Affine extensions of supersymmetry: The finite case

Nuclear Physics B138 (1978) 3 1 - 4 4 © North-Holland Publishing Company

AFFINE EXTENSIONS OF SUPERSYMMETRY: THE FINITE CASE Y. N E ' E M A N **

Tel-Aviv University, Tel-Aviv, Israel and Center for Particle Theory, Department of Physics, University of Texas, Austin, Texas 78712, USA T.N. SHERRY * Center fi~r Particle Theory, Department of Physics, University of Texas, A ustin, Texas 78712, USA Received 31 January 1978

We examine the graded Poincar6 (GP) Lie algebra of supersymmetry with a view to constructing possible affine extensions of the algebra, i.e. extensions of the GP algebra which contain as a subalgebra the Lie algebra ga(4,c~). We restrict our attention in this paper to an examination of the finite extensions. We demonstrate explicitly that if we adjoin only a symmetric tensor generator to the GP algebra, then such a generator cannot generate all the deformations, in particular the shear, of the general affine group GA(4, Q~). Similarly, we show that adjoining the supersymmetry generator to ga(4,c~) cannot lead to closure of the resulting algebra, even in the trivial case. We further demonstrate that the GLA ga(4/4,c'R) does not contain the Lie algebra ga(4,Q~) represented over the entire superspace upon which ga(4/4, c~ ) is defined.

1. I n t r o d u c t i o n G r a d e d Lie algebras [ l a ] ** were i n t r o d u c e d in the physics o f particles a n d fields in several c o n t e x t s . In dual m o d e l s a n d strings [2] several i n f i n i t e - d i m e n s i o n a l G L A ' s were applied as " s u p e r g a u g e " c o n d i t i o n s in the c o n s t r u c t i o n o f the s p i n n i n g string a n d were essential in the r e m o v a l o f divergences, etc. In space-time field t h e o r y , s u p e r s y m m e t r y [3a] *** p r o v i d e d a n e w g e n e r a l i z a t i o n o f the s y m m e t r y c o n c e p t : ¢~Research supported in part by the United States-lsrael Binational Science Foundation. * Reserach supported in part by the US Energy Research and Development Administration, Grant No. E(4P-1)3992. ** The classification of simple GLA's has been given by Katz [lb]. See also ref. [1 c]. *** Wess and Zumino [3b] independently rediscovered the graded Poincar~ group as a subgroup of their graded-conformal SU(2, 2/1). This paper [3b] ushered in the study of supersymmetry. 31

32

Y. Ne'eman, T.N. Sherry/Supersymmetry

Bose-Fermi symmetry. The most interesting results have been in the simplification and partial reduction of ultraviolet divergences in various field theories ? and in the construction of Goldstone-type models in which supersymmetry is spontaneously broken by a massless spin J = ½ fermion ". Higgs-type models in which this fermion is replaced by the acquisition of mass by a J -- ~ field have also been constructed. Supersymmetry has also been applied to gravity, either in c~ 4,4 (xla, 0&) superspace, first introduced [6] as a representation for supersymmetry, or directly in space-time c~ 4. Though some insights have been derived in the first approach [7a] '~'~ it is the theory of supergravity [8] which emerged from the second which has supplied the most interesting results. Spectacular progress in the renormalization of gravity [9], positive-definiteness of the Hamiltonian [10], the harmonious cancellation of the difficulties relating to charged J = 23-fields [ 1 1], etc., represent applications to gravity, aside from the development of supergravity itself and its generalizations [12]. In a recent work, we have suggested a new group theory cum geometry approach to these theories [13]. In discussing gravity as a field theory, one is led to consider GA(4,c~ ), the general affine group of transformations in a four-dimensional Euclidean (temporarily neglecting the Minkowski structure) space-time. This is the linear subgroup of Einstein's group of general coordinate transformations (GCT) and carries its induced representations [14]. In a future development of the theory, it would seem natural to proceed to a Lie-bracket grading of this group's generator algebra. Moreover, a recent generalization of Einstein's gravity has been suggested [ 1 5] in the form of a gauge theory of GA(4, c~ ), the metric-affine theory of gravity. This has been further developed as a theory fitting the hadrons in particular [ 16]. It gives rise to intrinsic currents of spin, dilation and shear (forming together the hypermomentum tensor) which couple to the connection-field (the Yang-Mills field of intrinsic GL(4, cE ), the homogeneous part) in addition to the coupling of the energy-momentum tensor to the (metric) vierbein. To produce intrinsic shear, the hadrons have to belong to the recently constructed [1 7] band-spinor infinite representations of GL(3, oN), embedded in "polyfields" [16] constructed from infinite representations of GL(4, oR). As has recently been pointed out [ 18] such structures produce bona fide bivalued spinorial representations of GL(4, oR) and GA(4, ~ ) , relevant to gravity theory. Presumably, if we intend to construct the graded spin-extension of GA(4, c/,?), we should turn to such representations. Embedding the conventional J = ½ supersymmetry generators of the graded Poincar6 group in an infinite set with J = s, 9, etc., and looking for the minimal closure with GA(4, c~) might yield a four-dimensional analog of the 1 Wess and Zumino [4a] gave the first direct application to the improvement of renormalizability properties of known theories. For a general review see refs. [ 1,4b,6]. The first successful "stable" model with spontaneous breakdown was provided by Fayet and lliopoulos [5a]. For more recent comments see ref. [5b]. ~'~ This is a graded-Riemannian geometry. For a superspace with torsion see ref. [7b].

