theory [4]. It is also known that in four dimensions, after reduction from ten, the model describes SU(4) supergravity coupled to six N = 4 vector matter multiplets. However, neither the theory in ten dimensions, nor in four dimensions, are fully worked out. O u r interest in this model stems from the fact that N = 4 supersymmetry is the largest one that admits matter representations, with matter defined as consisting of particles of m a x i m u m spin 1 [5]. Thus the ten-dimensional theory provides an example of an unrestricted scheme coupling supergravity to an arbitrary n u m b e r of matter multiplets. This may be considered as an advantage over the N = 8 supergravity. Other uses of this model for the spinning string theory are advocated by Schwarz [5]. Taking the N = 8 supergravity as an example [6], it is natural to try to find the hidden symmetries that may be present. One can argue, guided by the work of C r e m m e r and Julia [6], that the reduction from ten dimensions will result in an obvious SO(6)~ocal× SL(6R)globa~ symmetry. The presence of the large n u m b e r of scalars (38 with duality transformations and 37 without), is, by the " e x p e r i m e n t a l " 1 Address after 1 January 1981: Northeastern University, Boston, MA. 403
404
A.H. Chamseddine / N = 4 supergravity
rules of extended supergravity models [7-9], an indication of a larger symmetry. We argue that the symmetry can be enlarged to the non-compact groups SO(6, 6 ) x SU(1, 1) globally and SO(6) x SO(6) x U(1) locally with a duality transformation to the antisymmetric field (but only SO(6, 6 ) x U(1) global and S O ( 6 ) x SO(6) local without). On the negative side it was shown in the calculation of the scalar-scalar and p h o t o n - p h o t o n interaction of 0 ( 4 ) s u p e r g r a v i t y coupled to self-dual 0(4) matter multiplets that the results are not finite, implying the non-vanishing of the/~ function, although it does vanish for pure 0(4) supergravity and pure 0(4) matter [10]. The plan of this p a p e r is as follows. In sect. 2, we start with the N = 1 supergravity in 11 dimensions and show the consistent truncation that leads to the N = 1 theory in 10 dimensions. In sect. 3 we reduce the theory to four dimensions and identify the physical fields. Since we did not obtain all the interaction terms here, we shall only give in sect. 4 the group theoretical arguments that led us to identify the groups.
2. N = 1 supergravity in 10 d i m e n s i o n s
We start with the now familiar N = 1 lagrangian in 11 dimensions [6], using the same notation and conventions. The fields are the elfbein e ~ A, the Rarita-Schwinger Majorana field OM, and the antisymmetric potential AA~Np. The supersymmetry transformation rules are: 8e~ A = --i~FA~bM ,
3tSAMN P = ~EF[M N ~//P] , 8 0 M = D M (tO) e + 1 ~ i (FNPORM -- 8FPQR8 N ) EFNPQR .
(2.1)
The definitions of the supercovariant connection ~)MABand field strength FNPQRa r e given in ref. [6]. Using part of the local SO(1, 10) symmetry we can bring the elfbein to the form a
Bit
where ~, ~, . . . . = 6, i . . . . . 9 are curved indices, and a,/~ . . . . = 0, 1 . . . . . 9 are flat indices. Note that we reserve the dotted numbers for curved indices and the undotted for flat ones. By splitting the eleventh index explicitly the other fields decompose as AMNP = (A,~o, A , o l f ) ,
(2.3) We are not interested in keeping all these fields, as this will correspond to the known N = 8 supergravity in four dimensions (and N = 2 supergravity in 10 dimensions), and we have to set some fields to zero. Due to the special properties of ten
405
A . H . Chamseddine / N = 4 supergravity
dimensions, we can impose the Majorana and Weyl ccmsitions simultaneously on the spinors. Thus each of the Majorana spinors 0., 01f and e, split into two MajoranaWeyl ones. We can show that keeping only the left-handed components of 0 . and e and the right-handed component of ~lf, and setting B . n and A . , . o to zero, form a consistent truncation. In other words, we now have YnO~, = 0 , ,
YH01~ = - 0 , f ,
The = e,
(2.4)
where F 11 = - / 1 1
(Fll) 2 = -1,
= i'Yll ,
('~11) 2 =
+1.
