Consistency of covariant quantization of the GS-string

Volume 214, number 4

PHYSICS LETTERS B

1 December 1988

C O N S I S T E N C Y OF C O V A R I A N T Q U A N T I Z A T I O N O F T H E G S - S T R I N G R. K A L L O S H and M. R A H M A N O V Lebedev Physical Institute, Leninsky prospect 53, 117924 Moscow, USSR

Received 29 July 1988; revised manuscript received 29 August 1988

Some basic features ofcovariant quantization of the heterotic string theory in the Green-Schwarz form are discussed. In particular, it is explained how the origin of new variables and new symmetries introduced by the authors is related to the irreducibility of K-symmetry. The manisfestly space-time super-Poincar6 covariant gauge fixed action with vanishing conformal anomaly is presented.

1. The problem of covariant quantization of the Green-Schwarz superstring attracts much attention. The purpose of this paper is to explain and to develop some essential points of our formalism presented in ref. [ 1 ] and also to give some c o m m e n t s to the approach of Nissimov, Pacheva and Solomon (NPS) [2-4]. The main result of our paper [ 1 ] was the generalization of the method of quantization of the GS string developed in ref. [ 5 ]. This generalization allows to maintain s p a c e - t i m e Lorentz covariance as well as the two-dimensional conformal s y m m e t r y of the theory. The action for the heterotic string theory in the Green-Schwarz formulation obtained in ref. [5] in the gauge gap =P~,~p, ~'+ 0 = 0 is given by ~1

,~= OzX,UO~Xl~.~],$kOzOk..[_~t .jl_ ~rep.gh.

"

( 1)

In the manifestly supersymmetric gauge the action [51 is ~ = OzX~O~x~ + rbkOzOk + ZCz~OzOa

dl-~zaOzCa "~-*~'-[- ~ep.gh

•

(2)

The original SO ( 1, 9) 0" spinors were presented in ref. [ 5 ] as pairs of SO (8) spinors 0 a and 0 a. The SO (8) s y m m e t r y was further broken down, and only the S U ( 4 ) × U ( 1 ) s y m m e t r y remained manifest, since 0 a was decomposed into Ok and qk. To form the

four-component objects ~/k, Ok without breaking the Lorentz group we needed a particular kind of auxiliary variables [ 1 ]. The gauge fixing performed in ref. [ 5 ] is useful for the analysis ofmultiloop string amplitudes. It was also generalized to a curved superspace background [ 6 ]. However, the superspace curvature was constrained to be flat in two directions [ 6 ]. There is the hope that with the new variables introduced in ref. [ 1 ] it will be possible to construct the unconstrained superspace background along the lines of the harmonic superspace approach. The theory of harmonic superspace was elaborated in a series of papers by Galperin, Ivanov, Kalitzin, Ogievetsky and Sokatchev [ 7 ] in d = 4, N~< 3. For the heterotic string theory it would be desirable to have a harmonic superspace with N = 1 supersymmetry in d = 10. The new variables introduced in ref. [ 1 ] are some unconstrained two-dimensional fields, and only when the classical equations of motion are satisfied they become constrained harmonic fields. In the q u a n t u m theory this means that there exist some gauges where we have unconstrained fields and ghosts, and other gauges where new variables satisfy harmonic constraints. The main idea of introducing auxiliary (gauge) variables preserving the physical content of the first quantized superparticle or superstring theory, is to introduce the same n u m b e r ( m ) of gauge symme-

#~ In the notations ofref. [5]. 549

tries (or first-class constraints ~2) as the number (n) of new degrees of freedom, Saux= J dz[Piqi--AATg(p, q)] ,

(3)

i= l, ..., n ; A = 1, ..., m, and [ T A, T B l e B = f a B C T c .

(4)

This auxiliary action has no physical propagating degrees of freedom if n=m.

(5)

In the two-dimensional theory with appropriate transformation properties under reparametrization of the new fields [ 1 ] the condition n = m is necessary also for the absence of a conformal anomaly. 2. As a simple example we will now consider vector harmonics for the bosonic string. Following Sokatchev [8 ], we use vector harmonics which parametrize the coset space G / H = S O ( l , 9 ) / S O ( l , 1 ) × S O ( 8 ) by 100 real variables uM,'U={U+,U , U--,U, ua,u},a= 1, ..., 8, which are subject to 55 constraints U +,UU +-'u= U+'U u - ' u -

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Volume 214, number 4

1 =U+,uU a,u

= ua'uub'u --(~ab=o .

