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PICTURE CHANGING OPERATORS IN

arXiv:hep-th/9202087v1 26 Feb 1992TIFR-TH-92/14February, 1992PICTURE CHANGING OPERATORS INCLOSED FERMIONIC STRING FIELD THEORYR. Saroja and Ashoke Sen⋆Tata Institute of Fundamental Research, HomiBhabha Road, Bombay 400005, IndiaABSTRACTWe discuss appropriate arrangement of picture changing operators requiredto construct gauge invariant interaction vertices involving Neveu-Schwarz statesin heterotic and closed superstring field theory.

The operators required for thispurpose are shown to satisfy a set of descent equations.⋆e-mail addresses: RSAROJA@TIFRVAX.BITNET, SEN@TIFRVAX.BITNET1

A closed bosonic string field theory based on non-polynomial interaction hasbeen constructed recently [1 −4] . In this paper we shall discuss the constructionof gauge invariant field theory for heterotic and type II superstring theories basedon the same principles.

However, our analysis will be confined only to the Neveu-Schwarz sector of the theory; the construction of a similar field theory includingRamond sector states involves extra complications due to problems involving zeromodes of various fields.Although the field theory for fermionic strings is notcomplete without Ramond sector, we can get some insight into the theory just bylooking at the field theory involving the Neveu-Schwarz states. In particular, sincefor the heterotic string theory, all the bosonic fields come from the Neveu-Schwarzsector, the field theory involving Neveu-Schwarz states can be used to study thespace of classical solutions, as well as the geometry of the configuration space ofthe theory.We shall, for convenience, restrict our discussion to heterotic string theory only,the analysis for superstring theory proceeds in a similar manner.

Let H denote theHilbert space of states in the Neveu-Schwarz sector in the −1 picture. These arethe states created by the matter operators, and the ghost oscillators bn, cn, βn, γnacting on the state e−φ(0)|0⟩≡|Ω⟩, where φ is the bosonized ghost, related to β,γ through the relations β = e−φ∂ξ, γ = eφη.

Here η and ξ are fermionic fields ofdimensions 1 and 0 respectively. |0⟩denotes the SL(2,C) invariant vacuum in thecombined matter ghost theory.

As in the case of bosonic string theory, a generaloff-shell string state |Ψ⟩is taken to be a GSO projected state in H annihilated byc−0 ≡(c0 −¯c0)/√2 and L−0 ≡(L0 −¯L0)/√2, and created from |Ω⟩by an operatorof total ghost number 3. Following the construction of bosonic string field theory,we shall look for an action of the heterotic string field theory of the form:S(Ψ) = 12⟨Ψ|QBb−0 |Ψ⟩+∞XN=3gN−2N!

{ΨN}(1)where {A1 . .

. AN} and [A1 .

. .

AN] are multilinear maps from N-fold tensor productof H to C (the space of complex numbers) and H respectively, satisfying relations2

identical to the corresponding relations in bosonic string field theory:{A1 . .

. AN} ≡(−1)n1+1⟨A1|[A2 .

. .

AN]⟩(2){A1 . .

. AN} = (−1)(ni+1)(ni+1+1){A1 .

. .

Ai−1Ai+1AiAi+2 . .

. AN}(3){A1 .

. .

(L−0 Ai) . .

. AN} = 0 = {A1 .

. .

(b−0 Ai) . .

. AN}(4)(−1)n1{(QBA1)A2 .

. .

AN} =NXi=2(−1)Pi−1j=2(nj+1){A1 . .

. (QBAi) .

. .

AN}−Xm,n≥3m+n=N+2X{jk},{il}(−1)σ({il},{jk}){A1Aj1 . .

. Ajm−2c−0 [Ai1 .

. .

Ain−1]}(5)Here ni denotes the ghost number of the state |Ai⟩. The sum over {il}, {jk} ineq.

(5) runs over all possible divisions of the set of integers 2, . .

. N into the sets {il}and {jk}.

(−1)σ({il},{jk}) is a factor of ±1 which is computed as follows. Startingfrom the ordering QB, A2, .

