Equivalence of the Kallosh and Carlip quantizations of the Green-Schwarz action for the heterotic string

Volume 205, number 2,3 PHYSICS LETTERS B 28 April 1988

EQUIVALENCE OF T H E K A L L O S H AND CARLIP Q U A N T I Z A T I O N S OF T H E G R E E N - S C H W A R Z A C T I O N F O R T H E H E T E R O T I C STRING ~

Gerald GILBERT i and Desmond J O H N S T O N 2 California Institute of Technology, Pasadena, CA 91125, USA

Received 22 December 1987

We show that a particular choice of gauge fermion in the quantization of the Green-Schwarz action presented by KaUosh is equivalent to the Fermi-light-cone quantization of Carlip, and is anomaly-free. The technique circumvents recently discovered (and currently unresolved) ambiguities inherent in the use of the Neveu-Schwarz-Ramond form of the string action. Although the method of Carlip may in principle be used to compute amplitudes to arbitrary order in the string loop expansion, for most practical calculations no simplification over the use of ordinary light-cone gauge is observed.

At present it is entirely unknown whether the heterotic string [ 1 ] is intrinsically weakly coupled or intrinsically strongly coupled. In either case it seems eminently worthwhile, nevertheless, to persevere in the program of computing higher-order quantum corrections to semi-classical amplitudes for string processes. The overwhelming majority o f research papers devoted to this problem have utilized the Neveu -Schwarz -Ramond [2 ] form of the classical string-coordinate action as the basis for calculation. Yet this method is plagued with unresolved ambiguities which have as of this writing effectively brought progress in calculating quantum effects to a halt.

In the Neveu-Schwarz-Ramond picture the world- sheet traced out by the string admits manifest two- dimensional supersymmetry. A consequence o f this is that the Feynman-Po lyakov [ 3 ] integrals defining quantum amplitudes are to be performed over the odd and even moduli describing the underlying super- Riemann surface. The mathematics o f such surfaces is not sufficiently developed to allow an authentic superfield treatment giving an integral over the su- perspace o f even and odd moduli. Therefore, many physicists have, as a stop-gap measure, advocated the

This work was supported in part by the US Department of En- ergy under Contract No. DE-AC0381-ER40050. Weingart Fellow in Theoretical Physics.

2 Work supported by an SERC/NATO Postdoctoral Fellowship.

use of an iterated integral representation for the amplitudes in which integration over the odd moduli is performed first [4 ]. However, due to the local two- dimensional supersymmetry of the action one may freely choose a basis for the super-Beltrami differentials which are dual to the odd moduli o f the world- sheet. It has been shown [ 5 ] that this freedom gives rise to an ambiguity: the domain of the remaining integration over the even moduli will vary as the basis for the super-Bertrami differentials is varied. With respect to a fiducial basis for these differentials, the integration set for the ordinary moduli will be sup- plemented by some even nilpotent quantity drawn from the Grassmann algebra for the odd moduli.

This ambiguity is difficult to resolve at present. However, all compact Riemann surfaces of genus two are hyperelliptic surfaces for which we can always give explicit, closed form expressions for the abelian dif- ferential forms and the prime form, in terms of a local coordinate on the world-sheet [ 6 ]. This in turn allows us to write explicit expressions for the two-loop

amplitudes. For this case a resolution of the integration ambiguity has been proposed which follows from requiring the vanishing of on-shell Green functions involving at least one null state. It has been shown that this requirement (which is, incidentally, a necessary, but not necessarily sufficient, condition for two-loop unitarity) is enough to select a particular,

0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (Nor th-Hol land Physics Publishing Division )

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presumably "correct" basis for the super-Beltrami differentials [ 5 ].

Unfortunately, due to our incomplete understand- ing of the mathematics of Riemann surfaces of higher genus, we cannot at present resolve the integration ambiguity when there are more than two handles on the world sheet. Thus the calculation of anything at higher order in the string coupling in the Neveu-Schwarz-Ramond picture (including even the cosmological constant) is, for now at least, tech- nically out of reach.

