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N. Berkovits, M.T. Hatsuda, and W. Siegel1

arXiv:hep-th/9108021v2 28 May 1992Stony BrookStony BrookStony BrookStony BrookStony BrookStony BrookStony BrookStony BrookStony BrookStony BrookStony BrookAugust 1991ITP-SB-91-36THE BIG PICTUREN. Berkovits, M.T.

Hatsuda, and W. Siegel1Institute for Theoretical PhysicsState University of New York, Stony Brook, NY 11794-3840ABSTRACTWe discuss the conformal field theory and string field theory of the NSRsuperstring using a BRST operator with a nonminimal term, which allowsall bosonic ghost modes to be paired into creation and annihilation opera-tors. Vertex operators for the Neveu-Schwarz and Ramond sectors have thesame ghost number, as do string fields.

The kinetic and interaction terms arethe same for Neveu-Schwarz as for Ramond string fields, so spacetime super-symmetry is closer to being manifest. The kinetic terms and supersymmetrydon’t mix levels, simplifying component analysis and gauge fixing.1 Work supported by National Science Foundation grant PHY 89-08495.Internet addresses: nathan, mhatsuda, and siegel@max.physics.sunysb.edu.

1. INTRODUCTION1.1.

OverviewIn the Neveu-Schwarz-Ramond treatment of the superstring,the bosonic ghost of local world-sheet supersymmetry is neces-sary for the construction of spinor vertex operators and space-time supersymmetry. Unfortunately, the zero-mode of this ghostcauses an ambiguity in the cohomology of the Becchi-Rouet-Stora-Tyutin operator.

Fixing this ambiguity by choice of a “picture”leads to improper ghost number assignments, which must be fixedby the insertion of picture-changing operators. In the resultingstring field theory these picture-changing operators appear in thekinetic terms in a way which mixes levels, preventing a compo-nent analysis of the gauge-invariant lagrangian.The OSp(1|2)-invariant “vacuum” is not annihilated by one of the negative-“energy” ((mass)2) bosonic ghost oscillators (γ1/2), so ghost statesof arbitrarily negative energy can be created.In a previous paper [1] such complications were avoided bya general method which can be applied to any BRST formula-tion with unpaired, unphysical bosonic modes.

The addition ofa nonminimal term to the BRST operator, corresponding to thechoice of a Lorentz gauge (any gauge involving one time derivativeon the gauge field), introduces an additional bosonic mode whichcan be paired with the other to form creation and annihilationoperators, defining the vacuum unambiguously. After this minorchange, the BRST analysis is standard: The BRST cohomologyis unique, the physical states have the right ghost number, andb0 = 0 is sufficient to fix the gauge.In this paper we construct NSR conformal field theory andstring field theory along the lines outlined in [1].

The first-quanti-zation corresponds to gauges with a propagating world-sheet grav-itino. All (unintegrated) vertex operators in the conformal fieldtheory have ghost number one.In particular, even though theNS Hilbert space is essentially the same (up to nonminimal fields)as in the usual treatment, the vertex operators are unique, areclosely analogous to those in the Veneziano string, and differ fromthe usual choice because of a new unique choice of “vacuum.”We show the equivalence of general Neveu-Schwarz N-point treeamplitudes to the usual results by applying conformal field the-ory to our formulation in an explicit calculation.

We also discussamplitudes involving massless spinors.As in NSR light-cone string field theory, the treatment of theNS and R string fields is now identical, except for the usual dif-ference in boundary conditions on the world sheet: Both stringfields are fermionic (the tachyon “ground state” is an unphys-ical fermion), and have Φ†QΦ kinetic terms without “picture-changing” insertions, allowing component analysis and simple gaugefixing. The (NS)3 vertex is essentially the same as in Witten’soriginal version of NSR superstring field theory [2], except for thenonminimal coordinates.However, unlike that original version,the NS(R)2 vertex is now the same as the (NS)3 one, except thatagain the boundary conditions differ.

The same supersymmetryoperator transforms NS to R fields as R to NS.1.2. OutlineIn the following section we describe the relation between BRSTfirst-quantization with vertex operators and Zinn-Justin-Batalin-Vilkovisky second-quantization.

Both methods can be applied toparticles as well as strings.In particular, in Yang-Mills theorythere is a “vacuum” state from which the physical states can beobtained by applying the vertex operator, just as in string theory.In section 3 we consider the relation between these two meth-ods in more detail for particles: Yang-Mills and the Dirac spinor.In particular we examine the nonminimal fields which will beused in the massless sector of the NSR string later.We showhow this allows the choice of harmonic oscillator boundary con-ditions, which provide a unique picture for the cohomology, sothat the free Dirac spinor action does not require picture-changinginsertions. The fields obtained for this super-Yang-Mills system(Yang-Mills+spinor) resemble those found in the manifestly su-persymmetric version of 4D N=1 super-Yang-Mills.

The operatorformalism for interacting Yang-Mills is also described.The first-quantization of the free NSR string with nonminimalcoordinates is described in section 4. By starting with the hamil-tonian formalism the desired result can be obtained without fieldredefinitions and with little modification from the method usedfor the Dirac spinor.

The conformal weights for the nonminimalcoordinates are arbitrary, but match between the fermions and thebosons, guaranteeing conformal anomaly cancellation. (Similar re-marks apply to choices of weights for ghosts in the Green-Schwarzstring.

)Interactions for the first-quantized theory are treated in sec-tion 5 by the method of conformal vertex operators. We describesome general relations between the integrated and unintegratedforms of these operators which apply for our generalizations of theconformal gauge.

In the Neveu-Schwarz sector, the vertex opera-tors can be written without the use of bosonization. The physicalones are independent of the nonminimal coordinates, and so canalso be written in world-sheet superspace.

To treat Ramond ver-tex operators, and to relate our picture to the usual OSp picture,we give the form of bosonization corresponding to our harmonicoscillator boundary conditions.In section 6 we give the supersymmetry operator.Since inour picture all unintegrated (integrated) vertices have ghost num-ber 1 (0), so all string wave functions/fields have ghost number−12, the same supersymmetry operator can be used to transform2

from NS→R and R→NS. This is possible because, as in manifestlyspacetime supersymmetric formulations of particle theories, thesupersymmetry generator contains the analog of both the “∂/∂θ”and “p/θ” terms, whereas the old formulation was more like thecomponent approach in that these two terms appeared in the twoseparate transformations (i.e., in two different pictures).

As in theold formalism, the supersymmetry transformation is linear in thestring fields.We discuss scattering amplitudes in section 7, and calculatethe general NS amplitude (explicitly evaluating the ghost matrixelements). In practice, the calculations do not differ much from theFriedan-Martinec-Shenker-Knizhnik treatment.

The main concep-tual difference is that the FMSK approach required two picturesfor any type of vertex operator, as determined by ghost num-ber conservation (modulo the anomaly). On the other hand, inour formalism the vertex operator is always in the same picturebut contains terms similar to those appearing in both pictures ofthe FMSK approach (as does the supersymmetry operator, dis-cussed above).

However, only one type of term is chosen from anyparticular vertex operator because of the way the ghosts appear.Although NS amplitudes are slightly simpler than in the usual for-mulation, amplitudes involving spinors are somewhat messier intheir ghost dependence, and we just set up the formalism. (How-ever, the main problems in both formalisms are evaluating matrixelements of physical spin operators and deriving massive fermionvertex operators.

)In section 8 we discuss the superstring field theory. The NSand R string fields appear identically, except for the usual integervs.

half-integer mode numbers. Bosonization is unnecessary, sincethe NS and R Hilbert spaces can be treated as independent.

Bothstring fields are fermionic (but GSO picks out the right parts),both have the BRST operator as kinetic operator without picture-changing insertions, both use the same supersymmetry operator(as discussed earlier), and both interaction terms (NS(R)2 and(NS)3) have the same operator insertion ψ · p + · · ·. (Picture-changing exponentials, which appear in the super-Riemann surfaceformulation, cancel when one uses unbosonized ghosts and vacuuain the physical picture.) This suggests that this form is one stepcloser than the old formalism to a covariantly quantized Green-Schwarz formalism.In the final section we state our conclusions, including a fewconjectures based on some of the more detailed results discussedbelow.2.

SECOND QUANTIZATION VIA FIRST2.1. BRST cohomologyIn general, first-quantized relativistic systems can be describedcompletely by a (hamiltonian) BRST operator Q and a ghost-number operator J.Q is of the form (ghost)×(constraint) +(ghost terms), where the ghosts and constraints are unphysicaloperators.

Any of these operators which could have continuouseigenvalues must be paired up as creation and annihilation op-erators, except for the constraint p2 + M 2. (p2 + M 2 is treateddifferently because it is not a true constraint: In the classicalmechanics lagrangian, its Lagrange multiplier is restricted to bepositive, so the propagator is ∼1/(p2 + M 2), not δ(p2 + M 2).

)This may require the introduction of extra variables as nonmini-mal terms of the form ab, where a is a commuting c-number and banticommuting, so a can be paired with the unpaired boson thatappears in the minimal part of the BRST operator [1].The physics of the system is then described by the cohomologyof Q: states |ψ⟩which satisfy the equation of motion Q|ψ⟩= 0modulo the gauge transformation δ|ψ⟩= Q|λ⟩. These states canbe classified by their ghost number, which can take four values:−12 for the physical states, −32 for the “physical gauge parame-ters,” + 12 for the antifields of the physical states, and + 32 for theantifields of the physical gauge parameters.