The truncation is consistent because 6B~ al = - i e F l l O~ = f:6~ = O,
(2.5)
3-
~eFt,,vOol= O,
6A.~o =
and these vanish due to the condition (2.4). Moreover, the handedness of 011 and 011 is preserved as the quantities o3,~ H, o3~f~0, and F~,xo all vanish. We shall write the new transformation rules after the reduction of the theory, as some field redefinitions and scalings are needed. As the dimensional reduction technique is explained in detail in ref. [6], we shall only indicate the necessary steps so that the reader can keep track of the field redefinitions, and without falling into repetition. We note that, apart from the equivalent theory containing the opposite handed spinors, the truncation (2.4) is unique. The bosonic lagrangian in 11 dimensions is -1VllR
(to ) - ~ VllFMNpOF
M~PO
2
+ (-~
e
M~ M
"'" ~FMr..~FM~...M~AM~M~oM~ .
(2.6) To reduce the gravitational sector we define the Weyl scaling for the 10-bein: c~
1
c~
el0~, = q~e~, ,
(2.7)
and we obtain 1
I
132
- z V x l R (to) = -ZVloRlo(to) + ~(Z) (01o~ log
~)2.
(2.8)
The remaining parts of (2.6) give 1 117 --3/2 /a,;< vA 0o'1-', IU' i~ VlOq~ gioglogxol'~,,olf.~',
(2.9)
where F,~oxf = 30[,A~o]lf •
(2.1o)
@v~r~o~,
(2.11)
For the fermionic kenetic term
406
A.H.
Chamseddine
/ N = 4 supergravity
we need a Weyl-scaling and a field redefinition to diagonalize the 0 , and 4'1f contributions. This is achieved with Xll = e1111Olf
X,~ = e~,~O., (2.12) XIO :
3 --1/16 2-"~ ~ Xll
a [~X-
-1/16 0,10
= ~
,
elog
i
--~r,
x11
) ,
so that (2.11) gives the normalized form: 1•
--
-~tV10010~F
txvp
1 •
~.
DAblOo+ ~lVloXloF D,XlO .
(2.13)
To cut the story short, the interaction terms are easily calculated with no further definitions. We then collect all terms and forget about the orginal theory. As we are now in ten dimensions, we can restore the notation that indices at the beginning of the alphabet denote flat coordinates, and the later characters curved ones, e.g.,
A , B , C , . . . = O , 1. . . . . 9, (2.14)
M , N , P , . . . = 6 , i . . . . . (9. Thus the N = 1 lagrangian in ten-dimensions is given by 1 1• MNP 1 K2.5~1o= -~VR(oo)-~tVtpMF DN(i(w +¢3))oe+lv(o~ck)(oM4o)
1.
-
+~tV~F
M
1 A 1 V DM($(Og+¢O)) X + T5 e x p t.: z. ~. .p ) F MNPF
MNP
(2.15)
+ Ai V[~MXMN"ORON - i'/20~t (F Me°n - 3gMtPFOm)x](FeoR + #eOR)
+ ½~ v(o,~ + b ~ , ) ( ~ % ) , where x M N P Q R = INMNPQR __ 6gMtPFOgR~N,
(2.1 6)
4, =~ log ,p.
(2.17)
All the quartic terms are hidden in the supercovariant quantities, and the distinction between to and 03 is a consequence of the second-order formalism for which the to equation of motion is not supercovariant. The connection is O)MA B = ¢.OOMAB(e ,
Oe) + ~t[-ONF 1. N P MABOP -- XFMAB)(
+ 2(~MFBOA -- [ffMFAOB+ ~BFM~A) --liq~(OMFABX --2eMtA~B]X+tffNFNMAnX)],
(2.18)
A . H . Chamseddine / N = 4 supergravity
407
while the supercovariant connection is
&MAn = OJOMAB(e,Oe)+ i [ 2 ( ~ F B O A - OMFAOB + OBF~OA) 2.
--
-- ~l x/2eM[a~bn]X -- ~6XFMABX] .
