Volume 160B, number 1,2,3 PHYSICS LETTERS 3 October 1985

COVARIANT QUANTIZATION OF SUPERSTRINGS

Daniel FRIEDAN 1, : Enrzco Ferret and James Franck Instttutes and Department of Physws, Unwerstty of Chicago, Chwago, [L 60637, USA

and

Emil MARTINEC 2 Department of Physics, Princeton Unwerstty, Prtnceton, NJ 08544, USA

Received 17 May 1985

We present a manifestly Lorentz covanant formulation of supersymmetnc stnng theory In part:cular, we construct the fermion vertex and the supersymmetrygenerators m a BRST quanUzatlon using the techmques of superconformal field theory

L Introduction. The fermionic has been covariant formulation. The original covariant proposed as a unified theory of matter interacting formalism gave scattering amplitudes only for through gauge and gravitational forces. Attractive processes involving zero or two spacetime ferm- features of include the rich massless ions and an arbitrary number of bosons. The first particle spectrum - containing a , gauge calculation of a four-fermion amplitude used the vectors and chiral fermions - spacetime supersym- light-cone gauge [7]. The four-fermion amphtude metry, and most hkely finiteness as well. String was eventually calculated using the covariant theory is remarkably restricted. Internal con- formalism *x, but there was no effective description sistency forces the string to be supersymmetric, the of the multifermion amphtudes. In particular the dimension of spacetime to be ten and the funda- vertex operator describing spacetime fermion mental gauge group, if any, to be SO(32) or emission was never fully constructed. A local field E s × E 8. transforming as a spacetlme spinor was con- The original formulation of fermionic string structed [10], but its scaling dimension or confor- theory was Lorentz covariant [1,2]. The covariant mal weight was D/16 = 5/8, while a physical formulation was used in a number of interesting vertex operator must have dimension one. applications, but it was never actually finished. In this letter we complete the covariant con- The covariant formalism was used to discover the struction of the fermionic string following the critical dimension D = 10, spacetime supersymme- program described in refs. [4,11]. We start with try [3], to formulate the calculation in the Ramond-Neveu-Schwarz model, which de- string theory [4] and to find the resulting restric- scribes the world-sheet of the string in terms of a tions on possible gauge groups [5]. The proof of superconformally invariant (1 + 1)-dimensional and the explicit finiteness calcula- quantum field theory. Following a suggestion of tions were done in the light-cone gauge [6]. Goddard and Olive, we use the Faddeev-Popov Supersymmetry has never been proved in the ghosts resulting from fixing the covariant super- i Supported in part by U S Department of Energy grant ,1 The four-fermlon amplitude was developed m a senes of DE-FG02-84ER-45144, and the Alfred P Sloan FoundaUon calculations referred to in ref [8] For further references on 2 Supported in part by O S. Department of Energy grant earher developments in dual theory see the reviews in ref DE-AC02-76ER-03072 [9]

