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The spinning superparticle. II. A spectrum in ten dimensions
Volume 202, number 3
PHYSICS LETTERS B
10 March 1988
T H E S P I N N I N G SUPERPARTICLE. II. A S P E C T R U M I N T E N D I M E N S I O N S J. K O W A L S K I - G L I K M A N NIKHEF-H, P.O. Box 41882. 1009 DB Amsterdam, The Netherlands
Received 19 November 1987
We present the quantization of the spinning superparticle model in ten space-time dimensions. The resulting wave function represents two spin 3/2 multiplets for type I and spin 5/2 multiplets for type II theories.
In a recent p a p e r [ 1 ] (hereafter called I) we have f o r m u l a t e d a new superparticle m o d e l called the spinning superparticle. It can be considered as a global s p a c e - t i m e SUSY extension o f the massless spinning particle, or the local world-line SUSY extension o f the massless superparticle. The q u a n t i z a t i o n o f the m o d e l p e r f o r m e d in I was restricted to four s p a c e - t i m e dimensions. The spinning superparticle m o d e l can be also considered as a d i m e n s i o n a l reduction o f a d = 2 doubly graded sigma m o d e l ([ 2,3 ]) with rigid superspace as a target. Therefore one expects the spectrum o f our model in ten s p a c e - t i m e d i m e n s i o n s corresponds to the massless part o f the spectrum o f the spinning superstring. It is possible to quantize the spinning superparticle covariantly by introducing auxiliary variables as in refs. [4,5 ], however since the m a i n goal o f this p a p e r is to identify a spectrum o f the theory, in what follows I will restrict myself to the much simpler lightcone gauge quantization. The lagrangian o f the system reads as follows:
a n d 0 are M a j o r a n a - W e y l spinors ~2, V a n d ~, are the einbein and one-dimensional gravitino, respectively. The lagrangian (1) is invariant under local one-dimensional supersymmetry, Siegel s y m m e t r y and its bosonic counterpart as well as global s p a c e - t i m e supersymmetry. The explicit form o f t r a n s f o r m a t i o n laws has been presented in I. F r o m the canonical m o m e n t a corresponding to ( 1) 0 L = 0 J ( " = (1/V)(o), - i ~ ' A , + V ~ F , O ) = p , ,
(3)
OL/O0" = - iO~ p , F }~ =_ N ° ,
(4)
aL/00 ~ = 0 = 7rg,
(5) A
OL/O]l ~ = ½iAI, = n , ,
(6)
OL/Of'=O=n v ,
(7)
O L / O ~ = O = n ~' ,
(8)
one finds a p r i m a r y h a m i l t o n i a n H e = ½Vp 2 + i ~ ' p , A ~ - V p ~ F ~ O
+ Gp+zUG +fn ~ +shY+an
~ ,
(9)
L = ½[(1 / V) ¢oVkt, - iAuA" - (2i/V)AMo, where + 2 6 F . 0(¢o" - i~uAV)] .
(1) G , =n,--o + i O a p , F } , , ~ 0 ,
(10)
where "1 oY ~ = X ~ - i 6 F " O
,
(2)
~ The conventions are listed in the appendix. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
,2 Here I consider only type-I theory. The quantization oftype-II theories in which 0 and 0 are two M-W spinors can be performed along exactly the same lines, and the result will be discussed later. 343
Volume 202, number 3
PHYSICS LETTERS B
~u=ztuA -- ½iAu ~ 0 ,
(11)
are together with ~zv~0, ~ ' ~ 0 and n o ~ 0 the primary constraints of the model. The set of secondary constraints consist of pZ~0,
(12)
p~,AU~O,
(13)
p~,~F" .~O ,
(14)
and there are no tertiary constraints. The constraints (11 ) are easily recognized as standard fermionic second-class constraints, and one can use a Dirac bracket instead of a Poisson bracket and put ~ strongly equal to zero. As a result one obtains the Dirac bracket {A,,, A~} =iqu . .
(15)
The constraint ~r°~ 0 and p~,~Fu 0 ~ 0 form a mixed first- and second-class set. As has been shown in I, one can remove the whole 0 dependence from the theory by adding the gauge fixing condition
{A,,Aj}*=id,j
10 March 1988 (21)
.
