H ( Ff^L°9 . INTERNATIONAL CENTRE for \ | THEORETICAL PHYSICS

$H ( Ff^L°9 . INTERNATIONAL CENTRE for \ | THEORETICAL PHYSICS$

IC/87/420

H ( ff^l°9 . INTERNATIONAL CENTRE FOR I 'V. ': \| THEORETICAL PHYSICS

HARMONIC SUPERSTRING AND COVARIANT QUANTIZATION OF THE GREEN-SCHWARZ SUPERSTRING

E. Nissimov

S. Pacheva

and INTERNATIONAL ATOMIC ENERGY S. Solomon AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION

T T

IC/87/420

ABSTRACT International Atomic Energy Agency Based on our recent work, we describe a new generalized model of space- and United Nations Educational Scientific and Cultural Organization time supersymmetric strings - the harmonic superstring. It has the same physical content as the original Green-Schwarz (GS) superstring but it is defined on INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS an enlarged superspace: the D=10 harmonic superspace and incorporates additional fennionic string coordinates. The main feature of the harmonic superstring model is that it contains covariant and irreducible first-class constraints only. This allows for a straight- HARMONIC SUPERSTRING AND COVARIANT QUANTIZATION forward manifestly super-Poincaie'covariant Batalin-Fradkin-Vilkoviski (BFV) OF THE GREEN-SCHWARZ SUPERSTRING * - Becchi-Rouet-Stora-Tyutin (BRST) quantization. The BRST charge has a re- markably simple form and the corresponding BFV hamiltonian exhibits manifest Parisi-Sourlas O5p(l,l|2) symmetry.

E. Nissinsov ** In a particular Lorentz-covariant gauge we find the covariant vertex opera- International Centre for Theoretical Physics, Trieste, Italy, tors for the emission of the massless states which closely resemble the covariant "classical" expressions previously guessed by Green and Schwarz. S. Pacheva Institute of Nuclear Research and Nuclear Energy, Boul. Lenin 72, 1784 Sofia, Bulgaria

and

S. Solomon Physics Department, Weizmann Institute, Hehovot 76100, Israel.

MIRAMARE - TRIESTE December 1987

•' To appear in "Perspectives In String Theories" (Proceedings of the Copenhagen Workshop, Denmark, October 1987). '•'"'< Permanent address: Institute of Nuclear Research and Nuclear Energy, BouJ. Lenin 72, 1784 Sofia, Bulgaria. -1-

T due to a number of difficulties, (described in Sec. II) computations in this formulation were possible until the present work only in the Lorentz-non- covariant light-cone formalism (and restricted to the particular kinematical condition fc+ = 0) [7,8]. In the present report we review and further develop our recent work on the covariantly quantized GS superstring [9]. In particular, we use the general- I. Introduction ized formulation in which the supersymmetry is linearly realized [10,11]. In the last section ue discuss some novel material: the covariant vertex operators and The most fundamental role of space-time supersymrnetry in string theory scattering amplitudes. Throughout the report we rail our model "the harmonic (anomaly cancellation, finiteness, low-energy phenomenology, etc.) is appreci- superstring" with the understanding that.in fact it describes precisely the same ated since a long time (see ref. [1] for detailed discussion and a long list of refer- physical system as the GS superstring. ences). We expect the present approach to generalize to the D=ll supernvmbrane Each one of the existing formalisms describing superstrings- the Ramond- [12] whose covariant quantization meets similar problems as the original formu- Neveu-Schwarz(RNS) formulation [2,1] and the Green-Schwarz (GS) formula- lation of the GS superstrings (cf. Sec, II). tion [3,1] , has its own advantages and difficulties. The plan of the report is as follows . The world-sheet supersymmetric RNS formulation (the spinning string) Sec. II contains the basic notations and properties of the original GS super- allows to exploit the powerful machinery of D=2 superconformal field theory string as well as a review of the main obstacles to its covariant quantization. [4] and of modern algebraic geometry (with its supersymmetric generalization: In Sec. III. the basic definitions and properties of the D=10 harmonic super-Riemann surfaces, supermoduli etc.) [5]. These techniques allowed the computation of some superstring partition functions and amplitudes beyond the superspace are introduced. In particular, the structure of the Lorentz-covariant. tree level. The main disadvantage of the RNS formalism is the unnatural non- harmonic SO(8) algebra is revealed. geasetric appearance of space-time supersymmetry. Supersymmetry is realized in Sec. IV provides the construction of the harmonic superstring model and the RNS formalism by taking direct sums of quantized spinning strings with dif- presents its covariant algebra of irreducible first-class constraints. The covariant ferent sets of boundary conditions and by subsequent truncation of the spectrum relation between the GS and RNS superstrings from the point of view of the (the GSO projection)[6,l]. In particular, the description of space-time fermions harmonic superstring model is also briefly discussed. in the RNS formulation is extremely involved. Sec. V presents the Becchi-Rouet-Stora-Tyutin (BRST) charge [13,14] The main advantage of the GS formulation is that space-time supersym- and the Batalin-Fradkin-Vilkoviski (BFV) hamiltonian [14] of the harmonic super- metry is realized in a manifest manner from the very begining. Unfortunately, string. In particular, the manifest Parisi-Sourlas [15] OSp(l, 1|2) symmetry [16] of the latter is exhibited.