(2.19)
The supercovariant quantities FpoR and/gM& are
FpoR = FpOR -- 3v~i exp (& )(O~pFoOm - x@OcpFoRjv~ -- 7~2FpoRX) , /3~
= O~b +'f~0MX •
(2.20) (2.21)
Finally, the new transformation rules take the simple form
8eMA = --i~FA~oM ,
6AMN = exp (&)(igFtM~I'N1+ ~X/~gFMNX),
(2.22)
6qtM = OM(t3 )e + ~ exp (" ¢ )(FPORM + 96P FOR)eFpoR , 8X = xf~i19~cb(Free) + ~x/~i exp (--0)(FPORe)FpoR. Note that 4tM and A" preserve the conditions (2.4) under these transformations. Moreover, A ~ N is subject to the abelian gauge transformation 6AMN
= OMeN
-- ON~M.
To summarize, we have found the N = 1 supergravity lagrangian and transformation rules in 10 dimensions. The fields are the lO-bein eMA, the Majorana-Weyl lefthanded gravitino 4'M and right-handed spinor ,I', the antisymmetric tensor A M N , and the scalar &. As mentioned in the introduction, all these fields are predicted by the spinning string model in 10 dimensions [1].
3. Reduction to four dimensions
We know what to expect in four dimensions: N = 4 supergravity coupled to six N = 4 vector matter multiplets with a manifest SO(6) - SU(4) global symmetry and thus no duality transformations are needed. The only problem is to identify the physical fields, as for the dimensional reduction it is necessary to m a k e m a n y field redefinitions. Using part of the gauge invariance SO(1, 9) we bring the 10-bein to the form a
era"
"
(3.1)
where m, n, p . . . . = 4, 5 . . . . ,9, a, b, c . . . . = 4, 5 . . . . . 9, and we are left with the local symmetry SO(1, 3) × SO(6). The Weyl-scaling for the vierbein is e 4 . '~ = A 1 / 4 e . ~ ,
(3.2)
408
A.H. Chamseddine / N = 4 supergravity
where /11/2
=
(det ema).
(3.3)
The gravitational sector gives 1 1 g,O vo" -¼V~oR =-ze4R4 + ~e4V~g4 g4 (G4,,~a G4,,,,b )nab
1
-ige4g4
I,~v
(O,g,,,,,)(3,.g
nan
1
tzv
) + ~ e 4 g 4 (30 logA)(3~ l o g A ) ,
(3.4)
where a
gmn
b
= era en
G4ed3 a =
"qab
(3.5)
2e4[~"e4/~]~3~,(B~bebre)e,.~
= G4ed3mem
a .
This describes one graviton, six vector fields* B~', and 21 scalars* g,... The antisymmetric tensor A ~ N decomposes into three types: A~.~ describing a pseudoscalar [1], six vector fields* A,.., and 15 scalars* A.,.. U n d e r internal changes of coordinates: x m = x 'm + ¢ m ( x ) these fields transform as
3Am. = 0
(3.6)
8A.m = O.¢"A~..
Also 3BZ
= o~,~".
(3.7)
The appropriate potentials and field strengths which are invariant under internal changes of coordinates are: n
A ~ =A~,,~-B~, A.m
(3.8)
A~.~=A.~ + 2 B i g mA~]~ + B~, ,. B~ , A,,,. , and f n~.~a = eam(23b.m r~]m+ G4~.v'mnm) ,
F4.~. = 3(0t.A'~pl -
m
(3.9)
t
G4[.v mg]m). M
N
Pr-~
The definitions (3.9) are related to the quantities found from eA en e c PMNe by F,,o~, = A3/4F4aBv,
(3.10)
F,~o,, = A1/2Fa,~a.
With no further difficulties, the bosonic lagrangian takes the form K
b. . . . ic = - a e 4 R 4 ( w ) 1
+~gea~/~G4~,~, G4 ~
-ize4(O~.g.,.)(3 g
mn
gmn
1 )+~e4(0g logA)(3" logA)
1
+ ise, exp ( - 2 & ) [ A F a ~ , , , F 4
I.xvp
--
+ 3~/AF,~mF,
+ 3 g " P g "a (G,A,,,~)(O'~Apq)]. *
+~e4(0~b)(0 ~b)
The parities of these fields are discussed at the end of this section.
tzr;
,g
mn
(3.11)
A.H. Chamseddine / N = 4 supergravity
409
The field strength F4.~p was shown in ref. [1] to describe a pseudoscalar. This can easily be seen by writing F . . o = e . ~ o ~ B ~ and introducing a Lagrange multiplier B. This will produce many interaction terms when taking the fermions into account. Although the SO(6) global symmetry is manifest, we have to identify the six supergravity vectors and the six matter vectors, A study of the supersymmetry transformations of the [ermion fields and of their interaction shows that the physical field strengths are a combination of F4,,aa and G 4 ~ . We therefore define the new vector potentials 1 t , ~ . , = x/-~(At. ~ + ~B .m), p
1
(gravity), (3.12)
r
(matter),
with the corresponding field strengths =
"{" 2 G 4 t x v m ) ,
(3.13) ~t..m = x/~(F4.vm
- ~1 G 4 ,. v m ) ,
where B!
