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On the Background Independence of String Field Theory: III. Explicit Field Redefinitions

Given two conformal field theories related to each other by a marginal perturbation, and string field theories constructed around such backgrounds, we show how to construct explicit redefinition of string fields which relate these two string field theories. The analysis is carried out completely for quadratic and cubic terms in the action. Although a general proof of existence of field redefinitions which relate higher point vertices is not given, specific examples are discussed. Equivalence of string field theories formulated around two conformal field theories which are not close to each other, but are related to each other by a series of marginal deformations, is also discussed. The analysis can also be applied to study the equivalence of different formulation of string field theories around the same background.


INTRODUCTION
A complete formulation of closed bosonic string field theory has been given in the last few years in terms of non-polynomial interactions [1] [2] (see also ref. [3]), and quantization of this string field theory has also been carried out [4 − 8] . Such a field theory can be formulated not only in the background of flat space-time, but also in the background of any arbitrary conformal field theory [9] following the formulation of refs. [10] [11]. The natural question to ask is whether the string field theory is background independent, i.e. given the string field theory action around two different conformal field theory backgrounds, whether we can find a redefinition of string fields which relates these two actions.
The question was partially answered in two previous papers [12] [13]. In these papers we studied the relationship between string field theories formulated around two neighbouring conformal field theories, − CFT and CFT ′′ , − related by a marginal perturbation. Let Ψ andΨ denote the string fields corresponding to string field theories formulated around CFT and CFT ′′ respectively, and S(Ψ) and S(Ψ) be the corresponding string field theory actions. It was shown in ref. [12] that there is a classical solution Ψ cl of the equations of motion (∂S/∂Ψ) = 0 such that if we defineΨ = Ψ − Ψ cl andŜ(Ψ) = S(Ψ) − S(Ψ cl ), then the kinetic operator ofŜ(Ψ) is related to the kinetic operator ofS(Ψ) by a similarity transformation.
Furthermore, the linearized form of the gauge transformation ofΨ is also related to that ofΨ by the same similarity transformation. The analysis was carried out to the first order in the perturbation which relates CFT and CFT ′′ . In other words, ref. [12] established the equivalence between the quadratic terms in the actionŝ S(Ψ) andS(Ψ). On the other hand, in ref. [13] we analyzed the physical S-matrix elements in the two theories described by the actionsŜ(Ψ) andS(Ψ), and showed that they are the same. The analysis was carried out for all three point amplitudes and all N-point tachyonic amplitudes.
In this paper, we shall show with the help of the results of refs. [12] and [13] that one can construct explicit redefinition of string fields which converts the action S (Ψ) toŜ(Ψ), and hence to S(Ψ). The analysis is carried out completely for the cubic vertices in the two theories, using a method similar to the one used in ref. [14].
(The analysis for the quadratic terms was already done in ref. [12]). We also discuss some specific features that appear in the analysis of the higher order vertices, and show, in special cases, how explicit field redefinitions may be found which relate the vertices inS(Ψ) andŜ(Ψ). But a general proof of equivalence of these two vertices is not given.
Assuming that a complete set of field redefinitions can be found which relate S(Ψ) andS(Ψ) to first order in the perturbation parameter λ, one can ask if the analysis can be extended beyond first order in λ and hence can be used to relate string field theories around backgrounds that are not necessarily close to each other but can be obtained from each other by a series of marginal deformations. Intuitively it is clear that the string field theories constructed around two such backgrounds will also be related by field redefinition, which can be built by combining successive field redefinitions which relate two nearby conformal field theories. We give a general algorithm for finding these finite field redefinitions in terms of the infinitesimal ones.
The plan of the paper is as follows. Sect.2 contains a precise formulation of the problem that we are going to study, as well as its relationship with the work of ref. [12]. In sect.3 we prove the existence of field redefinitions that relate the cubic vertices of these two theories, and also give the general algorithm for constructing these field redefinitions. Sect.4 contains a discussion of the corresponding analysis for higher point vertices. Sect.5 deals with the case where the two conformal field theories are not necessarily close. We also use the result of this section to construct the classical solution in string field theory representing the perturbed conformal field theory to second order in the perturbation parameter. We conclude in sect.6 with a discussion of our results and some comments. The three appendices contain some of the technical results needed in the analysis of sects. 3, 4 and 5.