(For the Venezianostring, see [3]. In general, there could be zero-momentum statesin the cohomology at J other than ± 32, but not for the theorieswe consider, which have only Yang-Mills as massless gauge fields.

)The physical gauge parameters have ghost number one less thanthe physical states because Q has ghost number one (δ|ψ⟩= Q|λ⟩),and “physical” means that they are the global symmetries whichsurvive gauge fixing [4]: In the field theory the gauge transforma-tion has the form δψ = Qλ+ψ∗λ−λ∗ψ (+ perhaps higher-orderterms in ψ), where the form of “∗” depends on how ψ representsthe global group. When Qλ = 0, there is no inhomogeneous piece,so that λ does not gauge away part of some gauge field.

On theother hand, when λ = Qf for some f, then there is another gaugeparameter λ′ = ψ ∗f −f ∗ψ (+ higher-order terms in ψ) thatproduces that same transformation (up to transformations of theform δψi = ǫijδS/δψj for graded antisymmetric ǫ, which are in-variances of any action). Thus the nontrivial global parts of thegauge symmetries are those in the BRST cohomology.

(For exam-ple, the parts of the vector ghost in the cohomology for gravitycorrespond to global Poincar´e transformations, while the globalparts of antisymmetric tensor gauge transformations do not ap-pear in the cohomology. )3

2.2. ZJBV quantizationThe antifields are the same ones that appear in Zinn-Justin-Batalin-Vilkovisky quantization [5]: The second-quantized formal-ism which follows from BRST first-quantization [2] is equivalent toZJBV second-quantization [6].

Specifically, if we expand a generalstate as |ψ⟩= |ψ+⟩+ |ψ−⟩, where b0|ψ+⟩= c0|ψ−⟩= 0, c0 is theghost for p2 +M 2, and b0 is its canonical conjugate ({b0, c0} = 1),then |ψ+⟩corresponds to a field while |ψ−⟩corresponds to anantifield. The Hilbert-space inner product is nonvanishing onlybetween fields and antifields, since it includes integration over c0,and this inner product is therefore fermionic, and corresponds tothe ZJBV “antibracket” ( , ): Writing the field as a state |Φ⟩,(⟨A|Φ⟩, ⟨Φ|B⟩) = ⟨A|B⟩(2.1)represents the obvious bracket between two |Φ⟩’s, but with theunusual property of being fermionic because of the anticommuta-tivity of integration over c0.

(As usual, the Hilbert-space innerproduct ⟨A|B⟩can be written in a coordinate representation asRA†B.) Any state and its “dual” antistate (relative to the innerproduct: for ⟨ψ+i|ψj−⟩= δji , |ψi−⟩is the antifield to |ψ+i⟩) have op-posite ghost number (J is antihermitian) and opposite statistics.The “S” operator of the ZJBV formalism isS = 12⟨Φ|QΦ⟩+ 13g⟨Φ|Φ ⋆Φ⟩(2.2a)(+ perhaps higher-order terms), and both contains the equationsof motion and generates the BRST transformations (as does Qfor the corresponding free first-quantized theory).

S is bosonic, asfollows from the fermionic nature of both Q and the inner product.It can also be written asS = 12⟨Φ|QΦ⟩+ 13g⟨V3|Φ⟩|Φ⟩|Φ⟩,(2.2b)where |V3⟩is some three-particle(/string) state.The analog ofQ2 = 0 is(S, S) = 0,(2.3)which by a perturbation in g implies: (1) Q2 = 0, (2) Q is dis-tributive over the ⋆product (i.e., ⋆and |V3⟩are Q-invariant), and(3) the ⋆product is associative (up to higher-order terms). It alsoimpliesδΦ = (A, Φ),A = (S, B)⇒δS = 0(2.4a)for any B, since then δS = ((S, B), S) ∼((S, S), B) = 0.

Thisgeneralizes to finite transformations asf[Φ] →eLAfe−LA,(2.4b)where Lαβ ≡(α, β) is the Lie antiderivative. A is thus the genera-tor for gauge transformations: The usual field-independent gaugeparameter Λ appears asB = ⟨Λ|Φ⟩⇒δΦ = QΛ + g(Φ ⋆Λ −Λ ⋆Φ).

(2.4c)For the interacting theory to be nontrivial, we must also haveS ̸= eLΨ 12⟨Φ|QΦ⟩e−LΨ(2.5)for any fermion Ψ cubic (and perhaps higher order) in Φ. Thisimplies that |V3⟩is also not Q on something, so it is in the BRSTcohomology for three-particle(/string) states.In the first-quantized formalism, the Fermi-Feynman gauge isfixed by choosing as hamiltonianHF F = {Q, b0} = 12(p2 + M 2). (2.6)More general gauges can be chosen by generalizing the gauge-fixing fermion b0:H = {Q, b0 + f0},(2.7a)where f0 is independent of b0 and c0 since the b0 term alreadytakes care of fixing that one gauge invariance, or equivalentlyH′ = U −1HU = {Q′, b0},Q′ = UQU −1,U = ec0f0,(2.7b)where f 20 = 0, and H′ = H if [H, f0] = 0.

The equivalent proce-dure in the second-quantized formalism isSF F = S|b0=0,(2.8)where “|b0=0” means to evaluate at b0|Φ⟩= 0 (i.e., set antifieldsto zero), or more generallySgf = S′|b0=0,S′ = eLΨSe−LΨ,Ψ = −⟨Φ|c0f0Φ⟩. (2.9a)Note that this Ψ contains only fields, not antifields.

This can begeneralized: Terms in Ψ linear in antifields induce field redefini-tions, and terms quadratic modify the BRST transformations byterms proportional to the field equations. However, the less gen-eral case expressed in (2.9a) will generally be sufficient.

It canalso be written without a unitary transformation asSgf = S|b0+f0=0. (2.9b)In the Feynman rules, instead of eliminating the antifields we canachieve the same effect by using (b0 + f0)/H as the propagator.The gauge-invariant action is obtained from S by the restric-tion J|Φ⟩= −12|Φ⟩, which picks out the physical fields, but alsoincludes antifields with ghost number −12, which are Nakanishi-Lautrup-type auxiliary fields.4

2.3. Vacuum and vertex operatorsAll the statements above in this section apply equally well toparticles and strings, bosons and fermions.

As an example, con-sider Yang-Mills theory. The operator Q can be obtained by thefirst-quantized methods of the covariantized light-cone (as for anyother free theory) [7].

The part of the cohomology at J = −32 isgiven by the gauge parameter for global internal symmetry trans-formations, which is represented in the wave function (field) bythe zero-momentum Faddeev-Popov ghost. (In going from gaugetransformation to BRST transformation, gauge parameter is re-placed by ghost.) In open string theories Yang-Mills is the onlymassless (i.e., unbroken) gauge field, and therefore this state isalso the only state in the cohomology with J = −32.

The state atJ = + 32 is the antifield of the Faddeev-Popov ghost at zero mo-mentum. These two states are the only states in the cohomologiesof these theories which are Poincar´e invariant, since the states inthe cohomology satisfy the mass shell condition (and thus mustbe massless for p = 0), and these are the only massless scalarsin the cohomology.

In the Veneziano string theory, this J = −32state is the one invariant under the Sp(2) subgroup of the world-sheet conformal group. However, in the Neveu-Schwarz-Ramondstring theory, we will find that this is not the one invariant un-der the OSp(1|2) subgroup of superconformal transformations.

Infact, our choice of boundary conditions of the wave function inthe space of the nonminimal variables we have added (and theirpartners) eliminates the OSp(1|2)-invariant state from the Hilbertspace.We will find that, at least for string field theory, it is moreimportant to have a state in the cohomology with the right ghostnumber, and the NSR state with J = −32 plays the same roleas in Veneziano string theory, even though it is now not Sp(2)invariant. For example, the first-quantized analog of (2.2-5) is todefine interactions via vertex operators byQint = Q + gV,Q2int = 0,(2.10a)which implies similar conditions upon perturbation in g. In fact,such expressions can be derived from (2.2) by expanding Φ aboutan on-shell background and keeping just the terms quadratic inΦ, 12⟨Φ|QintΦ⟩.

(This gives the usual V ’s up to conformal trans-formations.) In particular, V must be in the operator cohomologyof Q:Q2int = 0⇒{Q, V } = 0,Qint ̸= e−λQeλ⇒V ̸= [Q, λ].

(2.10b)The last relation is just the statement that V is not pure gauge:A unitary transformation on Qint is equivalent to a gauge trans-formation on V . (Consider the corresponding statement on theYang-Mills covaraint derivative, ∇= p + gA.) The infinitesimal(abelian) gauge transformation is thus δV = [Q, λ].States are then defined as|ψ⟩= V |0⟩(2.10c)in terms of a vacuum which must be in the BRST cohomologywith J = −32:J|ψ⟩= −12|ψ⟩⇒J|0⟩= −32|0⟩,Q|ψ⟩= 0⇒Q|0⟩= 0,|ψ⟩̸= Q|λ⟩⇒|0⟩̸= Q|λ′⟩.