Y. Ne'eman, T.N. Sherry/Supersymmetry

33

Ramond-Neveu-Schwarz spinning string [2] GLA, perhaps the complete spinningspace-time GLA. However, it seems worthwhile to try first the finite possibilities. The generators of GA(4, c~ ), a non-compact Lie group, can be classified in terms of its SO(4) subgroup. We identify this SO(4) group with the Lorentz group, neglecting temporarily the Minkowski structure of space-time. The generators of GA(4, c~) are, in turn: the generators of SO(4), the antisymmetric Juv; the translation generators Pu; and a 2nd-rank symmetric tensor Tuv representing deformations, either by shear (the traceless J = 2 part) or by dilation (the trace). In this paper we wish to discuss how one should proceed to extend ga(4,c~ ), the Lie algebra of GA(4, c-~ ), to a finite graded Lie algebra. We shall address this problem from three different angles. To begin with we consider the Z 2 graded spinor extension of the Poincar~ algebra, GP, i.e., supersymmetry [3]. We adjoin to this algebra a 2nd-rank symmetric tensor generator Tuv and examine what restrictions the graded Jacobi identity [1] places on its commutation relations with the other elements of the algebra when we require the minimally extended algebra to close on itself. We find that this requires the spin-2 part of Tuv "not to see" the supersymmetry generator, while the spin-0 part behaves as the sum D + E of the scalar generators of the superconformal algebra SU(2, 2/1) [3]. This is consistent with the Cartan theorems [18], stating the inexistence of (finite) spinors in GL(n, c/q). Furthermore, the spin-2 part of Tuv cannot generate the shear deformations. Approaching the problem from a slightly different point of view, we next examine the Lie algebra ga(4, Q~), and attempt to extend it by adjoining a Lorentz spinor generator S. Again, postulating closure of the resulting graded Lie algebra, we use the graded Jacobi identities to restrict the commutation and anticommutation properties of S. The result is more startling in this case; the algebra cannot close without the addition of other generators, even in the simplest case when the supersymmetry generators anticommute. We next examine the graded Lie algebra of the group GA(4/4, c~). This is the linear subgroup of GCT in the Salam-Strathdee superspace [6,7]. The Bose sector of ga(4/4,'0~) consists of two ga(4, c~) algebras, one generating the affine transformations in x space, and the other generating the affine transformations in 0 space. Our purpose is to find the closure, within ga(4/4, c£), of an algebra containing the generators J, P, and S, and a symmetric tensor T. We identify the smallest closed subalgebra of ga(4/4, c'R ) which achieves this result. However, again we find that the symmetric tensor T cannot be identified anymore as a deformation-generating operator. T does generate "orbital" deformations in the space-time sector, but not in the spin sector. Again, this is consistent with the requirement of band-spinors for "intrinsic" deformations. Furthermore, the resulting algebra contains extra generators besides those already considered. We now proceed to exhibit our results explicitly. In sect. 2 we discuss the attempt to extend the GP algebra. In sect. 3 we briefly consider the minimal-spinor-

Y. Ne'eman, T.N. Sherry / Supersymmetry

34

extension of ga(4, oR), and in sect. 4 we examine the GLA ga(4/4,c~ ), and identify the relevant subalgebra. In this paper we use the notation of Corwin, Ne'eman and Sternberg [1].