.., = (B. a ean ) g .! , . ,
!
G4~.~m = G4.v !
n
l
grim,
i a t
g.,. = e .~ Ca.,
[see eq. (3.28)].
The assymetry in the kinetic terms for the vector fields due to the presence of exp ( - 2 5 ) will be removed by the use of the scaled internal metric g'~., and thus taking the simple form
le4x/~ exp (-O)(3:~..,.S~"". + ~..,~(~ i~v.)g irnn .
(3.14)
These relations are uniquely determined, as they also should emerge from the requirements that the supergravity fermionic transformations involve ~.v,. while the matter fermionic transformations are proportional to ~.~,., and also from the vector interaction terms. We now turn our attention to the fermions and we first note that our spinors are 32-components subject to the Majorana-Weyl conditions, and thus every spinor gives four independent spinors in four dimensions. We can represent the spinors with the SO(6) notation 0.A and Xa, where A is an SO(6) spinor index taking eight values and the Weyl conditions (2.4) take the form: ( F11)AB~IMB
= i'/50MA,
(3.15) (Fll) AB~
= --i~/sXA ,
and the SO(6) generators are the 8 × 8 matrices (Fab)~B. Also the SU(4) notation is
410
A . H . Chamseddine / N = 4 supergravity
possible by the following representation:
~A
= \--iYSOMK/
'
XA =
iYSXr
'
where K = 1. . . . . 4 is an SU(4) index• In the following we shall keep the SO(6) notation with the spinors subject to (3.15), and this can easily be converted to the SU(4) notation by using (3.16) and the explicit representation of the (Fa)A B matrices• • 3 The fermions will involve a two-stage diagonalization, first separating the spln-~ • 1 fields from the spln-~ fields, then taking care of the mixed spin-~ fields. We make the following scalings and redefinitions: ~a = t~Mea M ,
~ = e,~e~ ~ ,
0~4 = A-1/80~, X4 = A - 1 / 8 X , ---1/8
~k.4=zl
ot[.l,
1
a
e4. w.,~+~3'J" ffa),
(3.17)
where F a = 3/5 ® ( F a ) A B .
Then the fermionic kinetic lagrangian becomes K 2 ~ k i n = __l i e 4 ~ 4 ~A "~t~~OD~(to) ~14 oA
1. 1 a b ab u --~le4[~ta4(~l-" 1" q-~l )'y D ~ , ( t o ) ¢ A b 4 - X 4 " y " D ~ ( t o ) X 4 ] .
(3.18)
Although the X4 field is diagonalized, its transformation under supersymmetry is proportional to a mixture of ~ r . ~ and ~ , ~ , and thus can neither be identified with the gravity spinor nor with a matter spinor. The spinors that transform properly are X~ and A, defined by X4 = l v / 2 ( i F a x a - x / 2 A ) ,
(3.19) 0 ~ 4 = X . - ~1F . F
~,go - ~ i 4 ~ r o x
bringing the spin -1 fields to the form 1•
-
v
--
i,
gle4[XaA'Y D v ( t o )XaA "~-X a ' ~ D , , ( t o ) A A ] .
(3.20)
The definition of A is unique, and although the Xa diagonalization is defined up to orthogonal transformation of the 24 degrees of freedom, the required transformation for Xa restricts this freedom.
411
A.H. Chamseddine / N = 4 supergravity
The interaction terms arise from three sources: 1 .v r7 r, MNPr'-,RS~ --gl VllIMI 1 IffpO)NR$ 1 ,~r r - r ' , N R S
~l v x.t
(3.21)
XtONRS
+ i V exp (--,#)FPou[¢Mx MNeORoN -- i ' / 2 ~ ( F
~''°R - 3g~tPF°R1)X].