-2- -3-

...*.i... * «•>»- - :t Sec, VI is devoted to the construction of the vertex operators for emission of the massless states of the open harmonic superstring in a particular Lorentz-covariant gauge. The last, Sec. VII, contains our conclusions and speculations. II. Problems of the Standard GS Superstring All spinor notations and conventions are explained in the appendices of refs. [9,23,24], The original form of GS superstring action reads [3,1]: Throughout this report, we shall always work in the hamiltonian (phase- m mB space) formalism. Also we take for definiteness D=10 although the present con- s s = "nsn ng£( G (2.1) struction can be straightforwardly extended to other space-time dimensions (D=3,4,6),where classical GS superstrings are known to exist [3]. where

Here gmn(T,£) ( in,n=0,l) is the 2-D world-sheet metric and (A — 1,2) are the superstring coordinates . In the case of the II B theory "U = &Aa a™3 Dotn D=10 Majorana-Weyl (MW) spinors with equal (left-

handed) chiralities . In the case II A,they have opposite chiralities: 6X = 6\a,

02 = $% (= C*Pd2g). We always use D=10 a -matrices with undotted indices (cf. [9,23,24]). For definitjeness we shall consider explicitly the II B superstring theory.

The world-sheet metric gmn is a Lagrange multiplier which gives rise to the trivial bosonic first-class constraint:

m Ps " = 0 (2.2)

After gauge-fixing of (2.2) by imposing the condition }„,„-»,„„ = 0, one arrives at the following hamiltonian form of the GS action (2.1) [17,18]:

SCS =

-5- -4- In (2.3) \A)AAa denote bosonic (fermionic) Lagrange multipliers and the con- (ii) The presence of the second-class constraints, even after their covariant straints are given explicitly as follows: and irreducible disentangling, leads to highly complicated Dirac brackets among

the superstring coordinates XP,8Aa. This ruins the initial simple geometry of A TA = n^ — 4i( — l) d'AaD'A = 0 (reparametrization constraints) i (2.4) the embedding superspace. In order to achieve simple canonical Dirac brack- ti A i eta among X ,6Aa^ we had to impose in ref. [9] a covariant gauge-fixing of the where II* = P" + (~l) [X'f + 2i6Aa> ffA\ ; fermionic «;-gauge invariance (the first-class part of the constraints D°i (2.5) )). Thus, although acting in a manifestly Lorentz-covariant way, the supersymmetry transformations were realized nonlinearly. To realize the supersymmetry linearly Their Poisaon brackets (PB) algebra reads: we introduced [10,11] additional auxiliary pure gauge fermionic string coordi- (2.6) nates of the RNS type which helped to transform the second-class constraints of (2.3) into first-class. - V) , (2.7) The most important feature of the harmonic superstring model, is that (2.8) it possesses covariant and irreducible first class constraints only. Thus the co-

