Nuclear Physics B129 (1977) 39-44 © North-Holland Publishing Company
SUPERGRAVITY AS A GAUGE THEORY OF SUPERSYMMETRY A.H. CHAMSEDDINE International Centre for Theoretical Physics, Trieste, Italy P.C. WEST Laboratoire de Physique Thborique de l'Ecole Normale Sup~rieure, Paris, France Received 28 September 1976 (Revised 10 June 1977)
In a new approach to supergravity we consider the gauge theory of the 14-dimensional supersymmetry group. The theory is constructed from 14 × 4 gauge fields, 4 gauge fields being associated with each of the 14 generators of supersymmetry. The gauge fields corresponding to the 10 generators of the Poincar~ subgroup are those normally associated with general relativity, and the gauge fields corresponding to the 4 generators of supersymmetry transformations are identified with a Rarita-Schwinger spinor. The transformation laws of the gauge fields and the Lagrangian of lowest degree are uniquely constructed from the supersymmetry algebra. The resulting action is shown to be invariant under these gauge transformations if the translation associated field strength vanishes. It is shown that the second-order form of the action, which is the same as that previously proposed, is invariant without constraint.
1. Introduction It has been known for a long time that Einstein's theory of gravity could be regarded as the gauge theory of the Poincar6 group [1]. The purpose of this paper is to consider an extension of this formulation. We construct the gauge theory of the 14dimensional supersymmetry group and thereby construct a theory of supergravity. The idea of space-time dependent supersymmetry transformations is contained in ref. [2]. This approach to supergravity has two advantages. Firstly, because the number of gauge fields is the product of the number of generators of the group to be gauged and the dimension of the space-time, it can be chosen to be relatively few in number. Secondly, the field transformations and the Lagrangian of lowest degree are determined uniquely from the Lie algebra of the group. Perhaps the most consistent way to formulate gauge theories which are associated with space-time is in the context of fibre bundles. We will use the fibre bundle formalism and refer the reader to Nomizu and Kobayashi [3] for its definition and properties and to Trautman and Cho [4] for its application to gauge theories. However, 39
40
A.H. Chamseddine, P.C. West / Supergravity
we also outline the construction of the theory within the framework of the more common gauge theory formalism [5]. In sect. 2 we define the principal fibre bundle to be used and introduce a connection in the bundle. From these constructions we define the 4 × 14 gauge fields, denoted lt~e, Bu ab and ~u ~, corresponding to the 14 generators of the supersymmetry group. (Jsing these definitions we calculate their transformation laws and their associated field strengths. At the end of sect. 2 we outline the same steps in the context of the more common gauge theory formalism. In sect. 3, after identifying the vierbein and the Rarita-Schwinger spinor we construct the Lagrangian of lowest degree under the assumption that the translation associated field strength vanish. In fact, the Lagrangian implies that the condition adopted is satisfied by demanding that the connection field satisfies its equation of motion (i.e. is on "mass shell"). This theory of supergravity, when in its second order form, is the same as that proposed recently by several authors [6]. We hope this paper will clarify the origin of this Lagrangian and the algebra of the gauge transformations. 2. The field transformations and the field strengths We consider the 18-dimensional fibre bundle P whose base manifold is the usual four-dimensional space-time, M and whose fibre space is diffeomorphic to the symmetric group, c~. We denote by Pe, Jab, Sc~ the 14 linearly independent left-invariant vector fields on S which form its graded Lie algebra, c3 the graded Lie algebra being [Jab, Jccl]= i(•cbJaa - naaJcb - 7?bdJac + 11caJdb) , [Jab, ee] = i(eal?be - Pbrlae) ,
[Sco Jab] = l (Oab )af3Sfl , {Soo 8(3 } = - ( T e c)c~#ee ,
(1)
where oab = ½i[Ta, 7hi. We assume there exists a connection, F in P. Given a coordinate basis, Ou0a = 1 ...4) on M satisfying [Ou, a~] = 0 , we construct their corresponding horizontal lifts, bu on P (the covariant derivatives). However, ~u unlike 0 u, no longer commute, the commutation relations being [bu, bul = - c u v e p e - RuuabJab - DuvaS~ ,
(2)
where Cur e, R,,,, a~ and Duv '~ are covariant quantities and will later be identified with field strengths.