This gives a large number of terms, the easiest part of it being the spin 3--spin 3 interaction; the other spinors give complicated contributions because of definition (3.19). These, as in the N = 8 supergravity, are grouped once the operating symmetry is completly identified. H e r e we shall only write the gravitinos interaction, hoping to find a compact form for the spinor interactions when dealing with the hidden symmetries. 3 • 3 The spin ~-spln ~ interactions are: te__e~A1/4exp (--q~)oq~'KXa04,(F v.v•A + 2 g ~Kg vX)/~ a I/t4v 442 _
_
l ie4A 1/2 exp (-~b)F4~,,,o~'F"tk ° 1 •
m
n
--
+gte4 exp (-d~)ea eb (0oAm,)~b4,F 1 •
rn
--
+gle4(ea 3oeb,,)~b4~,F
~vO
F
ab
~vO
F
ab
(3.22) ~b4v
~ba~,.
The matter scalars interact proportionally to (Fab)AB which generates the SO(6) group. We can therefore write the gravitinos kinetic term as 1 •
--
-- ~te4~b4~AV
I~vO
B
(OvSA -- Q~AB)6oB,
(3.23)
where 1
m
Q~A B = ~[e,, O~eb,~ + exp (--d~ )e~eb'~O~A~,](Fab)A B .
(3.24)
There are two independent contributions in (3.24) and one is led to suspect that the local symmetry is S O ( 6 ) x S O ( 6 ) . However, this cannot be proved except after writing all interactions terms and identifying the global symmetry. In sect. 4 we shall argue for this assumption, from different considerations. To conclude this section, we give the supersymmetry transformation rules for the new fields. The vierbein transforms according to the universal law when the compensating transformation is taken into account: 8 e 4 , '~ =
--i~4aY°~lwA
(3.25)
,
where e4 = A 1 / 8 e .
(3.26)
The gravity and matter vectors transformations are ~ a
~-~ A - 1 / 2
8~',.,,~ = A -1/2
exp
(q~)(-ie4Fa~b.4
-
x/~g4yu,raA ) ,
exp (~b)(21-ig4yt,Xa+ . . . ) ,
(3.27)
412
A . H . Chamseddine / N = 4 supergravity
where !
D
tm
t
__
tm
s ¢ ~,,~ - sg~,,,,e a , ~ ~,,, - ~3~,,,e a .
The transformation rules of ~ and e,. ~ raise a little complication, and it turns out that these should be scaled in order to give the proper transformations. In particular we have to define exp (~b') =/I-1/2 exp (~b), (3.28) e m,a = exp (l(b )ema giving 8 e r,n a = e mrb~~ - - l E. 4- 1 r,a Xb)\ ,
&b' = " ~ g 4 h , 6Am.
(3.29)
= l"e m'~e n , b E- l. [. a X b ]
•
Actually the scaling (3.28) will absorb some of the inconvenient factors of exp (&); in particular, (3.24) now reads Q~,A B
lr
trn
=zte.
m ~ a
eb O ~ r n n + e ~
tm
t
0.ebm](ff
ab
)A
B
•
Finally, the spinors transform as 1.
6A = - ~ t , c / 2 ( y t~Xa = x / 2 A 1/4 1
-~[e~
vm
g.u
I"
a
"~ e4)~Tg~a - ~ i A 1/2 exp (-~b)(r"~°e4)/64~,,o - ½ i ( y % 4 ) ( O ~ , & ' )
exp (-~b) (~,ua (Y I ' * u e 4 ) ;n
!
eb O ~ , g m . ] ( y
~
-- 1y
+...,
I~ ( I . b e 4 ) F P 4 , a b
b
F e4)+''',
(3.30)
A 1/4
6~,4 = D~, (o3)e4 - ~ 1 al/2
+~zi
exp ( - d ~ ) ( y ~ x % . F a e 4 ) ~ a
exp (-4~)(F
VKA
^
__
lzr~ab
x~t
~,e4)F4~a ~-~tt
e4)r4..b
+" • • ,
where F'4~,ab = e'ame'b" O ~ , A , . . .
To obtain the correct parity assignments, the following chiral transformations are needed: (e, A, t#~,, t#a) ~ ( i y s ) - a / 2 ( e ,
(3.31)
A, ~9~., t~a) .