Now, the following two major problems obstructed until recently the variant BFV-BRST quantization procedure [14] applies directly and the whole progress towards the covariant quantization of the GS superstrings (2.1) (or, super-Poincare invariance is manifestly preserved (see Sees. IV and V). equivalently (2.3) ): To conclude this section let us point out that Sieget's superstring [22] (i) The fermionic constraints DJJ(O (2.5) form a mixture of first-class which is physically equivalent to the GS superstring [18] also contains covariant constraints (generators of the fermionic K-gauge invariance [3,1] ) and of second first class constraints only. However, these constraints once again form a re- class constraints (cf. eqs. (2.S) and (2.4) ) which cannot be separated [19] in ducible set with an infinite level of reducibility [20,21]. The formalism proposed a Lorentz-covariant and functionally independent (irreducible according to the in [21] to eliminate the higher ghosts generations within the BFV treatment ex- BFV terminology [14] ) way. The covariant separation proposed in refs. [17,18] plicitly breaks Lorentz invariance since it introduces constant light-like vectors is inconsistent since it leads in fact to reducible sets of constraints with an in- in the BFV-BRST action which arc not dynamical degrees of freedom. finite level of reducibility [20,21] (i.e. with an infinite number of generations of a ghosts for ghosts). The problem of covariant and irreducible disentangling of D A (2.5) was recently solved in ref. [9] by extending the usual D=10 stiperspace to include additional bosonic harmonic coordinates with a simple geometrical meaning (see Sec. III).

-6- -7-

•«*•«•'"••• the composite Lorentz vectors :

(3.3)

Eire identically light-like. Thus the vectors uj (3.1) together with uj (3.3) real-

ize the coset space 3Ofg&ff III. D=10 Harmonic Superspace and Covariant SO(8) Algebra This type of harmonic coset space was previously proposed in refs. [25] where the light-like vectors u* were taken as elementary (not biline&rs in D=10 The D=10 harmonic N-superspace of refs. [23,24,9] is parametrized by MW spinors). We stress however, that the presence of the bosonic Lorentz- (^.^."a'.uj) where (i",^)^ = 1,...,JV) are the ordinary N-extended spinor harmonics va ' (3.1) is crucial for the covariant and irreducible disen- superspace coordinates and the additional bosonic coordinates-harmonics ac- tangling of the fermionic GS constraint (2.5) as well as for the construction of cording to their geometrical meaning (see below)- axe defined as follows: the Lorentz-covariant harmonic SO(8) algebra described below. (i) v * are two D=10 (left handed) MW spinors, n The following harmonic differential operators (which preserve the kinemat- (it) u£ are eight (a=l,...,8) D—10 Lorentz vectors, ical constraints (3.1) ) will play an important role in the sequel: which satisfy the kinematical constraints: Dab = u 9 ~»» (3.4) 4--1T — U 2 ''flu dv 1

(3.1) (3.5)

= Ca (3.6) The group 5O(8) x 5O(l, 1) acts on u£,i>« * as an internal group of local rota- 2 tions where uj belong to atiy one of the three inequivalent 8-dimensiona] repre- Here and below, the following short-hand notations are used; 2 sentationa ((v),(s),(c)) of S0(8), whereas va carry charge ±\ under 5O(l,l). In the last line of (3.1) Cah denotes the invariant metric tensor in the rele- ah vant 5O(8) representation space (C is the unit matrix for (v) and it is the left A" = a"1, ...u a^ (3.7) (right) chiral charge conjugation matrix for (s),(c), respectively). for any Lorentz vector A''. Let us particularly stress that A*, A" are Lorentz Duo to the well known D=10 Ficrz identity (see e.g. [l]): scalars and they should not be confused with the vector components of A** which appear in the non-covariant light-cone formalism. The operators (3.4) - (3.6) represent the 50(1,9) algebra under commuta- )Bt = 0, tion: [Dab, Dcd] = CbcDad - CacDbd + C^D^ - CWDOC cd (7 )a6 = ^TV - 7 V)- [D°VP±C] = C6eZ>±a - Cac/?;fct ; [Dab,D-+] = 0 With the help of the harmonics (3.1) any left (right)-handed D=10 MW spinor (3.8) 0a (^°) Iaa-y be covariantly decomposed into two 8-component 5O(8) objects = CahD~+ + Dab !/>±2<1,(^±2a) in the following way:

From (3.8) one immediately recognizes Daf>,D~+ as generators of SO(8) x 50(1,1) whereas D±O are recognized as the coset generators corresponding to SO( 1,9) (3.12) SO(8)xSO(l,l) • The presence of harmonics (3.1) carrying Lorentz-spinor as well as or, inversely: Lorentz-vector indices allows to construct the following Lorentz-covariant 50(8) (3.13) matrices: In particular, for the solution of the D=10 Dirac equation />aP*l>p — 0 we get the (3.9) covariant relation (recall the notations (3.7) ):

:7CW+'* (3.14) (3.10) Let us stress on the fact that alt covariant harmonic SO(8) indices, appearing in

-IJ1/ -I [U the present formalism, belong to one and the same fixed 8-dimensional represen- Using (3.1) (3.2) one can easily show that the 8x8 matrices 7C = u^'', 7C = tation of SO(S)- either (v),(s) or (c). This is to be contrasted with the nan- u^ ; 7cii = u^7""u^, 7cd s u^""^ with 7",7^, 7^", 7^" defined in (3.9) , covariant light-cone formalism where all three types of SO(8) indices ((v),(s) (3.10) obey the same anticommutation (commutation) relations as the ordinary and (c)) do appear [1]. On the other hand, however, one can construct out of D=8 ,8x8 (T-matrices 7', 7' and 7^ = 7[I7J1, yij = 7['7J1> (!'J = L-8) the harmonic SO(8) matrices (3.9) the following three inequivalent Lorentz- respectively. Further, (3.3) and (3.10) satisfy the additional properties: invariant 8x8 representations of the 50(8) algebra:

(yabyd = g^ghd _ cadCbc (vector representation) (7±U = (T±U = 0,

a6 (3.11) (S*byd s l(7 )«' ((,,) _ spinor repi-e.seiitatiem)

-10- -11- (Cab)cd = ^ apiner representation)

Here the indices a,b label the SO(8) generators, whereas the indices c,d are 8x8 matrix indices. IV. The Harmonic Super-string As explained in ref. [24], it proves useful (in particular, for the second- quantization of the GS superatring) to introduce besides the harmonics v f, u° a We will use in the sequel the following generalization [10,11] of the har- (3.1) a second generation of harmonics: monic superatring action [9]:

a fc uinui = i '>'" = 0 , w*w' = C ' (3.15) S - SGS + Sharmotiic (4 realizing the coset space g . Here Ckl denotes the D=6 chiral charge conjugation matrix . tu*,tS* transform respectively as (4, j),(4,— j) under the SOS == JJdr + + J group 5f/(4) X 1/(1). In complete analogy with (3.4) - (3.6) one can introduce (4.2) the corresponding invariant harmonic differential operators representing the S0(8) algebra under commutation. The main characteristics of this harmonic superstriug action (4.1) are: 3 Finally we write down the action governing the (pure-gauge) dynamics of 1) it contains the harmonic space variables va , ujj from Sec. Ill; the harmonics v*i a, u£ (3.1) [23,24,9]: 2) it contains new formionic string variables i>A(O't

_1™ J-i -t-J-r,. _l 3) atl its constraints are first class and irreducible; 4) the supersymmetry is realized linearly; 5) it possesses a Sarger set of gauge iuvariances and it rrdt.ic.crs,in a particular

In (3.16) \ab,..., A+ denote Lagrange multipliers for the corresponding first- gauge, to the original GS action. class constraints «*"*,...,

4.1 _i,v _ I 4.1a =£>-<• _ (4.4)

-12-

-13-

T {TA((),D%(T,)}PB = same as in (2.7) (4.11) and D—h, D+a remaining unchanged (cf. (3.4) - (3.6) ). Here the following no- (4.12) tation is used (cf. (3.10) ): (4.13)

-a - cacb-b) (4.14) b One can easily verify that RA (4.5) satisfy the commutation relations of the (4.15) 50(8) algebra. vhere: In eq. (4.1) A , A^t, denote stringy Lagrange multipliers and the cor- A (4.16) responding constraints are all irreducible and first class. They are defined as As already explained in ref.[9], the harmonics vn ',uj, whose dynamics is de- follows; scribed by the action (3.16) , are pure-gauge degrees of freedom and, there- (4.6) fore, their independence on the world-sheet parameter £ does not spoil the reparametrization invariance of the physical superstring dynamics described by (4.1) . In fact, in the hamiltonian framework (in which we always work)