(2.10d)Since open strings include Yang-Mills at the massless level,we can also determine the normalization of these J = ± 32 states:An analysis of the BRST transformations of Yang-Mills (see e.g.,[7] and below) shows that in terms of the Yang-Mills ghost |0⟩atJ = −32 we have the Nakanishi-Lautrup field (the antifield of theantighost) c0|0⟩at J = −12, the antighost Qc0|0⟩at J = + 12, andthe antifield of the ghost c0Qc0|0⟩at J = + 32. From the factsthat the kinetic term ⟨Φ|QΦ⟩includes (Nakanishi-Lautrup)2 and(antighost)(ghost) and that ⟨field|antifield⟩= 1, we see that wehave the inner product⟨0|c0Qc0|0⟩∼1(2.11)for Yang-Mills and thus for open strings.The gauge fixed version of (2.10) isHint = {Qint, b0 + f0} ≡H + gW,W = {V, b0 + f0}.

(2.12)Unlike V , which is “gauge covariant,” W is gauge fixed, since itdepends explicitly on the choice of f0 (as does H).3. PARTICLES3.1.

Yang-Mills and OSp(D,2|4)We first consider the massless Dirac spinor and Yang-Mills aswarm-ups for the NSR string, and because we can illustrate somepedagogical details which for the string would be more compli-cated but not more enlightening.The method of adding equal numbers of commuting and anti-commuting dimensions to the light cone, enlarging the Lorentzgroup to an OSp group, can be used to derive first-quantizedBRST operators directly for arbitrary representations of the Poin-car´e group, without reference to classical mechanics actions [7].Adding 2+2 dimensions is a minimal prescription which is suf-ficient to deal with bosons, but 4+4 is necessary to deal withfermions [8], and is therefore useful for supersymmetry. The extra5

fields introduced by adding 4+4 instead of 2+2 are thus nonmini-mal, and our treatment of the NSR superstring will be analogous.For example, for Yang-Mills this means starting with a (D−2)-component vector |A⟩= Ar|r⟩(r = 1, ..., D −2) and extendingthe basis |r⟩(⟨r|s⟩= δrs) so that r runs over (D −2) + 2n com-muting values and 2n anticommuting.The transverse Lorentzspin operators M rs =[rs] ≡|r⟩⟨s| −|s⟩⟨r| are then gen-eralized in the same way.That means that M rs gets general-ized from SO(D-2) spin to OSp(D,2|4) (the graded orthogonalgroup for D space, 2 time, and 4 fermionic dimensions). (ThisOSp group, as well as the other OSp groups discussed in thissection, applies to both particles and strings, and should notbe confused with the OSp(1|2) subgroup of 2D superconformaltransformations discussed elsewhere in this paper, which appliesonly to strings.) We divide up this (D,2|4) dimensional basis into(D −1, 1|2)+(1,1|2) as |i⟩= (|a⟩, |α⟩) and |A⟩= (|±⟩, |α′⟩), andwrite the anticommuting values of the Sp(2) index α as α = (c, ˜c).“a” is the usual physical SO(D-1,1) index, and ± is an SO(1,1) in-dex.

We then have the generators of the corresponding spin groupOSp(D−1,1|2)⊗OSp(1,1|2): M ij =[ij) and SAB =[AB).M ij are the spin operators of the “minimal” case, adding 2+2 di-mensions to the light cone, corresponding to the usual quantiza-tion of the NSR superstring, while SAB are the “nonminimal” spingenerators, corresponding to the additional degrees of freedom wewill add for the NSR superstring.The BRST and ghost number operators then take the form,for arbitrary massless particles (generalization to the massive caseis easy),Q = c 12p2+i(M capa+S−c′)−M ccb,J = 12[c, b]+i(M c˜c+Sc′˜c′) . (3.1)In the case of Yang-Mills, we can then straightforwardly derivegauge transformations δ|A⟩= Q|λ⟩and the action12⟨A|QA⟩forthe free theory, with the expansion of the field |A⟩and some usefulequations:|A⟩=Aa|a⟩+ iC|0⟩−i ˜C|c⟩+ iC′|˜c′⟩−i ˜C′|c′⟩+ A+|+⟩+ A−|−⟩+ iBac|a⟩−Bc|0⟩+ ˜Bc|c⟩−B′c|˜c′⟩+ ˜B′c|c′⟩+ iB+c|+⟩+ iB−c|−⟩,⟨a|b⟩= ηab,⟨+|−⟩= 1,⟨c|0⟩= ⟨c′|˜c′⟩= i,J = (−1, 1, −1, 1)on(|0⟩, |c⟩, |˜c′⟩, |c′⟩),M cc|0⟩= 2i|c⟩,M ca|0⟩= −i|a⟩,M ca|b⟩= ηab|c⟩,S−c′|˜c′⟩= i|+⟩,S−c′|−⟩= −|c′⟩,(3.2)where the spin operators appearing in (3.1) vanish on other states.We have explicitly identified the state |˜c⟩for the Yang-Mills ghostas the “vacuum” |0⟩.

Details of the interacting theory (withoutthe nonminimal fields C′, ˜C′, and A±) can be found in [7]. Wewill return below to interactions by the method of external fieldsin order to stress the relation of this formulation of Yang-Mills tothe usual formulations of string theory.3.2.

First quantization of Dirac spinorFor the Dirac spinor (and similarly for NSR), there are threesimple ways to derive the first-quantized BRST operator: (1) Addextra dimensions to the light cone. This method works for bothparticles and strings.

(The other two methods work for stringsand a few kinds of particles, but not for Yang-Mills. )For thiscase, we add 2+2 dimensions for the x coordinates and 4+4 forthe ψ’s.

(2) Start with the constraint algebra + an extra abelianconstraint, and write the BRST operator for these constraints. (3) First-quantize the classical mechanics action.

We will describethis last method, since it most closely relates to the super-Riemannsurface approach, and may give some insight into it. The othertwo approaches are actually simpler, since the nonminimal termcan be treated separately.

In all cases, the result for Q will besimply to add a quadratic nonminimal term to the usual minimalQ.The easiest way to derive lagrangians for the mechanics of arelativistic system is to start with the hamiltonian form of thelagrangian, which simply states which variables are canonicallyconjugate and what the constraints are: For the spinor,LH = ( .x · p + i 12.ψ · ψ) −(g 12p2 + iχψ · p). (3.3)The gauge transformations of all the variables except the Lagrangemultipliers are given by commutators with the gauge generators(constraints), while the Lagrange multipliers transform exactlylike the time components of gauge fields (as derived e.g., by co-variantizing the time derivative by adding to it the latter term inLH).

(For a review and examples, see e.g., [9]. )Gauge fixing for reparametrizations works the same way asfor the spinless particle; here we focus mainly on the 1D super-symmetry.

The lagrangian BRST transformations on the physicalfields x, p, and ψ, the gauge fields g and χ, the ghosts c and γ,antighosts b and β, and NL fields B and µ areQx = icp + iγψ,Qψ = γp,Qp = 0,Qg = .c + 2γχ,Qc = −γ2,Qb = B,QB = 0,Qχ = .γ,Qγ = 0,Qβ = −µ,Qµ = 0. (3.4)6

The temporal gauge χ = 0 is then chosen byΨ =Zib(g −1) + iβχ⇒∆L = QΨ =ZiB(g −1) −ib( .c + 2γχ) −iµχ + iβ .γ⇒Lgf = ( .x · p + i 12.ψ · ψ + i .cb + i .γβ) −12p2,(3.5)where we have eliminated auxiliary fields by their equations ofmotion in the final result.To fix Lorentz gauges in a first-order formalism, we intro-duce a quartet of nonminimal fields (the defining representationof OSp(1,1|2), which includes BRST as a subgroup) by gauge fix-ing a trivial gauge invariance: We introduce a field ˜χ which is puregauge, and thus does not appear in the gauge invariant lagrangian,along with the correpsonding ghost ˜γ, antighost ˜β, and NL field˜µ:Q˜χ = ˜γ,Q˜γ = 0,Q ˜β = −˜µ,Q˜µ = 0. (3.6)We then consider the following gauges:temporal:Ψ1 =Zib(g −1) + iβχ + i˜β ˜χ“simple” Lorentz:Ψ2 =Zib(g −1) + iβχ + i˜β.˜χ“normal” Lorentz:Ψ3 =Zib(g −1) + iβ(χ −˜χ) + i˜β.˜χ,(3.7)which lead to the gauge fixing termsQΨ1 =ZiB(g −1) −ib( .c + 2γχ) −iµχ + iβ .γ −i˜µ˜χ + i˜β˜γQΨ2 =ZiB(g −1) −ib( .c + 2γχ) −iµχ + iβ .γ −i˜µ.˜χ + i˜β.˜γQΨ3 =ZiB(g −1) −ib( .c + 2γχ) −iµ(˜χ −χ) + iβ( .γ −˜γ)−i˜µ.˜χ + i˜β.˜γ,(3.8)and the gauge fixed lagrangiansL1 =( .x · p + i 12.ψ · ψ + i .cb + i .γβ) −12p2L2 =( .x · p + i 12.ψ · ψ + i .cb + i .γβ + i.˜γ ˜β + i.˜χ˜µ) −12p2L3 =( .x · p + i 12.ψ · ψ + i .cb + i .γβ + i.˜γ ˜β + i .χ˜µ)−[( 12p2 + i˜γβ) + iχ(ψ · p −2γb)].