The split of the fields to the different parities can be seen explicitly by using the form (3.16) for the spinors and the representation of the F,, matrices given by Fi=Y5®
a 0
i '
Fj
=Ys®
0
t
, (3.32)
i=4,5,6,
/=7,8,9
A.H.
Chamseddine
413
/ N = 4 supergravity
where (a i)kL and ([3J)kL are 4 × 4 antisymmetric matrices of SU(2) × SU(2) defined in ref. [2]. The important point is to notice from the supersymmetry transformations (2.22), (3.27), and 3.30) that A~i, B ~ (and thus ~ , ~.~) transform (after chiral transorfmations) like vectors, while A ' . i, B'. i (and thus ~ . j , ~ . i ) transform as pseudovectors: 8A ~i =iA --1/4~4 K ( O l i ) K L ~ J ~ L 8A'.i = iA --1/4 f a x ( B j)
KLY5
"1-" " • ,
~/txL "~"
(3.33) " " ,
and similarly for B~i and B " i. Also, the transformations (3.30) in terms of the SU(4) spinors for AK takes the form: 1.
8AK = --4t',/2(~/
b~v
e4L)~g~KL,
(3.34)
where ~T.vKL = (0~i)KL~r..i -- iys(~i)KL~,~j ,
(3.35)
similarly
while the spinors Xa give: XKL,M
= (0l i ) K L X i M
--
i'y5(~i)KLXiM.
(3.36)
Analogously the scalar fields gmn and Am. can be decomposed to give 18 scalars and 18 pseudoscalars in the bases (3.32). 4. C o m m e n t s on hidden symmetries and conclusion
Looking back at the spectrum, we have the N = 4 supergravity multiplet with the following fields: the graviton e. ~, four gravitinos O.K, six vectors and pseudovectors A~.i, A . i , four spinors AK, a scalar q~', and a pseudoscalar A.v (or B). The N = 4 matter multiplets give the fields: six vectors and pseudovectors B.i, B.j, 24 spinors XaK, and 36 scalars and pseudoscalars gin. and Am.. The pure N = 4 supergravity is known to have a dynamical local symmetry U(1) and a global SU(4) × SU(1, 1) for the equations of motion with the two scalars parametrizing the coset space SU(1, 1)/U(1), U(1) being the dynamical local symmetry extending SU(4) to U(4) [6]. The presence of matter is expected to enlarge the local and global symmetries especially with the large number of scalars present. To see this we note that there are regular patterns in extended supergravity models which can be taken as "experimental rules". The first rule is that the scalar fields [N(~b) in number] parametrize the homogeneous space G / H where G is the global and H is the local one [7, 8]: N(~b) = dim G - dim H .
(4.1)
414
A . H . Chamseddine / N = 4
supergravity
TABLE 1 Multiplicity of Bose fields in d dimensions d
10
9
8
7
6
5
4
4+
~b AM AMN gMN
1 0 1 1
2 2 1 1
5 4 1 1
10 6 1 1
17 8 1 1
26 10 1 1
37 12 l l
37+1 12 0 1
The 4 + corresponds to the theory with duality transformations in four dimensions
The second rule was first noticed by Cremmer [7] and, generalized by Julia [9], states that the rank and character X of G, and the rank r' of G / H all increase by one under dimensional reduction by one dimension. For our present purposes these two rules are sufficient to determine the groups G and H. To facilitate our job we construct two tables. Table 1, following Schwarz [8], classifies the multiplicity of Bose fields in d dimensions. The number of scalar fields satisfy, for this model, the formula N(&) = 1 + (10-d) 2 .
(4.2)
In the last column where we have indicated d by 4 + we have assumed a duality transformation changing AMN to a scalar. Table 2 is constructed, following Julia [9], by starting in d = 10 where we k n o w that N ( ¢ ) = 1 implies, G = U(1) and d i m H = 0. We deduce that r = r' = X = 1. Rule two then implies that
r=r'=x
= 1 +(10-d)
= ll-d.