FORMULATION OF THE PROBLEM
We begin this section with a precise formulation of the problem that we want to solve. Let CFT and CFT ′′ be two different conformal field theories, both with central charge 26, and hence both providing consistent background for the formulation of string theory. Let |Ψ and |Ψ denote string fields for the string field theories formulated around CFT and CFT ′′ respectively, and S(Ψ) andS(Ψ) be the actions of the corresponding string field theories. In order to prove background independence of string field theory, one needs to find a functional relationship of the form Ψ = f (Ψ) such that, with the gauge invariance, Let us introduce a basis of states |Φ n,r in the subspace H n of H of ghost number n, and annihilated by b − 0 and L − 0 . Similarly, let |Φ n,r be a basis of states in the subspaceH n ofH of ghost number n and annihilated by b − 0 andL − 0 . Then we may write, ⋆ This was done at finite value of the regulator which controls the short distance singularities. In our analysis we shall keep the regulator finite throughout our analysis, and set it to zero only at the very end. For a coordinate independent description see ref. [16].
We can now expand b − 0 |Ψ as, where ψ r and ψ ′ r are related by, rs ψ ′ s (2.10) Using eqs.(2.2), (2.3), (2.7) and (2.9) we may express S(Ψ) andS(Ψ) as, where, We now seek the following form of functional relationship between ψ ′ r andψ r : r |Φ 2,r must be a solution of the classical equations of motion derived from the action S(Ψ).
Let us define,ψ Using this, we may express eq.(2.11) as, where, where, and, (2.27) and more generally, We shall consider the case where CFT and CFT ′′ are related to each other by an exactly marginal perturbation. Also, in this section, we shall work to first order in the perturbation parameter λ. For definiteness, we take CFT to be a direct sum of a theory of some free scalar fields (one of which is time-like) and an internal conformal field theory; and take the marginal operator that relates CFT to CFT ′′ to be a dimension (1, 1) operator ϕ in this internal conformal field theory. It was shown in ref. [12] that if the two dimensional action for CFT ′′ is obtained by adding a term of the form −λ d 2 zϕ(z,z) to the two dimensional action of CFT, then It was also shown that to order λ, there is an operator S (called S in ref. [12]) which acts on states annihilated by b − 0 and L − 0 =L − 0 , and relates the BRST chargẽ Q B of CFTG ′′ toQ B defined in eq.(2.22) through the relation: Furthermore, S preserves the inner product between states: where |Φ 1 and |Φ 2 are any two states annihilated by b − 0 and L − 0 . Taking the matrix elements of both sides of eq.(2.30) between the states S|Φ 2,r1 and c − 0 S|Φ 2,r2 , and using eq.(2.31), we get, This is precisely eq.(2.26). Thus we see that S sr defined in eq.(2.33) provides a solution to eq.(2.26).
In the next sections we shall show how to obtain solutions of eqs. (2.27) and also discuss some general features of the other equations (2.28). For our analysis it will be more convenient to defineψ r as, Then the actionŜ(Ψ) given in eq.(2.19) may be expressed as, The relation (2.25) now takes the form: Eqs.(2.27) and (2.28) may now be writtten as, and,

ANALYSIS OF CUBIC TERMS IN THE ACTION
In this section we shall show that it is possible to choose suitable S sr1r2 ) so as to satisfy eq.(2.27) (or, equivalently, eq.(2.40)). We shall choose to work with the quantitiesS (1) sr1r2 and eq.(2.40), since the various equations we shall encounter during this analysis take a more compact form in terms of these variables. To begin with, we divide the basis states {|Φ n,r } into three sets: physical states {Φ n,kn }, unphysical states {|Φ n,αn }, and pure gauge states {Φ n,αn−1 }, satisfying the relations [13]:

Case I. All three indices correspond to unphysical states
In this case, let us choose, s is non-vanishing only when |Φ 2,r and |Φ 2,s are unphysical states, the only non-vanishing components ofÃ (2) are of the formÃ (2) α2β2 . We may now express eq.(2.40) as, From eq.(3.2) and the definition ofÃ (2) it is clear that the matrixÃ (2) α2β2 does not have any left or right eigenvector with zero eigenvalue. Hence it is an invertible matrix.