41
A.H. Chamseddine, P. C. West / Supergravity
Let w be the c3 valued connection form of the connection, P. Expanding w in terms of the basis of c3, (3)
w = w e e e + wabJab + ¢oaSc~
defines the forms w e, w ab and coa. For any given cross section o of P let the vector field on P denoted by o.(Ou) be the lift of Ou in Minduced by a. The gauge potentials are defined by lve = w e ( o . ( O V ) ) Bjb ~ u c'
,
= wab(a,(3U)) , =
w ~ ( o , (3#)) .
(4)
The gauge potentials corresponding to two cross sections o and a' of P are related, in the limit of infinitesimal group parameters, e c, w ab and ~ , by the equation: ~B# ab = ~#W ab + 2i(waaBua b - wbdB#da) , 51Uc = 3ue c + 2i(wCalu a -- eaBu ca) + ~7c ~
,
<--
~#a
In
O
= [~(()tz +~OcclD#
cd'~lOt
)J +W
ab
--
1
(l~Uab)
a
.
(5)
Expanding 3~ in terms of the usual local direct product basis and using the definition of the gauge potentials we obtain bg = 31z - l#epe - ntzabJab -- ~lzuSa .
(6)
To evaluate the field strengths given in eq. (2) we evaluate the commutator using eq.
(6): Cur a = 3lulv] a + 2illufBv] f l - ~uTa~kv, Ruvab = 3[uBu] ab _ 2iBiuaCBvlcb , 1
Dtav a = ~lv(g-M + ~Oat, Bal
ab)
.
(7)
The transformations of the field strengths are obtained from the Jacobi identities of eq. (2). For the transformations under supersymmetry we have: 5cuv a = ~3,a D~v , 5R~w ab = 0 , 8Duv~ =(rl~OabRl~ "--1 v a b ) ot •
(8)
A.H. Chamseddine, P.C West / Supergravity
42
The results obtained above from the fibre bundle are nothing more than those of the more usual ad hoc description of gauge theories. Consequently we could, apart from perhaps a loss of clarity, carry out the above steps in this more common framework. That is, to construct a gauge theory of the supersymmetry group we introduce by hand 14 × 4 gauge fields into the covariant derivative in the following way:
VU = ~u - luepe - BuabJab -- f S~Sc~ , Vu is of course none other than bu given in eq. (6). The field transformations of equation (5) can be obtained by demanding that V u acting on a matter field transforms in a covariant way. The field strengths are obtained exactly as before, that is evaluating V u V ~ - V~Vu •
3. The Lagrangian Having calculated the field strengths and the transformation laws of the gauge potentials we can construct a Lagrangian. This is easier to achieve if we make a geometrical identification first. That is l ue and qJu~ are to play the roles of the vierbein and a Rarita-Schwinger spinor respectively. We assume that the Lagrangian, 22 is of lowest degree and t h a t Claya = 0. We will show later that this constraint, is true if the connection field satisfies its equation of motion and hence is automatically satisfied in the second order formalism. Consequently, 22 being essentially constructed from the two remaining field strengths, Ruv ab a n d Duv c', is of the form
22 = g~VabRpvab + fuVo~Duvo~ ,
(9)
where gUVab and f u ~ are functions of leu, ~ u and Bu ab and Dum = C~Duu ft. Matters are considerably simplified if we use the fact that for ~a u = 0, 22 reduces to that of Einstein-Cartan theory (gravity alone), i.e.
gUVab = lltaUlbl u ,
(10)
where l = det lCx. Although there exist many possible terms which are invariant under the Poincar~ subgroup, we now show that invariance under the supersymmetry transformations restrict these to be just one. Varying the action under the supersymmetry transformations given in eqs. (5) and (8) and setting the result to be zero implies that
0 =fdx[SgUVabRuv ab + 5fU~Duvo~ - f tam]iOab~?" Ru. ab] . From eq. (10) we find that
(11)
43
A.H. Otamseddine, P. C West / Supergravity = l~(laUloVlch7 c - 7ClcUlahlb v - IcVlbXlaU7 c) 4X •
~u
(12)
By counting powers of ff in eq. (11) it is easy to see t h a t f u ~ must be linear in and so can be written in the form
where hUVx is a matrix and a function of lUa . Since the action must vanish identically for all possible values of the fields (subject to cauv = O) we choose, for the moment, only to examine terms linear in 4. n cd,,) hUVXDuv 0 = f dx [~-~']~WXab 4 h R u v ab + ~(O<-h + ~1 OcelJx -
-
(13)
~ x hUVXl Oab ~TRuvab ] .
