Volume 171, number 2,3 PHYSICS LETTERS B 24 April 1986

A M I N I M A L C O V A R I A N T A C T I O N F O R T H E F R E E O P E N S P I N N I N G S T R I N G F I E L D T H E O R Y ~

G.D. DATI~ 1, M. G U N A Y D I N 2.3, M. P E R N I C I , K. P I L C H 4 and P. V A N N I E U W E N H U I Z E N

Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11794-3840, USA

Received 10 February 1986

The BRST transformations of the string fields of the free open spinning string is analyzed. After expanding the string fields into ghost zero modes the BRST transformations are diagonalized and it is found that in the Ramond sector the physical fields are contained in only two string fields. A minimal covariant classical action for both sectors is constructed. The component expansion of the string fields yields the correct set of physical and Stueckelberg fields which are needed to construct a covariant local action for the corresponding massless and massive states. This is explicitly verified for the first three mass-levels.

During the past year there has been considerable interest in the problem of constructing a covariant field theory for strings. A convenient approach is to use the BRST formalism [ 1 - 7 ] , 1 . In this let ter we will use the BRST formalism to analyze the free open spinning string field theory. One can either start by constructing a BRST invariant quantum action [1] or one can directly construct a classical action [7]. In the former (quantum) case the kinetic operator K of the quarltum action is o f the f o r m K = [Q, O} where Q is the BRST charge and O is a suitably chosen ope- rator. The BRST transformation rules are used to identify the physical fields (including the Stueckelberg fields needed for the construction o f a local classical action), the ghost and antighost fields, and the auxili- ary fields needed for the ni lpotency of Q. The classi-

* Supported in part by NSF Grant PHY 81-09110 A-01. Present address: Weizmann Institute of Science, 76100 Re- hovot, Israel.

2 Present address: Lawrence Livermore Laboratory, Liver- more, CA 94550, USA.

3 Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA. 4 On leave from the University of Wrodaw, Wroclaw, Poland.

*l A large number of papers discussing other approaches to covariant string field theory exist, and we are studying their relation to the BRST approach. In this article we on- ly discuss the BRST approach.

cal action is obtained by dropping all but the physical fields from the quantum action. The arbitrariness in the choice of O corresponds to the freedom one has in choosing gauge-fixing terms, but the resulting clas- sical action is, of course, unique. In the lat ter (classi- cal) case the BRST operator is used for the construc- tion o f a classical action o f the form $QPcb (P is a projection operator [1,7] which we will discuss later). This action is invariant under the classical gauge trans- formations of the string field ~ , 8 ~ = QA with A an arbitrary string field parameter. One can eliminate part o f the classical fields by means of their algebraic field equations as well as by using part of the gauge transformations and show that at the free level both approaches become, in fact, equivalent [7]. The quantum approach ends up with a classical theory which contains fewer physical fields and gauge in. variances than the classical one. Whether the addition- al fields in the latter are indispensable for constructing local interactions is at this moment unknown.

The spinning string consists o f two sectors, the bosonic Neveu-Schwarz (NS) sector and the fermion- ic Ramond (R) sector. The treatment o f the NS sec- tor is analogous to that o f the bosonic string. Thus we will first briefly summarize the formalism of the open bosonic string.

The bosonic string, bo th in the open and closed

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sectors, has been analyzed by Siegel and Zwiebach [1 ], who used the quantum approach. The BRST operator for the open string expanded into the zero modes of the coordinate ghost and antighost fields reads

Q = (a/ac)H + cT+ + Q+. (1)

The nilpotency of Q implies the following relations:

[H, T+] = [H, Q+] = [T+, Q+]

= HT+ + Q+Q+ = 0 . ( 2 )

The operator T+ can be viewed as a part o f an SU(1,1) "isospin" algebra with generators T++ =- T+, T+_ = T_+ and T_ _ = T_ which are constructed from the ghost oscillators. Further Q+ together with an- other differential operator Q_ forms an isodoublet. Siegel and Zwiebach showed that the expansion of the string field ~ = ~b + c~ yields the classical fields in the sector o f q~ with the total isospin T = 0. [The iso- spin is the SU(2) generated by (T÷, T 3 , T_ ) where T 3 = iT+_ is ½ times the ghost number operator.] Choosing the kinetic operator with O = cO/Oc they found the classical lagrangian to be

J O = ½ ~ [H-l- 1 " -~I(Q_Q+ - a+a_)]er=o(O. (3)

An important step in the derivation of (3) was the ob- servation that the field ~ appears in the quantum ac- tion only in the combination T+ ~ so that after shift- ing ~ into ~ = ff - A¢ wi thA = T+IQ+, the field T÷ ~ is BRST inert and thus is an auxiliary field. The "inverse" operator T+ 1 satisfies T+ 1T+ = 1 - P T 3 =T and T+T+ 1 = 1 --PTa=_T.