In the Green-Schwarz picture [7 ], on the other hand, it is the embedding target space, not the world- sheet, which admits manifest supersymmetry. An important virtue follows from this: there are no odd partners adjoined to the even, ordinary moduli. (In the Neveu-Schwarz-Ramond picture the odd moduli appear as zero-modes of the world-sheet gravitino which cannot be removed by some combination of world-sheet symmetry transformations. There is no analog of this for the Green-Schwarz picture.) The Feynman-Polyakov integrals for quantum amplitudes are evaluated with respect to ordinary Riemann surfaces. The ambiguities attendant to the Neveu-Schwarz-Ramond formulation do not appear in this approach and we would like to take advantage of this fact.

However, certain difficulties in the Green-Schwarz picture have made its use in practical calculations problematic. Two features in particular make the Green-Schwarz string difficult to use. First, there are an infinite set of ghosts ("ghosts-for-ghosts") as a consequence of a special local fermionic symmetry of the Green-Schwarz action. Second, a term in the action quartic in ten-dimensional Majorana-Weyl spi- nor fields makes the path integral difficult to evaluate as it is not a gaussian integral.

One way to quantize the Green-Schwarz action is to impose light-cone gauge conditions on the string coordinates. This restricts use of the Polyakov form of the path integral to the zero- and one-loop amplitudes [ 8 ], as compact Riemann surfaces of genus two and higher admit no global conformal Killing vector field, and such a symmetry is required to allow consistent global imposition of the light-cone gauge. (The point is that without a globally well-defined conformal Killing vector field one is unable to specify a global " t ime" coordinate. As a result the condition

X + = p +r is not always satsfied. Of course, in using the Mandelstam ansatz, which employs non-compact surfaces [9 ] for the quantum amplitues there is no reason not to impose light-cone gauge conditions beyond one loop as a global " t ime" coordinate may be defined [ 10 ] ).

In an interesting paper Carlip [ 11 ] has presented a quantization of the Green-Schwarz action for the heterotic string. There he constrains the two-dimensional fields with Fermi- (or "half-") light-cone boundary conditions, i.e., the Majorana-Weyl (in ten dimensions) spinor fields, 0, satisfy F ÷ 0= 0, where /" + is the positive ten-dimensional light-cone component of the gamma-matrix, while the bosonic string coordinates (describing the "skin" of an ordinary non- super-Riemann surface) are left unconstrained. This results in setting to zero the quartic spinor term in the action yielding a gaussian path integral. To ad- dress the problem of the infinite replication of ghosts Carlip imposes in the path integral the constraint (~_ )2=0 , where ~_=_(O_XU-OFuO-O)Fu and 0_ -= 0/02. He showed that the resulting path integral is directly proportional (via an infinite constant) to the unconstrained path integral ~t. Yet the reasoning leading to this solution of the ghosts-for-ghosts problem seems unduly implicit, and the fact that the method works at all appears somewhat fortuitous.

In another interesting paper Kallosh [ 12 ], has presented a quantization of the Green-Schwarz action which utilizes a modification of the Batalin -Vilkovisky method [ 13 ] of quantizing a gauge theory. Using this approach one understands how to solve the problem of all-order ghost interactions, but unfortunately, the final expression given by Kallosh for the action apparently retains the complicated quartic spinor dependence. (Nevertheless, one finds free equations of motion for the fields. ) As a result it appears to be extremely difficult to evaluate the path integral, and furthermore it is not clear that the conformal anomaly cancels as it must to yield a sensible theory.

Here we show that the Batalin-Vilkovisky- Kallosh (BVK) quantization may be reduced, in fact,

~1 In addition, Carlip constrains the transformation parameter ~, of the local fermionic symmetry with (lf+~_/PuP~,)x=O.This resolves an ambiguity for fields satisfying (~_)2= 0, as other- wise both x and x+lf+l~_x' correspond to the same gauge transformation.

