Towards a unified theory of massless superfields of all superspins

Towards a unified theory of massless superfields of all superspins

27 February 1997 PHYSICS LETTERS B Physics Letters B 394 (1997) 343-353 Towards a unified theory of massless superfields of all superspins S. Jame...

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27 February 1997

PHYSICS

LETTERS B

Physics Letters B 394 (1997) 343-353

Towards a unified theory of massless superfields of all superspins S. James Gates Jr. a*‘, Sergei M. Kuzenko b,2, Alexander

G. Sibiryakov bT3

a Department of Physics, University of Maryland, College Park, MD 20742-4111, USA b Department of Quantum Field Theory, Tomsk State University, Z_eninAve. 36, Tumsk 634050, Rz~sia Received 4 December 1996 Editor: M. Dine

Abstract We describe the universal linearized action for massless superfields of all superspins in N = 1, D = 4 anti-de Sitter superspace as a gauge theory of unconstrained superfields taking their values in the commutative algebra of analytic functions over a one-sheeted hyperboloid in R3,‘. The action is invariant under N = 2 supersymmetry transformations which form a closed algebra off the mass-shell.

1. Introduction Off-shell superfield realizations of N = 1, D = 4 higher superspin massless multiplets were given previously [ 1] (see also [ 21) for Poincare supersymmetty and then extended to the case of anti-de Sitter (Ads) supersymmetry [ 31. The models obtained (both with integer as well as half-integer superspins) naturally form two dually equivalent series. Any cursory glance at the structure of these models suggests that there should exist a universal formulation in which massless models of all superspins occur as special cases. The use of such a formulation may be indispensable for constructing a consistent theory of interacting massless (super) fields of all (super) spins. In this respect it is worth mentioning a totally consistent system of equations for interacting massless fields of all spins, including the gravitational field, constructed by Vasiliev [4] and based on a geometric generating description for (properly generalized) higher spin gauge massless models in the AdS space [ 5,6]. Unfortunately, up to now it remains unknown whether these equations can be derived from an action functional. In the present paper we propose a generating formulation (wherein each superspin enters with multiplicity one) for the combined action of free massless superfields of all superspins in N = 1, D = 4 AdS superspace ZM = (PJP,&). W e realize this model as a gauge theory of unconstrained super-fields p( z”, qa) over ’ E-mail: [email protected]. ‘E-mail: [email protected]. 3 E-mail: [email protected]. O370-2693/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. Plf s0370-2693(97>00034-8

%I. Gates Jr. et d/Physics

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LettersB 394 (1997) 343-353

the Ads super-space and analytic in a four-vector variable 8 constrained by q”qa = 1. Usual higher tensor representation superfields emerge as coefficients in power series expansions in q. From a geometrical point of view, our formulation is based on enlarging the Ads superspace by a one-sheeted hyperboloid in R3,’ which is parametrized by q’. The bosonic sector of the final superspace can be realized as a homogeneous space of 0(3,2), i.e. the symmetry group of the AdS space. An 0(3,2) covariant realization under the constraints involves five-vectors yA and q A, A = 5,0,1,2,3, y*y*

= -r2 )

f&t

=o,

@qd

(1.1)

= 1~

where the indices are contracted with the use of the metric van = diag( - - + + +) . The constrained variables Y* = y* (x”) parametrize the Ads space, with - 12re2 the corresponding curvature. A remarkable feature of the manifold introduced is that spacetime (parametrized by y) and internal space (parametrized by q) originate, in a sense, on equal footing. This becomes obvious by rewriting ( 1.1) as 4*4&Z =O,

q+9-

= 1,

(1.2)

where q$ = (r-r y* f q”) / fi. The constraints 4+ + hq+ 9 4+ -+ 9- 2

q_ -+ r’q_

,

remain unchanged

under SO( 1,l)

A # 0,

and discrete transformations (1.3)

4- + 4+ )

(1.4)

which commute with the AdS transformations. Our present formulation for the combined action of massless multiplets of all superspins is described by two real scalar superfields X( z, q), Y( z, q) and a complex one U( z, q), and all gauge transformations arise from a complex parameter E( z, q). The most surprising thing is that the model obtained admits a considerable simplification by imposing the algebraic gauge condition Y = 0 which leads to an elegant formulation. What is more, the second real superfield X is auxiliary and can be eliminated without explicit supersymmetry breaking. Another remarkable feature of the proposed model is that it possesses an irreducible gauge algebra instead of the infinitely reducible gauge transformations present in the original higher superspin models [ 31. The approach under consideration enables us to combine all the Killing tensor superfields of the Ads superspace within a single superfield Y (z, q) satisfying a simple equation. This observation presents an exceptional possibility for the covariant study of the infinite-dimensional global symmetry of the higher-superspin actions which is likely to be described by the superalgebra proposed in [ 71.