⋆ Let M α2β2 be this inverse. Eq.(3.5) may then be satisfied by choosing, δ2β2γ2 is an arbitrary tensor which is symmetric in β 2 and γ 2 and satisfies: In particular, we may take L (1) to be 0, but eq.(3.7) represents the most general solution of eq.(3.5). As we shall see later in this section, L (1) needs to be adjusted appropriately in order to ensure that the coefficientsS (1) do not have any discontinuity as a function of the momenta of various external states.
Case II. One of the indices corresponds to a BRST invariant state, others correspond to unphysical states Since eq.(2.40) is completely symmetric in the indices r 1 , r 2 and r 3 , we may, without any loss of generality, choose: Eq.(2.40) then takes the form: This equation may be solved by choosing, where L ′(1) is an arbitrary tensor, satisfying, Case III. Two of the indices correspond to BRST invariant states, the ⋆ AlthoughÃ (2) α2β2 is an infinite dimensional matrix, it is block diagonal in the basis which are chosen to beL + 0 and the momentum eigenstates, with each block a finite dimensional matrix. Thus the results for finite dimensional matrices can be applied here.
Without any loss of generality we can take this to be the state |Φ 2,µ2 . Thus we take, We shall now show that for this choice of |Φ 2,µ2 , both, the left and the right hand side of eq.(3.16) vanish as a consequence of gauge invariance. This is intuitively obvious, since it involves on-shell three point amplitudes involving external pure gauge states; but we shall go through the proof in some detail, since it will be useful to derive a similar result in the next section. Let { Φ c n,r |} denote a basis of states conjugate to |Φ n,r , satisfying, Φ c n,r |Φ n,s ′′ = δ rs (3.18) Expanding the gauge transformation parameter |Λ given in eq.(2.5) as, we may express eq.(2.5) as, Invariance of the action (2.12) under the infinitesimal transformation given in eq. (3.20) gives,Ã etc. Let us now choose the indices r 1 and r 2 in eq.(3.24) such that, In this caseÃ (2) ν2s andÃ (2) τ2s vanish since Φ 2,ν2 |Q B = 0 = Φ 2,τ2 |Q B . Using eq.(3.21) we may now bring eq.(3.24) into the form: In an exactly similar way the gauge transformation (2.44) may be expressed in component form as, The gauge invariance of the actionS(Ψ) given in eq.(2.36) under the transformation (3.28) gives an equation identical to eq.(3.24) withB (1) andÃ (3) replaced byB (1) andĀ (3) respectively. Choosing the various external states as in eq.(3.26) we get, Thus we see that eq. redefinition. The equality of these two amplitudes was proved in complete detail in ref. [13]. Hence we reach the conclusion that eq.(3.16) is satisfied identically, thereby proving that it is possible to satisfy eq.(2.40) (and hence eq.(2.27)) by appropriately adjusting the coefficientsS (1) .

Continuity of S (0) , S (1) as functions of external momenta
The equations (3.6), (3.10) and (3.14), which determine the componentsS would then seem to indicate that the components ofS (1) change discontinuously at these special values of momenta. To be more concrete, let us consider the case when all the space-like flat directions have been compactified and have discrete momenta/winding number, so that the only continuous index is the momentum associated with the time like directions. Let us denote this by k 0 . If we consider the set of states in H with a fixed value of k 0 , then, for generic k 0 , all the states haveL + 0 = 0, and hence can be divided into pure gauge and unphysical states. In this case we can choose the basis of unphysical and pure gauge states in such a way that as we vary k 0 continuously, the unphysical states change smoothly into unphysical states, and pure gauge states change smoothly into pure gauge states. This continues till we reach some special values of k 0 (say k 0 c ) for which L + 0 = 0 states appear in the basis, so that the basis may contain physical states. In this case, the basis of states may be so chosen that as we approach k 0 c from a neighbouring value of k 0 , some of the unphysical states in the basis become physical, the rest remains unphysical; similarly some of the pure gauge states in the basis become physical, the rest remains pure gauge.
