- Email: [email protected]
Su(4) invariant supergravity theory
Volume 74B, number 1, 2
PHYSICS LETTERS
27 March 1978
SU(4) INVARIANT SUPERGRAVITY THEORY E. CREMMER and J. SCHERK Laboraroire de Physique Thdorique de l'Ecole Normale Supdrieure, Paris, France l and S. F E R R A R A CERN, Geneva, Switzerland Received 27 December 1977
We present a new supergravity theory which is invariant under four separate local supersymmetry transformations. The action is invariant under global SU(4) transformations realized on the fields without use of the equations of motion. In addition, the equations of motion are invariant under a non-compact global SU (1, 1) group. The equations of motion of this theory are shown to be equivalent to those of the previously derived SO (4) theory through a redefinition of the field variables involving duality transformations on the vector fields.
The dimensional reduction from 10 to 4 dimensions o f the supersymmetric Yang-Mills theory [1,2] yields a supersymmetric SU(4) invariant Yang-Mills theory, which has a vanishing/3-function at the one- and two-loop levels [3]. This suggests the possibility of constructing a supergravity theory having an SU(4) symmetry where the SU(4) transformations would leave the action invariant without use of the equations of motion. This is not the case o f the SO(4) supergravity theory [4,5], which is indeed U(4) invariant [6] but where the transformations other than SO (4) involve duality transformations which can be applied only to the equations o f motion. The spectrum of the theory consists of the g.raviton V,a , four Majorana spin 3/2 fields ~/~, three vector fields A~,, three axial fields B n, , four spin 1/2 fields ×z, a scalar field q~, and a pseudoscalar field B. It differs from the spectrum o f the SO(4) theory through the parity assignment of the spin 1 fields which are all vectors in the SO(4) theory. The resulting theory is explicitly invariant under the global SU(4) group and has a much simpler structure than the SO(4) theory. In particular, the non-polynomiality is confined to the scalar field q~, the B field appearing only through its derivative 3vB. It is also a non-polynomiality o f a very simple type, involving only exponentials of ~, and the action is not singular for any finite value o f the fields. ,i A~, × 'i , A ' , B ' ) Comparison between the SU(4) theory (V~, ~ i , A un ' Bun ' X i , O, B) and the SO(4) theory (V~, ~,~, reveals that as far as the equations o f motion are concerned the two models are equivalent as duality transformations and field redefinitions can be used to transform the equations of motion o f the models into each other. So this SU(4) supergravity theory may be viewed as a simplified version o f the SO (4) theory, where the SU (4) invafiance is made more transparent. Finally, the relation between the two theories has led us to discover that in each o f them there is a bigger group of transformations than the U(1) group extending SU(4) to U(4). The U(1) group is in fact part o f a non-compact SU(1, 1) group so that the total group o f invariance of these theories is SU(4) ® SU(1, 1). The appearance of the 1 Laboratoire Propre du C.N.R.S., associ~ ~ l'Ecole Normale Sup6rieure et ~ l'Universit~ de Paris-Sud. Postal address: 24 rue Lhomond, 75231 Paris Cedex 05, France. 61
Volume 74B, number 1,2
PHYSICS LETTERS
27 March 1978
SU(1,1) group, which is realized in a nonlinear fashion, is somewhat of a surprise and is linked directly to the presence of the spin 0 fields. In SO(2) [7] or SO(3) [8] supergravity theories, the invariance group is only U(2) or U(3) [9]. The SU(1,1) group appears also in the coupling of supergravity to the scalar multiplet [10]. Let us introduce the 6 real, antisymrnetric 4 X 4 matrices a n,/3 n (n = 1,2,3) of SU(2) ® SU(2) satisfying: (an, a m ) = ( / 3 n , / 3 m } = _ 2 f n m I ,
[otn,/3 m] = 0 ,
[otn,otm ] = 2enmpotp,
[/3n,/3m ] = 2enrnp/3p .