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to precisely the form given by Carlip upon selecting a different gauge-fixing condition from that chosen by Kallosh. We show that in the BVK quantization there is no conformal anomaly (in ten dimensions), no super-Poincar6 anomaly (if we further reduce the action to full light-cone gauge form in ten dimensions) and that the resulting path integral is gaussian. (In an appendix we also make some observations on the independence of physical amplitudes from the two null vectors that Kallosh introduces to "truncate" the proliferation of ghosts.) We also consider heterotic strings which are compactified in background yielding an effective four-dimensional lagrangian density with a gauge group containing at least one pseudo- anomalous ~2 U ( 1 ) group as a direct product factor. In this case we argue that a Fayet-Iliopoulos D-term arises at one string loop [14] (specifically, a non- vanishing squared-mass for (tree-level massless) scalars with non-zero anomalous U( 1 ) charge) as well as the consequent tadpole for the dilaton field at two loops. These effects have already been derived using the manifestly covariant Neveu-Schwarz -Ramond formulation as well as using the Green-Schwarz formulation fixed in full light-cone gauge (In the latter case, of course, only the one-loop mass-squared for scalars can be properly derived; the two-loop tadpole must be inferred from the one-loop result. )

Although we may thus demonstrate a certain con- sistency between the Neveu-Schwarz-Ramond and Green-Schwarz versions of string theory, we are, of course, not able to say that the two actions yield the same quantum theories. That remains an unresolved issue.

We now examine the proposal of Kallosh for the quantization of the action for the Green-Schwarz heterotic string, which is obtained by modifying the methods of Batalin and Vilkovisky to account for theories with an infinite system of ghosts-for-ghosts. We note that an earlier attempt [ 15 ] to covariantly quantize the Green-Schwarz string was shown [ 16 ] to suffer from a conformal anomaly in ten spacetime dimensions and therefore did not yield an acceptable

,2 There is an apparent gravitational anomaly if the value of the hypercharge summed over massless chiral fermions is non- vanishing. In fact, this "anomaly" is not present as a result of the Green-Schwarz counterterm which is added to the action for the ten-dimensional theory.

candidate for the quantum action. Here we shall not review the Batalin-Vilkovisky quantization, instead referring the reader to the original literature [ 13 ] for (abundant) further details.

The classical action for the Green-Schwarz heterotic string is given by

So= ~ d z d g [ ½ x / g g m " p U m P u ,

--Emn(OmXlt)(Or.uOnO)"l-N//glff f 'C-O_~l,I] , (1)

where m, n label world-sheet directions, ~mn is the two- dimensional antisymmetric tensor-density,/z labels ten-dimensional spacetime directions, ~u is a world- sheet Majorana-Weyl spinor, z - is a two-dimensional negative light-cone Dirac matrix, and

BUm = O , , , X u - O F U O , , , O . (2)

A key feature of the action in eq. ( 1 ) is the presence of a local fermionic symmetry in addition to the global super-Poincar6 invariance that might have been expected. This local fermionic symmetry has the transformations

a 0 = $ + x + ,

5X~ '=rFuSO, ~y++ = 8 g + 0 + 0 , (3)

,u a where ~,+ + = v /gg+ +, x+ = ($+/eae ,u )/¢, and + is a positive two-dimensional light-cone index. The difficulties with the quantization of the Green-Schwarz action arise essentially due to the nilpotency of the gauge generators evaluated on the mass-shell, i.e.,

( ~?+ ) 2= - 2 ~ S o / ~ y + + . (4)

The above condition is pathological in the Batalin -Vilkovisky quantization; to use the original au- thors' terminology, the theory is "o f an infinite stage of reducibility."

Kallosh's modification was to impose a constraint on the ghost fields which eliminated the infinite tower of ghosts-for-ghosts. She proved using mathematical induction, that the level at which this truncation is performed is irrelevant; i.e. she showed that a truncation at any finite level of ghosts-for-ghosts is equivalent to any other finite-level truncation. She also showed that any finite-level truncation in principle yields physical results which are in agreement with those obtained from the original theory (with an in-

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finite number of ghosts) constrained in the unitary gauge.