⋆ As a result, the coefficients ⋆ Note that the usual choice of gauge, b + 0 = 0, is not a good gauge choice for a general analysis, since the basis of physical, unphysical, and pure gauge states in this gauge collapses atL 0 + = 0 (k 0 = k 0 c ). In particular, at k 0 = k 0 c , new unphysical states may appear which do not satisfy the b + 0 = 0 gauge condition. In other words, if we choose a basis of unphysical and pure gauge states for k 0 = k 0 c , then it is not true that in the k 0 → k 0 c limit the set of unphysical (pure gauge) states divide themselves into unphysical (pure gauge) and physical states.
There, is, however, one case in which b + 0 = 0 is a good gauge choice; this is when at k 0 = k 0 c the onlyL + 0 = 0 state that appears is a tachyonic physical state. In this case, S (1) determined from eqs.(3.6), (3.10) and (3.14) may jump discontinuously as the momentum k 0 associated with one of the external states approach k 0 c . We shall now show that such a situation may be avoided by a judicious choice of the quantities L (1) and L ′(1) appearing in eqs.(3.6) and (3.10) respectively.
Let us start with eq.(3.6) and assume that the momentum associated with the state |Φ 2,γ2 approaches k 0 c , and also that in this limit the state |Φ 2,γ2 approaches a specific physical state |Φ 2,l (0) 2 . In this case eq.(3.6) gives a value ofS we want this value to agree with the answer given in eq.(3.10), we must demand that, On the other hand, if the momentum associated with the state |Φ 2,α2 approaches Thus the corresponding eigenvalue of M α2δ2 has a pole. In order that S (1) given in eq.(3.6) is finite in this limit, we must have, We may now ask if eqs.(3.31) and (3.32) are compatible with the constraint (3.7) that L (1) must satisfy. From these two equations we get, The right hand side of eq.(3.33) vanishes identically by eq.(3.11). Hence eqs. (3.31) and (3.32) are compatible with eq.(3.7).
in theL 0 + → 0 limit, an unphysical state gets converted to a physical state, without any other rearrangement of the basis. The examples that will be considered in sect.4 are all of this type, hence the b + 0 = 0 gauge choice in that section will not cause any problem in our analysis.
This also indicates that in order to extend our analysis of N -point vertices in sect.4 and that in ref. [13] beyond the tachyonic states, it may be more convenient to choose a gauge other than b + 0 = 0.
Next we consider eq.(3.10) and take the limit when the momentum associated with the state |Φ 2,β2 approaches the critical value k 0 c . Let us suppose that in this limit the state |Φ 2,β2 approaches the physical state |Φ 2,l (0) 2 . In this case eq.(3.10) gives us a value ofS . In order that this value agree with that obtained from eq.(3.14), we must have, On the other hand, if in eq.(3.10) we take the momentum associated with the state |Φ 2,δ2 approach k 0 c , then, in order thatS (1) approaches a finite value as k 0 → k 0 c , we must demand, We now see that eqs.(3.34) and (3.35) satisfy the constraint (3.11): Finally, note that in eq.(3.14), if we let the momentum associated with the state c , then, in order thatS (1) δ2µ2ν2 is finite in this limit, we must have,Ã which is part of the consistency condition given in eq.(3.16).
Similar remark also holds for the components S , we may express eq.(2.26) as, whose solution is given by, where L (0) is a tensor satisfying, In addition, eq.(2.30) (which, combined with eq.(2.31) is more restrictive than eq.(2.26)) gives a further constraint on K (0) : δ2µ2 ∼ λ.) As before, if the momentum k 0 associated with the state |Φ 2,β2 approaches k 0 c , then, in order that eq.(3.39) smoothly approaches eq.(3.40), we must have, On the other hand, if the momentum k 0 associated with the state |Φ 2,α2 approaches k 0 c , then, in order that the right hand side of eq.(3.39) is finite in this limit, we must have, From eqs.(3.43) and (3.44) we get, as is required by eq.(3.41).