2. Extension of the GP algebra [3] The generators of the GP algebra are, in turn, the generators of the Poincar6 Lie algebra, Juv and Px, and a Lorentz spinor generator S a. The commutation relations which characterize this algebra are

[Juv, J~p] = i~upJv~, + i~vxJ~p - i~u~Jvp - i~vpJ~x , [J,v, Ph]= -irluhPv + i~vxe~ ,

=0,

[e#,Sa] = 0 , (Sa, Sb) = --2(TXC)abPx •

(2.1)

This algebra is Z 2 graded, Juv and Px belonging to the even gradation and Sa belonging to the odd gradation. The graded Jacobi identity for a GLA is [x, [,v, z ] ] : [ [ x , y ] , z ] +@)kl[y, IX, Z ] ] ,

(2.2)

where x E L k , y E L I , k = even for even grading = odd for odd grading. We wish to adjoin a symmetric 2nd-rank tensor generator T~,v. Thus its commutation relation with Juv is given to us as

[Juv, Txp] = -irlux Tvp + i~?v~,Tup - i~TupT~v + irTvpTxu •

(2.3)

The form of the other commutation relations is open to us, so we write their generic form as

[Tuv, Px] = AuvxaP~ + Buvxu#Ja~ + Cu~,xa#Ta~ ,

[Tuv, Sa]

=

Duua b SO ,

[Tuv , Tap ] = EuvNoapa + F u v x o ~ J ~ + GUvXp~T~ .

(2.4)

35

Y. Ne'eman, T.N. Sherry / Supersymmetry

Here we have made use of the remaining property of a graded Lie algebra, namely ILk. LI] C Lk+ 1 ;

the commutation relations are consistent with the grading chosen. From their defining commutation relations (2.4), the undetermined coefficients have the following symmetry properties: A~v~, ~ = A v ~ c~ , B~vx a3 = Bvux u3 = _ B u v x 3 c~ , Cu.x~P = C.ux'~

= C u ~ ~ '~ ,

Dtava b =Ovtaa b , Euv~o c~, Fuvxp ' ~ and Guvx; c'3 are symmetric under (/l +-->v)

and (X +-->p) and are antisymmetric under (gw) +-+ (X0), Fuvxo ~t3 = - F u v x o ~ ~ , Guvxp ~¢ = Guvxp oc~

(2.5)

We further note that the most general form form Dora ~ is Duva a = rltav [a6ab + b(Ts)ab] .

(2.6)

Use of the Jacobi identity for [Tuv, (Sa, So}] leads us to the result BXo~ a3 = 0 , CXou ~ = O,

(2.7)

AXpu a = 2a~xp~u a .

(2.8)

All of the Jacobi identities are consistent with the form (2.6) for Duva °. The F-coefficient is shown to vanish from the Jacobi identity in the case IT.v, [Txo, e ~ ] ] ,

while we also derive the relation GuvxpU~r?a(s = 0 ,

(2.9)

from this same identity. Now, the form of the E- and G-coefficients can be seen from general covariance requirements. Without considering spinor quantities (e.g., 7 matrices), or constant vectors, we cannot construct a quantity such as Euvc4s x which has an odd number of vector indices. On the other hand, the G-coefficient

Y. Ne'ernan, T.N. Sherry / Supersymmetry

36

could have the form

Guvhp aft = al ~?uv~?xo~?~fl + a 2 ~ u v ( ~ x a ~ y + ~xfl~o a) + aarlXp(~uarlv ~ + rl#flrlv a) . However, from the symmetry requirements (2.5), we see at once that al = 0 and a 3 = - - a 2 = - - c , i.e.,

Guvxp '~- = c [~uv(~x %?o ~ + ~?x~/o~) - ~xo(~u %?v fl + ~ u ~ v a ) ] .