(3.9)The temporal gauge is χ = ˜χ = 0, the simple Lorentz gauge isχ = 0,.˜χ = 0, and the normal Lorentz gauge is χ = ˜χ,.˜χ = 0.The normal Lorentz gauge thus sets .χ = 0, so the gravitino ispropagating.3.3. BRST for Dirac spinorThe resulting hamiltonian BRST operators areQmin = Q1 = c 12p2 + γψ · p −γ2b,Qnonmin = Q2 = Q3 = Q1 + ˜γ˜µ.

(3.10)These are the usual BRST operators following from the hamilto-nian treatment of the constraints, where the Lorentz gauges differby just a nonminimal term. The gauge fixed lagrangians differonly by: (1) minimal vs. nonminimal fields in the term definingcanonical conjugates, and (2) a different choice of hamiltoniangauge fermion for the two Lorentz gauges: In the language of(2.7), f1 = f2 = 0, f3 = iβχ.

(From now on, we drop the e’s onχ and µ. )As discussed in [1], the assignment to the wave functionsof proper boundary conditions in the ghost coordinates allowsthe use of Φ†QΦ actions in the second-quantized theory, withoutthe use of picture changing operators.

In this case, the Hilbertspace is defined with respect to the bosonic ghosts by interpreting1√2(γ + i˜γ) and1√2(β + i˜β) as creation operators and1√2(γ −i˜γ)and1√2(β −i˜β) as annihilation operators. This has a natural in-terpretation in terms of OSp(D,2|4) [8]: (ψa, χ, µ; γ, β, ˜γ, ˜β) areDirac matrices for OSp(D,2|4), with a corresponding infinite-di-mensional spinor representation.

We then recognize Qnonmin asbeing of the form (3.1), where the OSp(D,2|4) spin operators arequadratic in these OSp(D,2|4) Dirac matrices. The usual physicalcomponents are those which are singlets under the Sp(2) subgroupgenerated by M αβ + Sα′β′.

The minimal ghosts, correspondingto adding just 2+2 dimensions to get OSp(D-1,1|2), give a spinorwhich does not contain such singlets.To analyze the cohomology of Q, we first make a field redefini-tion Φ′ = exp(iS+c′b)Φ, which is equivalent to the transformationQ′ = e−iS+c′ bQeiS+c′ b= c 12p2 + i(M capa + S−c′ + S+c′ 12p2) −(M cc + Sc′c′)b , (3.11)where in this case S+c′ = i˜γχ. This is the form in which Q directlyappears as one of the generators of OSp(1,1|2) with nonminimalfields, rather than the form (3.1), which is simpler when one ig-nores antiBRST [8].

(OSp(1,1|2) is an extension of BRST to in-clude antiBRST in a way which is useful for second-quantizationvia first-quantized BRST.) Eliminating the antifields (those notannihilated by b) by the equations of motion following from (3.11)generally leaves just the BRST-Sp(2) singlets: in this case, theSO(D,2) spinor representing (ψa, χ, µ).

(Similar remarks applyfor the vector.) The nonminimal term in the BRST operator thenallows the gauge choice χ = 0.

The remaining spinor field |s⟩,which is a representation of ψ and x only, satisfies1√2(γ −i˜γ)|s⟩=1√2(β −i˜β)|s⟩= χ|s⟩= b|s⟩= 0. (3.12)7

The antifield (c) part of the solution to the cohomology can theneasily be found in terms of this by re-solving Q = 0.Finally,inverting the transformation in (3.11), we find the solution to thecohomology for the original Q:|Φ⟩= (1 −γβ + icµβ)|s⟩= 32 −14(γ + i˜γ)(β + i˜β) + 12icµ(β + i˜β)|s⟩,p2|s⟩= ψ · p|s⟩= 0. (3.13)As usual, |s⟩can be expressed as a plane wave uα(k)e−ik·x|α⟩interms of the solution u to the Dirac equation and a basis |α⟩forrepresenting ψ.If we choose a coordinate basis for the wave functions (fields),the more convenient choices are ones in which the wave functionscan be chosen to be real.

In such a representation the creation andannihilation operators must be real (up to an overall phase), asfor the usual harmonic oscillator (a = x + ∂/∂x →|0⟩∼e−x2/2).In this case the real wave functions are Φ(γ, ˜β), so |s⟩∼eγ ˜β (orsimilarly for Φ(β, ˜γ)).3.4. Second quantizationIn general, Φ†QΦ actions have two particularly useful gauges:(1) a “Wess-Zumino”-type gauge, where only the usual physicalfields remain, which is useful for comparison with standard for-mulations of the field theory, and (2) a “Fermi-Feynman”-typegauge, where propagators are proportional to 1/(p2 + M 2), whichis useful for perturbation theory.

This is the same situation thatoccurs in the superfield formulation of 4D N=1 supersymmetricparticle field theories. In fact, some of the additional fields appear-ing at the massless level in the NSR superstring also appear in thesuperfield formulation of 4D N=1 super Yang-Mills [8]: If we elim-inate the ghosts (leaving just the physical, gauge, and auxiliaryfields which appear in the gauge invariant lagrangian) by lookingat just the SO(D,2) subgroup of OSp(D,2|4), then the SO(D,2)vector and spinor representing super Yang-Mills break up intoSO(D-1,1)⊗SO(1,1) representations as:SO(1,1) chargeSO(D-1,1) field+1 :A−(auxiliary scalar)+ 12 :λα(physical spinor)0 :Aa(physical vector)−12 :κα(gauge spinor)−1 :A+(gauge scalar)The engineering dimension of the fields is also given by the SO(1,1)charge up to a constant.To gauge fix the Φ†QΦ action for the spinor to a WZ gaugeit is sufficient to look at just the SO(D,2) spinor, consisting of λand κ above, since these are all that remain after eliminating theantifields by their nondynamical equations of motion.

In termsof the anticommuting coordinate µ, these can be represented as|Φ(µ)⟩= |λ⟩+ µ|κ⟩, where now1√2(γ −i˜γ)Φ =1√2(β −i˜β)Φ = 0.As can be seen by the gauge invariance δΦ = QΛ, these fieldscan only appear in the gauge invariant combination |λ′⟩= |λ⟩+iψ · p|κ⟩. In terms of this gauge invariant field (or equivalentlychoosing the gauge χ|Φ⟩= 0) the lagrangian reduces to just theusual ¯λ′ψ · pλ′.Gauge fixing to the FF gauge is much simpler: According to(2.8), just set b|Φ⟩= 0.

This gauge is just the covariantized lightcone one, which results from adding 4+4 dimensions to the light-cone Feynman rules, and gives results equivalent to the light-coneones by the usual Parisi-Sourlas arguments (at least for the case ofdoing one-loop calculations by putting this spinor in an externalfield).3.5. External fieldsThese BRST operators can easily be generalized to includeexternal fields: For example, for external Yang-Mills, the Diracequation constraint covariantizes to ψ · (p + A), and the Klein-Gordon equation constraint is just the square of this.

The resultis, for the minimal case,Qint = c[ 12(p + A)2 + 12σabFab] + γψ · (p + A) −γ2b,(3.14)where σab = 12[ψa, ψb] is the spin and Fab = [(p + A)a, (p + A)b] isthe Yang-Mills field strength. If we expand Qint in A as Qint =Q+V +· · · and choose Aa(x) to be a momentum eigenstate ǫae−ik·x(k · ǫ = 0), we find the vertex operatorV = ǫa(γψ + cp + ck · ψψ)ae−ik·x.

(3.15)For Yang-Mills in external Yang-Mills, BRST covariantizes toQint = c[ 12(p+A)2 + 12M abFab]+iM ca(p+A)a −M ccb. (3.16)(This expression also gives (3.14) if we substitute the OSp spinoperators for the spinor instead of the vector, as expected fromsupersymmetry.) The vertex operator is nowV = ǫa(iM ca + cpa + ckbMba)e−ik·x.

(3.17)(Again, this gives (3.15) with the appropriate substitutions.) How-ever, in the Yang-Mills case, there is an interesting analogy tostrings because of the occurence of the “vacuum” |0⟩in the spec-trum: We can write the physical Yang-Mills state in terms of thevertex operator acting on it,V |0⟩= ǫae−ik·x|a⟩,(3.18)8

and the 3-string vertex can be written as⟨0|V V V |0⟩. (3.19)Higher-point functions can also be treated, though not as conve-niently as in string theory, where duality allows all the diagramsto be expressed as a single propagator with insertions of 3-pointvertices.

Using the Fermi-Feynman gauge propagator b/p2, a dia-gram can be written as (using b|0⟩= 0)⟨0|V V bp2 V · · · bp2 V V |0⟩= ⟨0|V V 1p2 W · · · 1p2 W{W, V }|0⟩,W = {b, V },(3.20a)where in the present case of Yang-Mills, to lowest order in externalfields in each vertex,W = ǫa(pa + kbMba)e−ik·x(3.20b)consists of the usual orbital + spin interaction. If we eliminate theremaining ghost degrees of freedom in (3.20a), the result agreeswith that which follows from simply doing a perturbation expan-sion of Hint of (2.12).We can also easily check relations such as (2.11).