Using the explicit representation of ref. [1] one also discovers the following properties of these matrices which are of great use to prove the supersymmetry of the action and to put the equations of motion in a supercovariant form. Properties under duality transformations: i ~ijklotnt! = ankl'
1 ei]kl /3[~ = _/3n -£ kl" I Closure properties: O/n o~n =
i/ kl
f i k f j l -- f i l f j k
+
n
eiikl '
n
/3i1/3kl = f i k f ] l -- f i l f / k -- eqkl "
Under the global SU(4) group the fields transform as follows: f~i.u-- ~' [Arar + A'r/3 r +iY5~ n /3rn Anm]i/tp~, " f A 17 u -_ e nmr A r A um _ An" m B ~ ,
" fxi=-~l lArotr+A'r/3r_" 175otn -pm -/~ntn]ijX j,
R m + A m,, n A um , f B un = vnmrA, ~ --r-->
6dp = fiB = 5 Vau = 0 .
The action and transformation laws are constructed following the method described in ref. [5]. The action in 1.5-order formalism [1 l] reads as follows: ~=_(V/4K2)R(co)__~e~uvp-i~
~xy57uDv~oi +_~l VgUV[Ouc~av¢+exp(4K~)OpBOvB] + ½ i v 2 i T U D u x i
' V ( a " a n u u + R n B " U q exp ( - 2 K ~ b ) - ' +(KV/4x/~)exp(
[~i (. c uiju + --,1 f?.u.v'~, K~b)~,u , "b/ v-
K B O n U v A ~ , +BnUVB~v ) i -+u'gs(Ci] i - u p + ?u,,)ff]v] + (KV/4i)exp(_Kdp)~ipc~]u auv.ypX ] i]
~- iK exp (2 X(o) OoBeUUo° ~iu7 p ~ iv - 3 XVexp ( 2 K¢) b u B ~ ( i y S T u x i -- (KV/2x/'2) t ~ ( 0 , ¢ + exp (2 X¢) OrBiT 5 ) 7vTuX i ,
where: ] V =
al.!AnU, +l.y5/3;~Bnuv l] • ,
A pnv - _ OpA nv
OvA np ,
B pnv -_ O#B vn - OrB pn
and Auv = 1/2eu, At, o, and similarly for other expressions. This action is invariant under the following local supersymmetry transformations: 8~ = (1/x/2)e~× ~, 88 = (i/,fY) e×p ( - 2 K ~ ) g % × ' , 5Anu = (1/x/~)exp (K(b)[e-icLi]~u]
(i/x/2)gic~yu×/] ,
8 V~ = -iKg%~, 8B~ = (i/x/~)exp(K¢)[gi(J~]'yS~/~ +(i/vt2)gi3~]'yfVuX/]
/5~i = (i/x/'2-) g i(D uO + i75 ex p (2 K~b)huB ) 7 u + ½exp ( - K~b)YC'~'3 ca 3 - (3 K/2 X/2) YT5 X ]X i75 , 5 ~;iu = (1/ K ) ~i~ u _ ~1 1"exp (2 K ~ ) gi'y 5 DuB + (i/2 v " 5 ) e x p ( - KO ) gY d ~ j y u c a o
+ ¼ i K [X/75 7axJgiT5 7#7a + xiTaxJg/Tta'Ya - XiTsTaxIgJTsTuTa ] • The algebra of these supersymmetry transformations has been shown to close under commutation, using as usual the equations of motion. From the variations one defines, as usual, the supercovariant derivatives DuO, DuB, [guX i, Buy , as well as the gauge field strengths [5,12 l ff~v. The fermionic equations of motion are simply expressed in terms of these supercovariant objects
A,L-. 62
Volume 74B, number 1,2
PHYSICS LETTERS
27 March 1978
iT,/~uX i _ 3 K exp (2 K(~)IguB757"X i = O,
¢°ouv3, 5 3,, ~ ~0 + iKVexp ( - K(p) C~jV OuvT°XJ + K V x / ~ [ [ ) v ¢ + iDvB75 exp 2 Kq~]3,VT°x i = O . The fact that the B field occurs only through its derivatives is connected with its dimensional reduction from an antisymmetric tensor gauge field as discussed in refs. [1,13]. Let us now give the transformation laws on the fields which turn this theory into the SO(4) theory. As the Bun axial fields have to be transformed into vector fields, this involves necessarily a duality transformation, which can be carried out only for the equations of motion. For all other fields, simple field redefinitions are enough and do not involve the use of equations of motion, The correspondence is as follows. For the SU(4) fields A, B, let us introduce Z = K(A + iB) with 1 - 2 KA = exp ( - 2 K~). For the other O (4) fields A', B', let Z ' = K ( A ' + iB' ). The transformation (1 - Z) (1 - Z' ) = 1 is easily seen to transform the kinetic terms of the spin 0 fields of the two actions into each other:OuZ'3u2'/ (1 - Z ' Z ' ) 2 = 3 ~ Z ~ 2 / ( 1 - Z - 2) 2. 