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. 55 (North-Holland Physics Publishing Division) Volume 160B, number 1,2,3 PHYSICS LETTERS 3 October 1985 conformal gauge [12-14] to construct the BRST ing partners (0, 0), the action may be written in [15] operator, the fermion vertex and the space- terms of scalar superfields X~(z, Y, 0, 0) as time supersymmetry generator. The resulting formalism is Lorentz invariant, reparametrization Sgauge fixed = f d2z DX., (4) invariant, and supersymmetric. The choice of where D = 00 + 00z. For the heterotic string, 0 is superconformal gauge allows us to utilize the absent and D is replaced by 0~. The solution to techniques of two-dimensional conformal and the equations of motion is superconformal field theory [16,17[. The advantage is that the theory is expressed in terms of an X ~ = X~(z) + X~(~) + 0~(z) +0--¢(Y). (5) algebra of local conformal fields on the world From now on we focus on the z-dependent fields, surface, which is automatically defined on surfaces in particular Lp~'(z). If needed, the entire construc- of arbitrary global topology by the principles of tion may be duplicated for the Y-dependent pieces. conformal field theory. The conformal algebra also The two-point function is provides powerful methods for calculating correla- (X~'(zl,01)X*(z2,02)) = -g~log(z12), (6) tion functions. We will further explain and explore these aspects in a future communication [18]. where za2 = z a - z 2 - 0102. The time components X ° are negative metric 2. Fermionic strings, A Lorentz covariant, fields. The states created by X ° decouple [22] due reparametrization invariant action for the to the residual invariance of the gauge choice (2) fermionic string is obtained by coupling string under superconformal transformations (z, 0) --, fields X ~' and their world-sheet superpartners ~U' f(z, 0). These transformations are generated by to two-dimensional [19] the moments of the super stress-energy tensor s -- f d2z e [ lVaXtt, raSP __ ½i ~'yaVa~g T(z,O)= -½DX~'D2X ~'= Tv+OT B (7) = - + o(azx, a=x + + ½i(~a'ybva~t~)( ObX" -- ¼i~b~k")], (1) (8) where e,~ is a two-dimensional vierbein and Xa its T B is the ordinary stress-energy tensor and TF is gravitino partner. For the heterotic string [20t, the its dimension-3/2 partner. The conformal proper- pieces of the action involving the left-handed parts ties of string operators are an essential element of of X ~ and tp ~ are replaced by an action for an our construction. E 8 × E 8 or SO(32) current algebra .2. Reparametri- The coordinates (o, z), In z = ~- + io, describe zation invariance and local supersymmetry allow a cylindrical string world sheet which admits two the Lorentz covariant gauge choices possible spatial (o) boundary conditions for the fermion ~ ~': anti-periodic (Neveu-Schwarz [2]) e~ = e*3~, X~ = Y~', (2) boundary conditions describe spacetime bosons known as the superconformal gauge. One readily and periodic (Rarnond [1]) boundary conditions verifies that q~ and ~ decouple from the action (1) describe spacetime fermions. The ermssion of a due to its super-Weyl invariance, leaving the string state which is a spacetime fermion changes free-field action the boundary condition on the world sheet from NS to R or vice versa. The fermion field ~k" must Sga,.,~r,.,ea= f[~( OaX~)2-½i~yOq~']. (3) be double-valued around the operator describing this process, which therefore opens or closes a cut In the euclidean defined by the in ~p~'. It is called a spin field [17] (for a single complex coordinates (z, Y) and their anticommut- Majorana fermion ~, the corresponding spin field S would be the dimension-I/16 order or disorder operator of the Ising model [23]). The ,2 The use of rank 16 current algebras in incorporatinggauge symmetnes m stung theones was also suggested by Freund basic spin field S" has dimension D/16 = 5/8 [21] and transforms as a 32-component spacetime