2. Arbitrary

superspin massless models in the AdS superspace

The N = 1, D = 4 AdS superspace [ 81, with coordinates covariant derivatives DA = (Do, ‘Da, giy) 4

z M = ( xnt , F-‘, tip>, is specified by the algebra of

4 Our two-component notations and conventions coincide with those adopted in [29]. We consider only Lorentz tensors symmetric in their undotted indices and separately in dotted ones. A tensor of type (k, I) with k undotted and 1 dotted indices can be equivalently represented as ti(k, 0 = 1/l,(!+~) = (l/n,___ atti ,._.dl = V& ...ak)(til...iril. Following Ref. [ 61, we assume that the indices, which are denoted by one and the same letter, should be symmetrized separately with respect to upper and lower indices; after the symmetrization, the maximal possible number of the upper and lower indices denoted by the same letter are to be contracted. In paaicular 4n(k)$O(~) s r~5(~r...ak$ak+, ...ak+l) and %cSa(k) z cod(prr,

...olk_,). Given two tensors of the same type, their contraction

is denoted by 4

. S/Iz r#~~(~)~(‘)$~a(~~(~).

SJ. Gates Jr. et al./Physics {D&D/j}

=

4piq$,

[Dk,Dp~]

= -ip.

Letters B 394 (1997) 343-3.53

4

345

.Dp.

(2.1)

Here M, h? denote the Lorentz generators, the non-zero constant ,u of unit mass dimension determines the torsion (and hence the curvature) of the AdS superspace. The algebra of covariant derivatives is invariant under superspace general coordinate transformations and local Lorentz rotations DA

= [K,DAl

K: = -p2&

,

+

(k”Qy

+ ka(24kfa(*)

with superfield parameters Z@, k” kac2) The covariant the parameters kA, korc2)are constrained by k, = $I@k,a

derivatives

= K)

remain unchanged,

(2.2)

[K, DA ] = 0, provided

ka(2) = Vaka ,

,

@,,‘z,& = D?@

+ C.C.)

(2.3)

kak = 0 _

(2.4)

Eqs. (2.2)-( 2.4) define a Killing supervector Ic of the AdS superspace. The set of all Killing supervectors is known to form the superalgebra asp ( l(4). Now, we recall some results of [ 31 on the Lagrangian realization of higher-superspin massless multiplets in the AdS superspace. An important role in this approach is played by “transversal” and “longitudinal” linear superfields. A complex superfield I( k, I) subject to the constraint D’iYIYaCk)kCl) = 0,

@

(p - I - 2)l?( k, 1) =0 , (p -2)r(k,O)

is defined as a transversal DbGaCk)tiCl) =0,

linear superfield; w

is defined as a longitudinal

(p+Z)G(k,Z) linear superfield.

l>O; z=o;

=O,

(2.5)

a complex superfield

G( k, Z) subject to the constraint

=O.

(2.6)

Here we have used the notations (2.7)

For Z= 0 (2.6) coincides with the chirality constraint. For each superspin value greater 3 /2, there are known [ 31 exactly two dually equivalent superfield formulations called transversal and longitudinal. In the present paper we make use of the transversal formulation for a half-integer superspin s + l/2, s > 1, and of the longitudinal formulation for an integer superspin s, s 2 1. The dynamical superfield variables in these cases take the forms

vsil ={m,s),wVi =(H’(s-

l,s-l),l;(sI,s-

l,s-

l),G(s,s),G(s,s)},

where H and H’ are real unconstrained, superspin-( s + l/2) action reads

I transversal

l)},

S21

(2.8)

S>l

(2.9)

linear and G longitudinal

linear superfields.