1 ~i 1 r" For the spinor fields the transformation laws are" ~ = expi(½0 + ~ zr)')'5 ~u' xi = exp i(~ 0 + ~ ¢r)TSX ', where exp 2i0 = (1 - 2)/(1 - Z) = (1 - Z')/(1 - 2 ' ). which uv+ in the*uv For the vector and axial fields, the correspondence is the following. Denotingby Gijuvn thetartensor _ SO(4) theory gives the equation of motion of the F~ v fields, i.e. DuG~/v = 0, one has: eqjA n - (1/x/~)(Fi/ Fi~ ), =
-
-
t] -- rt
The first equation is a simple redefinition of the vector fields and can be carried directly on the vector potentials n = 0, i.e. that the equation of motion of the SO(4) Ate, AS.. The second, on the other hand, implies that D u B-uv gauge fields is satisfied. The complete proof of the equivalence of the two theories is very similar to the proof of the U(N) invariance of the SO(N) theory [9,6] and involves the following identities between supercovariant tensors:
F;
),
flijA~nvB = _(l/x/~-)(~b~.v- Fi/~,"v ^
where/ 8 = exp + 2K8 "" . Studying transformation properties of the action under homographic transformations of the Z variables led us to find that it is invariant under a group of SU(1, 1) transformations. Defining Y = 1 - 2 Z = exp ( - 2 K4~) - 2 i KB, the kinetic term of the scalar fields ¢, B can be written in the form 8 u Y S u Y / ( Y + ~-,)2, which is easily shown to be invariant under the transformation Y--* ( a Y + ifl)/(i'),Y+ 6), with a, fl, 7, 6 real and constrained by c~6 + fit = 1. These transformations form a non-compact group, namely SU(1, 1). To prove the invariance of the full theory, one needs to supplement this transformation law by those of the other fields of the theory. This is best expressed by giving the transformation laws for the three generators of SU(1,1) separately. (a) Y-* Y+ ifl, i.e. B--* B - fl/2K, other fields being unaffected. (b) Y-+ a 2 Y (B -+ a2B, ~ --* (p - (1/K) log e¢), Anu, B~ -+ -c~-1An --#, ~^ - 1 o~ on/ 2 , other fields unaffected. (c) In the infinitesimal form, the third transformation is given by: 6 Y = i X ( Y 2 - 1), Xreal
'
~AUV=XGU. v rt Art
66~ = - ½ iX exp ( - 2 K ¢ ) T S 6/u,
'
~ u v - ~ u- - vt , . U O r
tJz~rt
t
,
6xi=-3iXexp(-2KCp)Tsxi,
uv are defined such that the equations of motion of the Anu, B n" fields read DuGAn uv = D , G t~n ~v =0. where G Auv n , GBn This last transformation has been chosen to correspond in the SO (4) theory to the U(1) transformation extending SU(4) to U(4) [6]. As it involves a duality transformation it is an invariance of the equations of motion, not of the action while the two previous ones are directly invariances of the action. The invariance of the theory under a non-compact group does not imply the existence of ghosts as can be verified by inspection of the action, and the group is realized in a non-linear fashion. In the coupling o f supergravity to the scalar multiplet [10] a class of supersymmetric actions has for kinetic terms of the scalar fields the expression 63
Volume 74B, number 1, 2 2-
1
~ V [(0taA)2 +
where Z =
PHYSICS LETTERS
27 March 1978
(DUB)2]~[1 - ~,K2(A2+B2)] 2 = (V/2~,K 2) OuZOuZ/(1 - Z 2 ) 2
)~I/2K(A + i B ) .