(2.10)

Furthermore, this form is consistent with the restriction (2.9), and in fact with all the other Jacobi identities. We group together our results as follows:

[Tlav, Sa] = "(Itav[aSab + b(T5)ab] Sb ,

[Toy, TXo ] = 2c[~?uvTno - nxoTuv ] .

(2.11)

From the second of these equations we see at once that the spin-2 part of Tuv, the traceless symmetric tensor T(2)u v = Tuv - ¼~uvT does not "see" the supersymmetry generator S a. On the other hand, the spin-zero part, T(O)u v = ~ u v T , where T = ~'PTxo, does, as is seen from [T(0)uv,

Sa] = rllsv(a6ab + b(~/5)ab) Sb •

From this we see at once that T(O)u v is just the sum of the two spin-0 operators D and E of the superconformal algebra SU(2, 2/1):

T(O)uv = ~?uv • aD + ~ bE

,

(2.12)

where [D, Sa] = ~iSa and [E, Sa] = 3i(TS)abSb [1]. Furthermore, from the third of eqs. (2.11) we see that Tuv does not generate deformations, even if e were to vanish. The commutation of two deformations should yield the Lorentz generators, and we have already seen that in the third commutation relation of (2.4), the F-coefficients are forced to vanish. 3. Minimal extension of G(4,q~ ) The general affine group of transformations is the semi-direct product of the general linear group with the group of translations. Thus, as generators of the general affine group in four dimensions we have the antisymmetric rotation generators Juv, the symmetric deformation generators Tuv and the translation generators Pu" The Lie algebra of these generators is given by the following commutation relations:

[Juv, J~p] = i~lupJvx + i~vxJup - inu~Jvp - i~vpJux ,

Y. Ne'eman, T.N. Sherry / Supersymmetry

37

[Juv, PX] --- -i72uxPv + i~vxPu , [Juv, Txo] = -irluh Tvo - i~TuoTxv + i•xvTuo

+ i~2pvThu ,

[ Tuv, Txo] = iT?uxJv p - i~?uoJav + irlhvJuo - ir?ovJa u , [ Tuv, PX ] = -ir?ux Pv - i~?vxPu , [Pu,Pv] = 0.

(3.1)

We now consider extending this algebra in a minimal fashion by adding to it a Lorentz spinor generator, which we denote by S a, though it should not be confused with the generator of sect. 2. Thus, we have [j#v,, Sa] = --~(Ulav)abS 1 b .

(3.2)

The other commutation relations, using just the closure and the Z 2 grading, take the generic form

[Pla, Sa] = B # a b S b ,

(Sa, Sb} : Dab~Pa + E a b ~ J c ~

+ Falff~Ta# .

(3.3)

However, it is a straightforward exercise to check that these relations are not consistent with the graded Jacobi identities. This follows from the identity [So, [Tuv, Tap]] = [ [ S o Tuv], Txp] + [Tuv, [So Tap]]

(3.4)

and from the most general form for Aura b , namely Alava b = rl~v [a6ab + b(Ts)ab] •

With this form the right-hand side of (3.4) vanishes, whereas the left-hand side gives US

l i {rh, x ovo - 77uooxv + ~?XvOup --~Tpvaau}aaSb , (3.5) and this does not vanish. We are forced to conclude from this result that the assumption of closure of the graded Lie algebra after minimal extension by the addition of a Lorentz spinor generator was false. Already from sect. 2, we can see that this result is possible, for there we had closure but Tuv could not generate the deformations.