The most convenient point to carry out the truncation is at the first ghost level. We impose the constraints on the (bosonic) ghosts for the local fermionic symmetry by requiring that

~c+ =0, ~)lffc+ =c+ , (5a,b)

where $ = NuFu, Nff= MuFU, and the ten-dimensional vectors N and M satisfy

N2=M2=0 , M.N=+ 1. (6a,b)

The effect of the above conditions is to halve the number of ghost components to eight from sixteen, which restores the correct counting of degrees of freedom. Note that the condition

([/,+g_/pup~, ) . x = 0

imposed by Carlip is, in effect, a covariant version of eq. (5a) because it too eliminates half of the degrees of freedom in the gauge parameter (and hence for the ghosts). The idea of introducing two null vectors has also been suggested as a means of carrying out the Dirac quantization of the superparticle (and perhaps the superstring as well) by Brink et al. [ 17 ], who em- ploy them to separate first- and second-class constraints.

For convenience in the calculation of the anomaly we make a different choice of gauge fermion from that of Kallosh, who selected

q/= ~'~n ~ ~, ykt + ~+21ff)~/'O + 0, (7)

where ~'~" is the diffeomorphism anti-ghost, ~ + is the local fermionic symmetry anti-ghost, and

~kml n 1 k l l k kl = ~ ( ~ , ~ . + ~ , . ~ . - r t rtm.), (8)

where r/kt is the flat, two-dimensional metric tensor. Instead ofeq. (7), we take

~b ¢, = ~ m n ~ kmlnYkl .~_ ~+ l~[~¢n+ 0 ( 9 )

as the gauge fermion, where we have introduced one component, n+, of a two-dimensional vector, in order that the gauge-fixing term should have the proper indices. This may be compared to a Yang-Mills theory for which we could choose either a covariant gauge fermion e=~OuAU (as in Kallosh's choice) or a non- covariant gauge fermion ~u' =~nuAU, where 6 is the

Yang-Mills antighost, A u is the gauge field and n u is some vector.

The Batalin-Vilkovisky procedure entails substituting

~b~ -= 0~'/O~bA (10)

(where O *, • represent all of the (anti-) fields and A is a generic index) into

S~-- Smin + ~*mn ~ r~ln l~ kl-~- ~*+ ~J~i]r + . (11)

In the above equation

Smi.=So

+ f dzdz[ ~b]6G.c.~)A

- 8 ( ~ *++ + ½+*+~+)0+0$#tc+

+ 4e* + $~/c + e+ )~/)~rO + 0

- (g*+ FuO+O - 2~ *+ P + + X~7 *+ + )6+ ~I~¢FU~¢l~lc+

+ (6*+X~al-'u)$+~¢Nffc+ ] , (12)

where the diffeomorphisms represented by the second term have not been written explicitly, and/7kt is an auxiliary field used to fix diffeomorphisms. Using eqs. (9) and ( 1 0) we find

0* =~ +~/$n+, ~_ =~ /$n+ 0, (13a,b)

and all other factors carrying an asterisk are identical to Kallosh's. Substituting eqs. ( 1 3) and ( 1 2) in eq. ( 1 1 ) now yields

S=So

+ J dz de{~mn ~r~nHkl+ n+ 6~¢N/II-I +

+ ~ k l ~ l l n (VM~n --gmnVk~ k)

- + + o + o ¢ tc +

+~+~/$n+ [ (0k0)~k+~+$~/C+ 1}. (14)

Upon integrating out H k/we find

S=So +/7' +NffSn+O+~--O_~_

+ ~ + +0+ ~+ + ~+A//~r¢ +~r~/n + c+ , ( 1 5 )

where

H'+=H+-Om(~+~")+80+(~++g+)n + , (16)

and we have chosen n +n += 1. In eq. (15) we have

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taken y= r/. Inspection of eq. ( 15 ) reveals that with our algebraic gauge-fixing condition on the local fermionic symmetry the ghosts decouple. Integration over/7 ' + enforces the constraint h//~0=0, which is sufficient to remove all the quartic terms in 0 in the action. We note that h//~0 is very close to the gauge condition imposed by Carlip ( F + 0 = 0 ) but the result is the same in that half of the components of 0 are set equal to zero.