56 Volume 160B, number 1,2,3 PHYSICS LETTERS 3 October 1985

Majorana spinor. On the other hand, the emission Goddard and Olive suggested that the operator of spacetime bosons maintains the boundary might come from the Faddeev-Popov ghosts. condition on q,~ and can be described by The Faddeev-Popov (super) determinant com- operators which are the 0 integrals of superfields. pensates for fixing the intrinsic super metric on the For instance, the z-dependent part of the emission world surface. This superdeterminant is the vertex for a massless vector of momentum k and jacobian for the change of variables polarization ~ is [24] 3e~' = V,fl$ m, 3Xa = Va3c, (10) VB(k, Lz)= f dOe 'k xDX~,~'(k) used to factor out the super-reparametrization group [12,14]. The jacobian may be represented by =e 'k x(~)[a~X~,+i(k'~b)q%]~"(k), (9) a path integral over a conjugate pair of free ghost superfields with k" ~ = 0. A subsector of this NS-R string theory is CZ = c~ + O'l °, Bze = flze + Obzz, (11) spacetime supersymmetric; it is obtained by whose action is projecting onto the states of even world surface fermion number (-1) r= 1. In the R sector, Sghost = f d2z d2OB~o[)C ~. (12) (- 1) F acts as ~'n, so the even subspace corre- sponds to left-handed Majorana-Weyl spinors (we From this action we find the ghost super stress- use the conventions [)% "t~]+= -g~, Sa = S~%a, energy tensor %~ = -ca~, ,/~ = yf~). The lowest energy states now describe a massless ten-dimensional super- Tghost = -- (D2B)C + ½(DB)(DC) - 3B(D2C), multiplet. (13) In order to calculate fermion scattering ampli- tudes we shall have to compute spin field correla- and the two-point function (C(zt, 8t)B(z2,/t2) ) = tion functions. In principle these can be calculated 012/z12, where//12 =//1 -/92. It is the full stress- as the free energy of double-valued fields ~ ~ with energy tensor T,~ = Tmatter(X, ~b) + T~ost which cuts on the world surface connecting pairs of spin generates conformal transformations. In the fields. More practical methods [25] employ the algebra of conformal transformations resulting SO(1,9) currents ~p~k~(z) and either the null from the operator products [13] vector [26] or vertex operator [27] construction of c/2 2Tz~ 1 --4 --+ 3zTz: + ... , their spinor representation. For instance, in the (z-w)" (z-w) 2 z-w vertex operator construction for the Wick rotated SO(10) currents, one begins by bosonizing the ten (14) ~P~ to obtain five scalar fields O, which parametrize the coefficient c of the Schwinger term receives the maximal torus of SO(10). The spin fields are contributions of then the 25 exponentials :e '~ °:c,, with a= cx=D, c~=D/2, Cb, C=-26, Ca,r=ll. (+~>..., + ½) corresponding to the spinor weights of SO(10). % generates a two-cocycle in commuta- In D = 10 spacetime dimensions c = 0 so there is tion relations. This construction is analogous to no anomaly in reparametrizations. that used for the E 8 × E 8 and SO(32) currents of The key point to realize is that the world sheet the heterotic string. gravitino X~ and the spinor superconformal parameter ~ must obey boundary conditions 3. The ghosts. Vertex operators for physical corresponding to the NS and R sectors in order state emission must have dimension one to make that the action (eq. (1)) be well-defined and locally their z-integrals reparametrization invariant. The supersymmetric. Thus the correct fermion emis- vertex operator S~e ~k X(z) will not work since it sion vertex introduces a cut in all the world sheet has dimension 5/8 + k2/2 = 5/8. We need a spinor fields, and will therefore involve a spin field dimension-3/8 operator to complete the vertex. E for the (commuting) spinor ghosts flzS, 7 ° in