(4.3)
TABLE 2 The groups G and H d
10 1
1
2
u(1)110
3
so
s 0 ( 3 , 3) x u ( 1 ) 16
s-o-8) -g-dN 6
sTZ; 2 3
2
1
4
SO(2, 2) × U(1) 7
U(1) x U(1)
:/
3
4+
6 5
5 S0(4,4)
xU(1)~
6 6 SO(5, 5) x U(1) 46
7 7 SO(6, 6) x U ( 1 ) 67
sTig 20 so-6
so(4) x so(4)
s-6N 30
5
Our notation for each blockis
r • x[ r = rank G G dim H dim G H,, I where X = character G . r' = rank of G / H
7
7 S O ( 6 , 6 ) x S U ( 1 , 1) 69
iT)31
A . H . Charnseddine / N = 4 supergravity
415
B u t t h e c h a r a c t e r is also given b y = d i m G - 2 d i m H = 11 - d .
(4.4)
It is n o w a trivial m a t t e r to solve eqs. (4.1) a n d (4.4) (two e q u a t i o n s with two u n k n o w n s ) to d e t e r m i n e d i m G a n d d i m H , a n d with t h e k n o w l e d g e of t h e r a n k s r a n d r' identifies, a l m o s t c o m p l e t e l y , G a n d H [11]. A s s e e n f r o m t h e t a b l e , w h e n d = 4 t h e s y m m e t r y g r o u p s a r e G = SO(6, 6) x U(1) a n d H = S O ( 6 ) × S O ( 6 ) w i t h o u t d u a l i t y t r a n s f o r m a t i o n s , a n d G = S O ( 6 , 6) × SU(1, 1) with H = S O ( 6 ) × SO(6) × U(1) w h e n using d u a l i t y t r a n s f o r m a t i o n s . T h e e m e r g e n c e of S U ( 1 , 1) is e x p e c t e d f r o m t h e w o r k of ref. [4]. N o t e how, f r o m d = 10 to d = 4, t h e g r o u p U(1) factors o u t r e g u l a r l y . F i n a l l y , it r e m a i n s to b e s e e n h o w t h e s e g r o u p s a r e r e a l i z e d on t h e p h y s i c a l fields. O n e can c o n j e c t u r e with C r e m m e r a n d Julia [6] t h a t t h e local s y m m e t r y S O ( 6 ) × SO(6) x U(1) m a y b e r e a l i z e d d y n a m i c a l l y in a n a l o g y with t h e C P "-1 m o d e l in two d i m e n s i o n s [12]. It m a y b e i n t e r e s t i n g to study, in a n a l o g y with Ellis et al. [13], a s u p e r u n i f i c a t i o n m o d e l b a s e d on this n e w s y m m e t r y . It is also w o r t h n o t i n g t h a t this m o d e l in f o u r d i m e n s i o n s ( w i t h o u t d u a l i t y t r a n s f o r m a t i o n s ) is free of t r a c e a n o m a l i e s [14]. I w o u l d like to t h a n k Prof. B. Z u m i n o a n d D r . P. F a y e t for useful c o n v e r s a t i o n s .
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253 E. Cremmer, J. Scherk and S. Ferrara, Phys. Lett. 74B (1978) 61 E. Cremmer, B. Julia and J. Scherk, Phys. Lett. 76B (1978) 409 J. Scherk, in Recent developments in gravitation, Carg~se (1978), ed. M. Levy and S. Deser (Plenum Press) J.H. Schwarz, in Orbis Scientiae (Coral Cables, 1978) 431 E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141 E. Cremmer, Proc. Europhys. Study Conf. on Unification of fundamental interactions (Erice, 1980), ed. J. Ellis, S. Ferrara, and P. van Nieuwenhuizen, to be published J. Schwarz, Phys. Lett. 95B (1980) 219 B. Julia, Ecole Normale Sup6rieure preprint LPTENS 80/16 (1980), Nuffield workshop in Supergravity (Cambridge, 1980), ed. M. Ro~ek and S. Hawking, to be published M. Fischler, Phys. Rev. D20 (1979) 396 R. Gilmore, Lie groups, Lie algebras, and some of their applications (Wiley, New York, 1974) A. D'Adda, P. Di Vecchia and M. L/ischer, Nucl. Phys. B146 (1978) 63 J. Ellis, M.K. Gaillard and B. Zumino, Phys. Lett. 94B (1980) 343 H. Nicolai and P. Townsend, CERN preprint TH.2959 (1980)