This class o f t h e o r i e s d e p e n d i n g u p o n t h e a r b i t r a r y p a r a m e t e r ~. is i n v a r i a n t u n d e r an S U ( 1 , 1 ) g r o u p realized as follows:
Z= (aZ' +~)/(cZ'+~),
lal 2 - Icl 2-- 1,
•o
=exp(i/4?~)750(Z"2')¢o'
¢ --¢
× = exp(i/4),)(4),- 1)750(Z , Z ) x ,
t
with exp iO(Z', 2') = (cZ'+ d)/(c2'+ a). For c = 0 one recovers the usual chiral invariance already known for these models. In the limit where 2, goes to zero where one recovers the first discovered coupling o f supergravity to a scalar multiplet [ 14], the SU (1,1) group reduces to the E (2) group consisting of the chiral transformation (rotation in the A, B plane) and the two translations on the A and B fields. Although this invariance is present at the classical level, it is spontaneously broken at the quantum level since the vacuum is not invariant under field translations. The generalization of this invariance might also be helpful to generate larger supergravity theories, in particular to be able to write in closed form the SO(8) theory [15]. The authors wish to thank A. Neveu for an enlightening discussion about SU(1,1) invariance.
References [1] F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253. [2] L. Brink, J.H. Schwarz and J. Scherk, Nucl. Phys. B121 (1977) 77. [3] E.C. Poggio and H.N. Pendleton, Phys. Lett. 72B (1977) 200; D.R.T. Jones, Phys. Lett. 72B (1977) 199. [4] A. Das, Phys. Rev. D15 (1977) 2805. [5] E. Cremmer and J. Scherk, Nucl. Phys. B127 (1977) 259. [6] E. Cremmer, J. Scherk and S. Ferrara, Phys. Lett. 68B (1977) 234. [7] S. Ferrara and P. van Nieuwenhuizen, Phys. Rev. Lett. 37 (1976) 1669. [8] D.Z. Freedman, Phys. Rev. Lett. 38 (1976) 105; S. Ferrara, J. Scherk and B. Zumino, Phys. Lett. 66B (1977) 35. [9] S. Ferrara, J. Scherk and B. Zumino, Nucl. Phys. B121 (1977) 393. [10] E. Cremmer and J. Scherk, Phys. Lett. 69B (1977) 97; A. Das, M. Fishier and M. Rocek, ITP-SB-77-15, ITP-SB-77-38. ]11] P. Townsend and P. van Nieuwenhuizen, Phys. Lett. 67B (1977) 439; P. van Nieuwenhuizen, ITP-SB-77-55 preprint. [12] S. Deser, J.H. Kay and K.S. Stelle, Phys. Rev. Lett. 38 (1977) 527. [13] D.Z. Freedman, preprint CALT 68-624. [14] S. Ferrara, F. Gliozzi, J. Scherk and P. van Nieuwenhuizen, Nucl. Phys. B 117 (1976) 333; S. Ferrara et al., Phys. Rev. D15 (1977) 1013. [15] B. DeWit and D.Z. Freedman, Leiden preprint (1977).