To facilitate the calculation we make a particular choice for the vectors N and M, which necessarily spoils covariance. However, according to KaUosh the theory is independent of the constraints (N and M) so the end result ought to be independent of our choice. If we assume this to be the case (see appendix) we take

N~=½(1, 0 ..... 0, 1), (17a)

M~=½(1,0, . . . ,0 , - 1 ) . (17b)

The usual light-cone projection operators are given by v/2N and ,,/2M. We define as well the SO (8)- projected spinors

o~=-½F-F+O, f l = ½ F + F - O . ( lga,b)

We could, if we chose to, use the identity

/7'+ F - F + n+ O=x/~II '+ VF + n+ O, (19)

where T denotes transpose, to change integration variables in the path integral f rom/7 '+ to/-/,+T, as the associated jacobian determinant is non-singular. The effect of this would be to set F + 0 = 0 upon integration over H '+a'.

Integrating o u t / 7 ' + in eq. (15) thus yields the action

I - 0 + 0 _ X + 2 ( 0 _ X ) ( 0 + X - ) S= dz d~[ X i i +

- 2i(0_X + ) (flF-O+fl) + 4 - -O_ l_

+4++0+¢+ + # r - O _ q / ] , (20)

which is the result obtained by Carlip. Now, following Carlip, we evaluate the partition function for this theory fo find (in ten spacetime dimensions)

Z = j [ ~tr] [ (det V~ ) (det ~7~/2 )16 (det Vz_I ) --5]

× [ ( d e t V ° ) - 5 ( d e t V ~ - ~ ) ( d e t D ) 4] , (21)

where we retain the integral over the conformal re-

scaling function, a, and where we have placed the left- (right-) moving sectors in the first (second) set of square brackets. The operator D is given by

D=gZ~O~X+ Oz, (22)

where X~- is the solution of the equation of motion for X +

AX + = 0 , (23)

where A is the scalar Laplace operator, and t t V z _ ( gze ) - nOz ( gz~ ) n, z _ z~ V n - - g Oe, (24a,b)

If we enquire what is the variation of In Z with respect to the conformal rescaling ~gz~= 8a'g2z, we find, following Alvarez [ 18 ], for the left-moving sector

[ - 13/24~z+ 16(1/48zr) - 5 ( - 1/24zt) ]

X ~ dz d ~ g x / ~ t r = 0 , (25)

where R is the scalar curvature of the world-sheet, and we have preserved the factor order ofeq. (21 ) for the coefficients. For the right-moving sector we find

[ - 5 ( - 1/24~r) + ( - 13/24x) + 4 ( 1 / 1 2 r 0 ]

× f d z d z R x / / g ~ a = O , (26)

where the factor order ofeq. (21) is retained again. We have shown that with the gauge-fixing given in

eq. (9), the action proposed by Kallosh does not suffer from a conformal anomaly in ten spacetime dimensions. The action ofeq. (20) is compatible with the imposition of full light-cone gauge, for which the super-Poincar6 algebra is non-anomalous in ten dimensions, so we see that non-manifest super-Poin- car6 invariance is maintained for Kallosh's action once a particular choice of N and M is made. It is tempting to use Carlip's action to attempt to calcu- late physical quantities, and indeed, Carlip [ 19 ] has made some efforts in this direction. However, to sue- cessfully use eq. (20), one must integrate out X +, X- , and substitute Xd- to the left of the (flF-O+fl) factor. Unfortunately, the only solution of AX + =0 on a compact Riemann surface is X + = constant, and so the fermions will decouple. We are relegated to consid- ering non-compact surfaces, just as in the Mandelstam approach to the (full) light-cone gauge. Similarly, if we compute amplitudes by inserting the covariant Green-Schwarz vertex operators [20 ], we find that

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it is necessary to consider the singular limit k +- ,0, where k + is the positive light-cone component of the external particle momentum, to give a tractable calculation. Thus no simplification over the use of full- light-cone gauge is observed in Carlip's half-light-cone gauge.

We have argued that the Green-Schwarz action for the heterotic string fixed in Fermi-light-cone gauge offers no advantage in higher-loop calculations over the action fixed in full-light-cone gauge: in both cases one obtains identical expressions. It is useful to check that, at low orders of perturbation theory for which explicit results may be obtained, Carlip's action at least yields results consistent with those derived using full-light-cone gauge. The efforts of Carlip in this connection (alluded to in the preceding paragraph) focussed on the uncompactified heterotic string at zero temperature and on the thermal partition function for free heterotic strings. Here we will consider heterotic strings compactified in a special class of background for which a Fayet-Iliopoulos D-term ought to arise at the one-loop level. The relevant compactifications are those for which the low-energy gauge-group includes at least one U( 1 ) factor, the hypercharge of which, summed over massless chiral fermions, does not vanish.