(4.7) the reparametrization invariance is accounted for by the presence of the first- class constraints (4.6) , satisfying the correct Virasoro algebra ( (4.10) , (2.6)). where T , D"^ are the same as in (2.4) , (2.5) . Let us repeat our warning that A Moreover, let us stress that nothing prevents us from taking the harmonics all SO(8)x SO(1,1) indices ±,n, b... are internal (i.e. they are inert under space- ±1 va 5 >«£ to depend atso on £ by a straightforward generalization of (3.1) , (3.4) time Lorcntz transformations; cf. (3.7) ). - (3.6) , (3.16) . In the latter case, however, the expressions for the modified super- The action (4.1) is manifestly super-Poincare invariant/where the superstring constraints (4.6) - (4.7) and their PB algebra (4.10) - (4.13) become symmetry generators in the hamiltonian framework read; very Long. Let us now demonstrate that the harmonic superstring (4.1) describes the same physical degrees of freedom as the original GS superstring (2.1) . Introducing the second generation of harmonics iu*,u>£ (3.15) as addi- The PB algebra of the first class constraints (4.3) , (4.4) , (4.6) , (4.7) takes tional (pure gauge) degrees of freedom (for details, see [ 9,24]), we can impose now the form: the following gauge fixing condition: = same as in (2.6) (4.10) = 0 (4.17)

-15-

-14- corresponding to the following (one quarter) part of the fermionic constraints and substituting (4.24) into (4.19) instead of D^>a (4.20) , G+»* (4.21) , we fy\ (4.7): immediately deduce the exact equivalence of the action (4.19) with the original GS action (2.3) . = G' (4.18) The fermionic generators D%(Q (4.7) which generalize the Siegel-type The gauge condition (4.17) reduces (4.1) to the action: local fermionic K-symmetry of the initial GS action [3] :

= •{ /* (4-25) J — 7 (4.19)

-i. -+LO -i +±k (4.26) where TA is the same as in (2.4) and the fermionic first-class constraints are:

(4.20) (4.27) a + h i»J( (4.21) rW. s ^a"a+v- 5) (v- 2<7 a <7ce'A)

Now, following the analysis in rcf.[2M.we can replace the first-class constraints act as ordinary local N=2 space-time supersyrnmetry generators on 6Aa (4.25) ,

(4.21) in (4.19) with the set of second-class constraints: but they act nonlinearly on X? and i'A (4.26) - (4.27) . To conclude this section, let us briefly discuss the covariant relation be- G+Jk(eq.(4.2\)) , (4.22) tween the standard GS [3] and RNS [2] superstrings in the framework of the

- IJ) (4.23) present formalism. The RNS (closed) superstring action writen in the hamiltonian form reads without affecting the physical content of (4.19) . Then, grouping together (4.20) and (4.22) into the Loreniz-covariant mixture of first- and second- class constva.ints: SRNS = I dr I" df[P^ar-Y" + + MAJA)] H-28)

It contains only the first-class constraints (A=l,2): (4.24)

fA = V\ (4.29) = D"A v.q. (2.5)

-16- -17- = 0. (4.30) On the other hand, imposing in the harmonic suporstring action (4.1) thfi covariant gauge fixing conditions: vhcre:

SAa=0 (4.37)

Adding to (4.28) the (completely decoupled) harmonic action (3.16) : corresponding to the fermionic constraints (the generalized K gauge-invariance) D°L we get; Slot — S[WS + ^harmonic (4-31) S = SGS + Sharmo,lic, (4.38) and imposing in (4.31) the covariant gauge-fixing condition (cf. notation (3.7)): (4.39)

•/:,

8S where TA are the same as in (4.34) , and Sharmonic h the form of (3.16) with Dab,D~a substituted by: corresponding to the super-reparametrization constraints JA = 0 (4.30) we get the action: ab a b sb = D + d£R A = D' , eq.(4.7) , (4.40) + ^harmonic (4.32) a (4-33) b' = (4.41)

Here the rpparametrization constraints read: and D •*",£>"'"'* remaining unchanged. The comparison of eqs. (4.33)- (4.36) with (4.39)- (4.41) establishes the (4.34) covari&nt relation between the GS and RNS superstrings in the context of the manifestly super-Poiiicare'covariant harmonic superstring formalism. (4.33) and obey the same PB algebra as in (2.6) , while Sharmonic is of the form with (4.34) exactly coincides with the GS action (2.3) after covariant gauge- (3.15) with Dab,D~a substituted by: fixing of the K-gaugc invariancc [9]. Therefore the harmonic superstriug (4.1) (4.35) simultaneously contains both the (2.3) mid (4.33) as different gauge choices:

0Aa = 0 reduces (4.2) to (4.33) (4.36) while:

+ +a and D~ ,D remaining unchanged. iJ}A = 0 reduces (4.2) to (2.3)

-18- -19-

at1 However, due to the different gauge fixings, Sharmonic Bnd S harmonic are not identical. They correspond to the reduction of the 50(1,9) representation to different inequivalent representations o£ SO(8). The physical content of the (4.1) (4.32) and (4.38) models is of course V. BRST Charge and BFV Hamiltonian of the Harmonic Super- always the same and identical to the original GS system. string In conclusion, the action (4.1) provides a kind of "unification" of GS and RNS superstrings * where, however, one should notice the different represen- The fermionic constraints D^ (4.7) form an open algebra, i.e. the coeffi- tations of the "non-orbital" harmonic SO(8) rotations in (4.35) - (4.36) and cients of the constraints in the right-hand side of the PB's are functions of the (4.40) - (4.41) by ^A a:nd &A = i(7a6)cJ'/>M (4-5) , respectively (see [lj canonical variables. For such systems, the Faddeev-Popov methods of quanti- ma for the non-covariant light-cone gauge relation between the GS and RNS super- zation do not apply and the BFV-BRST charge QBUST y contain in general strings). higher powers of the ghosts when the correct [14] procedure is applied. Fortunately, in the case of the harmonic superstring these potential com- plications do not materialize and the resulting QBRST

~~QBUST — Qha lie ~^~ Qittring ~t~ Qtibciian ' (5-1)~~

~~Qhar~~

~~(5.2)~~

~~-17 -b di]ab~~

+ Recently, a new superparticle action was proposed [26] possesing both global space-time as well as world-line super.symmetry, It contains, however, more physical degrees of freedom than the usual superparticle [27] and, therefore, describes completely different superiiniltiplets.

~~-20-~~

-21- Wnbelian = » + » a . a a , a a (5.3) does not pose any nortnal ordering problems for their consistent definitions. This is due to the simple fact that all «+, Q+ modes commute trivially among themselves: iting = £ T - 4i(-l)V*-±- + x'A & J—ir OCA + HcA [a+>a+l = fl5n+rn,0(u+u ") = 0 of. (3.2), (3.3). (5.6) 6 (5.4) Ac In order to second quantize the harmonic superstring and with the particular objective of exhibiting the Pariei-Sourlas O5p(l, 1|2) symmetry [16] of the The variables appearing in (5.2) - (5.4) are organized as follows : hamiltonian (5.5) , we expand the canonical variables in normal modes: ' Lagrange multiplier ghost antighost of the constraint

h-Aa(.t) XAaiO (5.7) ab Aat Vab b A+- 7,+- £>- + 6 1 A+" ri+a (5-8) Now, for the BFV hamiltonian [14]: (5.9) n=l with the specific choice Y = T~ gauge-fixing function, where c A0 denote the zero modes of the reparametrization ghosts cA(£) , we get:

~~(5.11) n=l~~

In eq. (5.5) Lo,..., LQ "" are precisely the zeioth (ghost) Virasoro generators [1]. Let us particularly stress that the appearance in the expressions (4.3) , (4.4) , (4.7) and in QBRST (5-1) of fractional and inverse powers of the quantized operator: n=l "An

~~(5.14)~~

~~-22- -21- In terms of the normal modes variables the Hgpv becomes: VI. Vertex Operators a 2TT f + X' X'a~~

In this section we briefly discuss the construction of the vertex operators "4*£. for emission of massless states of the GS superstring. So far we are able to do this only in a formalism where the fermionic K invari nee is covariantly gauge- , dl An fixed [9]. According to the discussion in Sec. IV, the latter gauge precisely cor-

responds to the covariant gauge QAa = 0 for the harmonic superstring (4.1) , i.e. the relevant action is given by S (4.33) . For simplicity,we shall consider in this section the case of open super- (5.15) strings One recognizes in this expressions the OSp(l, 1|2) invariant "squares" corresponding to the following subspaces: with the usual identifications [1]:

~~ti V \b forfG [0,T] (6.1) (-f)/or{6 [-*,(~~

~~(t)for(E [0,T]~~

This LorRiitz-covariant separation of the pure-gauge modes eliminated by the As a result of the gauge fixing 9 = 0 (4.37) the super-symmetry generator Parisi-Sourlas mwhaiiisrn relics heavily on the use of the harmonic variables Ali (4.9) acts now nonlinearly: (3.1):

Q l'"(0, (6.2) J — (6.3) Following the guess by GS [3] we consider an ansatz for the massless states vertex operators of the form (the subscripts "B" , "F" stand for bosons and where (cf. (3.7) ): fermions of the SYM multiplet, respectively): Q±\a = v±liaaQ {Qa as in (4.8)), (6.8) ei< o = F"(O - x'"(t) = 4= £ < ' (6.4) VF(F;k) = Uv+Y^iv-k (6.9) and (7b)ac is the same as in (3.9) . Here, we reapeat our remark from Sec, V that there are no normal where V^ is the same as in (6.4) . We will occasionally use the shorthand: offering problans in defining fractional and inverse powers of the quantized operator V+ (6.4) (cf. eq. (S.6) ). (6.10) The inassless states of the open GS auperstring form on shell the D=10 + + + N=l Super-Yang-Mills (SYM) multiplet. They are described by bosonic The matrix fvmctions g^\g^ are assumed to depend only on V = v i pv \ and i/)a and are determined from the requirement that (6.8) , (6.9) transform (fermionic) wave functions C(fc), Fa(k) which satisfy [lj: into one another under the (nonlinear) supersymmetry triuisformations given by 2 2 * C(fc) = 0 (i.e. t =0); (6.2) , (6.3) :

~~(6.5) (6.11) itt « C»i + **>~~

~~-kJ c~(*) - k~c+(k) + kau.k) = o, (6.7)~~

~~C+(k) = 0 {choosing X(k) = -t^ tn (6.6) ) SssC{k) = (6.13) /- (3-14)), where~~

~~-26- -27- > (6.14) where K is essentially LQ:~~

(6-21) are expressed in terms of harmonic components (3.7) , (3.13) , (3.9) , (3.10) (we used here the notation: T^v = i(k"C - fc"C)). With these ingredients one estimates scattering amplitudes using the following Inserting (6.2) , (6.3) , (6.8) , (6.9) , (6.13) , (6.14) into (6.11) , (6.12) , techniques. we get in complete analogy with [3] the result (written in matrix form): A typical scattering amplitude A^ consists in an "in" state \N >, an "out" state < 11, an arbitrary number of vertex operators for emission of on- 9B ~ c shell particles f(Q\ ki\V)eikx, and propagators A. E.g.: gp = (M"3)jm/t(Mi); (6.15) =< (6.22) Mab =2k+{r+r1Rab, When acting directly on the "in"-"out' states the {V+)' operators (a € R) M"~ = 2(7>+)-1.Racfc = -AT0, (6.16) e contribute simply only with their zero modes. To get CP+)* to act on the "in"- all remaining elements of M/£" = 0; "out" states one has to "drag" them through the propagators and .through the exponentials in (6.22) i.e. to commute them until they act on the states. In the (6.17) procecs of "dragging" an arbitrary function h of the operator V~^ through the propagators and exponentials one uses the commutation formulae: where (j )cli is the same as in (3.10) . With the values (6.15) for SJ.-,B, the formulae (6.8) , (6,9) completely (6.23) define the vertex operators for the the emission of the massless states which con- - T2)] stitute a D = 10 SYM multiple),: and + iTK !< (6.24) + 6 18 P {£)e = e" V+d + r). Vfl«;*) = P \ VOIP* ~ ( - ) We used the notations:

~~(6.25) In outer to estimate scattering amplitudes we have to define the propagator:~~

~~(6.26)~~

~~-29-~~

T' The above commutation relations are directly obtainable from the canonical fact the nonpolinomial dependence of /(£; k\k'} on k' is exactly due to the k'+ component of k' and it results in arbitrarily high inverse powers of the jfc+ com- commutation relations of the Fock space operators: ponents of the external momenta appearing in the amplitude. However these a a n n> mi — *m+n- (6.27) terms cancel due to the harmonic invariance;