. .

AN, bring them into the order Aj1, . .

. Ajm−2, QB, Ai1, .

. .

Ain−1,using the rule that QB is anticommuting, and Ai is commuting (anti-commuting)if ni is odd (even). The sign picked up during this rearrangement is (−1)σ({il},{jk}).Using eqs.

(2)-(5) and the nilpotence of the BRST charge QB, one can show thatS(Ψ) given in eq. (1) is invariant under a gauge transformation of the form:b−0 δ|Ψ⟩= QBb−0 |Λ⟩+∞XN=3gN−2(N −2)!

[ΨN−2Λ](6)where |Λ⟩is an arbitrary GSO projected state in H created from |Ω⟩by an operatorof ghost number 2, and annihilated by c−0 and L−0 .Thus, in order to construct a gauge invariant action, we need to constructmultilinear maps {A1 . .

. AN} from HN to C, satisfying eqs.(2)-(5).

We look for3

an expression similar to the one in bosonic string theory [3] [5]:{A1 . .

. AN} = −ZR(N)d2N−6τ⟨f(N)1◦(b−0 PA1)(0) .

. .

f(N)N◦(b−0 PAN)(0)KN⟩(7)Here P is the projection operator δL0 ¯L0. f(N)iis a conformal map that maps theunit circle into the ith external string of the N-string diagram associated withthe N-string vertex.

f(N)i◦(b−0 PAi) denotes the conformal transform of the field(b−0 PAi) under the map f(N)i. τi are the modular parameters characterizing theN-string diagram. The region of integration R(N), and the N-string diagram ischosen in such a way that at the boundary ∂R(N) of R(N), the N-string diagram isidentical to an m string diagram and an n(= N +2−m) string diagram glued by atube of zero length and twist θ.

One particular example of such N-string diagramsis provided by polyhedra with N-faces, the perimeter of each face being equal to2π. In this case τi correspond to independent parameters labelling the lengths ofeach side of the polyhedron.

The region of integration R(N) over τ is such that itincludes all such polyhedra with the restriction that the length of any closed curveon the polyhedron constructed out of the edges should be greater than or equal to2π. (Such polyhedra were called regular polyhedra in ref.

[2]).Finally we turn to the description of the operator KN appearing in eq.(7). Forbosonic string field theory KN was given by Q2N−6i=1(ηi|B), where(ηi|B) =Zd2z(ηzi¯z b(z) + ¯η¯ziz ¯b(¯z))(8)ηzi¯z , ¯η¯zizare the beltrami differentials, which tell us how the components gzz,g¯z¯z of the metric, induced on the sphere by the N-string diagram, changes as wechange τi [3].

In the case of fermionic string theories, such a choice of KN will givevanishing answer for {A1 . .

. AN} due to ghost number non-conservation.

This isremedied by introducing appropriate factors of picture changing operators [6 −8]4

in the definition of KN [9]. We shall make the following choice of KN:KN =2N−6Xr=0C(r)N ∧Φ(2N−6−r)N(9)where C(r)Nis an r-form (in the 2N −6 dimensional moduli space spanned by τi)operator, given by,C(r)Ni1...ir =rYk=1(ηik|B)(10)Φ(r)N , on the other hand, is an r form operator satisfying the ‘descent’ equation:d(E)Φ(r)N = [QB, Φ(r+1)N}(11)where d(E) denotes derivative with respect to τi, acting on the explicit τ dependentfactors in the expressions for Φ(r)N , and [ } denotes a commutator (anti-commutator)for odd (even) r in eq.(11).

Φ(0)N takes the form:Φ(0)N =XαA(α)(τ1, . .

. τ2N−6)X(w(α)1 (τ)) .

. .

X(w(α)N−2(τ))(12)where X(z) = {QB, ξ(z)} is the picture changing operator [6].The sum overα in eq. (12) runs over a finite set of values.

For each value of α we have a set ofpoints w(α)1 (τ), . .

. w(α)N−2(τ) on the N-string diagram, and an weight factor A(α)(τ),satisfying the normalization condition Pα A(α)(τ) = 1.