Furthermore,unlike the spin-0 and 12 cases of M ij for (3.16), for self-interactingYang-Mills Q2int = 0 only when the external field is on shell.4. FIRST QUANTIZATION OF NSRAs for the particle, we start with the hamiltonian form of thelagrangian.

In the case of the NSR (nonheterotic) string, knowingthe commutation relations and constraints, we can directly writeLH = ( .x·π+i 12.ψ±·ψ±)−g∓( 12p2± ± i 12ψ′± · ψ±) + iχ∓ψ± · p±,(4.1)where p± ≡(π ± x′)/√2 and ′ ≡∂/∂σ.This is the simplestform from which to derive hamiltonian BRST operators: It avoidsmessy field redefinitions needed to simplify the more nonlinearexpressions obtained from quantizing the usual second-order la-grangians in the usual Lorentz gauges (see [10] for the case of theVeneziano string). It is also the simplest way to derive the usualsecond-order lagrangian: Simply eliminate the auxiliary field π byits equation of motion (in both the lagrangian and transformationlaws).

We then find the usual result, including quartic fermionterms:L = −e−1(e+x) · (e−x) ∓i 12ψ±e∓ψ±± ie−1χ±ψ∓· e∓x −12e−1χ+χ−ψ+ψ−,(4.2)where we use (one-component) Weyl spinor notation, all deriva-tives have been collected into the zweibein (in a particular Weylscale and local Lorentz gauge) ase± ≡e±m∂m = g±∂1 ± ∂0,e ≡det eam = g+ + g−, (4.3)and the fermions have been rescaled by convenient powers of e.Gauge transformations can be derived by the same methodsas for the particle. Here we will instead write down the BRST op-erator directly from the constraints; the relation to the lagrangianapproach follows the analysis of section 3.2.

As there for the par-ticle, the addition of a nonminimal term gives the same result,up to a unitary transformation, as gauge fixing the gravitino in aLorentz gauge, so that one of the nonminimal fields can be inter-preted as a propagating gravitino. The result isQ =Zc( 12p2 + 12ψ′ · ψ + c′b + 32γ′β + 12γβ′) + γψ · p −γ2b + ˜γµ,(4.4)where now R≡Hdz/2πi and ′ ≡d/dz.

(As usual, we havedefined ρ = τ + iσ and conformally transformed from ρ to z,where z = eρ near the end of a string, or on the free string.) Theoperators are normalized so that the singular term in the operatorproduct b(z1)c(z2) is 1/(z1 −z2), and the same for ψψ, χµ, βγ,and ˜β˜γ, while for pp it is 1/(z1 −z2)2.

The Virasoro operators canbe defined by the generalization of (2.6,7): If we define them as{Q, b} as in (2.6), the nonminimal fields will be invariant under thecorresponding conformal transformations. We therefore choose, asin (2.7), the Virasoro operatorsL ≡{Q, b + f} = {Q, b + 12 ˜β↔∂χ + w(˜βχ)′}= 12p2 + 12ψ′ · ψ + 2c′b + cb′ + 32γ′β + 12γβ′+ (w + 12)˜γ′ ˜β + (w −12)˜γ ˜β′ + (w + 12)χ′µ + (w −12)χµ′,(4.5)where we have chosen f so that the nonminimal fields are confor-mal with weights 12 ± w. L0 is still of the form p2 + M 2, but nowM 2 depends on the mode numbers of the nonminimal coordinatesas well as the minimal ones.

As in section 3 for the gauge-fixedlagrangian, this corresponds to modifying the gauge fixing condi-tion: [e± −(w −12)ω±]χ± = 0, where ω± ≡∂me±m is the Lorentzconnection (actually ±ω±). Although the conformal weights 12 ±wfor the nonminimal fields are arbitrary, we choose w = 1 so thatγ ± i˜γ and β ± i˜β have definite conformal weight.This procedure can be applied in arbitrary theories to givearbitrary conformal weights to nonminimal fields.

In particular,the conformal weights assigned in [11] to the nonminimal fields inthe Green-Schwarz formulation of the superstring are consistentby such a procedure with those later chosen in [12], which explainswhy they both gave a vanishing conformal anomaly. (The formerchoice has the advantage that the Virasoro operators commutewith the BRST-Sp(2) that rotates ghosts into antighosts.

)The boundary conditions in the Neveu-Schwarz sector withrespect to the ghosts follow from just treating the positive-energyoscillators as creation operators and the negative-energy ones asannihilation operators. For the bosonic oscillators, this prevents9

the occurence of states of arbitrarily negative energy. The sameholds for the Ramond sector, except that there are now also zero-energy modes, which are just the modes of the spinor particledescribed in section 3, and are therefore treated the same way.For example, in the NS sector, the state of lowest energy (i.e.,(mass)2) is the tachyon, which satisfiesb0|t⟩= 0,ζn|t⟩= 0∀n > 0∀oscillators ζ(4.6a)The Yang-Mills ghost, which we’ll treat as the first-quantized “vac-uum,” is given by|0⟩= β−1/2|t⟩,(4.6b)and the physical Yang-Mills field as usual by|a⟩= ψa−1/2|t⟩.

(4.6c)In the R sector, the massless spinor satisfies ζn|s⟩= 0 for all n > 0for all oscillators ζ, while the ζ0 act as described in section 3.3.5. CONFORMAL FIELD THEORY5.1.

NS Vertex operatorsThe particle Yang-Mills vertex was obtained in section 3.5by covariantizing the derivative p, and this method can also beapplied to strings. Another way is to consider [Q, xa], which willpick out the same terms at k = 0, multiply by e−ik·x, and addcorrections perturbatively in k until a BRST invariant expressionis obtained.

An equivalent way is to consider [Q, pa], and first pulloffan overall derivative before following the same procedure. Thisis characteristic of a general relation between two types of vertexoperators in string theory: vertex operators which are integratedand have ghost number zero, and those which are local on theworld sheet and have ghost number one, which are more useful inthe second-quantized approach.

The relation is:[Q, W] = 0,W =ZW⇒[Q, W ] = V ′⇒{Q, V } = 0. (5.1)The last equation follows from the fact that {Q, V } must have van-ishing derivative but is local by construction, and therefore mustvanish.

W is then the integrated form of the vertex, while V is thelocal version. This W is the same one which appeared in (2.12): Itis just the contribution of the interactions to Hint = L0 +W.

Theambiguity δW = λ′, which has the form of a (1D) gauge trans-formation, is equivalent to the BRST-type gauge transformationδV = [Q, λ]. Without loss of generality, W can be chosen to haveconformal weight 1 with respect to L ≡{Q, b + f}, so W is con-formally invariant.

Then V has conformal weight 0. We can theninvert the derivation (5.1) of V from W as (see (2.12))W =Z(b + f), V⇒[Q, W ] =ZL, V= V ′.

(5.2)This works for general conformal coordinates (with the Rand ′for that coordinate): In particular, for the σ coordinate RL = L0(but in the z-plane RL = L−1), etc. Remember that W dependson the gauge choice, fixed by f.This appears more subtly in(5.1): Since L also depends on f, the notion of “locality” in theexpression W = RW depends on how conformal transformationsare defined.

The class of f’s defined by (4.5) implies that W belocal in the nonminimal coordinates as well as the minimal onesto transform simply under the corresponding conformal generatorsL.In the Veneziano string theory W is some operator constructedfrom x, and V is just cW (e.g., W = pe−ik·x for the vector). Thiscan be generalized easily to the NS string.

We can again use (5.1),or we can go to world-sheet superspace:[Q, W] = 0,W =Idz2πi dθ W⇒{Q, W} = −DV⇒{Q, V} = 0⇒[Q, W ] =Zdθ DV = V ′⇒V = V|θ=0,(5.3)where D = ∂/∂θ + θ∂/∂z. This is just a reflection of the factthat V differs at different points in superspace only by a BRSTvariation, the analog of the statement that W can be chosen su-perconformal covariant with weight 12 (so W is a superconformalinvariant).Thus physical NS vertex operators can always be expressed asV = −Zdθ CW[k, X(Z)]for some W,Z = (z, θ),C = c −θγ,X = ˆx + θψ,DX = ψ + θp,p = ˆx′,x(z, ¯z) = i[ˆx(z) −ˆ¯x(¯z)].

(5.4)We have here performed a minor slight-of-hand: By the aboveanalysis, we should actually get CDW−12 (DC)W for V, but thisevaluated at θ = 0 (for V ) becomes just −Rdθ CW by redefiningC to have an extra 12 in its θ term.As an example, we have the Yang-Mills vertex [13] (cf. (3.15)):W(k, Z) = ǫa(DXa)ek·X⇒W =Idz2πi ǫa(p + k · ψψ)aek·ˆx,10

V = −ǫaZdθ (CDX)aek·X = ǫa(γψ+cp+ck·ψψ)aek·ˆx. (5.5)5.2.

Bosonization and OSp(1|2)Because of our choice of boundary conditions, our bosoniza-tion of the bosonic ghosts differs from the usual [13]:1√2(γ −i˜γ) = ηeφ,1√2(β + i˜β) = −ξ′e−φ,1√2(γ + i˜γ) = ¯ηe¯φ,1√2(β −i˜β) = −¯ξ′e−¯φ,χ = e−eφ,µ = eeφ. (5.6)(There are also some Klein transformations/Jordan-Wigner fac-tors/cocycles, which we have carelessly omitted.) As usual, eα·ζhas conformal weight 12α · α + w · α, and operator products satisfyeα1·ζ(z1)eα2·ζ(z2) = eα1·ζ(z1)+α2·ζ(z2)(z1 −z2)α1·α2, where α · ζ ≡αiζi and 12α·α+w·α ≡ηij( 12αiαj +wiαj), ηii = +1 for bosonizedfermions and ˆx and −1 for bosonized bosons, and wi = 0 for phys-ical spinors and ˆx and 1 for these ghosts.