It may at first fight appear that the general argument presented by Carlip [ 19 ] to show that the one- loop cosmological constant vanishes also indicates that no one-loop D-term (and consequently no two- loop tadpole for the dilaton field) appears. However, there is no problem in demonstrating the proper re- suits if one is careful to correctly count fermion zero- modes in the functional integral.

To expose the presence of a D-term we should compute the one-loop correction to the squared- masses of (tree-level massless) scalars which are charged with respect to the "anomalous" U ( 1 ) group. The derivation is actually indistinguishable from the full-light-cone derivation. I f we study, say, the spin (32)/Z2 version of heterotic string theory, we first introduce vertex operators for the scalars as

V( k ) =hii( X)~tr~/a( oX~ + kuOX + lTF~O)exp( ik.X) . (27)

Here we have considered the S U ( 3 ) ® O ( 2 6 ) de- composition of the various fields: the 32 left-moving fermions q/decompose as 3, 3, 26(~G ~r, ~a), and

the Green-Schwarz fermions 0 decompose as 3, 3, 1, 1(0% 0'~ 0 °, 0~); h,~X) is a closed harmonic (1,1) form on the internal space appropriate to the scalar field and k is the external particle momentum. In- spection of the index structure in eq. (27) now reveals that the fact that we are using Fermi-light-cone gauge is entirely irrelevant to the subsequent analysis. We thus proceed as in the full-light-cone case, which reveals a one-loop squared-mass of the f o r m

j d2ztr( O°(O)OO(O )JL (O ) ) , (28) m2oz

where JL is the left-moving current, the zero-mode of which is the U ( 1 ) charge operator Q. Now, following the analysis of ref. [ 14] we may reduce eq. (28) to the form

m2oc ~ c A t r (Q A) , (29) A

where the {c A} are (lattice) vectors of length x/~. Since the trace of the hypercharge is non-vanishing by construction the claim of the appearance of a Fayet-Iliopoulos D-term is verified.

Conclusions. In this paper we have examined the method of quantization proposed by Kallosh for the Green-Schwarz heterotic string. The final form given by Kallosh requires explicitly breaking Lorentz invariance.

Within Kallosh's approach, when one makes the particular gauge choices required to eliminate the quartic spinor dependence and so give an apparently useful form of the action, one inevitably recovers precisely the same action that Carlip derived. In Car- lip's derivation the string coordinates are constrained in Fermi-light-cone gauge, and thus it would appear that one might be able to perform explicit computations to arbitrary order in the topological se- ries expansion of quantum amplitudes.

One can show that there is no conformal anomaly. However, it does not appear to be the case that this form of the action leads to any real simplification as compared to full-light-cone quantization. It is true that it is not legitimate to impose light-cone gauge conditions on compact surfaces at two or more loops. As Mandelstam noted more than a decade ago, one must therefore introduce singular points on the higher-loop world-sheets and then in principle computations may be carried out. However, the calculations are not particularly simple. Carlip's action, can

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at first sight be used on any compact Riemann surface. However, the difficulties of the Mandelstam ansatz are, unfortunately, apparently preserved here as shown in the text. We must solve AX + = 0 on a compact surface, leading to a constant value for the solution X~-, which causes the fermions to decouple. Any useful calculation thus seems to require solving AX + = 0 on a non-compact surface. Furthermore, in calculating a multi-particle S-matrix element the vertex functions as well as Xff depend in a complicated way on 0, and the only apparent method of eliminat- ing this dependence (i.e., taking the limit k + - , 0 ) is a singular procedure. Thus, the calculation of a generic scattering amplitude seems quite problematic.

We have shown that in a particular background with anomalous U ( 1 ) factors, the method allows the com- putation of a Fayet-Iliopoulos D-term. However, this calculation is essentially identical to that which one performs in full-light-cone gauge to derive the same result and no simplification occurs.