Φ(r)N ’s, on the other hand,are constructed as linear combinations of products of X’s, ∂ξ’s and (ξ(P)−ξ(Q))’s.⋆Besides the descent equation (11), Φ(r)N are also required to satisfy the followingboundary conditions. As has been indicated before, the boundary ∂R(N) of R(N)consists of several pieces; each piece corresponds to gluing two vertices with less⋆Appearance of extra factors proportional to ∂ξ when the locations of the picture changingoperators are moduli dependent, was discussed in ref.

[10].5

number of external states (m and n = N + 2 −m, say) along one of the exter-nal strings from each vertex with a certain twist θ. We demand that on such acomponent of ∂R(N),Φ(r)N∂R(N) =rXs=0Φ(r−s)m∧Φ(s)n(13)Here Φ(r)N∂R(N) denotes the component of Φ(r)N tangential to ∂R(N).

Finally, Φ(r)Nshould be invariant under permutation of the external strings. Besides these re-strictions, the choice of Φ(r)N (i.e.

the quantities A(α), w(α)i, and the correspondingquantities appearing in the expressions for Φ(r)N for r ≥1) is completely arbitrary.†We shall now show that the quantities {A1 . .

. AN} constructed this way arenon-zero in general, and satisfy eqs.

(3)-(5) with [A1 . .

. AN−1] defined througheq.(2).

To see that {A1 . .

. AN} is non-zero in general, we only need to note thatthe contribution to the right hand side of eq.

(9) from the r = 2N −6 term is ofthe form2N−6Yi=0(ηi|B)XαA(α)(τ)X(w(α)1 ) . .

. X(w(α)N−2)(14)Besides providing the appropriate factors of (ηi|B) as in the case of bosonic stringtheory, we now also have the correct number of picture changing operators.

Thus,if each |Ai⟩is created by a ghost number 3 operator acting on |Ω⟩, at least thecontribution from the r = 2N −6 term in eq. (9) to {A1 .

. .

AN} will be non-zero.As we shall now see, the other terms are necessary for {A1 . .

. AN} to satisfy eq.

(5).Verification of eq. (2)-(4) with the definition of {A1 .

. .

AN} given in eq. (7) isstraightforward, so we turn to the verification of eq.(5).

Using eq. (7), both sides ofeq.

(5) may be expressed as integrals (over τi) of appropriate correlation functionsin the conformal field theory.In the left hand side of eq. (5), we may express†Although we shall try to choose Φ(r)N in such a way that they are continuous inside R(N),this is not a necessary constraint, since, as can be seen from eq.

(11), a discontinuity of Φ(r)Ninside R(N) may be compensated by a δ-function singularity in Φ(r+1)N.6

(QBA1) as the contour integral of the BRST current around the location of theoperator b−0 A1. We may now deform the BRST contour and shrink it to a point, inthe process picking up residues from the locations of various other operators.

Theresidues from the locations of the operators b−0 Ai (i ≥2) give rise to the first set ofterms on the right hand side of eq.(5). We are now left with terms proportional to[QB, KN].

Residues from the terms proportional to b(z), ¯b(¯z) in (ηi|B) generatesa factor ofRd2z(ηzi¯z T(z) + ¯η¯ziz ¯T(¯z)) ≡T (1)i. Thus we may write,[QB, C(r)N } = T (1) ∧C(r−1)N(15)where C(r)N has been defined in eq.(10).

Since insertion of a T (1)iinside a correlationfunction generates the τi derivative of the correlation function, with the derivativeacting on the implicit τi dependence of the correlation function due to the depen-dence of the string diagram on τ, we see that the terms proportional to [QB, C(r)N }in [QB, KN] give rise to a term of the form:ZR(N)d2N−6τd(I)B(16)where,B = (−1)n1⟨NYi=1(f(N)i◦b−0 Ai(0))2N−6Xr=1C(r−1)N∧Φ(2N−6−r)N⟩(17)and d(I) denotes the τ derivative acting on the implicit τ dependence of the cor-relation function but not on the explicit τ dependence appearing in Φ(r)N . On theother hand, terms proportional to [QB, Φ(r)N } in [QB, KN] may be analyzed usingthe descent equation (11), and gives an answer similar to eq.