57 Volume 160B, number 1,2,3 PHYSICS LETTERS 3 October 1985

addition to the spin field S a. E is most conve- function (ep(z),l,(w)) -- ln(z - w). niently constructed using an exponential represen- yz=-a~, Tzz=-l(a/oa~,-Qa~ep). (20) tation of the fl, ~, algebra. In order to do this we reconstruct the algebra of ghost operators by The integrated anomaly can be interpreted as the decomposing the c = 11 spinor ghost system into existence of a background charge coupled to ~b of two subsystems, one with c = 13 and the other magnitude X" Q/2. with c -- - 2. The former is obtained by bosoniz- The difference T_ 2 = T(fl, 3') - 7"13commutes ing the ghost current corresponding to ghost with j~ and has c- -2. It may be written in fermion number terms of a pair of anticommuting free fields ~l(z) of dimension 1 and ~(z) of dimension 0 as j; = (15) T 2 = (Oz~)~. (21) This current satisfies the anomalous conservation law The free field theory for ~(z), ~(z) has the two-point function (~(z)n(w)) = (z - w) -t. The 0,Z = ~I/~QR (2), (16) exponential e ~ is a conformal field of dimension with R ~2) the intrinsic scalar curvature on the - ½a(a + 2); the dimension-3/2 and - 1/2 fields world sheet and Q = 2. This anomaly is reflected flz0, go are represented by in the operator product expansion BzO = e-*Ozli, ~,0 = e*n. (22) Tzz(B,'y)jw-Q(z-w)-3 +(z-w)-Zjz. (17) Of course ~ and ~ can also be represented as exponentials of a free scalar field, of opposite On the general world surface with g handles the signature to 4~ and with background charge integrated anomaly gives the Riemann-Roch Q = -1. We observe that the conjugate fields index e -*/2 and e *: have dimensions 3/8 and - 5/8, # zero modes (V) - # zero modes (B) respectively, and are therefore candidates for inclusion in the fermion vertex as the spin field F.. = X [dim (fl) - 1/2] = X" Q/2, (18) Since ( has dimension 0 there is always a single where X is the Euler characteristic X = 2 - 2g. On constant zero mode in ~ for any world sheet the sphere the anomaly is related to the two topology. Note that the zero mode of ~ does not globally defined superconformal spinor zero appear in the fl, "y algebra; only derivatives of modes which have been omitted from the appear. This is important, because the irreducible Faddeev-Popov determinant. A careful treatment algebra of BRST-invariant operators will only of ghost zero modes in the involve factors of 0~. We are only interested in is given in ref. [28]; the supersymmetric case will correlation functions of such physical operators, be sketched in ref. [18]. so we can dispense with the zero mode of ~, and From the ghost current j~ we construct another calculate expectation values in the vacuum 10) of stress-energy tensor the smaller algebra. Alternatively we can carry out the same calculation in the larger Hilbert space 7"13 = - ½( JzJz - O O~jz), (19) where the zero mode acts, but then the vacuum is which has the same operator product (eq. (17)) doubly degenerate because of the zero mode. The with j~ but c = 13. This two vacua can be written 10) and ~(0)10), with current and stress-energy tensor can be repre- (010) = 0 but (0[~(z)[0) = 1. One consequence sented by a scalar field .3 q~ with two-point is that a factor of ~(z) must be included in correlation functions to absorb the zero mode. The location of this extra ~(z) on the world sheet is ,3 Thas modafication of the scalar field was first thscussed m ref [29]. It has been extensively developed in the context irrelevant since only the constant zero mode of conformal field theory m ref [30] D. F would hke to contributes. thank Vl S. Dotsenko for pointing out the role of the The Hilbert space of the spinor ghosts fl, ,/is anomalous current m these scalar field systems rather exotic because it derives from a first order