The open string in general, i.e., including interac. tions, was subsequently analyzed by Witten [7]. He

• t2 started from the lagranglan 22 = ½~QPI"3 = - 1/4 ~ where 7"3 = T3 - ¼ [c, 0/0c] is the total ghost number operator (including the zero modes). The operator P]'3 = - 1/4 projects out the physical fields in the sec- tors T 3 = 0 in ¢ and T 3 = - 1 / 2 in ~O. The lagrangian in terms of ~ and ~ reads

.12= ½(-~HPT3 =0~ + 2~ Q+PT 3 =0 ~b

-- ~T+PT 3 : -1 /2 ~b). (4)

Eliminating PT 3 -- - 1 / 2 ~b by its field equations (which is possible since it has nonvanishing 7') one obtains the lagrangian ~ = ½dP(H + Q+T+IQ+ )PT3=Odp. The gauge transformations with A = ?~ + c~" read

5¢ = -Q+~" + T+;k, 5 ~b = a+;k + H~', (5)

and can be used to gauge away all T :/: 0 parts of ~b. The resulting action is that of ref. [1] in (3) [7,8].

Let us now turn to the discussion of the spinning string. As a first quantized two-dimensional field theory defmed on the string world-sheet it contains the following fields: XU(o, r), C(o, O, -C(o, r), •U(o, z), S(o, r) and S(o, r). The bosonic and fermionic fields X u and ?~u are vectors of the ten-dimensional space- time, and their transverse polarizations describe the physical modes of the string. The remaining fields, anticommuting C and ~" and commuting S and S- are, respectively, coordinate and supersymmetry ghosts and antighosts. The coefficients of the Fourier expan- sion of the fields X u, ..., S- in o (-Tr ~< o <~ or) at ~" = 0 are the oscillators aUn,Cn and -(n wi thn = +1, +2, ..., dUn, s n and~ n for which n = +1, +2 .... (respective-

+1 4.3 ". ly n = -~-, -.~-, ...) as a consequence of the periodic (respectively antiperiodic) boundary conditions on ?~u, S and S- in the R (respectively NS) sector. The os- cillators with positive (negative) n are absorption (creation) operators. The zero modes, if present, will be denoted by x u , pU = i~u, d~ = (l /V~-)7 u, c, ~/~c, ~/~s and s, respectively.

The first quantization of the spinning string using the BRST formalism has been exhaustively discussed in many recent papers [9] where explicit expressions in terms of the oscillators can be found for the BRST charges in both sectors. In what follows we will only need the general structure of these charges.

As we have already mentioned the construction of the string field theory for the spinning string in the NS sector proceeds in complete analogy with the one for the bosonic string. Due to absence of the zero modes in ?~u and the supersymmetry ghost fields S and S, the BRST charge has the same form as in (1), with the only difference that the operators H, Q+ and T+ contain in addition the oscillators o f these fields * a In particular the "isospin" can be extended to the S- ghosts. The (c_ n, c--n) and (s_ n, s--n) form isodou- blets while a_n and X-n are isosinglets. One can now

*2 We discuss here only the case of the free string. .3 And, of course, the dimension of the spacetime is 10.

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repeat all the steps we have discussed above and find that the physical fields of the NS sector are in the T = 0 part of ~ and the classical action is given in (3) [2, 3,5].