./i- has a vanishing Dirac bracket with all canonical variables and therefore can be omitted in what follows; all other canonical brackets are not modified. From (20) we see that At may be represented after quantization as D-- 8 ?)-matrices. Having removed a gauge freedom associated with local supersymmetry and bosonic Siegel symmetry we turn to the constraints (10). It is well known that this set of constraints contains first- and second-class constraints, which can be separated only in a noncovariant way. Using a fixed light-cone frame one can split a 16-component ten-dimensional Majorana -Weyl spinor into two eight-component eight-dimensional spinors: 0~--, (0 a, 0 ~) . In terms of these spinors a first-class part of (10) is i ob -- ~ p _ pi y a/,Tr
zr.o ,.~ O ,
and the second-class one is G,i _ ~o,~ _ ipi yiat, O b _ i x / ~ p _ O,i ,.~ O .
P~,=(--Po,P9,Pi),
i = 1, ..., 8 .
(16)
In fact (16) and (14) are equivalent to the single set of constraints 0.~0 which together with zro~0 form a second-class set of constraints. After implementing the Dirac bracket procedure one can make 0 and zr~ strongly equal to zero. In the next step we choose a noncovariant gauge to break the local supersymmetry associated with the first-class constraint (13): A+~0.
(17)
Using (15) we find that {p.A,A+}=-ix~p_
(18)
,
and define a Dirac bracket {A, B}* = (A, B}
+{A, A + } { p u A ~', B } ) .
+ , B}
(19)
After making a rescaling
one finds that 344
+p~Aqp_
(23)
Since {G a, G b } = 2,,/2ip_ ,
(24)
one defines a Dirac bracket {A, B}** = {A, B}* - (1/2x/2ip_){A,
Ga}*{G a, B}*.
(25)
In order to remove "off-symplectic" double-star brackets one makes a redefinition n Oa ~'ffOa + iOapiyiaa ,
(26)
X i__,)~i + i0a faa 0, ,
(27)
7~°a--.~°a:Ga ,
(28)
o ~ - - , ~ = (,,/5p_ ) "2o ~ .
(29)
In terms of these variables one has canonical brackets
+ (l/ix/2p_)({A,p~,A~'}{A
A----,.4- =A-
(22)
,
(20)
{)~', p/}**=d},
(30)
{z7°", 0b}** = - d g ,
(31)
{ X - , p _ }**= 1.
(32)
In the dotted spinors sector the only nonvanishing bracket reads
Volume 202, number 3
PHYSICS LETTERS B
{tTa, fib}**= ½i~ab.
(33)
ff~ has a vanishing bracket with all canonical variables (which is obvious, since it is itself a second-class constraint) and therefore can be put strongly to zero. The formula (33) can be understood in two equivalent ways: either (as it is usually done) one splits 0 ~ into two S U ( 4 ) spinors and interpret them as canonical coordinates and momenta, or one realizes (33) after quantization as a Clifford algebra and represents
1( (7,,)~ 0 '7)
Oa--,(Oa)a~= ~
.
(34)
In our analysis of the spectrum we will follow this second possibility. N o w we remove the first-class constraint (22) by fixing the light-cone gauge: 0 ~ 0. After substituting definitions of tilde variables ( 2 6 ) - (29) into (22) we have
p2Oa-xf2p_ na ~O , 0~0
,
which is equivalent to ~r~.~0,
(35)
0~0,
(36)
and ~" and 0 ~ are removed from the theory by using the Dirac bracket method (as it was in the case of the ¢~variable). To quantize the theory we replace Dirac brackets by i times (anti) commutators and find a representation of canonical variables as operatos acting on the Hilbert space of wave functions. F r o m the canonical (anti) commutators
10 March 1988
The remaining constraint p2 ~, 0 is a first-class one and, on the quantum level correspond to the condition on the wave function [~=0.