(16) with d(I) replacedby d(E). Thus if d ≡d(I) + d(E) denotes the total τ derivative, the contributionfrom the term proportional to [QB, KN} that appears in the analysis of the left7

hand side of eq. (5) may be written as,ZR(N)d2N−6τdB =Z∂R(N)d2N−7τB(18)Let us now consider a specific component of the boundary ∂R(N) of R(N) thatcorresponds to gluing of two lower order string diagrams of m and n = N + 2 −m external states, with a twist θ.

Let τ(1) and τ(2) be the modular parametersdescribing these lower order string diagrams. Then,d2N−7τ∂R(N) = d2m−6τ(1)d2n−6τ(2)dθ(19)C(r)Nθa1...ar−1∂R(N) = (ηθ|B) r−1Xs=0C(s)m ∧C(r−s−1)na1...ar−1(20)Using the boundary conditions given in eqs.

(13) and (20) it is easy to see that, 2N−6Xr=1C(r−1)N∧Φ(2N−6−r)Nθτ 1(1)...τ 2m−6(1)τ 1(2)...τ 2n−6(2)∂R(N)=(ηθ|B) 2m−6Xr=0C(r)m ∧Φ(2m−6−r)mτ 1(1)...τ 2m−6(1) 2n−6Xs=0C(s)n∧Φ(2n−6−s)nτ 1(2)...τ 2n−6(2)(21)Standard manipulations identical to the one for the bosonic string theory can nowbe used to show that the contribution to the right hand side of eq. (18) is identicalto the second set of terms to the right hand side of eq.(5).

This completes theproof of eq. (5).We shall now indicate the basic steps involved in the calculation of Feynmanamplitudes in this field theory.

The external states are taken to be physical statesof the form b−0 |Ai⟩= c¯cVi(0)|Ω⟩≡c¯c ˆVi(0)|0⟩where Vi is a dimension (1/2, 1)superconformal primary field in the matter sector. Since the contribution from8

various Feynman diagrams can be brought into the form of the contribution fromelementary N-point vertex given in eq. (7) using standard techniques [11], we shallanalyze the contribution to A(1, .

. .

N) from the elementary N-point vertex only.In analyzing this contribution we shall use the familiar expression for ηzi¯z , ¯η¯zizinterms of quasi conformal deformations [12] vz, ¯v¯z, and write,(ηi|B) =IC(dzδvzδτi b(z) + d¯zδ¯v¯zδτi ¯b(¯z))(22)where the contour C encloses all the points zi, as well as the locations of all theoperators appearing in ˆΦ(r)N . δvz (δ¯v¯z) is analytic (anti-analytic) inside the contourC but not outside C. We can now deform C and shrink it to a point, picking upresidues from the locations of various operators in this process.In the case ofbosonic string theory, the only possible residues are picked up from the locationsof the vertex operators b−0 Ai = c¯c ˆVi.This removes the c¯c factors from N −3of the vertex operators and generates appropriate measure factors that convertsthe integration over τi to integration over the locations of the (N −3) vertices.In the present case, however, the integration contour can pick up residues fromthe locations of the picture changing operators inside Φ(r)N also.

For any r formoperator O, let us define an r + 1 form operator δO as,(δO)i1...ir+1 = A ICdz δvzδτi1 b(z) + d¯z δ¯v¯zδτi1 ¯b(¯z)Oi2...ir+1(23)where A denotes antisymmetrization in the indices i1, . .

. ir+1, and C denotes acontour enclosing the locations of all the operators in O.

Using eqs. (7)-(10) thecontribution to A(1, .

. .