The Ramond sector is more complicated and more interesting. In addition to the anticommuting zero mode c of the coordinate ghosts, there is also a com- muting zero mode s of the supersymmetry ghosts. Consequently, the expansion of the string field • into the zero modes yields an infinite series

• = ~b(s) + c~P(s) = ~ _ (sn~n + csn¢n). (6) n O

Since • depends on s, one must know how to "inte- grate" over s in order to define an inner product in the space of string fields. This has been discussed re- cently in refs. [2,3,7,10]. We have found that for the construction of a classical action one can bypass this problem by first looking at the BRST transformations of the fields in the expansion (6). As we shall show, in a suitable basis, the BRST transformations are diagonal and the physical fields can be easily identi- fied. It turns out that they are contained in only two string fields, denoted below by ~0 and ~1" In terms of these string fields one can find a gauge invariant ac- tion which yields correct local actions for the massless and massive x-space fields contained in the string fields.

The BRST charge Q(R) for the Ramond sector has the following form

Q(R) = (a/ac)H + cT+ + 0.+,

I"+ = T+ - s 2, 0.+ = Q+ + sF + (b/as)K+, (7)

where H = ½p2 + ... is the hamiltonian, F = (1/V~-)~ + ... is the Di rac -Ramond operator, Q+ contains terms linear in p, and T+ and K+ are purely algebraic operators constructed from the ghost oscillators. As before T+ can be extended to an SU(2) algebra with the ghost number operator T 3. The total ghost num- ber operator, including the zero modes, is ~r 3 = T 3 - ¼[c, a]~c] +{{s , a/as}. The nflpotency of Q(R) im- plies the following identities:

FF = H, IF, T+ ] = 2K+,

HT+ + K+F + Q+Q+ = o, (8a)

K+K+ = [K+, T+] = {K+, F]- = {K+, Q+]- = [K, H]

= [H, T+] = [H,F] = [H,Q+] = {F,Q+]-

= [Q+, T+] = 0. (8b)

The BRST transformations of the string fields in (6) are

a,(s) -- , [O+,(s) - ~'÷,(s)].

6 ~b(s) = e[Q+ $(s) + HOp(s)], (9)

where e is the constant anticommuting BRST param- eter. We now observe that the presence o f the s 2 term in T+ allows us to perform a shift o f the field ¢J(s) into $(s) such ~ a t the whole field ~(s) is BRST inert, rather than T+ ~k(s) as in the case of the bosonic string. We first introduce the operator s "-1 def'med by ~- lsn = s n - l ( 1 - fin,0)" Then a left inverse operator ~,~-1 can be defined as follows

fr;-1 = _ ~ (T+)n0-1)2(n+l). (10) n--0

Note that the series in (10) evaluated on any state generated by acting with a finite number o f the crea- tion operators on the vacuum has only a finite number of nonvanishing terms. It is straightforward to check that

~r+l~r+ = 1, ~r+l'+ 1= l+ (Po+P1)Tr+l s 2, ( l l a , b )

where the projection operators Pn are defined by Pnsm m m ~(s) is def'med analogously = 6 n s . The shifted field

to the bosonic case by ~(s) = ~0(s) - ~r+l d+~(s). Using the identities in (2) (with hats) and (11), the BRST transformation rules in the new basis become

a~'(s) = 0,

tS~b($) = e(1 - T+T+I)Q+q~(s)- eT+~'(s). (12)

Observing that (1 - ~ r + f f l ) ~ + = 0 and (1 - fr+fr~-l) 2 = (1 - ~+~,~-1) it is natural to make yet another change of basis from (~b0, ~b 1) to (¢0, ~1), where

~b" 0 + S~l = (1 -- ~ ' + l ) ~ b ( $ ) . (13)

It is remarkable that the same combination o f ¢(s) ap- pears on the RHS of (12) (since Q+ commutes with

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~r+), so that we have actually decoupled two string fields q~o and ~1 from the others by the above field redef'mition. The BRST transformations now read

6r~ 0 = e(Q+q) 0 + ~ {T+, F } ~ 1), (14a)

6~1 --e ( F ~ + a + ~ l ) , (14b)

C~n=et~"n_2, n>~2; ~i~'n = 0 , n>/O, (14c,d)

where ~n = ~n + T+~bn+2 + "" which fo rn = 0, 1 coin- cides with (13).

We will now identify the minimal set o f physical fields contained in the string fields ~n and ~b~ n. As we do not have a quantum action, we cannot follow the procedure used by Siegel and Zwiebach in the case of the bosonic string. Neither will we use the Witten ap- proach as we do not want to go into the details of the before mentioned s integration. Instead we will com- bine elements of these two approaches and derive a gauge invariant action, whose correctness we will af- terwards verify by expansion into x-space fields.