64
PHYSICS LETTERS
27 March 1978
SU(4) INVARIANT SUPERGRAVITY THEORY E. CREMMER and J. SCHERK Laboraroire de Physique Thdorique de l'Ecole Normale Supdrieure, Paris, France l and S. F E R R A R A CERN, Geneva, Switzerland Received 27 December 1977
We present a new supergravity theory which is invariant under four separate local supersymmetry transformations. The action is invariant under global SU(4) transformations realized on the fields without use of the equations of motion. In addition, the equations of motion are invariant under a non-compact global SU (1, 1) group. The equations of motion of this theory are shown to be equivalent to those of the previously derived SO (4) theory through a redefinition of the field variables involving duality transformations on the vector fields.
The dimensional reduction from 10 to 4 dimensions o f the supersymmetric Yang-Mills theory [1,2] yields a supersymmetric SU(4) invariant Yang-Mills theory, which has a vanishing/3-function at the one- and two-loop levels [3]. This suggests the possibility of constructing a supergravity theory having an SU(4) symmetry where the SU(4) transformations would leave the action invariant without use of the equations of motion. This is not the case o f the SO(4) supergravity theory [4,5], which is indeed U(4) invariant [6] but where the transformations other than SO (4) involve duality transformations which can be applied only to the equations o f motion. The spectrum of the theory consists of the g.raviton V,a , four Majorana spin 3/2 fields ~/~, three vector fields A~,, three axial fields B n, , four spin 1/2 fields ×z, a scalar field q~, and a pseudoscalar field B. It differs from the spectrum o f the SO(4) theory through the parity assignment of the spin 1 fields which are all vectors in the SO(4) theory. The resulting theory is explicitly invariant under the global SU(4) group and has a much simpler structure than the SO(4) theory. In particular, the non-polynomiality is confined to the scalar field q~, the B field appearing only through its derivative 3vB. It is also a non-polynomiality o f a very simple type, involving only exponentials of ~, and the action is not singular for any finite value o f the fields. ,i A~, × 'i , A ' , B ' ) Comparison between the SU(4) theory (V~, ~ i , A un ' Bun ' X i , O, B) and the SO(4) theory (V~, ~,~, reveals that as far as the equations o f motion are concerned the two models are equivalent as duality transformations and field redefinitions can be used to transform the equations of motion o f the models into each other. So this SU(4) supergravity theory may be viewed as a simplified version o f the SO (4) theory, where the SU (4) invafiance is made more transparent. Finally, the relation between the two theories has led us to discover that in each o f them there is a bigger group of transformations than the U(1) group extending SU(4) to U(4). The U(1) group is in fact part o f a non-compact SU(1, 1) group so that the total group o f invariance of these theories is SU(4) ® SU(1, 1). The appearance of the 1 Laboratoire Propre du C.N.R.S., associ~ ~ l'Ecole Normale Sup6rieure et ~ l'Universit~ de Paris-Sud. Postal address: 24 rue Lhomond, 75231 Paris Cedex 05, France. 61
Volume 74B, number 1,2
PHYSICS LETTERS
27 March 1978
SU(1,1) group, which is realized in a nonlinear fashion, is somewhat of a surprise and is linked directly to the presence of the spin 0 fields. In SO(2) [7] or SO(3) [8] supergravity theories, the invariance group is only U(2) or U(3) [9]. The SU(1,1) group appears also in the coupling of supergravity to the scalar multiplet [10]. Let us introduce the 6 real, antisymrnetric 4 X 4 matrices a n,/3 n (n = 1,2,3) of SU(2) ® SU(2) satisfying: (an, a m ) = ( / 3 n , / 3 m } = _ 2 f n m I ,
[otn,/3 m] = 0 ,
[otn,otm ] = 2enmpotp,
[/3n,/3m ] = 2enrnp/3p .