Finally, if we take the momentum k 0 of the state |Φ 2,α2 in eq.(3.40) to approach k 0 c , then, in order that the right hand side of this equation is finite in this limit, we must have,Ã This equation was shown to be satisfied identically in ref. [12].
These examples illustrate, how, by adjusting the parameters L (0) , L (1) and L ′(1) , we may maintain the continuity of the solutions S (0) and S (1) as function of the momenta k 0 of various external states.
Note that in eq.(3.40), by taking the limit where the momentum k 0 associated with the state |Φ 2,α2 approaches k 0 c , we can evaluate K is obtained from eq.(3.42), but it is easy to check that this agrees with the expression for K was found in ref. [13] (eq.(3.60)) from the analysis of three and higher point functions. Thus we must ensure that these two results agree with each other. A similar situation arises also in case ofS (1) k2µ2ν2 . As we shall see in the next section, analysis of four and higher point functions determines the quantitiesS (1) k2µ2ν2 ; hence we must ensure that this value agrees with the value ofS obtained by taking the k 0 → k 0 c limit ofS (1) δ2µ2ν2 , where k 0 is the momentum associated with the state |Φ 2,δ2 .
A general argument showing that this must be the case may be given as follows.
As we have seen, by adjusting L (1) , L ′(1) and L (0) , we can ensure that none of the components of S (0) and S (1) has any singularity as a function of the momenta of the external states. Another set of consistency conditions are obtained by starting from eq.(3.42) and going to a limit when the state |Φ 2,α1 =Q B |Φ 1,α1 approaches a physical state |Φ 2,m (0) 2 after appropriate normalization. This would happen if in this limit |Φ 1,α1 reduces to a physical state so thatQ B |Φ 1,α1 vanishes in this limit unless it is divided by an appropriate normalization factor before we take the limit. A consistency condition is then obtained by demanding that K agrees with the answer given in eq.(3.60) of ref. [13]. The analysis of appendix A shows that this is indeed the case. (Since physical states of ghost number 1 appear only in the zero momentum sector, we need to study only the k = 0 sector for this analysis.) There is another related point that we must mention before concluding this section. In proving the equality of the physical amplitudes calculated from the actionsS(Ψ) andS(Ψ), we had assumed in ref. [ there is a physical state at zero momentum which cannot be expressed in this form. This is the zero momentum dilaton state, given by, Since we must show the equality of the two sides of eq.(3.16) for all states, we must also prove this for the case where one or more of the external states correspond to the state |D given in eq.(3.47). We have shown in appendix A that this is indeed what happens, thereby completing the proof that eq.(3.16) is satisfied for all values of µ 2 , ν 2 and τ 2 .