(It’s 32 for c and b, butwe don’t bosonize those here. )Another way to look at the NS vertex construction is in termsof the OSp(1|2)-invariant vacuum |OSp⟩: Although it is not inthe Hilbert space, we can formally write states in terms of it, then“picture-change” the vertex operators back in terms of the Yang-Mills ghost vacuum |0⟩.

In the original NSR formalism withoutnonminimal coordinates, the relation between these two vacuuawas simply |OSp⟩= X|0⟩in terms of the picture-changing oper-ator X, or |0⟩= Y |OSp⟩in terms of the inverse picture-changingoperator. One way to define X is as {Q, ξ} (not a true BRSTcommutator, since ξ0 is not really defined in terms of β and γ).However, in (5.6) we now have both a ξ and a ¯ξ, and therefore thecorresponding modification to this picture changing is:X ={Q, ξ} ≡:1√2eφ([β, Q] + iµ) :=eφ 1√2(ψ · p + iµ) + 12cξ′ + 12η′e2φb + 12(ηe2φb)′+ 34c′ ¯ξ′eφ−¯φ + 12c(¯ξ′e−¯φ)′eφ −¯ηeφ+ ¯φb,X ={Q, ¯ξ} ≡:1√2e¯φ([β, Q] −iµ) :,Y =√24 (3e−φ−eφ −i√2cξ′e−2φ −¯ηξ′e−2φ+ ¯φ−eφ + 12cc′ξ′ξ′′e−3φ−eφ) ,Y =√24 (3e−¯φ−eφ + i√2c¯ξ′e−2 ¯φ −η¯ξ′eφ−2 ¯φ−eφ + 12cc′ ¯ξ′ ¯ξ′′e−3 ¯φ−eφ)⇒iY Y = 1√2c(ξ′e−2φ−¯φ + ¯ξ′e−φ−2 ¯φ)e−eφ,: iXX : = : eφ+ ¯φ+eφ[β, Q] : ≡: δ(β)[β, Q] : δ(˜β)µ= eφ+ ¯φ+eφψ · p+i 1√2e¯φ+eφ cξ′ + e2φη′b + (e2φηb)′+ h.c.,[β, Q] = ψ · p −2γb −cβ′ −32c′β,|OSp⟩= : XX : |0⟩,|Φ⟩= V |0⟩= V0|OSp⟩⇒V = : V0XX :.

(5.7)The ambiguities in the normal ordering have been fixed by requir-ing that : XX : is the inverse of Y Y . Thus, : XX : can be writtenas the product of the usual minimal picture-changing operatorwith a trivial nonminimal picture-changing operator.

In practice,it is frequently easier to evaluate XV as [Q, ξV ], etc. Both V andV0 are in the BRST cohomology and have conformal weight zero,as do X and X, but their ghost numbers differ since X has J = 1.Then NS vertex operators take the formV0 = ce−(φ+ ¯φ+eφ)Ω,W0 ≡{b0 +f0, V0} = e−(φ+ ¯φ+eφ)Ω+c 1√2(iξ′e−2φ−¯φ−2eφ+h.c.

)Ω;(5.8a)V = −[ bG, cΩ] = γΩ+c{ψ·p, Ω} ,W = {ψ·p, Ω}for physical Ω,bG ≡ψ · p −γb −c′β,(5.8b)where Ωhas conformal weight 12. In relation to (5.4), Ω= W|θ=0and {ψ · p, Ω} = Rdθ W. The expression bG is not [β, Q], but isthe OSp(D-1,1|2)-invariant generalization of the light-cone vertexinsertion ψ · p, which differs by the same factors of 2 as in (5.4).

(Similar observations have been made in the covariantized light-cone form of the NSR three-string vertex [14], which differs fromthe string field theory vertex only by a conformal transformation). (5.4) is thus reproduced, since the bG commutator has the sameeffect as our modified θ integration.

For example, for the masslessvector Ω= ψe−ik·x reproduces (5.5).We can extend the prescription for picture changing to inte-grated vertex operators: Introducing one factor of X at a time,W2 ≡Q,ZξW1=Z(XW1 −ξV ′1) =Z(XW1 + ξ′V1),{Q, XW1 + ξ′V1} = XV ′1 + X′V1 = (XV1)′⇒V2 = XV1 = [Q, ξV1],W2 = XW1 + ξ′V1 = {Q, ξW1} + (ξV1)′. (5.9)The fact that no explicit η’s, ξ’s, or φ’s, or the equivalentδ(β)’s or δ(γ)’s, appear in V , as opposed to V0, is a reflectionof the fact that the vacuum |0⟩is in the same “picture” as thephysical states.

The exponentials appearing in V0 are those whicharise in the path-integral formalism due to Rφ-type terms in theclassical mechanics action, when the world-sheet curvature R isconcentrated at the point of the vertex insertion. In the physi-cal picture, these picture-changing factors just cancel the similar11

factors in the picture-changing operators : XX :, leaving the op-erator bG as a vertex insertion. In that sense there is no picturechanging in this formalism, and the picture-changing operatorsdo not appear in the expressions actually used: In the operatorformalism for conformal field theory, we use expressions such as(5.4) directly; similarly, in the operator expression for the stringfield theory 3-string vertex it is bG that appears and not : XX :.In the NS sector these expressions are slightly easier to use thanthe vertex operators of other pictures, since no exponentials orδ-functions occur.5.3.

R vertex operatorsWhile the nonminimal coordinates were not needed for thephysical NS vertex operators, they are crucial in the Ramondstring. The first step in finding R vertex operators is to compareboundary conditions in the NS and R sectors.

From the discussionof section 4, looking at physical states in each sector, as well as atthe OSp(1|2)-invariant vacuum, we haveannihilation operators for n ≥:|OSp⟩NSR1√2(γ −i˜γ)321201√2(γ + i˜γ)321211√2(β −i˜β)−121201√2(β + i˜β)−12121χ32120µ−12121c211b−100where we have chosen the “first” term of R vertex operators: inthe language of section 3.3, the “field” (b0 = 0) part before uni-tarily transforming from the OSp(1,1|2) representation (3.11) backto the usual BRST representation (3.1). For the NS sector, thischange in ghost boundary conditions from the OSp(1|2)-invariantvacuum to physical bosonic states is represented by the ghost fac-tors in (5.8a).

For the massless spinor, we find|s⟩= ce−(3φ+ ¯φ+3eφ)/2uαSαe−ik·x|OSp⟩,(5.10)and using the relation (3.13),V0 =√24 [3ce−(3φ+ ¯φ+3eφ)/2 −c¯ηξ′e−(5φ−¯φ+3eφ)/2+ i√2cc′ξ′e−(5φ+ ¯φ+eφ)/2]eiπ/4uαSαe−ik·x. (5.11)The final vertex operator is thenV =eiπ/4uαe−ik·xi 1√2ce−(φ−¯φ−eφ)/2Sα −12ce−(φ−¯φ+eφ)/2pαβSβ−hηe(φ+ ¯φ)/2 + ¯η′e(−φ+3 ¯φ)/2 −12cc′ξ′e(−3φ+ ¯φ)/2+ 23 ¯ηe−φ/2 e3 ¯φ/2′+ 12bc¯ηe(−φ+3 ¯φ)/2ie−eφ/2Sα,(5.12)where pαβ ≡γaαβpa.Interestingly enough there is another picture where the vertexoperators are simpler, and also hermitian.

As in the other pictures,the vertex operators have conformal weight zero and are in theBRST operator cohomology, but differ in ghost number.This“hermitian” picture is related to the OSp picture and the physicalpicture bybV = XV0 = XV 0;V = X bV,V = X bV. (5.13)This should be contrasted with the NS vertices, where the “in-termediate” vertices are the ones which are nonhermitian.

In thehermitian picture the massless spinor vertex is simplybV = eiπ/4ce−(φ+ ¯φ+eφ)/2uαSαe−ik·x. (5.14)Unlike the NS case, for the R string this picture is the one wherethe vertex operators resemble those of the minimal version of theNSR superstring.

In fact, if we were to redefine the R string fieldΦ by Φ →Y ( π2 )Φ, where XY = 1, all vertex operators would besimple in the OSp picture, but then the field theory action wouldbe like Witten’s, with Xmin →XX and Ymin →Y Y .Thus,the nonminimal coordinates ˜γ, ˜β, χ, µ have effectively allowedelimination of Ymin from the R kinetic term by allowing us totake its “square root.”The reality condition for string fields works a little differ-ently for the nonminimal coordinates. Consistency with conformalfield theory (or CPT in the first-quantized theory) requires thatcomplex conjugation correspond to the conformal transformationσ →π −σ (z →−1/z).