N) from the elementary N-point vertex may be written as,gN−2ZR(N)d2N−6τ2N−6Xr=0rXs=01s! (r −s)!Dδs NYk=1c¯c ˆVk(zk, ¯zk)∧δr−sΦ(2N−6−r)NE(24)where δs denotes s successive operations of δ and zi = fNi (0).9

So far our description has been independent of the choice of coordinates of themoduli space. We can now simplify our analysis by choosing the moduli parametersτi to be identical to the coordinates xi (1 ≤i ≤2N −6) defined as follows:xi = z i+12for i odd;xi = ¯z i2for i even(25)In that case, for odd i,δvzδτi =1 at z = z i+12 ;δvzδτi = 0 at z = zj for j ̸= i + 12δ¯v¯zδτi =0 at z = zj for all j(26)and, for even i,δvzδτi =0 at z = zj for all jδ¯v¯zδτi =1 at z = z i2;δ¯v¯zδτi = 0 at z = zj for j ̸= i2(27)Let us now define,ˆΦ(s)N =sXr=01r!δrΦ(s−r)N(28)and,L(s)N = 1s!δs NYk=1c¯c ˆVk)(29)so that,L(2N−6−r)Ni1...i2N−6−r =δδCi1.

. .δδCi2N−6−rNYi=1c¯c ˆVi(zi, ¯zi)(30)where,Cj = c(z j+12 )for j odd;Cj = ¯c(¯z j2)for j even(31)10

Then eq. (24) takes the form:A(1, .

. .

N) = gN−2Zd2N−6xi2N−6Xr=0⟨L(r)N ∧ˆΦ(2N−6−r)N⟩(32)By techniques identical to those used in the case of bosonic string field theory onecan show that the integral over xi runs over the full moduli space when we addthe contribution from all the Feynman diagrams.Using the relations {QB, b(z)} = T(z), {QB,¯b(¯z)} = ¯T(¯z), the fact that theinsertion ofRd2z(ηzi¯z T(z) + ¯η¯ziz ¯T(¯z)) in a correlation function generates the (im-plicit) derivative of the correlation function with respect to τi, and the descentequation (11) we get,[QB, δrΦ(l)N } = rd(I)(δr−1Φ(l)N ) + d(E)(δrΦ(l−1)N)(33)Using eq. (33) one can easily verify that ˆΦ(r)N defined in eq.

(28) satisfies the descentequation:dˆΦ(r)N = [QB, ˆΦ(r+1)N}(34)where d denotes total derivative with respect to the xi’s.We now turn to the question of comparing the amplitude given in eq. (32) withthe one calculated from the first quantized formalism.

In order to do so we shallfirst prove the followingLemma: If we have two sets of operators ˆΦ(r)N and ˆΨ(r)N , both satisfying the descentequations given in eq. (34), and, if,ˆΦ(0)N −ˆΨ(0)N = {QB, χ(0)N }(35)for some χ(0)N , then,Zd2N−6xi2N−6Xr=0⟨L(r)N ∧(ˆΦ(2N−6−r)N−ˆΨ(2N−6−r)N)⟩= 0(36)11

The lemma is proved in the following way.From eq. (35) and the descentequation for ˆΦ(r)N , ˆΨ(r)N , we get,{QB, ˆΦ(1)N −ˆΨ(1)N } = {QB, dχ(0)N }(37)This gives,ˆΦ(1)N −ˆΨ(1)N = dχ(0)N + [QB, χ(1)N ](38)for some χ(1)N .

Repeating this, we get the general equation,ˆΦ(r)N −ˆΨ(r)N = dχ(r−1)N+ [QB, χ(r)N }(39)Finally, using the definition of L(r)N and the commutation relations,[QB, ˆVi(zi, ¯zi)] = ∂∂zic ˆVi(zi, ¯zi)+ ∂∂¯zi¯c ˆVi(zi, ¯zi){QB, c ˆVi(zi, ¯zi)} = ∂∂¯zi(¯cc ˆVi),{QB, ¯c ˆVi} = ∂∂zi(c¯c ˆVi),[QB, c¯c ˆVi] = 0(40)we get,[QB, L(r)n } = dL(r−1)N(41)Using eqs. (39), (41), and the fact that the expectation value of a BRST exactoperator vanishes, we can bring the left hand side of eq.