4. ga(4/4, t'~) and its graded subalgebras In this section we shall be examining the graded Lie algebra of GA(4/4, ~ ), and some of its closed subalgebras. GA('4/4, c~) can be viewed as the (pseudo) group of

Y. Ne'eman, T.N. Sherry/Supersymmetry

38

general affine transformations in a space which has four Bose dimensions and four Fermi dimensions, a space commonly referred to as superspace. This is the linear subgroup of the GCT on superspace. The Bose sector of ga(4/4, c~ ) is ga(4, c~) 1 × ga(4,t~)2, where GA(4, c~)1 is the group of affine transformations on the Bose space, and GA(4,c~)2 is the group of affine transformations on the Fermi space. The SO(4)I subgroup of GA(4, c~ )1 is also defined over the Fermi space; in fact this is the defining feature of the variables of that space. We choose the variables of the Fermi space to transform as spinors under the SO(4)l subgroup. We can realize the generators of the algebra ga(4/4, c)~) by the differential representations

Juv = i(xu3v - xvc3u) GA(4, c~)1 acting on the Bose space,

Tvv = i(xuO v + xuOv)

Lab = iOa ~

=-iOa~b1 GA(4, c~)2 acting on the Fermi space,

N a=-i-~a-

-ioa

J

Qua = xu ~ a -~ XvOa odd generators, causing transitions from Bose to Fermi dimensions and vice versa. (4.1)

Rav= OaO~

Here 0a is an anticommuting Majorana spinor, which is thus real by our conventions. We use right-differentiation, so that

~a(ObOc) = Ob6ca _ Oc6ba ,

(4.2)

where {0b, 0~} = O.

From this realization of the generators we find the communication relations of the graded Lie algebra to be: (i) the commutation relations (3.1) for J, T and P; (ii) for the remaining generators

[Lab, Lca]= i6adLcb -- i6cbLaa , [L~b, N~]

=

iSaJb ,

39

Y. Ne'eman, T.N. Sherry/Supersymmetry [J/sv, Ra~.] = iT?v),Rau - ir?uhRav , [Tuv, Qha] = i~?vhQ~a + iT?uxQva , [ Tu~, Rah] = -i~?vxRau - ir?uxRav , [Pu, Qxa] = rluxNa , [Lab, Qxc] = i6acQXb , [Lab, R e x ] = - i 6 c b R a x ,

v + T u . ) _ ~ir?uuLba I {Qua, R o . }. = gl6aO(ju ]" ,

(4.3)

and all others vanish. There are in all 72 generators. For our own purposes it will be more convenient to work with a slightly different set of generators. Rather than the 16 generators Lab used above, we use the equivalent set of Lorentz-covariant generators v.~ = (Co.S)~bL~b , V . = (CTu C)abLab , AI~ = (CT,'), 5C)abLab , F s = (C'~s C)aaLab ,

F = (C1C)abLab.

(4.4)

(Here Cis the charge-conjugation matrix.) These are tensor, vector, axial-vector, pseudo-scalar and scalar generators, respectively, under the action of Lorentz transformations. The commutation relations among this set of generators are [V.u, Vxo] = 2It/up Vux + r/ux V.p - rTuxVvo - ~vpVua] , [ V u, V.] = - 2 V u ~ ,

[a., A~] = - 2 v . ~ ,

[v.~, v~] = 2(~x~v. - ~ x . v , , ) , [Vuv, A~.] = 2(r~xuA u - r/xuA~),

40

Y. N e ' e m a n , T.N. S h e r r y / S u p e r s y m m e t r y

[ V u, Av] = -2irTuvFs , [ Vu, Fs] = - 2 i A , ,

[Au, Fs] = +2iVu,

(4.5)

and all others vanish. The commutation relations with the other generators of the algebra can be easily evaluated from eqs. (4.3). As can be seen from the 1st relation of (4.5), gt 1. Vuv furnishes a representation of the SO(4)1 generators Juv, over the 0 part of superspace. Thus we have 2 representations of 2 SO(4) groups over 0 space: SO(4) 1 ... ½iVu v

=

1 -_ . .lOa]abOb uv -~(Cou~C)abOaO~ •

,

mn

8 0 ( 4 ) 2 ... l ( L m n _ Z n m ) = lOa l a b Ob "

It is straightforward to see that, since det l m n = 0, while detj uv --/=0, these are not equivalent representations of an S0(4) group. We can now identify the GP subalgebra of global supersymmetry, which acts on both sections of the superspace. We denote the generators by script letters, and they are ~uv = J#v

CJa = - C a ~

+1.