Using the explicit representation of ref. [1] one also discovers the following properties of these matrices which are of great use to prove the supersymmetry of the action and to put the equations of motion in a supercovariant form. Properties under duality transformations: i ~ijklotnt! = ankl'
1 ei]kl /3[~ = _/3n -£ kl" I Closure properties: O/n o~n =
i/ kl
f i k f j l -- f i l f j k
+
n
eiikl '
n
/3i1/3kl = f i k f ] l -- f i l f / k -- eqkl "
Under the global SU(4) group the fields transform as follows: f~i.u-- ~' [Arar + A'r/3 r +iY5~ n /3rn Anm]i/tp~, " f A 17 u -_ e nmr A r A um _ An" m B ~ ,
" fxi=-~l lArotr+A'r/3r_" 175otn -pm -/~ntn]ijX j,
R m + A m,, n A um , f B un = vnmrA, ~ --r-->
6dp = fiB = 5 Vau = 0 .
The action and transformation laws are constructed following the method described in ref. [5]. The action in 1.5-order formalism [1 l] reads as follows: ~=_(V/4K2)R(co)__~e~uvp-i~
~xy57uDv~oi +_~l VgUV[Ouc~av¢+exp(4K~)OpBOvB] + ½ i v 2 i T U D u x i
' V ( a " a n u u + R n B " U q exp ( - 2 K ~ b ) - ' +(KV/4x/~)exp(
[~i (. c uiju + --,1 f?.u.v'~, K~b)~,u , "b/ v-
K B O n U v A ~ , +BnUVB~v ) i -+u'gs(Ci] i - u p + ?u,,)ff]v] + (KV/4i)exp(_Kdp)~ipc~]u auv.ypX ] i]
~- iK exp (2 X(o) OoBeUUo° ~iu7 p ~ iv - 3 XVexp ( 2 K¢) b u B ~ ( i y S T u x i -- (KV/2x/'2) t ~ ( 0 , ¢ + exp (2 X¢) OrBiT 5 ) 7vTuX i ,
where: ] V =
al.!AnU, +l.y5/3;~Bnuv l] • ,
A pnv - _ OpA nv
OvA np ,
B pnv -_ O#B vn - OrB pn
and Auv = 1/2eu, At, o, and similarly for other expressions. This action is invariant under the following local supersymmetry transformations: 8~ = (1/x/2)e~× ~, 88 = (i/,fY) e×p ( - 2 K ~ ) g % × ' , 5Anu = (1/x/~)exp (K(b)[e-icLi]~u]
(i/x/2)gic~yu×/] ,
8 V~ = -iKg%~, 8B~ = (i/x/~)exp(K¢)[gi(J~]'yS~/~ +(i/vt2)gi3~]'yfVuX/]
/5~i = (i/x/'2-) g i(D uO + i75 ex p (2 K~b)huB ) 7 u + ½exp ( - K~b)YC'~'3 ca 3 - (3 K/2 X/2) YT5 X ]X i75 , 5 ~;iu = (1/ K ) ~i~ u _ ~1 1"exp (2 K ~ ) gi'y 5 DuB + (i/2 v " 5 ) e x p ( - KO ) gY d ~ j y u c a o
+ ¼ i K [X/75 7axJgiT5 7#7a + xiTaxJg/Tta'Ya - XiTsTaxIgJTsTuTa ] • The algebra of these supersymmetry transformations has been shown to close under commutation, using as usual the equations of motion. From the variations one defines, as usual, the supercovariant derivatives DuO, DuB, [guX i, Buy , as well as the gauge field strengths [5,12 l ff~v. The fermionic equations of motion are simply expressed in terms of these supercovariant objects
A,L-. 62
Volume 74B, number 1,2
PHYSICS LETTERS
27 March 1978
iT,/~uX i _ 3 K exp (2 K(~)IguB757"X i = O,
¢°ouv3, 5 3,, ~ ~0 + iKVexp ( - K(p) C~jV OuvT°XJ + K V x / ~ [ [ ) v ¢ + iDvB75 exp 2 Kq~]3,VT°x i = O . The fact that the B field occurs only through its derivatives is connected with its dimensional reduction from an antisymmetric tensor gauge field as discussed in refs. [1,13]. Let us now give the transformation laws on the fields which turn this theory into the SO(4) theory. As the Bun axial fields have to be transformed into vector fields, this involves necessarily a duality transformation, which can be carried out only for the equations of motion. For all other fields, simple field redefinitions are enough and do not involve the use of equations of motion, The correspondence is as follows. For the SU(4) fields A, B, let us introduce Z = K(A + iB) with 1 - 2 KA = exp ( - 2 K~). For the other O (4) fields A', B', let Z ' = K ( A ' + iB' ). The transformation (1 - Z) (1 - Z' ) = 1 is easily seen to transform the kinetic terms of the spin 0 fields of the two actions into each other:OuZ'3u2'/ (1 - Z ' Z ' ) 2 = 3 ~ Z ~ 2 / ( 1 - Z - 2) 2. 