Finally, we make a summary comment on the current status of the first-quantized path integral for her- erotic strings.

(1) Neveu-Schwarz-Ramond action: At present we know of no unambiguous prescription for the calculation of any amplitudes beyond two loops.

(2a) Green-Schwarz action in light-cone quantization: (i) The Polyakov ansatz of summing over compact world-sheets is incompatible with light-cone gauge beyond one loop. (ii) The Mandelstam ansatz of summing over non- compact world-sheets yields awkward expressions, although progress has been made in the bosonic case by exploiting the equivalence to the Polyakov ansatz [10].

(2b) Green-Schwarz action in Batalin- Vilkovisky-Kallosh quantization: When the quartic spinor term of this action is eliminated, Caflip's Fermi-light-cone action is obtained. No simplification over light-cone gauge seems to appear. ,

We would like to thank Mike Douglas for useful conversations and John Schwarz for reading the manuscript.

Appendix

In the calculation of the conformal anomaly we have made a particular choice for N and M which breaks Lorentz covariance. Kallosh, as mentioned above, gives a general argument that the partition function will be independent of the way in which the ghosts are constrained. We can also see this by con- sidering the Nielsen identity [21] for the local fermionic symmetry. We begin with the Ward identity for the 1PI effective action for the a-model of the string coordinates

O= f dzdZ( 8 ~ 8F 8F 8F 8F 8F + +

8F 8F 8F 8F 8F ) + + + n ' + (A. 1 )

where F i s the effective action, we take semi-classical fields and the {p} couple to the BRS variation of the subscripted fields. In eq. (A. 1 ) we have assumed that the local fermionic symmetry is a good quantum symmetry (it is, of course, a good classical symmetry only in three, four, six or ten spacetime dimensions with appropriate choices of properties for the fermionic coordinates).

We shall represent eq. (A. 1 ) by the shorthand no- tation S(F) = 0. We now introduce a BRS variation that acts on N (or M) by

8nRsN# = ~t/l , , (A.2)

where qu is a Grassmann vector, and ~ is the global BRS parameter. A compensating term to preserve BRS symmetry in the action is appended as

S'= f dzd~[(+n+NI~O], (A.3)

which results in a new Ward identity:

quOF/ONu + S( F) =0. (A.4)

If we differentiate with respect to t/and then take t /~0 we find

OF/ON~, + S( -OF/O~l~ ) = O . (A.5)

Writing this out explicitly, we have

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J" z~ aF aF(0`,) aF aF(0`,) o= dzd ~-~ ~p~--~-+ a~ ap.

aF aF( Co , ) 8F aF( C~ , ) aF aF(Co`,) + - - + - - - - + a0 ado a~,++ ape ÷ aa ap,~

8F( (~").'~ 8 F (A .6 ) + n ' + aa + ) + a - ~ '

where (9`,-=~+n+dlffF"0. Now, to compute amplitudes we inser t the cova r i an t ve r tex opera to r s in the

pa th integral and set the sources for the t w o - d i m e n -

s ional f ields to zero. Thus , we have

8/" a F 8/" a F a F 0=/7'+=aX ~ - a0- a0-ad+ -a~++" (A.7)

The cova r i an t G r e e n - S c h w a r z ver tex opera to r s are

(up to i r r e l evan t t e r m s ) inva r i an t u n d e r the local

f e r m i o n i c symmet ry , so we m a y freely inser t t h e m

in to the W a r d iden t i ty to f ind

OF( Vt ...Vp )/ON,, = 0 , (A .8 )

where we have used eq. (A .7 ) and F(V~...Vp) is the

genera t ing func t iona l wi th p c o v a r i a n t ve r tex opera-

tors inser ted. A s imi la r e q u a t i o n is ob t a ined for Z:

OZ( V,...Vp ) /ONv=O , (A .9 )

wi th equa t ions ana logous to eq. (A .8 ) and eq. (A .9 )

for the vec to r M.

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Equivalence of the Kallosh and Carlip quantizations of the Green-Schwarz action for the heterotic string