58 Volume 160B, number 1,2,3 PHYSICS LETTERS 3 October 1985 action for Bose fields. The SL(2, Iq) invariant NS string theory means that vacuum state 10)sh of the ghosts is annihilated by 2 (26) the superconformal generators QBRST = 0. Physical states satisfy dz Ln=t~--zn+lT. gh°st n=0, +1, ~r 21ri .... QBRsTIphys) = 0. (27) dz Operators such as [QBRsT, O], O any operator, Gn=g~--zn+l/2T ~a°st n = +_1/2, (23) )v 2~ri z, , automatically commute with QBRST but create null states. The superspace integrals of dimension-I/2 but is not a highest weight state (i.e., not superfields (e.g., eq. (9)) commute with QBRST and annihilated by all the lowering operators of the are appropriate vertices for boson emission. We fields b, e,/3, 7- Rather the highest weight state is find that the operator e-*(°)c ~(0) 10), which has L 0 eigenvalue (energy) Vl=e-*/2S,,e 'k Xu"(k), 7.ku=0, (28) -1/2; this is the tachyonic NS vacuum state. Similarly, the highest weight state in the Ramond is physical; this is to be expected since we have sector is e-*/2(°)c~(0)10), which has energy - 5/8. shown that it creates a physical R vacuum state. When we take the tensor product of this state with Since q~-charge is conserved in interactions we the states of energy 5/8 created by the spin need a second piece of the fermion vertex with operator S~, we find the massless R vacuum states. opposite q~-charge in order to have nonvanishing The coherent state operators :e-~*(z): act to amplitudes; it is redefine the "Bose sea level" by filling a given set of energy levels of the/3, 3' system. For instance, V2 = [e*/2Savfl.( OzX ~ + ¼ik. ¢~) 10)~ is annihilated by the Fourier components +e3*/2"qbzzS~]e ik Xu"(k), 7.ku=O. (29) y, = fdz2rriz"-3/27(z) (24) The second term will not contribute to expectation values due to b-charge conservation. Note that for n >I 3/2 whereas e-*(°)10)~ is annihilated by although S a has the opposite chirality compared 7, for n >i 1/2. In contrast to the Fermi case, these to S~, V2 differs from Vx by carrying one more different sea levels are unitarily inequivalent unit of @charge so that the overall value of (- 1) r representations of the fl, ~, algebra. Note also that remains the same. One might at first think that V2 e-~(°)c~(O)lO) carries one unit of @charge and is a null operator since it can be written therefore has (-1) F= -1; the tachyonic NS ground state will be eliminated upon projecting V2 = [QnRsa', ~Vl], (30) onto even (total) sheet fermion number. by taking the contour integral of the BRST current over a contour winch circles the operator ~Vl(z ). 4. Verttces. We are now ready to assemble the However ~ is not part of the/3, 7 algebra and various pieces of the fermion vertex. A physical therefore this description is just an artifact of our vertex in either the NS or R sectors is one which representation. Nevertheless this presentation of respects BRST invariance, which is to say it V2 will be a useful tool. In particular it im- commutes with the BRST charge mediately implies that V2 is BRST invariant. The 1 full fermion vertex is now QBRsT= "~'~i ~)dzdOCZ(rmatter + l Tghost) VF = l(V 1 + V2). (31)

-r~ dz [ Z[T~(x, tl/)+ ½T~(b,c,/3,7)] This vertex is a local dimension-one conformal - ;r 2,ri tc field so amplitudes constructed using it are +7°[T~a(x, tl/)+½Tzo(b,c,/3, y)]}. (251 manifestly dual. We can picture Vt acting to create a cut in the world sheet spinors and Vz acting to Reparametrization invariance in the quantum close a cut. Their operator product factorizes on