One needs also the commutation relations which permit to drag the exponen- = 0 (6.32) tials one through another in order to be able to use the property that annihilation (creation) operators annihilate the "in" ("out") massless states: where Dab,D +,D±a are the invariant harmonic differential operators (3.4) 2 - (3.6) . Eqs. (6.32) also ensure that uJJ,i>a do not appear in the final re- (6.28) sult which is therefore a Lorentz and harmonic invariant depending only on the

where Lorentz-covariant components of the external momenta fcf{i = 1,..., JV) and on the wave functions C(i),.Fa(k) and not on their harmonic components: (6.29) n=l A4 = Kinernatical factor (Ci, fc,; , k3 • ,«,/o (6.30) (6.33)

After annihilating all the bosonic Fock space creation (annihilation) opera- Here s,t are Mandelstam variables. Formula (6.33) was first derived in the tors on the "out" ("in") states, one is left with expressions which depend only light^cone gauge formalism with the restriction fc+ = 0 for the external momenta on the external momenta and the fermionic Fock space operators (we skip the [1,71. 6(kl + k2 + k<3 - kt) factor):

~~JO * (6.31)~~

Next, one has to address the fermionic operators ij>a which are left in the expressions for gs,F (6.15) . Oi.e has to drag them one through the other until they act on the appropriate "in"-"out" states. These Lorentz covariant computations closely resemble the light-cone-gauge formulae except that the a, + , — indices are internal, Lorentz-ccvariant, and there is no restriction to the k+ = 0 case, In

-30- -31- gauge (where the generalized ic-invariance is fixed) and the diagranret ic rules for VII. CONCLUSIONS the scattering amplitudes are established (Sec. VI). In spite of the progress described in (i)-(iv) above, there are important The main conclusions we can draw from the material presented in Sees. II- problems still to be solved: VI above are as follows: (a) The immediate task is to construct the vertex operators in the formal- (i) The long-standing problem of covariant and irreducible disentangling of ism in which the generalized K-invariance is not fixed. the fermioiiic constraints in the GS superstring action (2.3) is solved by means (b) The next important task is to apply the present formalism for compu- of introducing additional (pure-gauge) bosoiiic degrees of freedom with simple tation of loop diagrams, especially (potentially) anomalous ones. geometrical meaning - Lorentz-spinor and Lorentz-vector harmonics (Sec. III). (c) After making the global super-Poincare invariance manifest, the gen- (ii) The problem with the presence of (the covariantly disentangled) second eralized fermionic s-gauge invariance generated by the constraints ZJJ (4-7) is class constraints, which ruined the linear realization of the space-time auper- still nonlinear (cf. (4.25) • (4.27) ). One could probably linearize it by intro- symmetry in the GS superstring, is solved by introducing extra fermionic string ducing further auxiliary string variables which would make the structure func- coordinates. They allow to convert the set of second class GS constraints into a tions in the PB relations constant. physically equivalent covariant and irreducible set of first-class constraints ISec. IV). (iii) The BFV-BRST quantization procedure is then sraightforward and leads to a manifestly super-Poincare invariant quantum superstring theory. The BUST charge takes a very simple form and is of rank one. The corresponding ACKNOWLEDGMENTS BFV hamilkmian is bilinear, trivially diagonalissable in terms of normal modes Two of the authors (E.N. and S.S.) would like to thank Professor imd exhibits the Parisi-Sourlas OSp{l, 1|2) symmetry which, at the second- Abdus Salam, the International Atomic Energy Agency and UNESCO for quautizrd level eliminates the unphysical gauge degrees of freedom (Sec. V). hospitality at the International Centre for Theoretical Physics,

~~(iv) The Lorcntz-covanant supersymmetric vertex operators for emission of Trieste, where part of this work was completed. (E.N. and S.P.)~~

~~(open string) mnaslcss states iuf cxpliritly constructed in a particular covariant acknowledge the hospitality of Professor Frishman at the Einstein~~

~~Centre for Theoretical Physics and of the High Energy Theory Group at~~

~~the Weizmann Institute, Rehovot, during the first part of the present~~

~~study.~~

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~~-36-~~