Including the Jacobian factors appropri-ate for the conformal weights, which are just phase factors, thiswould mean Φ[ζ(σ)] = Φ[eiπdζ(π −σ)]. However, this would con-flict with the reality condition for the nonminimal coordinates, asdescribed in section 3.3 for the zero-modes: Under complex conju-gation, γ0 ˜β0 should not change sign, but they would since γ ˜β hasweight d = 1.

(Only the product of their phase factors mattersbecause of GSO projection.) We therefore introduce an appropri-ate BRST and conformally invariant “charge conjugation matrix”ω asω(˜γ, ˜β, χ, µ)ω−1 = −(˜γ, ˜β, χ, µ),ω(η, ¯η, ξ, ¯ξ, φ, ¯φ)ω−1 = (¯η, η, ¯ξ, ξ, ¯φ, φ),ωeφω−1 = eφ + iπ,ωQω−1 = Q,ωXω−1 = X,(5.15a)and use it to impose reality conditions on the string fields andvertex operators:Φ[ζ(σ)] = ωΦ[eiπdζ(π −σ)],V = ωV ω−1.

(5.15b)ω is just a “twist” operator which introduces phase factors. Itsaction in (5.15b) is similar to that of the usual Dirac charge con-jugation matrix on spinors and γ-matrices.12

6. SUPERSYMMETRYJust as the integrated, k = 0, Yang-Mills vertex in the phys-ical picture gives the momentum operator, the integrated, k = 0,massless spinor vertex in the physical picture gives the generatorof supersymmetry [13].

This is because the massless vector andspinor are related by supersymmetry in the same way as p0 andq. Another way to understand this is from the Green-Schwarz for-malism, where the Yang-Mills field and the massless spinor are re-lated to the gauge fields for derivatives with respect to the Green-Schwarz coordinates xa and Θα [15].

Thus, the supersymmetrytransformations are given byδΦ = ǫαqαΨ,δΨ = ǫαqαΦ,qα = ωˆqα = ˆqα†ω−1,(6.1)for R field Φ and NS field Ψ, where ˆqα is taken from the physical-picture integrated vertex operator W for the massless spinor asW|k=0 = uαˆqα. ω allows the supersymmetry transformation tobe written in a more symmetric form: On the space ΨΦ, wherethe supersymmetry generator has the form ( 0AA†0 ), we have A =A† = q.To evaluate the closure of the supersymmetry algebra, wewriteˆq =Q,ZξcW=Q, ξ( π2 )ZcW+Q,Z{ξ −ξ( π2 )}cW= X( π2 ) bW +Q,Z{ξ −ξ( π2 )}cW.

(6.2)The second term is a true BRST variation, since ξ0 drops out. (In [2] this manipulation was performed for p0.However, thesame manipulation can be performed for one of the supersym-metry transformations given there for the minimal theory.

Thus,supersymmetry does not mix levels there either, even though theR kinetic term in the minimal theory does.) We therefore have{qα, qβ} = ˆq(α†ˆqβ)≈X( π2 )X( π2 ){ bWα, bWβ}= X( π2 )X( π2 )˜pαβ≈p0αβ,(6.3)where “≈” means up to Q-commutator terms, ˜p ≡Re−(φ+ ¯φ+eφ)ψis W for the k = 0 Yang-Mills field in the OSp picture, and wehave used the fact that X commutes with bW.

The Q-commutatorterms give two types of terms: When acting on a field, {Q, A}Φ =QAΦ+AQΦ, the first term is a gauge transformation and the sec-ond term is proportional to the field equations (the trivial gaugesymmetry of the form δφi = ǫijδS/δφj mentioned in section 2.1).Such trivial non-closure terms are common in component formu-lations of supersymmetric theories. If this theory had been de-rived from a manifestly supersymmetric formalism (such as a co-variantly quantized Green-Schwarz string), the first term wouldhave resulted from going to a Wess-Zumino gauge by eliminat-ing nonderivative-gauge degrees of freedom (similar to A+ or κof section 3.4), and the second term would have resulted fromeliminating auxiliary fields (similar to A−in section 3.4) by theiralgebraic equations of motion.For this closure to be maintained at the interacting level (see(2.2a,2.4c)), the interacting contributions to these two terms mustcancel [2]:{Q, A}Φ = QAΦ + AQΦ= [QAΦ + Φ ⋆(AΦ) −(AΦ) ⋆Φ] + A(QΦ + Φ ⋆Φ)⇒A(Φ ⋆Φ) = (AΦ) ⋆Φ −Φ ⋆(AΦ),(6.4)which means that the operator A must be distributive over the ⋆product, just as Q is.7.

AMPLITUDESAs an example of a conformal field theory calculation, considerthe three-Yang-Mills vertex:⟨0|Va(k1, z1)Vb(k2, z2)Vb(k3, z3)|0⟩=Zdθ1 dθ2 dθ3 ⟨0|C(Z1)C(Z2)C(Z3)|0⟩× x⟨0|Wa(k1, Z1)Wb(k2, Z2)Wc(k3, Z3)|0⟩x,(7.1)where |0⟩x is the vacuum for just the physical coordinates X.The ⟨0|CCC|0⟩matrix element can easily be evaluated, sincethe only nonvanishing contributions come from⟨0|γ1/2c0γ−1/2|0⟩= 1,(7.2)since γ−1/2|0⟩is the tachyon. From the usual expansion for a fieldof conformal weight d, ζ = Pζnz−n−d, we then have⟨0|C(Z1)C(Z2)C(Z3)|0⟩= θ1θ2z3(z1 +z2)+cyclic permutations .

(7.3)We now consider the generalization to the N-point amplitude.We need to include contributions from the string propagators: Thepropagator is (b0+f0)/L0, as for the Veneziano string, except thatfor the Veneziano string f0 is usually taken to be 0. For physicalNS vertex operators the f0 won’t contribute anyway because ofthe absence of nonminimal coordinates in these operators.

Thecalculation is similar to the Veneziano case [16]: The separationof the vertices in the complex plane, and the integration over that13

separation, accounts for the 1/L0’s. The factors of b0 + f0 appearas contour integrals of b + f surrounding certain sets of vertices,corresponding to those on either side of that propagator in thestring field theory Feynman diagram (those inside the contour andthose outside).

Since these contours can be expressed as sums ofcontours around individual vertices, we can consider without lossof generality a complex plane with such contours surrounding eachof N −3 of the N vertex insertions. (We use the fact that b0|0⟩=f0|0⟩= 0.) Each contour integral gives simply the commutator{b0 + f0, V } = W (see (5.2)).

(Note that b0 + f0 and L0 are bothintegrals over ρ = τ + iσ, but the result can be converted to thez-plane since dρ W (ρ) →dz W (z) and V (ρ) →V (z).) These stepsare also clear by direct manipulation in the operator formalism.We thus have the amplitudeA =Zdρ4 · · · dρN ⟨0|V V V (b0 + f0)V · · · (b0 + f0)V |0⟩=Zdz4 · · · dzN ⟨0|V V V W · · · W |0⟩=ZdNθ dz4 · · · dzN ⟨0|C1C2C3|0⟩· x⟨0|W1 · · · WN|0⟩x .

(7.4)All steps except the last (i.e., substitution in terms of Rdθ using(5.4)) are the same as for the Veneziano model. (A similar, but lesssimple evaluation occurs for the particle (3.20).) This calculationapplies to arbitrary physical NS vertices.

(Of course, unphysicalstates, including ghost and auxiliary states, involve more generalexpressions, including dependence on the nonmimimal fields. )The F = x⟨0|W · · · W|0⟩x matrix element is the same as thatobtained from the “old covariant” approach [17].

There the am-plitude was written asAoc = (z1 −z3)(z3 −z2)Zdθ3 dθ4 · · · dθN dN−3z F.(7.5)This amplitude is Sp(2) invariant, and invariant in particular un-der the cyclic permutation of the first three Z’s. If we plug (7.3)into (7.4), noting that the two θ’s just kill two of the θ integrals,we findA =z3(z1 + z2)(z1 −z3)(z3 −z2) + cyclic permutationsAoc = Aoc,(7.6)reproducing the old result.Except for the manifest world-sheet supersymmetry (whichholds only for the NS sector anyway), this calculation is similar tothat using the minimal coordinates [13].

Because there the OSpvacuum was used, two V0 vertices were needed, the rest were V ’s(see section 5.2; this is easily derived using the minimal analog of(5.7)). However, since the OSp vacuum satisifes ⟨OSp|ccc|OSp⟩∼1 instead of ⟨0|γγc|0⟩∼1 we obtain the same result, since the cterm (the only term) of V0 is essentially the same as the γ termof V .Using the techniques described in section 5, R vertex operatorscan be constructed and used to calculate amplitudes.

The explicitexpressions for the R vertex operators in the physical picture aremessier than in the usual, minimal formalism. (See e.g., (5.12).

)However, many of the terms do not contribute, since there mustbe a balance in the numbers of each of the different kinds of ghostfactors. On the other hand, calculations can also be performed inmixed pictures, as in the usual approach, with simple expressionssuch as (5.14).

It is clear that working with our new nonmini-mal coordinates does not offer any computational advantages tothe usual approach when applied to conformal field theory. Themain advantage in the conformal field theory is conceptual: Allstates and all vertex operators can have the same ghost number, soquantization of the NSR string requires no concepts that weren’talready understood from the Veneziano string.