(6)

To make all new variables pure gauge ones it is necessary to find a 45-parameter gauge symmetry in order to satisfy the n = m rule (5). The new symmetry which does not break the harmonic constraints (6) is the local SO ( 1, 9) acting internal space. The generators of this symmetry are

also want these fields to be propagating and simultaneously to be pure gauge variables. With the purpose of constructing two-dimensional conformal field theory we start with all 100 unconstrained fields uMu. The action for these fields will contain the constraints (6) with Lagrange multipliers. The theory will acquire the "metric" G MN ( z ) = ug,uu~,,rr '~ . The set of harmonic constraints (6) can be represented now as a condition for the vanishing of the "gravitational field", hMU(z) =_G M N ( z ) _ ?]MN.~. O .

(7)

We need also the inverse matrix U'UM, which is defined by the orthogonality and completeness conditions uM,UU'UN=~MN,

rl,u, = UuMUM,, .

For the fields u g,u which are not subject to harmonic constraints one can modify the symmetry generators so that they will commute with the harmonic constraints, LMN = -- UM"u O/ OuN'U -1- GNu Oll OUMu ,

(8)

[LMu, h eQ] = 0 .

(9)

Note, that for the l g u generators

[lMN, hPQ l ~heQ. The classical action of auxiliary fields is Sn= I dzdae

l MN= __ uM,UO/ Ou N,U'~- uN,Uo / ouM,U .

In ref. [ 1 ] we defined the generators of the Hsubgroup by H and the generators of the coset space G / H by F. In this case it means that l + - - H + - , lab- H °b and l -+a _ F -+a. With the help of harmonic variables it is possible to form the light-cone and transverse variables from the usual d = 10 vectors, e.g., X +-= U +-,uS "u, X a = U a ,uX'u.

The new fields ug,u must depend on the world-sheet coordinates, since otherwise the constant fields u -+,u, ua,u would break the space-time Lorentz group. We

X(pzM'UOgUMu--½]t2MNLzMN--I]2MNhMN),

where LzM N = V'UMPzN'u -- U'UNPzM'U ,

and the Lagrange multipliers have the properties ~tMN=l~ [MN], ].~MN=I~(MN). This action is reparame-

trization invariant, it has 45-dimensional invariance under the action of LzMg and 55-dimensional invariance under the action of h My. The algebra of the modified generators LMNis [ LMN, L e a ] = GMeLNQ "1-perm.,

~2 One may replace s first-class constraints by 2s second-class constraints.

550

i.e. the structure functions become field dependent.

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We may also present this algebra in the form

The covariant derivatives acting on 0 -+1/2,a are

[Lmu, LeQ ] = rlmeLuQ + perm. + r/raN,eQ. Rsh Rs .

Oz 0 + l/2,a= -- ( 1 / G +- )~:~ l/2yay +_OzO.

In this form the algebra has constant structure functions, but it is closed only on-shell, since the last term is proportional to 5SI~/5lt. The classical action of the bosonic string with harmonic variables which allow in principle some lightcone type quantization without violating the spacetime Lorentz group is the following: S = j d e dr eDzXMDzXNGMN +Saux = f dadze X ( -- D z X + D z X

The crucial point in choosing the set of internal symmetries for the heterotic superstring in the GS formulation is the following. We take the constraint y+xz=0

on the parameter of the fermionic x-symmetry, as it was proposed in ref. [ 5 ], to pick up eight irreducible symmetries form 16 reducible ones. However, now ~,+ is not a constant; we take ~+ = u + u ( z ) y u. The solution of this constraint is

- - DzX-DzX + + D~X~D,X ~

X z = ( 1 / 2 G + - ) ~ , + 7 -'`~ . + PzM OZu M - ½,~~.MNLzmN -- ½•MNh MN ) ,

where fiMN = ]2MN + 2DzXMDzXp( 1 + h ) F~ .