(36) into the form:Zd2N−6xid 2N−7Xr=0L(r)N ∧χ(2N−7−r)N(−1)r(42)which vanishes after integration over xi. This completes the proof of the lemma.Let us now choose,ˆΨ(0)N = X(z3)X(z4) .

. .

X(zN)(43)Using the fact that ˆΦ(0)Nhas the form Pα ˜A(α)({xi})X(w(α)1 ) . .

. X(w(α)N−2) withPα ˜A(α) = 1, we see that ˆΦ(0)N −ˆΨ(0)N is BRST exact.

Thus by the above lemma,12

we can replace ˆΦ(r)N by ˆΨ(r)N in the expression for A(1, . .

. N) given in eq.(32).

Aset of ˆΨ(r)N satisfying the descent equation are given by,ˆΨ(r)Ni1...ir =rYk=1∂ξ(z ik+12 )NYj=3j̸= il+12X(zj) for ik odd and 3 ≤ik + 12≤N −3 for all k=0 otherwise(44)Using this we get the following expression for A(1, . .

. N):A(1, .

. .

N) =gN−2ZN−3Yi=1d2zi2Yi=1ˆVi(zi, ¯zi)NYi=N−2(c¯c ˆVi(zi, ¯zi)X(zi))N−3Yi=3ˆVi(zi, ¯zi)(X(zi) −c(zi)∂ξ(zi))(45)where d2zi ≡d¯zi ∧dzi. In the above equation, ˆVi(zi, ¯zi) corresponds to integratedvertex operator in the −1 picture [6].

Also,ˆVi(zi, ¯zi)(X(zi)−c(zi)∂ξ(zi)) = {QB, ˆVi(zi, ¯zi)ξ(zi)}−∂ ˆVi(zi, ¯zi)c(zi)ξ(zi)−¯∂ ˆVi(zi, ¯zi)¯c(zi)ξ(zi)(46)correspond to integrated vertex operators in the zero picture. Finally, ˆVi(zi, ¯zi)c(zi)¯c(¯zi)X(zi)correspond to unintegrated vertex operators in the zero picture.

Thus we see thatthe right hand side of eq. (45) has precisely the form expected from the analysis ofthe first quantized theory.We now turn to the problem of determining Φ(r)N satisfying eqs.

(11) and (13).We shall discuss one particular construction, but one should keep in mind thatthis construction is in no way unique, and there are (infinitely) many other choicespossible. In order to prescribe Φ(r)N in a way that avoids the divergences associatedwith collision of picture changing operators [13],⋆we shall find it more convenient⋆Removal of such divergences in open string field theory has been discuss by several au-thors [14 −16]13

to take the N-string diagram not just a regular polyhedra with N faces, but regularpolyhedra with N faces with tubes of a certain fixed length l0 attached to eachof the faces; together with diagrams corresponding to such regular polyhedra withm1, m2, . .

., mr+1 = N + 2r −(m1 + . .

. + mr) faces, joined by tubes of lengthl1, .

. .

lr (0 ≤li ≤2l0) and twist θ1, . .

. θr (0 ≤θi < 2π).

(Such vertices have beenused in ref. [17] for a different purpose.) We shall denote by R(N)ethe componentof R(N) for which the corresponding string diagram is a regular N-hedron withtubes of length l0 attached to its faces.

R(N)cwill denote the component of R(N)for which the corresponding string diagram is given by two or more such regularpolyhedra connected by tubes of length ≤2l0.We shall first discuss the construction of Φ(0)N for points inside R(N)e. In thiscase the picture changing operators are taken to be at the mid-points of the edges ofthe polyhedron. Since the number of edges (3N −6) of an N-hedron is larger thanthe number of picture changing operators (N −2), there are several possibilities.We average over all configurations, with the weight factor for a given configurationbeing proportional to Qi f(si), where si is the length of the ith edge, the productover i runs over all the (N −2) edges containing the picture changing operators,and f(si) is a smooth function of si satisfying the constraint:f(si) = 0 for si ≤η;f(si) = 1 for si ≥2η(47)where η is a small but fixed number.