1 ~i 1 r" For the spinor fields the transformation laws are" ~ = expi(½0 + ~ zr)')'5 ~u' xi = exp i(~ 0 + ~ ¢r)TSX ', where exp 2i0 = (1 - 2)/(1 - Z) = (1 - Z')/(1 - 2 ' ). which uv+ in the*uv For the vector and axial fields, the correspondence is the following. Denotingby Gijuvn thetartensor _ SO(4) theory gives the equation of motion of the F~ v fields, i.e. DuG~/v = 0, one has: eqjA n - (1/x/~)(Fi/ Fi~ ), =
-
-
t] -- rt
The first equation is a simple redefinition of the vector fields and can be carried directly on the vector potentials n = 0, i.e. that the equation of motion of the SO(4) Ate, AS.. The second, on the other hand, implies that D u B-uv gauge fields is satisfied. The complete proof of the equivalence of the two theories is very similar to the proof of the U(N) invariance of the SO(N) theory [9,6] and involves the following identities between supercovariant tensors:
F;
),
flijA~nvB = _(l/x/~-)(~b~.v- Fi/~,"v ^
where/ 8 = exp + 2K8 "" . Studying transformation properties of the action under homographic transformations of the Z variables led us to find that it is invariant under a group of SU(1, 1) transformations. Defining Y = 1 - 2 Z = exp ( - 2 K4~) - 2 i KB, the kinetic term of the scalar fields ¢, B can be written in the form 8 u Y S u Y / ( Y + ~-,)2, which is easily shown to be invariant under the transformation Y--* ( a Y + ifl)/(i'),Y+ 6), with a, fl, 7, 6 real and constrained by c~6 + fit = 1. These transformations form a non-compact group, namely SU(1, 1). To prove the invariance of the full theory, one needs to supplement this transformation law by those of the other fields of the theory. This is best expressed by giving the transformation laws for the three generators of SU(1,1) separately. (a) Y-* Y+ ifl, i.e. B--* B - fl/2K, other fields being unaffected. (b) Y-+ a 2 Y (B -+ a2B, ~ --* (p - (1/K) log e¢), Anu, B~ -+ -c~-1An --#, ~^ - 1 o~ on/ 2 , other fields unaffected. (c) In the infinitesimal form, the third transformation is given by: 6 Y = i X ( Y 2 - 1), Xreal
'
~AUV=XGU. v rt Art
66~ = - ½ iX exp ( - 2 K ¢ ) T S 6/u,
'
~ u v - ~ u- - vt , . U O r
tJz~rt
t
,
6xi=-3iXexp(-2KCp)Tsxi,
uv are defined such that the equations of motion of the Anu, B n" fields read DuGAn uv = D , G t~n ~v =0. where G Auv n , GBn This last transformation has been chosen to correspond in the SO (4) theory to the U(1) transformation extending SU(4) to U(4) [6]. As it involves a duality transformation it is an invariance of the equations of motion, not of the action while the two previous ones are directly invariances of the action. The invariance of the theory under a non-compact group does not imply the existence of ghosts as can be verified by inspection of the action, and the group is realized in a non-linear fashion. In the coupling o f supergravity to the scalar multiplet [10] a class of supersymmetric actions has for kinetic terms of the scalar fields the expression 63
Volume 74B, number 1, 2 2-
1
~ V [(0taA)2 +
where Z =
PHYSICS LETTERS
27 March 1978
(DUB)2]~[1 - ~,K2(A2+B2)] 2 = (V/2~,K 2) OuZOuZ/(1 - Z 2 ) 2
)~I/2K(A + i B ) .