We hope that the sample considered above may give some insight into more complicated superstring harmonic variables. 3. In this section we will describe harmonic variables appropriate to GS-string theory [ 1 ]. We will use the commuting MW-spinors ( v + 1/2)= introduced in refs. [2-4 ], U+l.~=~+l/2~,ulfl±l[2

(U±)2---~0 •

(10)

Now the internal group generators LMN will be modified, L+_ --*H+_ I~-- 1/2..y + _ 0 / 0 0 - - 1/2 = 2 1uG+ 1/2,~, I + _ Va/a,-~ l V V + 1/2_t_ 1-~v

L±~ ~ F x a = -- U± aO/Ouau+ ½~+l/2~+_a0/0/.7+ 1/2 + ½v--l/2~±aO/OV--1/2 ,

L~b--" Hab = -- UaUO/OubU+ UbuO/Ou ~ + ½~+ 1/2~)ab 0/0/~+ 1/2 + ½if-- 1/2~)ab O/OV-- 1/2.

(11) Using v ± l/2, we define 0 ± 1/2a as follows: 0± l/2a= _ ( l / G +-)O~ l/2yay±O, O=~aV+ 1/20 -- 1/2,a+ ~a U -- 1/20+ 1/2,a .

(12)

(13)

The x-symmetry transformation is now ~O=g~y+~,-Xz/2G +-. For a consistent quantization this irreducible x-symmetry must form an onshell closed algebra with all gauge symmetries of the theory. This property is satisfied by H+ _, Hab, F_a, but not by F+a, since the action of the F÷asymmetry breaks the property (12) and adds to eight irreducible x-symmetries (12) another eight symmetries ~ ~- y + x, making it reducible again. Fortunately, the set of generators (11 ) forms a dosed algebra even without F+a. This gives us a possibility to exclude the F+a generators and therefore to preserve the irreducibility of x-symmetry (12), (13). Moreover, eliminating F_a from the set ( 11 ) we gain an additional 28 Spin (8) generators, since the Tab 0/0~ pieces decouple. Therefore the set of internal symmetry generators forming an on-shell closed algebra and preserving the irreducibility of x-symmetry consists of H ÷ _ , Hap, F_~ generators in eqs. (11 ) and the generators gab = O + 1127ab 0/0/7+ I/2. The total number of these generators is 1 + 8 + 28 + 28 = 65. Since two of the 55 constraints (6) are solved by the introduction of v + 1/2 [ 2-4 ], see eq. (10), the number of harmonic constraints is now 53. Therefore the n = m rule, which in this case is 1 1 2 = 5 3 + 5 9 , implies that we need 59 symmetry generators. This means that six symmetries (65-59) are superfluous. The corresponding six generators can be eliminated from 28 H~b, and we arrive naturally at S O ( 8 ) / S U ( 4 ) × U ( 1 ) , since 2 8 = 1 + 6 + 5 + 1 5 . Thus, instead of using the coset S O ( l , 9 ) / S O ( 1 , 1) 551

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× S O ( 8 ) we must use SO(I, 9)/SO(1, 1 ) × S U ( 4 ) X U ( 1 ). The auxiliary variables are now v + 1/2, uAu, u'¢u, A, .4= 1, ..., 4, u A ~, uSu are 4 + t/2, 7~_~/2-spinors (u s - (UA)*), where _+½ means U(1 ) helicity. The transformation properties under the SU (4) group are u~-,[exp(½co°p°)]Aaua,

i=1 ..... 6.

i,

[H, KAB] = --KAB ,

[H, Ksa] = K s a , [F i, F_s] = - p i s a F _ n , [F i, Ksa] = - P ~ - c - Kca + p ~ c K c s ,

The properties of p°-matrices and also the SU(4) × U ( 1 ) notation are that of ref. [9], the matrix C kk ofref. [ 1 ] being equal to ~kk. The number of harmonic constraints remains unchanged, l + 16+ 1 6 + 2 0 = 53. These constraints are

[F i, KAa] =pigcKcA, [H, F-A ] = 1 F - a , [H, F - S ] = ½F-s, [H+_, F_A] = --F_A ,

~+ l/2~uU+ 1/2~7-- 1/27.u U -- 1/2+ 1

[H+_, F - s ] = - F - s ,

~ ~ q- l /2~,UV ++-I / 2 u A I.t~ uAI.tUBI.t

=uauuaU-8~a=O.