~tVlav ,

b - l (,~t2)ab,Rbl a

= - C a b N b - ½(,It . R ) a .

(4.6)

These generators satisfy the GP commutation relations of eqs. (2.1). We notice that ('y" R)a is one of the spin-½ projections of Rau. For the present this is the only projection of Rau that we have need of. We can further identify a 2nd-rank symmetric tensor generator 7uv with respect to ~uv. In this there are two arbitrary parameters A and B:

cj'uv = Tuv + 2~?lsv(A6ab + B('YS)ab ) Lab.

(4.7)

The commutation relations with the generators of (4.6) are

[ C~uv, ~7ao ] = -i~?uh Srvo - irluo~'?,v + irlxvc.Juo + ir?pv~rhu , [ 7uv, c5,,] = -irluv[-2AS,, b + 2B(z's)aa]CJb + irluv(A + 1)(7" R ) a , [ sruu, ~X] = -i~Tuxgu - irlvx~u ,

(4.8)

Y. Ne'eman, T.N. Sherry / Supersymmetry

[ cJuu, ~rKo] : i~uh[~vp - ½iV~o]- i~Tup[~xu

41

1.

+ ir~xv[guo - ½iVuo] - iTIou[~x # - ½iVxu].

(4.9)

From (4.9) we see that ~ruu does not generate the deformations, as on taking the commutation of two 5r transformations we are not left with a pure ff transformation. We further notice that ~uv, 5~u, CJa and 5ruu do not close to form a subalgebra of ga(4/4,cR). In the case A = -¼, we need to consider also the generator Vx~, while in the general case (A ¢ - 1 ) we should also include (7 " R)a. The commutation relations of Vxu with 9 , ~ , d and 3"are as follows: [7.~, vxp] = i I n . . v ~ , [7~,

+ n~x v . o - n . x v~. - n~,~v.7,] ,

vxp] = o ,

[ CJa, VhO ] = --i(Ohp)a b cJ b + (('Yh)abRbp -- (~[p)abRbh) •

(4.10)

Clearly then we also need to include the generator Rbo, even in the case A = -~-. Its commutation relations are 1

[ ~#u, Rah] = i~vhRau - irluhRau -- ~(°~v)aeRch , [ ~ruu, Rah] = -irluxRau - irluhRau - 2irluu(ARax - B(3~s)aeRch), ( d a, Rex} = - C a c ~ x ,

[Ra., v~ o ] = - i ( o ~ . ) , ~ R c .

.

(4.11)

Grouping together the sets of commutation relations (2.1), (4.8), (4.9), (4.10), (4.1 l) and the first ofeqs. (4.5), we see that the42 generators ~uv, 5ruv, ~u, Vuv and Rau close to form a graded subalgebra of GA(4/4, cR ). What we have constructed is the smallest graded subalgebra of GA(4/4,c'R) which contains both the GP algebra of simple supersymmetry and a second-rank symmetric tensor generator (under Lorentz transformations). However, we have seen explicitly that this symmetric tensor generator does not generate deformations, and so we cannot consider the algebra we have found to be the supersymmetric extension of ga(4,cR).

5. Discussion

In this paper we have re-examined the algebra of supersymmetry with a view to extending it in the direction of the general affine transformations. The extension of the GP algebra to the graded conformal algebra SU(2, 2[N) has been known since shortly after supersymmetry itself was introduced [3]. Our attempts in this