(42)
Since the variable 0 a is represented as a matrix, a wave function is an element of the vector spaces on which this matrix acts. It follows from the representation (34) that the wave function consists of two parts 7J,•- a vector in the 8s representation of SO(8), and ~v _ an SO (8) spinor in representation 8c. This is exactly the spectrum of the N = 1 superparticle in ten dimensions. In our case, however, we should also take into account an action o f A i. Therefore the full wave function consists of four terms:
~Jai, ~-Idti, ~Jaa, ~Jha •
(43)
The wave functions in (43) belong to reducible representations of SO(8) and we must decompose them into irreducible pieces. It can be done most easily by using the rules of multiplication o f SO (8) representations (see refs. [6,7]). It is easy to see that the representation of (43) is nothing but the (8v+8c) × (8s+ 8c) representation (8c is a second inequivalent spinor representation of SO (8)), where the first multiplier corresponds to A ~and the second to the 0 a action. For the irrep decomposition one finds (8v +8c) × (8s +8c) = 1 + 2 8 + 3 5 c + 8 s +56~ +8c +56c +8~ + 5 6 v . (44)
[X", p~]_ = - i O ~ ,
(37)
[A,, AS] + = ~ o ,
(38)
The irreps (45) correspond to two chiral gravitino multiplets with opposite chiralities (see ref. [ 7 ]). It is clear now how to extend the above analysis to type II theories. In those cases we have two 0 variables which satisfy the following anticommutation relations:
[0 ~, 0hi+ =½oab ,
(39)
[0 'a, 0 '~] + =-~c~~/' ,
(45)
[02a, 02b] + =-½Oa~ ,
(46)
[0 ~, 02~'] + = 0 ,
(47)
one finds p~ = i0/0X ~' , 1 ((yi)~b0
A ' = --~
(40) (7~/')
,
and a representation of 0 a is given in (34).
(41)
for type IIA theory (0 variables have opposite chiralities), or [0,a, 0,b] + =-½~ab ,
(48) 345
Volume 202, number 3
PHYSICS LETTERS B
[ 0 2 a 02/'~] + = l(~h[) ,
(49)
[01~, 02~] + = 0 ,
(50)
for type IIB theory (both 0 variables have the same chirality). In type IIA theories the wave function is therefore in the representation ( 8v + 8c ) × ( 8s + 8c) × ( 8s + 8v ), where the first factor comes from the representation o f A ' and the second and third from the ones o f theta variables. Accordingly in type IIB theories the wave function belongs to the (8v+ 8c) X (8s+ 8c) × (8s+ 8c) representation. In both cases one finds fields o f spin higher then 2 (spin 5/2 multiplets) in the spectrum, which, however, may be not so disastrous in string theory as it is in the case o f point-like theories since string theories already contain massive higher spin excitations. More dangerous is probably a presence o f two gravitons in the spectrum, since there is no obvious way to understand which one corresponds to the gravitational force. It does not mean however that the spinning superparticle model is not interesting anymore - even if it is dubious whether it is realistic, it may provide a useful tool for deeper understanding of superstring theories [ 8 ]. I would like to thank Dr. J.W. van Holten for useful discussions and reading the manuscript. This work is part o f the research program o f the stichting FOM.
Appendix The F matrices satisfy
346
10 March 1988
[F u, F"] + =2~/l" , where q,J=Oo, i,j= 1.... ,8, ~/+- = - 1. Our representation of 3 2 × 32 F matrices is F~=(
0
a/~6 )
0
' 0
/-11 = ( 1;6
__116
) '
where ~" ~,~ =
(~,,.)~t
and •iabyjl,b + ~jM,?ibb =
26o ~ab .
References [ 1 ] J. Kowalski-Glikman, J.W. Van Holten, S. Aoyama and J. Lukierski, Phys. Lett. B 201 (1988) 487. [2] R. Brooks, F. Muhammed and S.J. Gates Jr., Nucl. Phys. B 268 (1986) 599; Class. Quant. Gray. 3 (1968) 745. [ 3 ] J. Kowalski-Glikman and J.W. van Holten, Nucl. Phys. B 283
(1987) 305; J. Kowalski-Glikman, Phys. Lett. B 180 (1986) 359. [4] L. Brink, M. Henneaux and C. Teitelboim, Nucl. Phys. B 293 (1987) 505. [5] E. Nissimov, S. Pachewa and S. Salomon, Nucl. Phys. B 296 (1988) 462. [6] R. Slansky, Phys. Rep. 79 (1981) 1. [ 7 ] B. de Wit, in: Supersymmetry, supergravity and superstrings '86 (World Scientific, Singapore, 1987). [8] J. Kowalski-Glikman, J.W. Van Holten, S. Aoyama and J. Lukierski, in preparation.