ANALYSIS OF FOUR AND HIGHER POINT TERMS OF THE ACTION
In this section we shall sketch the analysis of eq.(2.41) for N ≥ 4, but shall not give a complete proof of existence of solutions to these equations. Let us start with eq.(2.41) for N = 4. Following the same analysis as in the previous section, one can show that this equation can be satisfied by adjusting the coefficientsS (2) when at least one of the indices r i correspond to unphysical states. In order to avoid repetition, we shall not go through the whole analysis, but only consider a single case as an example. Consider the case, In this case eq.(2.41) for n = 4 is satisfied if we choose, (1) where L (2) α2β2γ2µ2 is an arbitrary tensor which is symmetric in β 2 and γ 2 and satisfies the relation: In the last set of terms on the right hand side of eq.(4.5), the sum over s can be broken up into sum over physical, unphysical, and pure gauge states. Of these, the sum over pure gauge states vanishes, since, as we have seen in the last section, A sµ2ν2 vanish for such states. In the second set of terms on the right hand side of eq.(4.5) the sum over s and s ′ can be restricted to unphysical states, sinceÃ ss ′ vanishes for other set of states. Using eq.(3.14), and the fact that M α2β2 is the inverse ofÃ α2β2 , we may bring eq.(4.5) to the form: We shall first show that the above equation can be satisfied when one of the external states (say |Φ 2,µ2 ) is of the form of a pure gauge state. Let us take rα1 |Φ 2,r . In this case, in eq.(4.6), all factors ofÃ respectively. The left hand side of eq.(4.6) then takes the form: Choosing r 1 = ν 2 , r 2 = α 2 in eq.(3.24) we get, Using this and similar equations with ν 2 replaced by τ 2 , σ 2 , and the relation Ã (2) α2β2 M β2γ2 = δ α2γ2 , eq.(4.7) may be brought into the form: On the other hand, choosing r 1 = ν 2 , r 2 = τ 2 , r 3 = σ 2 in eq.(3.25), and using the relationsÃ Comparing the left hand side of eq.(4.10) with the expression given in eq.(4.9) we see that they differ from each other only in the fact that the sum over r in eq.(4.10) runs over all states, whereas the sum over β 2 in eq.(4.9) runs over unphysical states only. SinceÃ corresponds to a physical state. In other words, using eq.(4.10), the expression (4.9) may be brought into the form: Using a similar analysis, the terms inside the square bracket on the right hand side of eq.(4.6) may be brought into the form of eq.(4.11) withÃ (3) andB (1) replaced bȳ (1) respectively. Finally, the first three terms inside the parantheses in the right hand side of eq.(4.6) vanish by eq.(3.27) for the choice |Φ 2,µ2 = |Φ 2,α1 .
The equation (4.6) then takes the form: Using the consistency conditionĀ k2σ2τ2 vanish for all k 2 , and hence the last set of terms on the right hand side of eq.(4.6) vanish. Also, in this case, M α2β2 , being the inverse ofÃ (2) α2β2 in the subspace of unphysical states (which in this case coincides with the subspace of states which are not pure gauge) may be interpreted as the propagator of the gauge fixed theory, both, for the theory described by the actionS(ψ) and the theory described by the actionS(ψ).
As a result, the left and the right hand side of eq. This is precisely the problem that we analyzed in ref. [13], where we proved that the on-shell amplitudes in these two theories with arbitrary number of tachyonic external legs are indeed identical.
⋆ Although the proof of this result could not be extended to the case of arbitrary external states due to some technical difficulties, we believe that the result does hold for arbitrary external states, particularly in view of the fact that many of the S-matrix elements involving higher excited states ⋆ In ref. [13] we worked with the actionŜ(Ψ) instead ofS(Ψ), but since they are related by a linear field redefinition, they have the same on-shell S-matrix elements.
in string theory can be determined by studying the S-matrix elements involving tachyonic states near the poles.
We now turn to the case of special values of momenta, which allow physical external states to propagate in the s, t or u channel. We shall assume, for simplicity, that this happens only in one channel, say the µ 2 ν 2 → σ 2 τ 2 channel, although our analysis can be easily extended to the case where this happens in more than one channel simultaneously. Also, in order to avoid technical difficulties similar to the one that arose in the analysis of ref. [13] for arbitrary external states, we shall k2σ2τ2 survive, and this equation can be written as, It is shown in appendix B that ifĀ σ2τ2k2 vanish for all k 2 , then the left hand side of eq.(4.14) vanishes. Thus the left hand side of eq.(4.14) must be of the form:Ā for some tensor Σ (1) . Thus eq.(4.14) may be satisfied by choosing,  were left undetermined in the analysis of two point functions in ref. [12], but were determined during the analysis of on-shell three point functions in ref. [13].