Using Cu, a = 0, the form of h uvx, and keeping terms only linear in 4 we obtain from eq. (13) O = f dx [~-91UVXab 4 XR
_ ~ h ' tt'/'wx_]Uab,l,t,,pv~D 1 ab
-- ~ h " V h ( ~ h -- l oabghab ) Ouv] •
This implies that 12h uvh = --cJ~ laVhab oab = _l(TXou v _ 7uoXv _ 7%u x) = - 3i euvX°7o7 s .
Hence we have determined .~ to be .12 = llatilbVRuv ab + ~i euvx° ~ x Ts ToDuv •
However, we have only shown A? to be invariant when keeping terms linear in 4. We now show that .~ is completely invariant. We dropped two terms both cubic in if, one coming from the variation of 7o in the second term in the Lagrangian and the other coming from the Coy a condition. These terms are f d x ( - i l e ~ V h ° ) [ - ~ ' Y a T s D u v • ~ h 3'a 4 a + ~ 7 a 4o~X'raTsDu~l .
A Fierz reshuffle shows these terms to cancel. Hence, the above action is shown to be invariant under the gauge transformations of eq. (8) if we assume that the translation associated field strength, Cur a vanishes. This approach is consistent in so much as the equation of motion for the connection field, B~ ab implies that Cur a does indeed vanish. It follows from this that the action
A.H. Chamseddine, P. C West / Supergravity
44
is invariant without constraint when in its second order form, that is once the connection field, B uab has been eliminated in favour of the vierbein and the RaritaSchwinger spinor by use of its equation of motion. The action and the transformation laws in the second order form are the same as those of the supergravity theory already proposed [6]. Although the action in its first order form is the same as that proposed by Deser and Zumino [6] we differ in the transformation law for Bu ab. We stress that this does not imply that the action is invariant under two different sets of transformation rules for the gauge fields. This is because the action is invariant under the transformations o f Deser and Zumino but is only invariant under our transformations if Cur a = O. However, it is a result of the special nature of our condition, Cur a = 0 that the action, in the second order formalism is invariant. We expect it to be possible with the use of more fields to construct a theory which has linear transformation laws and an invariant first order action. It is easy from our approach to obtain the non-linear transformation laws of Deser and Zumino [6] which leave the action, A in first order formalism invariant. Namely, let us add a non-geometrical piece ~xBt~ab to the only auxiliary field, Bu ab . The change in the variation of the action due to AB~ b is
fax
~)._Z ~B~ ab
z~xBlab ~ f d x ( f b j f ABaab + CfafZ~kBba b _ CabfZ~kn?b )
We require this change to cancel the variation of the action under the gauge transformations, which is of the form afdx cuvaNa uv with
NataV = e41vPK-~'y5,~a l~pK . Equating coefficients of Cur a we can determine ABu ab in terms ofN~ ab, the result being that obtained by Deser and Zumino [6]. Further progress could be made choosing our base-manifold to be the 8-dimensional manifold, introduced by Abdus Salam and J. Strathdee and labelled by (Xu, 0a). The resulting gauge theory would contain many more gauge particles. We wish to thank Professor Abdus Salam, Dr. R. Delbourgo and Professor T.W.B. Kibble for helpful discussions.
References
[1] R. Utiyama, Phys. Rev. 101 (1956) 1597; T.W.B. Kibble, J. Math. Phys. 2 (1961) 212. [2] B. Zumino and J. Wess, Nucl. Phys. BT0 (1974) 39. [3] Kobayashi and Nomizu, Foundations of differential geometry, vol. 1. [4] A. Trautman, Reports on Math. Phys. 1 (1970) 29; Y.M. Cho, J. Math. Phys. 16 (1975) 2029. [5] E.S. Abers and B.W. Lee, Phys. Reports 9 (1973) 1. [6] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214; S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335; D.Z. Freedman, P. van Nieuwenhuizen, Phys. Rev. D14 (1976) 912.