From~the BRST transformations in (14) we infer that all ~b n fields are auxiliary fields, the q~n with n /> 2 contain the antighosts, while the ¢0 and ~1 fields contain physical fields and ghosts. This identification is not unique, but it leads to a minimal set o f physical fields. Note also that direct comparison with ref. [1 ] seems difficult because in our case there is a double expansion in isospin and in s. The physical fields have total ghost number ~r 3 = 0 [7]. This means that they are contained in the T 3 = 0 part o f '~ 0 and T 3 = - 1 / 2 of '~ 1 . Thus the only question which remains to be answered is whether there are any restrictions on the total isospin. (As we discussed above, both in the bosonic case as well as in the NS sector o f the spinning string the total isospin T of the physical sector is T = 0.) We shall argue that in order to construct a gauge invariant action one must take all isospins. Let Pphy,~.be the projector on the physical sector o f ¢0 and ¢1" The expansion into components at the zero mass level is ~0 = X + ... where X is the spin-l/2 field. In order that the string action reduces to the Dirac action for X it must contain a term of the form .6? = tP0FPphys~b 0 + ... . We can complete this action by re- quiring that it be gauge invariant under the gauge transformations of the physical fields. These are ob- tained from (14a,b) by first acting o n b o t h sides with Pphys and then replacing in the RHS e¢o (respective-

l y e~ ' l )wi th the gauge parameters GO (respectively ~1)- In order to prove that the resulting action is gauge invariant one must use the identities in (8). This is possible only if Pphy s can be "moved" through the operators which appear in the action. In particular one finds that Pphys must satisfy [Q+, Pphys] = [(F~T÷},P~hys] - 0 _ A.~ neither O: nor ~F,T+}~ are isosinglets ,4 Pphys cannot be a projection on a deffmite total isospin component. We conclude that the physical sector~ should be identified with PTa =0~b0

and PTa =- 1/2 ~1. It follows that the minimal classical gauge invariant

action for the R-sector of the spinning string is

+ ~oQ+PT3=_I/2~I. (15)

This lagrangian is independent of the ghost zero modes c and s. However we must still specify how the product of the string fields in (15) is defined. We con- sider them as the wave functions in the Fock space of the first quantized string. Then ~'0 and 71 are ex- panded into oscillators acting on the vacuum, with x- space fields as coefficients. In the NS sector those fields are spacetime tensors, while in the R sector they are tensor-spinor fields, in other words in the R sec- tor the spinor index of the vacuum is put on the spacetime fields. This means that in the BRST charge, the oscillators dUn, c n and F n, n 4: O, which anticom- mute with d~ and therefore contain T l l , will also act on this spinor index of the spacetime field. I f the field q~ has the expansion

~b a = pt°~(X) + )t~(X)at~_l + ~(x)d~U_l

+ a~(x)C_l + ~-a(x)~-_ 1 + ma(x)S_l

+ m-a(x)s-_ 1 + ...] la), (16)

where a is the spinor index, the conjugated field ~ is defined as ~ = ( a l IX(x) + (a~ 1)* Xz(x) + ...]. The * denotes here the hermitean conjugate and the bar on the spinor fields is the usual Dirac conjugation. The string fields ~o and 71 in (15) satisfy a reality condi- tion which can be obtained by requiring it to be pre- served by the BRST transformations. Then the corn-

*4 Note that in the bosonic and NS cases both H and Q2 are isosinglets.

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ponent fields in the expansion (16) are Majorana (i.e., real) or anti-Majorana (i.e., imaginary) spinor fields, and in the proof o f the gauge invariance o f the action (15) one must use t h e symmetry properties of the operators, like, e.g., 8~0F~0 = ~0FS~0 which are straightforward generalization of the analogous prop- erties for ordinary Majorana fields. We will further restrict the string field by choosing it to have even G-parity. This is necessary if we want to construct the spacetime supersymmetry [11]. The even G.parity se- lects, in the NS sector, fields with integer masses, while in the R sector it imposes a chirality condition on the spinors, depending on the number of oscilla- tors dU_n, S_n,S--n and the level in the expansion in the zero mode s [3].