The transversal

(2.10)

SJ. Gates Jr. et al/Physics

346

and the longitudinal S! = (-l)$

superspin-s /d”z

action is given by

E-l{

$H’“(s-1~‘(s-1?D’s(~2

+ _!._H~‘-+-‘)+‘) s+l + (s+U2_ 2

Letters B 394 (1997) 343-353

(DaDdG,Cs)ats)

,u,cLH’~H’+~G.GG+

- ~/J)D~H~~,_,~,~,_,~

_ DjiYDDaGnCs)irCs))

--$(G.G+G.C)

}.

(2.11)

Here d8zE-’ is the measure of the AdS superspace. Any of the models (2.10) and (2.11) describes the massless on-shell asp ( 1,4) -representation of the corresponding superspin and its conjugate representation. The action S&i,, is invariant under the following gauge transformations: s) = g(s, s) + ,!?(s, s) , s ~bwa(s)&(s) =cY(s-l)&(s-1) = 2(s + 1) 6H(s,

with a longitudinal

linear parameter

(2.12)

.

g( s, s) . The action S! is invariant under the following

gauge transformations

SH’(s-1,~l)=y(s-l,s-I)++-1,s~l), SG a(s)&(s)

(2.13)

with a transversal linear parameter y( s, s) . It was mentioned in [ 31 that the linear dynamical variables I’, G and their gauge parameters g, y can always be re-expressed via unconstrained superfields at the cost of introducing an additional gauge freedom, for both formulations (2.10) and (2.1 l), of infinite stage of reducibility. We are going to show that there exists a generating formulation in terms of unconstrained superfields for the unified theory of massless multiplets of all superspins with the action (2.14) s=l

x=1

Here the scalar multiplet

(superspin-0)

is ordinarily

described

by a non-gauge

(anti) chiral scalar superlield (2.15)

D&G=O,

Vo={G,G}, with the action So =

(2.16)

d’z E-‘GG. s

We also make use of the standard description

of the vector multiplet

(superspin-l/2)

by a real scalar superfield (2.17)

Vf ={H}, with the action Sf = $

J

d’, E-‘HV”(D2

- 4,z)DaH,

(2.18)

and the gauge invariance 6H=g+&

Dij,g=o.

(2.19)

S.J. Gates Jr. et al./Physics

Lefters B 394 (1997) 343-353

347

An important observation suggesting the existence of such a formulation is based on a property of the AdS superspace that a complex tensor superfield V(k,Z) possess a unique decomposition into its transversal and longitudinal parts U(k,Z)

=I’(k,Z)

+G(k,Z).

(2.20)

As a consequence, one can equivalently convert the constrained dynamical variables I?( s, s) (from the superspin-( s + 3/2) multiplet) and G(s, s) (from the superspin-s multiplet) into an unconstrained complex superfield U( s, s) and completely analogous for the gauge parameters y( s, s) and g( s, s) .

3. Generating

formulation

The system with action (2.14) can be reformulated as a gauge theory in AdS superspace with all the dynamical superfields as well as gauge parameters taking their values in the algebra of analytic functions over the one-sheeted hyperboloid in R3,’ qaqn = -;qaaqa&

(3.1)

= 1.

An analytic function 4(q) undotted and dotted indices

is completely specified by a set of tensor coefficients &(s)e(s) separately), s = 0, 1, . . ., that originate in the power series

(symmetric

(3.2)

= 2 &-r(s)&(s)qn(s)irts) t s=o

4(q)

in

where 4

cu(s&(s) _ a&. . -4&.

a&

(3.3)

s times The algebra possesses

the unique,

modulo normalization,

Lorentz-invariant

trace (3.4)

for arbitrary

analytic

functions

& = qativaD& )

4(q)

and $(q).

C.J= -qad’DJ&

Upon introducing

the second-order

differential

)

operators (3.5)

one finds the identity d8z E-‘&2++

tr I

= tr

s

d8z, E-‘(

C&5)+ ,

(3.6)

is valid for arbitrary superfields 4( z, q) and $( z, q) . Let us consider a linearized theory constructed from an unconstrained superfields X( z, q), Y( z, q) which all appear in the action functional

s=

;tr

Jd8zr1{$$,~+qp-

1),_qi)x-

-~X~-~Y(g~U+~PO)fqX(QUfeo)-o(P~~-2)u-uPu-~Po).