In writing down the above equation, we have used the gauge invariance of three and four point amplitudes to eliminate terms involvingS (2) α1rst andS (1) α1rs . We now consider, as before, only those configurations for which the onlyL + 0 = 0 states which can propagate in the intermediate state in the µ vanish. Hence this has the form: Note that we have taken the coefficient ofΓ (4) in eq.(4.22) to be the same tensor Σ (1) as the one that appeared in eq.(4.15). This is due to the fact that the coefficient of theΓ (4) term is expected to be determined solely by the structure of the states , |Φ 2,µ (2) 2 and the propagator M α2β2 , and hence is expected to be equal to the coefficient of theĀ (3) term on the left hand side of eq.(4.14) with appropriate external indices. ⋆ Using eq.(4.16) we see that the term involvingΓ (4) in eq. (4.22) ⋆ This is similar to the result of ref. [13] where the coefficient S k2l2 required to satisfy the equality of on-shell N -point amplitudes in the two theories turned out to be independent of N . is identical to the term involvingΓ (4) on the right hand side of eq.(4.21). Thus eq.(4.21) may be satisfied by choosing, The coefficientsS   To be more specific, let us now consider a one parameter family of allowed backgrounds, such that any two neighbouring members of the family are related to each other by a marginal perturbation. Let τ denote the parameter labelling these backgrounds. Then, given the conformal field theories at τ and τ + δτ , the two dimensional action of these two conformal field theories are related by,

BACKGROUNDS SEPARATED BY FINITE DISTANCE AND RELATED BY MARGINAL DEFORMATION
where ϕ(z,z) is a properly normalized dimension (1,1) primary field. f (τ ) is some function of τ that can be set to 1 by appropriately reparametrizing τ . For simplicity, we shall assume that this has been done.
Let {|Φ n,r (τ ) } be an appropriate set of basis states in the conformal field theory corresponding to a given value of τ , and b − 0 |Ψ(τ ) = r ψ r (τ )|Φ 2,r (τ ) be the string field. Identifying the conformal field theory corresponding to the point τ to CFT ′′ , that corresponding to the point τ + δτ as CFT, and using eqs.(2.10), (2.17), we see that ψ r (τ ) and ψ r (τ + δτ ) are related by a functional relationship of the form: Since ψ s (τ + δτ ) and ψ s (τ ) are expected to differ by an amount of order δτ (which was called λ in the previous sections), we may rewrite eq.(5.2) as, Note that we have included a τ dependence in T (N ) since these coefficients depend on the correlation functions in the conformal field theory labelled by the parameter τ .
Let τ 0 be some fixed reference point in the τ space. We shall now try to show that there is a field redefinition of the form: which relates the string field theory actions formulated around the points τ and τ 0 even when τ and τ 0 are not close to each other. Using eqs.(5.4) we get, On the other hand, replacing ψ ri (τ ) in the right hand side of eq.(5.3) in terms of {ψ ri (τ 0 )} given in eq.(5.4) we get, is a solution of the classical equation of motion in string field theory formulated around the point τ that describes the background labelled by τ 0 .
We shall now derive an expression for R (0) s (τ, τ 0 ) to order (τ − τ 0 ) 2 by solving eq.(5.9), and compare this with the results of ref. [19] where a general algorithm was given for constructing solutions of the classical equations of motion in string field theory to all orders in the deformation parameter for (nearly) marginal perturbation. Since for τ ≃ τ 0 , R (0) s ∝ (τ − τ 0 ), and since we are interested only in the value of R (0) s to order (τ − τ 0 ) 2 , we may express eq.(5.9) as, Let us denote by |ϕ the dimension (1,1) primary state in the internal conformal field theory representing the marginal deformation, and assume, for simplicity, that c 1c1 |ϕ is the only physical state |Φ 2,k2 ≡ |Φ 2,k2 (τ ) at zero momentum. If we take s = k 2 in eq.(5.11), we get, We shall now carry out a further reparametrization of τ such that the right hand side of eq.(5.13) is exactly given by k2 takes the form: Let us now concentrate on the components R β2 are the only relevant components to be computed. From eq.(5.11) we get, Note that the τ reparametrization needed to make eq.(5.14) an exact equation is of It thus remains to determine T β2k2 . It has been shown in appendix C that, Hence the classical solution to order (τ − τ 0 ) 2 may be written as, It can be easily verified that this agrees with the result computed in ref. [19] to this order up to a gauge transformation and possible reparametrization of τ . ⋆ ⋆ The notation for the choice of basis given in ref. [19] is different from the one used here. In order to translate the result of ref. [19] to the present notation, we must make the replacement Φ c n,kn | → Φ c 6−n,k6−n |, Φ c n,αn | → Φ c 6−n,α5−n |, and Φ c n,αn−1 | → Φ c 6−n,α6−n | for all n.