We will now discuss the component expansion of the R sector for the first three mass levels. As the ex- plicit formulas quickly become messy we only outline the main features which will be discussed in more de- tail elsewhere [12]. The general strategy is as follows. One first determines the transverse states for a given mass level, which, except for m = 0, combine into irreps o f SO(9). These SO(9) representations tell us which massive fields are present at a given mass level after fixing all the gauge symmetries. These massive fields are some linear combinations of the fields which appear in the expansion of the string fields without ghost oscillators and with ghost oscillators, similarly to the case of the bosonic string as first ob- served by Siegel and Zwiebach. These authors also found that at a given mass level the set of fields ob-

tained by expanding the physical sector of the string field is the same as the set one would obtain by con- sidering the fields corresponding to the irreps o f the little group of massive states as massless fields in one dimension higher and subsequently performing di- mensional reduction. Exactly the same algorithm holds for the spinning string and in the R sector we find that one must take all fields with the correct ghost number without any further restrictions on the total isospin. This shows that our projection opera- tors are the correct ones. The details are a straightfor- ward, although tedious, exercise in group theory. In table 1 we summarize the results for the first three levels.

At the m 2 = 0 level we find the left-handed spin- 1/2 field for which (15) reduces to the Dirac action. In fact this is how we derived (15). For m 2 = 1 we Fred, respectively, left. and right-handed fields X~(x), $u(x) in 40, and ~(x) , E(x) in ~1, which have the fol- lowing gauge transformations

SX z = iau~ + (1/V'2)7ut/,

~.'= i~.~ + (1/2x/~)~,.~,

~ = - ( i / , , / g ) i ~ - 2 n ,

8 ~ = -( i /V~)~rl + ½~. (17)

Performing the following field redefinitions

Table 1 Ramond sector with tensor-spinor fields. The SO(9) irreps of massive states are given in terms of Young tableaux which describe tensor-spinors which are -r-traceless. The last two columns list d = 10 x-space fields contained in ~o and ~1- These fields are not restricted by any trace conditions, and satisfy only symmetry requirements on the tensor indices. By • we denote a spinor field without any tensor indices.

(Mass) 2 SO(9) irreps PT3 =0~o PT3=_l/2"~I

SO(8) spinor • none [] []

CE]+B+UI+ • r-Tq+B+[]+ 2 x • D + 2 x -

~ +~ + [ZZEI~ E]E] + ~] + i-I'Tq 2 X rq-1 + 2 X ~ + 4 X F I

+ 2 x B + 2 x r - r - 1 + 3 x F 1 + 2 x K Z I + 2 X [ ~ + g x E I + 3 x -

+ 2 X • + 4 X •

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, 17 (1/3V~')T.g,

= - h V , , v ' -

g ' = ~ + ( l / x / ~ ) 7 " X , ~ ' = r n + ( l / x / 2 ) 7 " ~ 0 , (18)

we find the canonical set of the physical and Stueckel- berg fields which transform as

p t 8 ~ = i a ~ , 6 ~ =iat , rl, 6~-'= 3~/,

8 ~ ' = 3~. (19)

After eliminating the Stueckelberg fields a ' and m' we P

fmd that the left- and right-handed fields X~ and ~b' combine into a massive spin-3/2 field and (15)yields the canonical action. Similar results hold for the high- er mass-levels, except that the field redefinitions are more complicated. We would like to point out one subtlety which is encountered from the third level on- wards. Massless tensor-spinor fields satisfy the triple 7-tracelessness condition [13], and in order to obtain the correct field content from the higher dimension, this condition is essential. At the third mass level this applies to the third-rank symmetric tensor-spinor field.

The analysis in the NS sector is analogous, and is summarized in table 2.

We conclude with a few comments. In our action (15) the fermions appear in the usual Dirac action whereas in Witten's action (4) (with hats and proper integration over s) the fermions in Pc, -0@ would seem to appear with a Klein-Gordon action. Actual-

ly0 the @ field equation from (4) yields the proper Dirac equation for @, and the @ field equation is not independent but can be obtained from the @ field equation by multiplying by ir+lQ+. In the Witten string field equations QP~ = 0 with q~ = @ + c@, @ and @ can be shifted in the same way as above. The resulting field equations are then equal to our field equations ~ r T0 and @1, together with the field equations @n.,~ 0 for all n. The absence of a field equation for @n, with n ~> 2, is clear from the alge- braic gauge invariance under 6@ n = ~'n - 2 which fol- lows from (14c).