$yp-

complex

superfield

U( z,, q) and real

1)2y (3.7)

S.J. Gates Jr. et al./Physics

348

This action remains xJ=

unchanged

under the following

--@E, sO=-;ee,

Letters B 394 (1997) 343-3.53

gauge transformations sx=(P-l)E+(P-l)E,

6Y=&+E,

(3.8)

expressed in terms of an unconstrained complex gauge parameter E( z, q) . The corresponding generators are obviously seen to be linearly independent. The actions (2.14) and (3.7) can be used to define one and the same physical theory! This can be proven by making use of the following decompositions m

u = c ua(,)&(s)q’y(s)b(s) )

U(s, s) = T(s, s) + G(s, s) ;

(3.9)

s=o M Y = c Y,(,)a(sjqa(s)i’fs) J=o

,

Y(s,s)

=H’(s,s)

- H(s,s)

;

(3.10)

03

x = c Xa(s)&(s)qa(s)b(s) ) X(s,s) = (s+ l){H’(s,s) +H(s,s)};

(3.11)

s=o where (2.20) has been applied to (3.9). Now, calculating Similarly, representing the gauge parameter in the form

the trace in (3.7) reduces it to that given by (2.14).

(3.12)

Eq. (3.8) As can expressed additional

proves to be equivalent to the gauge transformations (2.12), (2.13) and (2.19). be seen from -( 3.10) and (3.11)) the expansions of the real q-analytic superfields X and Y are in terms of the coefficients H’( s, s) and H( s, s). This in turns implies that we may define two q-analytic superfields via the equations co

(3.13)

A' = cH',(,)~(,)q"(")~("), s=o

(3.14)

Consequently,

we find the results

x=

rzA'-A, A remarkable Y=O

1+ qa”![

dq”&

1[d't'dl.

(3.15)

feature of the theory in field is that it admits the simple gauge condition =+

e=ip,

p=p.

(3.16)

and in this gauge the action reduces to the form S= itr

JdbZE-‘{

- Fx2

+ ;x(

&u+

QO) - D(P + P - 2)U - UPU - OPO)

(3.17)

S.J. Gates Jr. et al. / Physics Letters B 394 (1997) 343-353

and the residual

gauge transformations

with p being a real unconstrained

[a&I

take the forms

su = +I, .

&m&I/L

SX=i(P-P),=-

349

superfield.

(3.18)

Here we have used the identity

=%wP-n.

(3.19)

An additional surprise is that X is nothing more but an auxiliary eliminated with the aid of the equation of motion

superfield.

In the gauge (3.16)

this can be

(3.20)

x=-+J+eo,.

As a result, the theory is completely described by the complex unconstrained superfreld U( z, q) . It is worth pointing out that any complex superfield has a unique decomposition into the sum of transversal and longitudinal linear ones (see (3.9) and (3.12) ) only in the Ads superspace at ,u # 0. Nevertheless the action (3.17) has a well defined flat limit. To see this, let us set /_L= ,G and make the replacements u*&J,

X + Xl&T

in (3.17) and (3.18). arrive at the action

Due to these re-definitions

X(Qu

S= ttr

P’fiP

+ @J) - u(D*

all the singularities

+ r)*)U

-

UD*U

-

at /_L= 0 disappear

uD*i!?

,

and in this limit we

(3.21)

and the gauge transformations 6X=

;(D*-

D*)p,

su = go,

80 = -;Qp,

(3.22)

(here Q = q”‘D,D&) in flat superspace. Note that the superfield X is no longer auxiliary. Although (3.21) is the formal Minkowski space limit of our anti-de-Sitter theory, it will require additional investigation before its full relevance is understood. It can be seen from (3.10) that the condition Y = 0 is equivalent to the condition H = H’. We may ask what is the significance of this? From (2.8) and (2.9), it should be clear that in the gauge Y = 0, the physical content of both the superspin-s and superspin-( s + l/2) multiplets are realized as component fields in a single real superfield H. In the superspace supergravity theory [ lo] associated with the N = 1, D = 4 heterotic string a similar phenomenon is known to occur. In the special @!FC supergeometry of [ lo], the graviton supermultiplet as well as axion supermultiplet all occur as components of the usual axial-vector supergravity prepotential Hati. The special PFFC supergeometry is also known in the literature as “the string gauge.” Thus, we conjecture that here the condition Y = 0 may be the analog of the string gauge. It should also be clear that the q-analytic superfield A( z,q) plays a role analogous to the string-field functional in linearized covariant string field theory.