PHYSICS LETTERS B
10 March 1988
T H E S P I N N I N G SUPERPARTICLE. II. A S P E C T R U M I N T E N D I M E N S I O N S J. K O W A L S K I - G L I K M A N NIKHEF-H, P.O. Box 41882. 1009 DB Amsterdam, The Netherlands
Received 19 November 1987
We present the quantization of the spinning superparticle model in ten space-time dimensions. The resulting wave function represents two spin 3/2 multiplets for type I and spin 5/2 multiplets for type II theories.
In a recent p a p e r [ 1 ] (hereafter called I) we have f o r m u l a t e d a new superparticle m o d e l called the spinning superparticle. It can be considered as a global s p a c e - t i m e SUSY extension o f the massless spinning particle, or the local world-line SUSY extension o f the massless superparticle. The q u a n t i z a t i o n o f the m o d e l p e r f o r m e d in I was restricted to four s p a c e - t i m e dimensions. The spinning superparticle m o d e l can be also considered as a d i m e n s i o n a l reduction o f a d = 2 doubly graded sigma m o d e l ([ 2,3 ]) with rigid superspace as a target. Therefore one expects the spectrum o f our model in ten s p a c e - t i m e d i m e n s i o n s corresponds to the massless part o f the spectrum o f the spinning superstring. It is possible to quantize the spinning superparticle covariantly by introducing auxiliary variables as in refs. [4,5 ], however since the m a i n goal o f this p a p e r is to identify a spectrum o f the theory, in what follows I will restrict myself to the much simpler lightcone gauge quantization. The lagrangian o f the system reads as follows:
a n d 0 are M a j o r a n a - W e y l spinors ~2, V a n d ~, are the einbein and one-dimensional gravitino, respectively. The lagrangian (1) is invariant under local one-dimensional supersymmetry, Siegel s y m m e t r y and its bosonic counterpart as well as global s p a c e - t i m e supersymmetry. The explicit form o f t r a n s f o r m a t i o n laws has been presented in I. F r o m the canonical m o m e n t a corresponding to ( 1) 0 L = 0 J ( " = (1/V)(o), - i ~ ' A , + V ~ F , O ) = p , ,
(3)
OL/O0" = - iO~ p , F }~ =_ N ° ,
(4)
aL/00 ~ = 0 = 7rg,
(5) A
OL/O]l ~ = ½iAI, = n , ,
(6)
OL/Of'=O=n v ,
(7)
O L / O ~ = O = n ~' ,
(8)
one finds a p r i m a r y h a m i l t o n i a n H e = ½Vp 2 + i ~ ' p , A ~ - V p ~ F ~ O
+ Gp+zUG +fn ~ +shY+an
~ ,
(9)
L = ½[(1 / V) ¢oVkt, - iAuA" - (2i/V)AMo, where + 2 6 F . 0(¢o" - i~uAV)] .
(1) G , =n,--o + i O a p , F } , , ~ 0 ,
(10)
where "1 oY ~ = X ~ - i 6 F " O
,
(2)
~ The conventions are listed in the appendix. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
,2 Here I consider only type-I theory. The quantization oftype-II theories in which 0 and 0 are two M-W spinors can be performed along exactly the same lines, and the result will be discussed later. 343
Volume 202, number 3
PHYSICS LETTERS B
~u=ztuA -- ½iAu ~ 0 ,
(11)
are together with ~zv~0, ~ ' ~ 0 and n o ~ 0 the primary constraints of the model. The set of secondary constraints consist of pZ~0,
(12)
p~,AU~O,
(13)
p~,~F" .~O ,
(14)
and there are no tertiary constraints. The constraints (11 ) are easily recognized as standard fermionic second-class constraints, and one can use a Dirac bracket instead of a Poisson bracket and put ~ strongly equal to zero. As a result one obtains the Dirac bracket {A,,, A~} =iqu . .
(15)
The constraint ~r°~ 0 and p~,~Fu 0 ~ 0 form a mixed first- and second-class set. As has been shown in I, one can remove the whole 0 dependence from the theory by adding the gauge fixing condition
{A,,Aj}*=id,j
10 March 1988 (21)
.