This construction guarantees that the picturechanging operators are always inserted at the mid-points of the edges which havelength ≥η, and hence two picture changing operators never collide inside R(N)e.†This completely specifies Φ(0)N inside R(N)e.†Note that we could also have, in principle, chosen the picture changing operators at thevertices, imitating the corresponding construction for open string field theory. But in thiscase special care (like moving the picture changing operators away from the vertices) isneeded to ensure that two picture changing operators do not collide, since for N ≥5, R(N)econtains configurations where the number of well separated vertices (say, by a distance≥η) is less than N −2.

Such a situation does not occur if we insert the picture changingoperators at the midpoints of the edges, since a regular N-hedron contains at least N edgesof finite length.14

Let us now turn to a point inside R(N)cwhich corresponds to two regular poly-hedra joined by a tube of length l. Let h be some fixed length ≤l0. For l ≥2h,we choose Φ(0)N to be simply the product of Φ(0)N on the two polyhedra.

This auto-matically ensures that Φ(0)N satisfies the boundary condition (13) at the boundaryl = 2l0 of R(N). For l ≤h, we choose the picture changing operators to be on theedges of the two polyhedra in such a way that Φ(0)N is independent of l and is iden-tical to its value at l = 0, where it is given by the previous construction of Φ(0)N forpoints inside R(N)e. This does not specify Φ(0)N completely, since a picture changingoperator inserted on the right boundary of the tube or the left boundary of thetube at the same angle generates the same configuration when the tube length iscollapsed to zero.

We remove this ambiguity by assigning equal weight factor toeach of these configurations. Finally, in the region h ≤l ≤2h, Φ(0)N is chosen tobe a linear combination of the expressions for Φ(0)N for l ≤h and l ≥2h such thatthe resulting expression smoothly interpolates between the values of Φ(0)N for l ≤hand l ≥2h.This construction can easily be generalized to points inside R(N)crepresentingstring diagrams where more than two polyhedra are joined together by tubes oflength ≤2l0.

Ambiguities similar to the one discussed above arises in the regionwhen some of the tubes are of length ≤h, since many different segments of edgesbelonging to different polyhedra may correspond to the same segment when alltubes of length ≤h is collapsed.Hence if the polyhedron obtained after thiscollapse has an insertion of picture changing operator on this segment, we needto decide how to distribute it over the various segments before the collapse. Onepossible consistent way is to distribute it only among the two segments which areat the two extreme ends of the chain of tubes connecting these different segments,with equal weight factor.This finishes the discussion on the construction of Φ(0)N .

Once Φ(0)Nis given,Φ(r)N may be obtained by solving the descent equations. We shall not discuss thegeneral case here, but as an example, give a specific solution for Φ(r)4 .

In this case15

Φ(0)4may be expressed asΦ(0)4= X(P)X(Q) +Xαf(α)(τ)X(P (α))(X(Q(α)) −X(R(α)))(48)where the sum over α runs over a finite set. The corresponding solution for Φ(r)4is,Φ(1)4=(dξ(P)X(Q) + X(P)dξ(Q)) +Xαdf(α)X(P (α))(ξ(Q(α)) −ξ(R(α)))+Xαf(α)dξ(P (α))X(Q(α)) −X(R(α))+ X(P (α))dξ(Q(α)) −dξ(R(α))Φ(2)4=dξ(P) ∧dξ(Q) −Xαdf(α) ∧dξ(P (α))(ξ(Q(α)) −ξ(R(α)))+Xαf(α)dξ(P (α)) ∧dξ(Q(α)) −dξ(R(α))Φ(r)4=0for r ≥3(49)where dξ(P) = ∂ξ(P)dP etc.To summarize, in this paper we have discussed the construction of a gaugeinvariant field theory for the Neveu-Schwarz (NS) sector of the heterotic stringtheory.

This construction can easily be generalized to include the NS-NS sector ofclosed superstring field theory as well. The Ramond sector, however, suffers fromextra problems due to the presence of zero modes of various operators, and cannot,at present, be treated by the same method.

We hope to come back to this questionin the future.16

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