42

Y. Ne'eman, T.N. Sherry /Supersymmetry

direction have not led to the more interesting physical solutions containing complete deformations. On the other hand, the cases we have examined might well be useful for less ambitious purposes. The general affine groups of transformations are characterized by their 2nd-rank symmetric tensor generators which on commutation yield the generators of their special orthogonal subgroup. Thus, as a first attempt, in sect. 2, we adjoined to the GP algebra a 2nd-rank symmetric tensor. Requiring that the resulting operators close on commutation, and using the graded Jacobi identities, we found that the adjoined tensor generator does not generate deformations. In fact, the analysis showed that the traceless (shear) or spin-2, part commutes with the supersymmetry generators, while the trace, or spin-0, part behaves like a linear combination of the scalar and pseudo-scalar generators of SU(2, 2/1) (i.e., the dilations and chirat transformations). On commutation amongst themselves, the symmetric tensor generators can at most reproduce themselves, or vanish. We examined essentially the same problem, from a different point of view, in section 3. There, we began with the general affine group GA(4, c~), and considered adjoining to its Lie algebra a generator transforming as a spinor operator under the SO(4) subgroup of GA(4, cR ). We demonstrated explicitly that the resulting algebra could not close: not even trivially with the supersymmetry generators anticommuting. We are thus led inexorably to consider larger graded Lie algebras. The example we examined in sect. 4 is realized most simply on superspace. It is the GLA of the linear subgroup of the general coordinate transformations on superspace, ga(4/4, c~ ). The Bose sector of this GLA consists of ga(4, 9~)1 X ga(4, c~)2 where ga(4, c~)t acts on the x space and ga(4, c'~)a acts on the 0 space. However, the 50(4)1 subgroup of GA(4, cR)l also acts on 0 space, defining the 0's as spinor coordinates. Making use of this defining feature of the superspace we identified the GP subalgebra of ga(4/4, c~). This algebra was extended to include the most general 2nd-rank symmetric tensor-generator. For closure, however, other generators of ga(4/4, 9~) were required. From the resulting commutation relations it is apparent that the symmetric tensor generators do not generate complete deformations. This analysis is telling us that we cannot represent the T~, generators of GA(4, 9~) linearly on the 0 space. In actual fact, there is no non-linear differential representation either. This follows from the result that the GCT group on 0 space is generated by a finite-dimensional GLA. There are 64 generators in all, represented as follows with the multiplicities and gradings shown: generators 0 OOa 0 00 aO0--

multiplicity

grading

4

-1

16

0

Y. Ne'eman~ T.N. Sherry/Supersymmetry generators

multiplicity

grading

OcOb bO----~a

24

+l

b OdOcOb - boa

16

+2

4

+3

3

OlOdOcOb

b

43

the grading operator being 0a O/bOa. The only possibility to represent Tu, with the correct covariant structure would utilize OaOcOo O/bOa, but these generators actually commute. This finding may well have intrinsic importance. On superspace, as it is conventionally defined, we cannot represent the group GA(4, c~). Thus, the full potential of gravity cannot be realized on such spaces. As we have seen there are defined on such spaces algebras more general than the GP algebra. It remains to be seen whether or not the algebra we have exhibited possesses double-valued representations of the band-spinor type. In the meantime we conjecture that the algebra does not possess such representations. Thus, we are led to the result that embedding space-time in a larger space in which one specific realization of spin is selected, automatically eliminates the possibility of holonomic spinors, intrinsic shear, etc., and thus restricts gravity in at least its spin degrees of freedom. At any rate, it seems apparent that if we restrict our attention to spin -1 generators alone, we cannot extend the GP algebra in the affine direction. Our next avenue of investigation is to examine GLA's with higher spin supersymmetry generators. In the light of the above discussion we do not restrict our attention to a finite spin. We adjoin to ga(4, c~) an operator which transforms as a GA(4,c~ ) spinor. Such spinors have recently been shown to exist [ 18], and they are infinite-dimensional representations of the group. However, as their structure is not fully known as yet, we shall have to content ourselves with an analysis of GA(3, c-~), for which the multiplicityfree spinor representations have been classified [ 17]. It is an intriguing possibility that the realization o f such a group of transformations as the symmetry group of a "superspace" would require a superspace with a countably infinite number of spinor sectors. Such a program of investigation offers more interesting possibilities, and will be reported elsewhere. One of the authors (Y.N.) would like to thank Dr. L. Smalley for raising the issue discussed in this paper. References [1] (a) L. Corwin, Y. Ne'eman and S. Sternberg, Rev. Mod. Phys. 47 (1975) 573. (b) V.G. Katz, Functional analysis and applications (USSR) 9 (1975) 91.

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1I, Ne'eman, T.N. Sherry / Supersyrnmetry

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