./i- has a vanishing Dirac bracket with all canonical variables and therefore can be omitted in what follows; all other canonical brackets are not modified. From (20) we see that At may be represented after quantization as D-- 8 ?)-matrices. Having removed a gauge freedom associated with local supersymmetry and bosonic Siegel symmetry we turn to the constraints (10). It is well known that this set of constraints contains first- and second-class constraints, which can be separated only in a noncovariant way. Using a fixed light-cone frame one can split a 16-component ten-dimensional Majorana -Weyl spinor into two eight-component eight-dimensional spinors: 0~--, (0 a, 0 ~) . In terms of these spinors a first-class part of (10) is i ob -- ~ p _ pi y a/,Tr
zr.o ,.~ O ,
and the second-class one is G,i _ ~o,~ _ ipi yiat, O b _ i x / ~ p _ O,i ,.~ O .
P~,=(--Po,P9,Pi),
i = 1, ..., 8 .
(16)
In fact (16) and (14) are equivalent to the single set of constraints 0.~0 which together with zro~0 form a second-class set of constraints. After implementing the Dirac bracket procedure one can make 0 and zr~ strongly equal to zero. In the next step we choose a noncovariant gauge to break the local supersymmetry associated with the first-class constraint (13): A+~0.
(17)
Using (15) we find that {p.A,A+}=-ix~p_
(18)
,
and define a Dirac bracket {A, B}* = (A, B}
+{A, A + } { p u A ~', B } ) .
+ , B}
(19)
After making a rescaling
one finds that 344
+p~Aqp_
(23)
Since {G a, G b } = 2,,/2ip_ ,
(24)
one defines a Dirac bracket {A, B}** = {A, B}* - (1/2x/2ip_){A,
Ga}*{G a, B}*.
(25)
In order to remove "off-symplectic" double-star brackets one makes a redefinition n Oa ~'ffOa + iOapiyiaa ,
(26)
X i__,)~i + i0a faa 0, ,
(27)
7~°a--.~°a:Ga ,
(28)
o ~ - - , ~ = (,,/5p_ ) "2o ~ .
(29)
In terms of these variables one has canonical brackets
+ (l/ix/2p_)({A,p~,A~'}{A
A----,.4- =A-
(22)
,
(20)
{)~', p/}**=d},
(30)
{z7°", 0b}** = - d g ,
(31)
{ X - , p _ }**= 1.
(32)
In the dotted spinors sector the only nonvanishing bracket reads
Volume 202, number 3
PHYSICS LETTERS B
{tTa, fib}**= ½i~ab.
(33)
ff~ has a vanishing bracket with all canonical variables (which is obvious, since it is itself a second-class constraint) and therefore can be put strongly to zero. The formula (33) can be understood in two equivalent ways: either (as it is usually done) one splits 0 ~ into two S U ( 4 ) spinors and interpret them as canonical coordinates and momenta, or one realizes (33) after quantization as a Clifford algebra and represents
1( (7,,)~ 0 '7)
Oa--,(Oa)a~= ~
.
(34)
In our analysis of the spectrum we will follow this second possibility. N o w we remove the first-class constraint (22) by fixing the light-cone gauge: 0 ~ 0. After substituting definitions of tilde variables ( 2 6 ) - (29) into (22) we have
p2Oa-xf2p_ na ~O , 0~0
,
which is equivalent to ~r~.~0,
(35)
0~0,
(36)
and ~" and 0 ~ are removed from the theory by using the Dirac bracket method (as it was in the case of the ¢~variable). To quantize the theory we replace Dirac brackets by i times (anti) commutators and find a representation of canonical variables as operatos acting on the Hilbert space of wave functions. F r o m the canonical (anti) commutators
10 March 1988
The remaining constraint p2 ~, 0 is a first-class one and, on the quantum level correspond to the condition on the wave function [~=0.