There does exist a superalgebra, namely Q(2), con- taining the isospin generators, K+ which with two further fermionic charges K 0 and K forms an SU(2) triplet, and the bosonic central charge N = {K+, K_} which counts the sums of levels of the ghost and anti- ghost oscillators. The lowest-weight states of a given irreducible representation of this superalgebra (states which are annihilated by T_ and K_) transform into each other under the action of Q. However, the physical states cannot be described as irreducible representations of this superalgebra. The reason is that the requirement of T 3 being fixed leaves then only states with T = 0 which together with [T3, K± ] = -+K± implies that N vanishes. Therefore in the ex- pansion of string fields only the nonghost oscillators would contribute and this would not provide enough fields for a local action.

The action in (15)without ~ojection operators is itself the BRST variation of ~0@1, but with the pro- jectors this is no longer the case. In the approach of

Table 2 Neveu-Schwarz sector with tensor fields. The Young tableaux of the SO(9) irreps describe traceless tensors. The d = 10 x-space fields satisfy only symmetry requirements, but no trace conditions. The • denotes a scalar field.

(Mass) 2 SO(9) irreps PT=0 without PT=O¢ with ghost operators ghost operators

0 SO(8) vector [] [] none

l ~ + []~] ~ +~ +[]Z] + 17

+FFTT+[] 2 × +WT1

+ 2 X[]~ +V1

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ref. [1] and ours, zero modes are treated differently from oscillators. This leads to nonlocalities on the world-sheet (see, e.g., the operator (O/ao) -1 in ref. [1 ] ). Whether this will cause problems for the con- struction o f interactions is unknown.

Note added. After this paper was typed, we re- ceived a preprint by Kazama, Neveu, Nicolai and West [14]. These authors develop differential forms for the spinning string which extend the formalism of Banks and Peskin [15] for the bosonic string. They start from the physical states and construct the minimal set of covariant fields needed for local field equations. Since they do not consider string fields which depend on ghost coordinates [1 ], they obtain infinite series o f string fields and infinite sets o f field equations. We believe that after resumming these series, and con- structing a corresponding action, their results will be- come equivalent to ours.

Subsequently, we received a preprint by Terao and Uehara [16], which contains a gauge invariant action with only two string fields. These authors start from a gauge-fLxed quantum action and perform a series o f truncations in order to obtain correct spectra. This procedure is not off-shell BRST invariant at inter- mediate stages but their Final results are off-shell BRST invariant and equal to ours. They cast their Final action in the form ~ Q ~ with xls depending on the two string fields.

References

[1] W. Siegel and B. Zwiebach, Berkeley preprint UCB-PTH- 85/30, and references therein.

[2] N. Ohta, Texas preprint UTTG-36-85. [3] H. Terao and S. Uehara, Hiroshima preprint RRK 85-32. [4] H. Ooguri, Kyoto University preprint KUNS 819 HE (TH)/

85/14. [5] A. LeClair, Phys. Lett. B 168 (1986) 53. [6] M. Ito, T. Morozumi, S. Nojiri and S. Uehara, Hiroshima

preprint RRK 85-34. [7] E. Witten, Princeton preprint (1985). [8 ] W. Siegel, private communication. [9] D. Friedan, S. Schenker and E. Martinec, Phys. Lett. B

160 (1985) 55; J.H. Schwarz, CALTECH preprint CALT-68-1304 (1985); N. Ohta, Texas preprint UTTG-3-85.

[10] D. Friedan, S. Schenker and E. Martinec, Chicago preprint (1985)

[11] F. Gliozzi, J. Scherk and D. Olive, Phys. Lett. B 65 (1976) 282; Nucl. Phys. B122 (1977) 253.

[12] M. Pernici and K. Pilch, in preparation. [13] J. Fang and C. Fronsdal, Phys. Rev. D18 (1978) 3630;

T. Curtright, Phys. Lett. B 85 (1979) 219. [14] Y. Kazama, A. Neveu, H, Nieolai and P. West, CERN

preprint TH.4301/85. [15] T. Banks and M. Peskin, SLAC preprint 3740 (1985). [16] H. Terao and S. Uehara, Kyoto-Hiroshima preprint

RRK-86-3.

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