[H, F i ] = - F

1 December 1988

(14)

Now we will look for a set of 59 generators which do not break the harmonic constraints and form the onshell closed algebra. They are

[KAB, Kcz~] = ~A~Ksc + ~BcKA~ -- JAcKBz~ -- ~Bz~KAc, [KAa, Kc~] = t~ABKac + JacKab , [KAB, Kc~ ] = OAOKBc -- JB~KAc , [Ksn, Kcz~ ] =OacKxz~- K~r~ .

H+_ = ½/7+ 1/2~+_ 0/0u+112+ ½/7-1/2y+_ 0/0/) -1/2 , All other commutators in this algebra vanish. Thus, we have explained the origin of three groups of 59 generators which are necessary for constructing the classical action for the heterotic string in the GS formulation, including the action for auxiliary fields. The Lorentz covariant quantization of this theory in conformal gauge was performed in ref. [ 1 ].

HO= _ UsupoxB OlOuau- UAuffOsB O/0Ua~ , H = ~U,~uO/Ou 1 _ A ~-~UA,,O/OuSu, 1 F_A=

U_#O/OUAIz+

IV-I/2~ ) AO/O~ -1/2

F _ s = - U_ ~O/OuSu + ½~ - 1/~7_s 0/0~- 1/2, F ~= UAup iAn O/OUau,

[ H ij, F k ] =t~ikFJ--t~JkFi ,

4. With a proper choice of gauge the theory considered above can be represented in a manifestly supersymmetric form. For the theory considered in ref. [ 5 ], in the manifestly supersymmetric gauge instead of the light-cone fermionic condition y +0= 0 the gauge fixing term ~zy + 0z0 should be introduced, see eq. (2). The Lagrange multiplier nza is related to the third ghost [ 10]. The field n~ is now propagating, as well as the x-symmetry ghosts. Generalization of this manifestly supersymmetric gauge (2) for the theory with additional variables [ 1 ] is straightforward. The resulting gauge fixed action is given by

[ H ij, F_~ ] =pOAgF_a ,

~e= L~osIg=p~

KA~ = ½~ + l /2yA~ 0 /0~ + 1/2, KSa = ~t7+ I/2~S/~0/0U+ 1/2, K~a = ½V + l /2yAaO/ O0+ 1/2.

(15)

The dimension of symmetries in these three groups is ( 1 + 1 5 + 1 ) + ( 4 + 7 ~ + 6 ) + ( 6 + ~ + 1 6 ) = 5 9 . The algebra of these generators at h = 0, i.e. when eq. (14) is satisfied, is the following: [ H °, H~Z]PB = -- (~ilHJk-- OeflkHil+ t~ikHJl+ o~JlHik,

[H is, F_s] = p ° ~ F _ a , [ H °, KAa ] =pi~cKc~ --p'JscKcA , [H ° , Ksa] =pO scKca_pO acKcs, [H °, KAa] = p ~ c K c a - P ° acKc~ , 552

+ ~ - 1/2~Dz 0 + 1/2s+ Rg- 1/2SOz 0 + l/2,4 + ~ - l / 2 A D z C + l / 2 S + ~":r-1/2SOzC+ l/2A + ~e..~.. +pzi O,qi+ C~Ao~c A + C~o,c~.

After a change of variables similar to that used in ref. [ 5 ] one obtains

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£#= D z X M D ~ X u + tbXDz 0 A + h e - 1~2ADz 0 + 1/2.~ "11-7~~-- 1 / 2 X D z O + I /2A .~. ~7.-- I / 2 A D z C + 1/2`1 -'l- ~ ~- l / 2"~D z C + l / 2A + ~rep.gh.

+Pzi Ozq i+ (?zA OzC A + ~ 5~Oz C .t z "~"~-nonli . . . . .