(This could al-ready be done for the NS sector even in the minimal formalism. )The reason that this does not simplify the treatment of the R sec-tor is that the conformal field theory approach requires that statesand vertex operators be expressed with respect to the NS vacuum.On the other hand, in the string field theory approach the NS andR strings can have their own separate vacuua, avoiding this prob-lem, so the treatment of the R string field is significantly simplerthan in the usual, minimal treatment.8.

NSR SUPERSTRING FIELD THEORYIn the string field theory, the separate treatment of the NSand R strings, together with the use of the same “picture” (i.e.,ghost number) for these two strings, allows the avoidance of bothbosonization and δ-functions of ghosts. This means that vertexoperators have simple forms, such as (5.4,5) for the NS string, and(3.13) for the massless spinor.

In the old formulation bosonizationand ghost δ-functions were avoidable in some terms in the action,but not for the R kinetic term.As described above, the gauge invariant kinetic terms for boththe NS and R string fields are just ⟨Φ|QΦ⟩, as in string field the-ory for the Veneziano string. This agrees with Witten’s originalversion of NSR superstring field theory [2] in the NS sector, exceptthat Q now has the nonminimal term.

Of course, this new Q is thesame one in both the NS and R sectors, except that the NS and Rfields satisfy different boundary conditions (or, equivalently, themode expansion of Q differs).The expression for the (NS)3 vertex is also the same as in Wit-ten’s version up to nonminimal stuff, since the NS sector neverreally had a picture-changing problem. (The apparent problemin the conformal field theory was due to the choice of a vacuumwhich did not correspond to any state in the string field theory.

)This means that there is still an operator insertion at the vertex,14

but it is not related to ghost-number problems: In fact, this ver-tex insertion, although called the “picture-changing operator,” hasdependence on non-ghost coordinates, in contrast to the so-called“inverse picture-changing operator” that appeared in the old ver-sions of the R-string kinetic operator [18,2], which depended onlyon ghosts.We therefore have a situation similar to light-conestring field theory [19], where the kinetic operators are both L0,and both vertices have a vertex insertion, similar to the one here.However, because the R-string kinetic operator is now thesame as for the NS string, the two vertices, (NS)3 and NS(R)2,are also the same. We can therefore effectively write the NS andR string fields collectively as a single field.Also, unlike previ-ous treatments of NSR superstring field theory, the same operatorperforms supersymmetry transformations on the NS and R stringfields, correctly changing the boundary conditions in both cases.The explicit expressions for these two vertices appearing inthe action (2.2) are therefore similar to those used for the (NS)3vertex in the minimal case [20-22].

Each vertex is defined as theproduct of the “naive” one with the factor iXX, inserted at the in-teraction point (midpoint). The “naive” one is that obtained e.g.,by functionally integrating eS over the infinitesimal strip connect-ing the 3 strings, and thus includes not only δ-functionals of thecoordinates but also eφ-type factors coming from the Rφ termsin the first-quantized action for the bosonized ghosts.

In termsof analytic variables, there are effectively two interaction points(σ = π/2 or −π/2), so the interaction vertex can be defined tohave X inserted at either point, and the same for X (so X and Xneed not be at the same point). In practice the simplest methodof analysis is that of Suehiro [22]: The 2D Green functions for thephysical variables and unbosonized ghosts, which define the vertexoperator, are evaluated with appropriate boundary conditions atboth the positions zr of the external strings and the interactionpoint z0 (and its mirror image ¯z0).

This means that the ρ-planeGreen functions are expressed in terms of z as just 1/(z−z′) timesappropriate powers of z −z0, z −¯z0, and z −zr (and of similarfactors with z′). In particular, the boundary conditions at z0 arechosen to absorb the eφ factors from X and X, so the remain-ing insertion at the interaction point is an expression of the formψ·p+· · ·, similar to bG of (5.8b), which can be expressed directly interms of the unbosonized ghosts.

This remaining insertion is alsoexpressed in terms of the Green functions. The mode expansionof the Green functions is then performed as by Gross and Jevicki[20].It is also interesting to analyze vertices which correspond toendpoint interactions, instead of midpoint ones.

Although suchinteractions do not give the correct field theory, they are equal tothe correct midpoint ones on the (free) mass shell. Such interac-tions occur on the real axis (the string boundary) in the z-plane,z0 = ¯z0, so there is a simplification in both the Green functionboundary conditions and the vertex insertion.

There is also a sim-plification in interpretation, since the Jacobian factors ∂ρ/∂z ∼p(z −z0)(z −¯z0)/ Q(z −zr) [16] simplify to (z −z0)/ Q(z −zr).The result is that the ghosts can be considered to have the sameconformal weights as the corresponding physical fields (c and bthe same as x and p; β, γ, ˜β, ˜γ, χ, and µ the same as ψ), leavinga vertex insertion of c bG. The result is covariant with respect toOSp(10,2|4) (or OSp(9,1|2) in the minimal case [14]), which shouldhave been expected since the endpoint interaction can be deriveddirectly by adding extra dimensions to the light-cone result [7],which also uses endpoint interactions.Just as the interaction term can be expressed as ⟨V3|Φ⟩|Φ⟩|Φ⟩,the supersymmetry transformation can be expressed as ⟨q|Φ⟩|Φ⟩,where one Φ is an R string and one an NS string.

This ⟨q| actuallyrepresents the operator ˆq of (6.1), and its mode expansion can becalculated in the same manner as the 3-string vertices, withoutbosonization or ghost δ-functions. The ω factor which appearedin q in (6.1) arises from the reality condition: The reality condi-tion on the string field can be expressed as ⟨Φ|Φ′⟩= ⟨I2|Φ⟩|Φ′⟩[23], where ⟨I2| has an additional factor of ω compared to thenaive value.

This is because the naive expression is just the usualtwist operator, expressed functionally as δ[ζ1(σ) −eiπdζ2(π −σ)]in terms of δ-functionals of coordinates ζ of conformal weight dbetween strings 1 and 2. However, all the nonminimal coordinateshave been assigned conformal weights 12 ±1, whereas their naturalweights are 12.

(A conformal weight of 12 requires no eφ insertions,since there are then no Rφ terms in the first-quantized action. )Equivalently, the OSp formalism discussed above, which holds forthe kinetic terms in all cases, gives the nonminimal coordinatesthe same weight12 as for ψ.

The ω factor compensates for thisby changing the relative sign of ζ1 and ζ2 for the nonminimalcoordinates, fixing the eiπd factor.Because the (NS)3 and NS(R)2 interaction terms now bothrequire an operator insertion, the usual contact-term divergences[24] are present in all four(or more)-string NSR scattering am-plitudes. This situation is similar to that of the non-supersheetlight-cone formalism, where operator insertions are also necessaryfor both (NS)3 and NS(R)2 interactions.

Other authors [25] havetried to eliminate these contact-term divergences in the string fieldtheory by considering kinetic terms of the form ⟨Φ|ZQ Φ⟩, whereZ is a BRST-invariant operator inserted at the midpoint. How-ever, if one restricts Φ to satisfy the usual boundary conditions(i.e., it is constructed out of the ground state with a finite set ofoscillator modes), it is not possible to gauge fix these types of ki-netic terms to the simple form L0 [26].

To be more specific, gaugetransformations of the type Φ →Φ + B for ZB = 0 are not al-lowed since B cannot be constructed out of the ground state witha finite set of oscillator modes. Without the use of these gaugetransformations, it is impossible to define a propagator for such15

formulations which avoids the divergences in the four-string am-plitudes offshell. On the other hand, if one were to allow arbitraryboundary conditions for Φ, the ⋆product would lose its associa-tivity, making the interaction term ill-defined.

At this point, it isunclear if some “compromise” set of boundary conditions can befound which would allow the gauge transformations necessary forsuch a formalism, but still preserve the associativity property ofthe ⋆product.9. CONCLUSIONSIn this paper we have given a new formulation of NSR super-string theory which avoids picture changing.

The main differenceis the use of nonminimal fields which solve the bosonic zero-modeproblem of the R sector, allowing the construction of the Hilbertspace and BRST formalism in a way more similar to that of theVeneziano string. In particular, in the string field theory the Ra-mond string is then treated in exactly the same way as the Neveu-Schwarz string except for boundary conditions.

In the conformalfield theory, we choose as the new vacuum the unique string fieldtheory state in the BRST cohomology with ghost number −32.As a result, vertex operators are unique, as opposed to the oldapproach, where picture changing was required even for the NSsector.It might be interesting to see if such methods would suggesta method for obtaining a covariant Green-Schwarz formulation.Since the GS and NSR formalisms are directly related in the lightcone by triality, a covariant form of triality might appear if an ap-propriate set of ghosts (that which closes supersymmetry offshell)is chosen. With regard to the OSp analysis of section 3, it shouldbe noted that the set of auxiliary scalars agreeing with those ob-tained by superfield methods for N=1 supersymmetry in variousdimensions is given by OSp(2,1|2) in D=3, OSp(4,2|4) in D=4,and OSp(8,4|8) in D=6, suggesting that perhaps OSp(16,8|16)might be appropriate for D=10 and thus for superstrings.

On theother hand, the SO(D,2) “physical” submultiplets occurring inthe present formalism are strongly suggestive of broken conformalinvariance, as commonly used in supergravity.ACKNOWLEDGMENTN.B. thanks Christian Preitschopf for useful discussions.M.T.H.

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