(42)
Since the variable 0 a is represented as a matrix, a wave function is an element of the vector spaces on which this matrix acts. It follows from the representation (34) that the wave function consists of two parts 7J,•- a vector in the 8s representation of SO(8), and ~v _ an SO (8) spinor in representation 8c. This is exactly the spectrum of the N = 1 superparticle in ten dimensions. In our case, however, we should also take into account an action o f A i. Therefore the full wave function consists of four terms:
~Jai, ~-Idti, ~Jaa, ~Jha •
(43)
The wave functions in (43) belong to reducible representations of SO(8) and we must decompose them into irreducible pieces. It can be done most easily by using the rules of multiplication o f SO (8) representations (see refs. [6,7]). It is easy to see that the representation of (43) is nothing but the (8v+8c) × (8s+ 8c) representation (8c is a second inequivalent spinor representation of SO (8)), where the first multiplier corresponds to A ~and the second to the 0 a action. For the irrep decomposition one finds (8v +8c) × (8s +8c) = 1 + 2 8 + 3 5 c + 8 s +56~ +8c +56c +8~ + 5 6 v . (44)
[X", p~]_ = - i O ~ ,
(37)
[A,, AS] + = ~ o ,
(38)
The irreps (45) correspond to two chiral gravitino multiplets with opposite chiralities (see ref. [ 7 ]). It is clear now how to extend the above analysis to type II theories. In those cases we have two 0 variables which satisfy the following anticommutation relations:
[0 ~, 0hi+ =½oab ,
(39)
[0 'a, 0 '~] + =-~c~~/' ,
(45)
[02a, 02b] + =-½Oa~ ,
(46)
[0 ~, 02~'] + = 0 ,
(47)
one finds p~ = i0/0X ~' , 1 ((yi)~b0
A ' = --~
(40) (7~/')
,
and a representation of 0 a is given in (34).
(41)
for type IIA theory (0 variables have opposite chiralities), or [0,a, 0,b] + =-½~ab ,
(48) 345
Volume 202, number 3
PHYSICS LETTERS B
[ 0 2 a 02/'~] + = l(~h[) ,
(49)
[01~, 02~] + = 0 ,
(50)
for type IIB theory (both 0 variables have the same chirality). In type IIA theories the wave function is therefore in the representation ( 8v + 8c ) × ( 8s + 8c) × ( 8s + 8v ), where the first factor comes from the representation o f A ' and the second and third from the ones o f theta variables. Accordingly in type IIB theories the wave function belongs to the (8v+ 8c) X (8s+ 8c) × (8s+ 8c) representation. In both cases one finds fields o f spin higher then 2 (spin 5/2 multiplets) in the spectrum, which, however, may be not so disastrous in string theory as it is in the case o f point-like theories since string theories already contain massive higher spin excitations. More dangerous is probably a presence o f two gravitons in the spectrum, since there is no obvious way to understand which one corresponds to the gravitational force. It does not mean however that the spinning superparticle model is not interesting anymore - even if it is dubious whether it is realistic, it may provide a useful tool for deeper understanding of superstring theories [ 8 ]. I would like to thank Dr. J.W. van Holten for useful discussions and reading the manuscript. This work is part o f the research program o f the stichting FOM.
Appendix The F matrices satisfy
346
10 March 1988
[F u, F"] + =2~/l" , where q,J=Oo, i,j= 1.... ,8, ~/+- = - 1. Our representation of 3 2 × 32 F matrices is F~=(
0
a/~6 )
0
' 0
/-11 = ( 1;6
__116
) '
where ~" ~,~ =
(~,,.)~t
and •iabyjl,b + ~jM,?ibb =
26o ~ab .
References [ 1 ] J. Kowalski-Glikman, J.W. Van Holten, S. Aoyama and J. Lukierski, Phys. Lett. B 201 (1988) 487. [2] R. Brooks, F. Muhammed and S.J. Gates Jr., Nucl. Phys. B 268 (1986) 599; Class. Quant. Gray. 3 (1968) 745. [ 3 ] J. Kowalski-Glikman and J.W. van Holten, Nucl. Phys. B 283
(1987) 305; J. Kowalski-Glikman, Phys. Lett. B 180 (1986) 359. [4] L. Brink, M. Henneaux and C. Teitelboim, Nucl. Phys. B 293 (1987) 505. [5] E. Nissimov, S. Pachewa and S. Salomon, Nucl. Phys. B 296 (1988) 462. [6] R. Slansky, Phys. Rep. 79 (1981) 1. [ 7 ] B. de Wit, in: Supersymmetry, supergravity and superstrings '86 (World Scientific, Singapore, 1987). [8] J. Kowalski-Glikman, J.W. Van Holten, S. Aoyama and J. Lukierski, in preparation.