In this equation all terms in ~o~*ine,r have the same property as those in the gauge ~+0=0: they do not give any contribution to the loop diagrams on the twodimensional surface. The conformal anomaly in this theory is absent since the contribution of rt, 0 fields is compensated by the contribution of the x-symmetry ghosts. 5. In the last few years there were many related works on superstring quantization by NPS [2-4]. Since there were controversial statements [ 1-4] about the problem of unitarity in their approach, we wish to comment on this issue. In the first-quantized theory the preservation of the physical content of the theory when new variables are added, as in eqs. ( 3 ) - (5), is ensured by the n = m rule. In the papers by NPS [ 2-4 ] to be discussed below, the classical superparticle or superstring actions were supplemented by the auxiliary part of the type of ( 3 ), (4) with n > m ~3. The consistency of this approach was studied in refs. [2-4 ] at the level of the second-quantized theory only. However, the main ingredient of the second-quantized theory is the operator QBRSTof the first-quantized theory. But in theories with a conformal anomaly (and here n # m leads to a conformal anomaly of the two-dimensional theOry) Q2BRST is not equal to zero. Therefore it is not clear whether it is actually possible to perform a consistent second quantization of the string theory starting from the inconsistent first-quantized theory. Leaving this problem apart, let us discuss the claim by NPS that in the second-quantized theory [2-4] new variables qi= (u, v, w) are pure gauge. Their argument is based on the requirement that the secondquantized superfield on shell does not depend on the harmonic variables u, v, w. Since the number ( m ) of equations DA~(qi .... ) = 0 is less than the number of variables (n), this statement can be true only for very special kinds of functionals ~(q~ .... ). The proof of ~3 In one of their early papers [ 11] the fields are used with the correct counting; however, the main results were obtained in refs. [2-4], where n~ m.

1 December 1988

unitarity in refs. [ 2,3 ] is based on the hypothesis that the superfields ~(qi .... ) have only analytical dependence on u, v, w, i.e. no negative powers of u enter; e.g. there are no factors (p +)-n, where p + =p~'u + j,. There is no proof of this hypothesis. It is confirmed only by the previous experience with the N = 2, 3, d = 4 theory [7]. However, unlike the N = 2 , 3, d = 4 theory, in d = 10 (or N = 4 , d = 4 ) p+ enters with negative powers in the C P T self-conjugacy condition, which expresses the reality condition imposed on ~. The constraints from the GS part of the action and the measure of integration also contain (p ÷ ) -n (see eqs. (4.1), (5.7), (5.16), (6.6) in ref. [3]). This makes the validity of the abovementioned hypothesis questionable and the proof of unitarity of the second-quantized theory contained in ref. [3] incomplete. Anyway, since in the more recent papers [ 4 ] NPS considerably modified their original theory and gave up their hypothesis of analyticity, we will not consider the problem of unitarity of their original theory [2,3 ] here anymore. In ref. [ 4 ] they make another assumption about the properties of the superfields. The second-quantized action depends on some superfields ~, S = f d X d O d u a u d v X l / 2 ~ ' ~ ( ~ ( ..., V +-1/2, ...)),

where qb is supposed to be independent of v+l/Etru X V -1/2,

l)+l/20"l.tvpv--l/2 ,

v+l/20"l.q...lz5 v - 1 / 2 ,

P -+1/2

× a~, ~5 v -+~/2. However, no definition of functional integration over the fields ~ with such properties does exist, since these properties of • are not related to any symmetry of the action. Therefore the second quantization performed in ref. [4] also seems incomplete. Let us analyze, however, the main result ofref. [4], which could be considered as an indirect confirmation of the validity of their approach, the derivation of Nilsson's constraints [ 12 ]. The second-quantized action depends upon some superfields ¢ = (y+~/2a, B a). Using the notation of ref. [ 4 ], the equations expressing the super-Yang-Mills connections through integration variables ¢ (see eqs. ( 5.11 ), ( 5.12 ) in ref. [4] ) can be written as follows: y+ l/2a= ½i( V + l/2Gaa-- )or 0 +A ''~ , n a = u a b t a 'It ,

(16)

where 553

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A' = / 2 - ~A/2+ ( 1 / g ) / 2 - ID/2. It was implicitly assumed in ref. [ 4 ] that the solution ofeqs. (16) has the following form:

A,~=2i(v+t/2a,O,~(1/O + )y+l/2a, Aa=u'~B,,-u

+j - ( B ,

Y) ,

(17)

where

A =D --lAD+ ( 1 / g ) l ) - t D O . Eq. (17) tells that a+o,A~=u+#~;=0.