4. Global symmetries It follows equation &Y=O,

from (3.18)

r=r,

that the parameters

=+

of global symmetries

(P - P)Y

= qakDad.Y = 0.

of the unified

model should solve of the

(4.1)

SJ. Gates Jr. et al./Physics

350

This proves to have the following

y

=

Letters B 394 (1997) 343-353

general solution:

~{t,(S)&(S) + la(s),(s)}qa(s)b(s) ,

(4.2)

s=o

where the transversal

l’s coefficients

f’s and longitudinal

L(s)&(s) = L(s)&(s) 7

ml(,)&.(s)

are constrained

= m%L(s)~(s)

=09

by (4.3)

and

-i

1Lu(s)ct(s)

-

9

n(s)ci(s)

Q+Za(,)ti(,) = z)a~~za(,)&(s) =0

(4.4)

9

respectively. The constraints in (4.3) and (4.4) coincide with Killing equations for tensor superfields and define a complete set of independent Killing tensor superfields. In the AdS superspace their solutions are parametrized by a finite number of constant parameters. The component content of the Killing superfields t( s, s) and 1( s, s) is given by the Killing tensor fields &(s)&(s) = L?(S)d.(S)If ‘%(S)&(S) =

satisfying

LY(s)ci(s) I 3

GY(s+l)&(s)

= Q&(s),(s)

&(s-l)&(s)

=

IT

~%(S)&(S)I)

(4.5)

the conditions

Vaa%(k)a(l)

= 0>

~a&~cT(k)&(l) = 0.

Here the symbol ‘I’ indicates the zeroth-order component in 19= 0, V’s denote the covariant derivatives of the AdS space. The space of Killing tensor superfields (4.3>, (4.4) may be endowed with the structure of a super-algebra. Given two tensors, for example both transversal t( s, s) and t(r, r), one can build a Killing longitudinal tensor l(s+r-2p+l,s+r-2p+l) forp=O,l,... , min (s, r) as their antisymmetric bilinear combination defined uniquely modulo a factor from the Eqs. (4.3). In general the following combinations are available: l(s,s)

andl(r,r)

t(s,s)

andt(r,r)

l(s,s)

and t(r,r)

combinetol(sfr-2p-l,s+r-2p-l), p =O,l,. ..,min(s-l,r-1), combinetol(s+r-2p+l,s+r-2p+l), ..,min(s,r), p =O,l,. combine to t(s+r-2p - l,s+r-2p ,min(s-l,r). p=O,l,...

(4.6) - 1))

Here we present manifestly only the first sector. For that purpose let us introduce the notations I”( s + ~1,s - n) , n=--s,-s+l,..., s,and1”+‘/2(s+n,~-n-1),n=-~,-~+1 ,..., s - 1, (1”) * = I-” for the derivatives of the longitudinal Killing tensor l”( s, s) E 1( s, s) by the recurrent relations

a.

a

p+w

a(s+n)ir(s-n-l)

=

-2l:(,+,)&(s-Jz) )

(4.9)

and their complex conjugates. Now if l( s, s) and I’( r, r) are two longitudinal Killing superfields, any longitudinal Killing superfield with s + r - 2p - 1 undotted and dotted indices (p = 0, 1, . . . , min (s - 1, r - 1) > that can be built of 1 and 1’ is proportional to

S.J. Gates Jr. et al./Physics

1~(s+r-2p-l)~(s+r-2p-l)

+liI

Letters B 394 (1997) 343-353

=

m-q ~a(r-p-_q-~l)P(p+n+l~iu(r-_p+q)~~p-~n)

q+l/2 n+q+lp(p+n+l)b(p-77) Qn+1/2lnfs-p+q)iY(s-p-q-l)

where the summation over q under the conditions constants /3 and a are equal i(-l)n (y4+V2 = n (p+n)!(p-n)!P”

-pun



q+lP %+w=

(4.10)

>

“=-p-l

Pq+l/2=

351

s - p - 1 > q > p - s and r - p - 1 > q 2 p - r and the n

2i( -l)n+’ (P+n+l)!(p--n)!;+t



[(s-p+q)!(s-p-q-l)!(r-p+q)!(r-p-q-l)!]-’.