This class o f t h e o r i e s d e p e n d i n g u p o n t h e a r b i t r a r y p a r a m e t e r ~. is i n v a r i a n t u n d e r an S U ( 1 , 1 ) g r o u p realized as follows:
Z= (aZ' +~)/(cZ'+~),
lal 2 - Icl 2-- 1,
•o
=exp(i/4?~)750(Z"2')¢o'
¢ --¢
× = exp(i/4),)(4),- 1)750(Z , Z ) x ,
t
with exp iO(Z', 2') = (cZ'+ d)/(c2'+ a). For c = 0 one recovers the usual chiral invariance already known for these models. In the limit where 2, goes to zero where one recovers the first discovered coupling o f supergravity to a scalar multiplet [ 14], the SU (1,1) group reduces to the E (2) group consisting of the chiral transformation (rotation in the A, B plane) and the two translations on the A and B fields. Although this invariance is present at the classical level, it is spontaneously broken at the quantum level since the vacuum is not invariant under field translations. The generalization of this invariance might also be helpful to generate larger supergravity theories, in particular to be able to write in closed form the SO(8) theory [15]. The authors wish to thank A. Neveu for an enlightening discussion about SU(1,1) invariance.
References [1] F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253. [2] L. Brink, J.H. Schwarz and J. Scherk, Nucl. Phys. B121 (1977) 77. [3] E.C. Poggio and H.N. Pendleton, Phys. Lett. 72B (1977) 200; D.R.T. Jones, Phys. Lett. 72B (1977) 199. [4] A. Das, Phys. Rev. D15 (1977) 2805. [5] E. Cremmer and J. Scherk, Nucl. Phys. B127 (1977) 259. [6] E. Cremmer, J. Scherk and S. Ferrara, Phys. Lett. 68B (1977) 234. [7] S. Ferrara and P. van Nieuwenhuizen, Phys. Rev. Lett. 37 (1976) 1669. [8] D.Z. Freedman, Phys. Rev. Lett. 38 (1976) 105; S. Ferrara, J. Scherk and B. Zumino, Phys. Lett. 66B (1977) 35. [9] S. Ferrara, J. Scherk and B. Zumino, Nucl. Phys. B121 (1977) 393. [10] E. Cremmer and J. Scherk, Phys. Lett. 69B (1977) 97; A. Das, M. Fishier and M. Rocek, ITP-SB-77-15, ITP-SB-77-38. ]11] P. Townsend and P. van Nieuwenhuizen, Phys. Lett. 67B (1977) 439; P. van Nieuwenhuizen, ITP-SB-77-55 preprint. [12] S. Deser, J.H. Kay and K.S. Stelle, Phys. Rev. Lett. 38 (1977) 527. [13] D.Z. Freedman, preprint CALT 68-624. [14] S. Ferrara, F. Gliozzi, J. Scherk and P. van Nieuwenhuizen, Nucl. Phys. B 117 (1976) 333; S. Ferrara et al., Phys. Rev. D15 (1977) 1013. [15] B. DeWit and D.Z. Freedman, Leiden preprint (1977).
64