(18)

The gauge-independent content o f eq. (18) is u +~,FI'=u + ,,,FF'""~5=O .

(19)

The superstring constraints are used in ref. [4] to show that F ~, F~'"~'5 are independent o f u, and therefore the complete set o f Nilsson's constraints follows eq. (19), F~=F~'"'~5=0.

A(~*°n) = A s ( Y ) + a - ~ p c +p , (21)

where X +p, Z +- are arbitrary functions. The last terms ( a - X + and u ~ Z + ) were missed in ref. [4]. To derive Nilsson's constraints with the help o f eqs. (21) one must a s s u m e that X+=Z+=0.

(22)

However, this equation is equivalent to Nilsson's constraints in the u + direction (see eq. (19) ). Thus, it is possible to derive Nilsson's constraints by the methods o f ref. [ 4 ] only if the validity o f some of these constraints is assumed ab initio. Such a derivation does not present any confirmation o f the consistency o f the approach o f r e £ [4].

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6. The main conclusion which follows from the present work is that there is not much freedom in choosing different schemes o f covariant quantization of manifestly supersymmetric string theories. The requirement o f consistency o f the first quantization o f string theory puts strong limitations on the possibilities to choose additional variables and their gauge symmetries. Though we cannot exclude that alternative schemes o f covariant quantization may exist, the requirement o f consistency and compatibility o f new gauge symmetries with the gauge symmetries o f the heterotic string in the GS formulation (reparametrization invariance, conformal invariance and x-symmetry) rather definitely points to G / H = SO ( 1, 9) / SO(I, 1)XSU(4)XU(1) [ll. We are grateful to I. Batalin, M. Grisaru, E. Ivanov, R. Mkrtchyan, A. Morozov, V. Ogievetsky, M. Henneaux, D. Sorokin and A. Van Proeyen for the interest in our work and useful discussions.

(20)

However, eqs. (17) do not actually follow from the original expression for Y+ ~/2a and B a, in terms o f A,, A~, since u + ~A' ~ and a +,~pA' a are not uniquely determined by eqs. (16) which contains only projected parts ofA,~, A ' ~ ( ( a - A ' ) , and ua~A'~). The general solution o f eqs. (16) is

A~g~n)=A~,(Y,B)+u-~,Z+ +u+~,Z - ,

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References [ 1] R. Kallosh and M. Rahmanov, Phys. Lett. B 209 (1988) 233. [2 ] E. Nissimov, S. Pacheva and S. Solomon, Nucl. Phys. B 296 (1988) 462. [ 3 ] E. Nissimov, S. Pacheva and S. Solomon, Nucl. Phys. B 299 (1988) 206. [4] E. Nissimov, S. Pacheva and S. Solomon, preprints WIS88/23, 24 (1988). [ 5 ] R. Kallosh, Phys. Len. B 195 ( 1987) 369; R. Kallosh and A. Morozov, Intern. J. Mod. Phys. A 3 (1988) 1943; Phys. Lett. B 207 (1988) 164. [6] M.T. Grisaru, H. Nishino and D. Zanon, Phys. Lett. B 206 (1988) 625. M.T. Grisaru, and D. Zanon, preprint BRX-TH-IFUM-344FT ( 1988), M. Tonin, preprint DFPD 9/88 ( 1988). [7 ] A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky and E. Sokatchev, Class. Quant. Grav. 1 (1984) 469; 2 (1985) 155, 601,617. [8] E. Sokatchev, Phys. Lett. B 169 (1986) 209; Class. Quant. Gray. 4 (1987) 237. [9] M. Green and J. Schwarz, Nucl. Phys. B 243 (1984) 475; M. Green, J. Schwarz and E. Witten, Superstring theory ( Cambridge U.P., Cambridge, 1987). [ 10] I. Batalin and R. Kallosh, Nucl. Phys. B 222 (1983) 139. [ 11 ] E. Nissimov and S. Pacheva, Phys. Lett. B 189 ( 1987) 57. [ 12 ] B.E.W. Nilsson, GiSteborgpreprint 81-6 ( 1981).