59 Volume 160B, number 1,2,3 PHYSICS LETTERS 3 October 1985 the physical NS boson vertices [31] o) b) ,Vt(zO Vl(k, z, u)V2(p,w, v) - (z - w)k'p-l~i'yP'u .(vl(zt) × [ OzX~' + i(k +p)" t~/~]e l(k-ep) X(Z) ,((z) .v2(z2) + .... (32) @ Duality creates an apparent problem: how can we arrange the vertices so that they occur in the (time) c) d) order Kl(z)K2(w)Vl(t)V2(u ) .... always taking us ,V2(z~) between the canonical NS and R Hilbert spaces? In the crossed channel we could not prevent the o((Z) .V1(z2) successive occurrence of V1V1 or V2V2. These ,EV1(z2) operator products factorize the amplitude on what seem to be unphysical states associated with non-canonical Bose sea levels. We resolve this problem by the following argument. Fig 1 Mampulations necessary to demonstrate eq (33). Consider any correlation function of fermion vertices. It is a sum of correlation functions each containing an equal number of Vx and V2 vertices (fig. lc). Finally replace the contour integral with by @ charge neutrality. We show that any such V2(zl) and move ~(gl) back to z (fig. ld) to obtain correlation function satisfies the identity the fight-hand side of eq. (33). This manipulation shows that there is a < "'" Vl(kl, u 1, z1) "'" V2(k2, u2, z2) .-. > remarkable redundancy in the representation of = (--- ua, ... v1(/,2, ... ). the fermion amplitude. In order to accommodate the fermion vertex we were forced to extend the (33) Hilbert space to include an infinite number of This allows us to rewrite all of the correlation inequivalent representations of the ghost algebra functions of V1 and V2 in the canonical order associated with the Bose sea levels. The fermion V1V2Vy2 ..., ensuring that multifermion ampli- vertex generates a closed subalgebra of the tudes factorize on only physical poles. enlarged matter-ghost system which describes all To go from the right-hand side of eq. (33) to the physical particles. As we have just seen all Bose left-hand side we calculate both in the large levels are equivalent under the algebra of vertex algebra which includes the ~ zero mode. Recall operators. We also note that the picture changing that a factor of ~(z) must be included in operation of the old NS dual model [32] can be correlation functions to absorb the zero mode. The implemented by the above operations along with location of this extra ~(z) on the world sheet is the identity irrelevant since only the zero mode contributes. Note that the calculation in the smaller algebra [QBRsT, ~e-q,q,~e~k x] = VB" (34) (without the zero mode) is manifestly BRST invariant, but in the large algebra it is not because The supersymmetry current is VF(k = 0). The QBRST ~l 0) 4= 0. Otherwise eq. (30) would imply operator product eq. (32) shows that contour that V2 decouples and all fermion amplitudes integrals of the supersymmetry current will change would vanish. a fermion to a boson vertex as the contour is To demonstrate the identity (33) perform the deformed through it. The potentially troublesome following manipulations. The left-hand side of eq. V1V1 and VEV2 operator products are removed by (33) is pictured in fig. la. First move ~(z) to z 1. the above procedure. The supersymmetry algebra Next write V2(Z2)= [QBRsT, 2~Vl(z2)], expressing is made more transparent by interpreting e ~/2S '~(z) the commutator as a contour integral as in fig. lb. as the dimension-0 superspace partner 6 ~ to Deform the contour to surround z I instead of z 2 X~(z), with e-q'/ES,(z) and P~= OzX ~ their