They satisfy the relations Pq+1/2

= P--4-1/2

9

&l/2

=

If

_&q-w --n

%+1/2

that ensure simultaneously the reality of I” and its I’(?-, r). Let us consider the lowest components of the describe important symmetries of the model: N = 1 supersymmetry [ 121, respectively. The N = 1 Ads transformations can be written in 6Y=KY,

6X=KX, where the operator

-q--1/2 = Ly-n-If2

-q-t-1/2

9

antisymmetry



with respect

to the replacement

Killing super-field Y, namely t and Z,&. They prove to Ads transformations (see, e.g., [2,11] ) and N = 2 global the form

6lJ=lCU,

K is expressed

I( s, s) f-f

(4.11)

in terms of special Killing

superfield

Y: (4.12)

where Y 3 q’Ykk,h. Another Killing superfield Y: &Y = 0,

symmetry

of the action (3.17)

involves

component

t of the

SIX = 2i[ P, t] RU + C.C.,

s,ci=~~p[P+P,tl~x+~~~(~-l)[~+~,t]RY+iq~~(~~t)~aR(U+0), where R denotes the reflection R+(q)

the scalar transversal

= 4(-q)

(4.13)

on the hyperboloid (4.14)

.

These are exactly the N = 2 supersymmetry transformations and O(2) rotations of the N = 2 AdS super-algebra, which were found in components in [ 121. Therefore we conclude that the action (3.17) possesses N = 2 AdS supersymmetry which is described in the covariant form in terms of the generating superfields X( z, q), Y(z, q) and U(z, q) with the Killing parameter Y (z, q) containing only t and Ink components. The N = 2 transformations turn out to form a closed algebra off the mass-shell. In the gauge Y = 0 a commutator of two transformations (4.13) is given by the sum of an AdS transformation (4.11) and a gauge variation:

[S,,6,,]X=IcX+i(jj-P)p,

[ti,,&]

u=Icu+

&I,

(4.15)

S.J. Gates Jr. et al./Physics

352

where K: is given by (4.11)

Y =4i

Letters B 394 (1997) 343-353

with

((in,t)(ij&‘) - (IDat)(

qa’,

p= ;Y(u+o).

(4.16)

In the work of [ 7] the superalgebra of higher spins and auxiliary fields shsa( 1) was proposed to play the role of the structure algebra of a higher spin theory. This algebra describes the global symmetries of the free action as well. As the spin content of our model coincides with that arising in the framework of [ 41, the superalgebra shsa( 1) can be considered as a global symmetry algebra of the action (3.17) too. It is then natural to identify the N = 2 Ads superalgebra of the transformations (4.11) and (4.13) with the finite-dimensional subalgebra of shsa( 1) . The fact that these transformations are expressed in terms of Killing superfields along with some observations at the component level allows us to suggest that all the global symmetries can be expressed in terms of Killing tensors (4.3) and (4.4). Then the (anti)commutator relations of the superalgebra of global symmetries could be described by the bilinear combinations (4.6)) (4.10) after appropriate normalization of the multiplicative factors. Most likely the Killing tensors entering the superfield Y (4.10) correspond only to the physical subalgebra of shsa( 1) (at equal powers of the Klein operators [7] ) because the number and the Lorentz structure of generators in this subalgebra coincides with those of the constant parameters in the solutions of (4.3) and (4.4). In closing this work, we also note that the issue of duality can also be investigated within the context of the universal action in (3.7) as there seems to be no fundamental impediment to performing a superfield duality transformation upon this action. Presumably such a duality transformation would replace the chiral scalar multiplet described in (2.15) and (2.16) by a chiral spinor superfield since the latter is known to describe the axion multiplet in the work on the N = 1, D = 4 supergeometry [ 101 that arises from the heterotic string. It would be an interesting exercise to show that the duality transformation when implemented on the generating formalism, does indeed interchange the two dually equivalent superfield formulations [ 31. But then it is worth expecting the appearance, in the generating formalism, of inverse powers of (2P - 1) which turn into numerical factors only upon passing to the components in q and decomposing the complex unconstrained superfields into their transversal and longitudinal parts by the rule (2.20).

Acknowledgements One of us (S.M.K.) thanks the Institute for Theoretical Physics at Hannover, where the present paper was completed, for hospitality. This work was supported in part by the Russian Foundation for Basic Research Grant No. 96-02-16017 and the Ioint DFG-RFBR Project Grant No. 96-02-001806. The research of S.J.G. was supported by the US NSF under grant PHY-96-43219 and NATO under grant CRG-930789. The work of A.G.S. was also supported by a Fellowship of Tomalla Foundation (under the program of the International Center for Fundamental Physics in Moscow).

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