60 Volume 160B, number 1,2,3 PHYSICS LETTERS 3 October 1985 canonical conjugates. Then we can write J Schwarz, Plays Lett. 37B (1971) 315, E Corrigan and D Ohve, Nuovo Clmento 11A (1972) Vv( k = O) - ½( 8/~0" + O¢'),2pV~') (35) 749. for the supersymmetry current. [11] D Fnedan and S H Shenker, NSF-ITP conference on In conclusion, we have constructed the fermion exactly solvable systems (September 1984), unpubhshed [12] A M. Polyakov, Phys Lett 103B (1981) 207, 211 vertex operator in a Lorentz covariant BRST [13] D Fnedan, 1982 Les Houches Summer School, eds J -B quantization using conformal field theory tech- Zuber and R Stora, Les Houches, Session XXXIX, niques, completing a covariant quantization of the Recent advances in field theory and statistical mechamcs superstring. We have proven that the scattering (North-Holland, Amsterdam, 1984) amplitudes of superstring bosons and fermions are [14] E Martinet, Phys Rev D28 (1983)2604 [151 M Kato and K Ogawa, Nucl Phys. B212 (1983) 443, Lorentz invariant, BRST invariant, supersymmet- S Hwang, Phys Rev D28 (1983) 2614; ric, and dual; further details will be presented see also K. Fujakawa, Phys. Rev. D25 (1982) 2584 elsewhere [18]. This formalism simplifies the [16] A Belavm, A.M Polyakov and A B Zamolodchlkov, calculation of fermion multipoint correlations; the Nucl. Phys B241 (1984) 333 four-fermion tree amplitude will be reproduced in [17] D Fnedan, Z Qiu and S H Shenker, Phys. Lett 151B (1985) 37 ref. [25]. The method generalizes to multiloop [18] D Fnedan, E Martmee and S H Shenker, m preparation corrections in a straightforward way; in particular; [19] S Deser and B Zummo, Plays Lett 65B (1976) 369, supersymmetry is apparent to all orders (at least L Bnnk, P Dive¢chm and P Howe, Plays Lett 65B to the extent that the expansion itself is sensible). (1976) 471 It should also be possible to give a superficial [20] D Gross, J. Harvey, E Martinet and R Rohm, Phys. Rev Lett 54 (1985) 502; Pnnceton prepnnt, and to be formulation of the superstring along the hnes published envisioned by Siegel [33], thus relating the NS-R [21] P.G O Freund, Phys Lett. 151B (1985) 161 quantization to the quantized version of the [22] R C Brower, Phys Rev D6 (1972)235, Green-Schwarz covariant theory [34]; the idea is P Goddard and C B Thorn, Phys Lett 40B (1972) 235 that the two should be related by a version of [23] L.P KadanoffandH Ceva, Phys Rev B3 (1971)3918 [24] C.Rosenzweig, Nuovo Clmento Lett 2 (1971) 924; Witten's non-abelian bosonization [35]. L Bnnk and F O Wmnberg, Nucl Phys B103 (1976) 445. We would like to thank T. Banks, J. Cohn, [25] J Cohn, D Fnedan, Z Qm and S.H Shenker, m VI. S. Dotsenko, P. Goddard, D. Olive, Z. Qiu, R. preparation Robin, W. Siegel, and E. Witten for their insights. [26] M. Halpern, Phys. Rev. D12 (1975) 1684; T. Banks, D. Horn and H. Neuberger, Nucl. Phys. B108 [1] P Ramond, Plays Rev D3 (1971)2415, (1976) 119; [2] A Neveu and J. Sehwarz, Nuel Phys B31 (1971) 86 V. Kac and I. Frenkel, Invent. Math. 62 (1980) 23, [3] F Gliozzx, J Scherk and D Ohve, Nuel Plays B122 G. Sehgal, Commun. Math. Phys. 80 (1982) 301. (1977) 253 [27] V Kac and I Frenkel, Invent Math. 62 (1980) 23, [4] D. Fnedan and S H Shenker, Aspen Center for Physics G Segal, Commun Math Phys 80 (1982)301 (July-August 1984) unpubhshed; Proc Santa Fe meeting [28] O. Alva.re-z, Nucl. Phys B216 (1983) 125 of the APS Division of Parucles and Fields [29] A. Chodos and C. Thorn, Nucl. Phys. B72 (1974) 509, (October-November 1984), to be published D Fatrhe, unpublished [5] M Green and J Sehwarz, Phys Lett. 149B (1984) 117, [30] B L Felgm and D B Fuehs, Moscow prepnnt (1983); Calteeh preprlnt V1 S. Dotsenko and V A Fateev, Nuel Phys B240 [FS12] [6] M Green and J. Schwarz, Nucl Phys B181 (1981) 502; (1984) 312 Phys. Lett 109B (1982) 444 [31] C B Thorn, Phys Rev D4 (1971) 1112; [7] S Mandelstam. Phys Lett 46B (1973)447. L. Bnnk, D. Ohve, C Rebbl and J Seherk, Phys Lett 45B [8] D Olive and J Scherk, Nuel Phys B64 (1973) 335; (1973) 379 E. Comgan, Nucl. Phys B69 (1974) 325; [32] A. Neveu, J Sehwarz and C B Thorn, Phys Lett 35B J Schwarz and C C. Wu, Phys Lett 47B (1973) 453, (1971) 529 E Corngan, P Goddard, D Olive and R A Smith, Nucl [33] W Siegel, Phys Lett 148B (1984) 276; Phys. Lett. 149B Plays B67 (1973) 477 (1984) 157, Phys Lett 151B (1985) 391,396 [9] S Mandelstam, Phys. Rep 13C (1974) 259, [34] M Green and J Sehwarz, Nucl. Phys. B243 (1984) 285 J Scherk, Rev Mod Phys. 47 (1975) 123 [35] E Wxtten, Commun. Math. Phys 92 (1984) 455 [10] C B Thorn, Plays Rev. D4 (1971) 1112,

61