DISCUSSION
In this paper we have shown that string field theories constructed around two nearby conformal field theories that are related by marginal perturbation are actually the same, in the sense that they are related to each other by field redefinition.
Although the result has been proved in complete detail only for the quadratic and cubic terms in the action, we have shown, by analyzing several special cases, that the result is likely to be valid even for the quartic and higher order terms in the action. Finally, we have also shown that the result holds beyond leading order in the perturbation parameter λ; as a result, string field theories formulated around two different conformal field theories which are not necessarily close to each other but are still connected to each other by a series of marginal deformations, are also related to each other through a field redefinition. In this case, the appropriate field redefinition is found by solving an infinite set of differential equations.
A question that naturally arises at this stage is whether it is possible to formulate string field theory in a way where the background independence is manifest.
For open string field theory such a formulation was given in reference [20] where it was shown that starting from a purely cubic action which is independent of the background, one can obtain string field theories around different backgrounds by shifting the field by a classical solution. The action expanded in terms of the shifted field has a background dependent kinetic term, but a background independent interaction term. Such a simple notion of background independence cannot, however, be implemented in non-polynomial closed string field theory, since, by the very nature of the non-polynomial interaction, a shift in the string field will modify all the interaction vertices.
We would like to point out that the analysis of this paper may be applied not only to study the equivalence of string field theories formulated around two different backgrounds, but also to study the equivalence of different formulations of string field theory around the same background.

APPENDIX A
In this appendix we shall try to verify eq.(3.16) when one or more of the external states |Φ 2,µ2 , |Φ 2,ν2 , and |Φ 2,τ2 correspond to a zero momentum dilaton and the rest of the physical states are of the form c 1c1 |Ṽ , |Ṽ being a dimension (1,1) primary state in CFT ′′ .
Let us define, The starting point of our analysis will be the identity: where k is an arbitrary momentum satisfying k 2 = 0, α µ n are the matter oscillators associated with the flat directions, and n µ is any vector. This shows that |s(k) has the form of c 1c1 |Ṽ + pure gauge state. Hence for amplitudes involving such states,Ã (3) andĀ (3) have the same values. If we now take the k → 0 limit of such amplitudes, then, if the k → 0 limit is smooth, and is identical to the amplitude for k = 0, this would imply thatÃ (3) andĀ ( has the form c 1c1 |Ṽ , and hence, by following the analysis of ref. [13] one can see thatÃ (3) andĀ ( Thus it remains to show that the k → 0 limit of the amplitudes involving the state |s(k) is smooth, both inÃ (3) andĀ (3) , and is identical to the corresponding amplitudes involving |s(0) . ForÃ (3) such amplitudes are obtained simply by taking the three point correlation functions of appropriate vertex operators. The k → 0 limit of such an amplitude is manifestly smooth. On the other hand, if we take |Φ 2,µ2 = |s(k) , then, The second term on the right hand side of eq.(A.4) has well defined k → 0 limit and is identical to the k = 0 answer. The first term, in principle, can have a value whose k → 0 limit does not agree with the k = 0 answer, since it involves the operator S = 1 + λK + O(λ 2 ). K is determined by the set of eqations discussed in ref. [12] [13], and the algorithm given there was not manifestly smooth in the k → 0 limit, since it involved treating the physical, unphysical, and pure gauge states on a different footing. We shall now demonstrate explicitly that the k → 0 limit of the operator K is actually smooth. Once this is established, it would imply that the first term on the right hand side of eq.(A.4) also has smooth k → 0 limit that agrees with the k = 0 result.
Thus we now need to show that the state K|s(k) reduces, in the k → 0 limit, to K|s(0 where, |ϕ being the same state that appeared in eq.(2.29), and ǫ is the short distance cut-off used to define the correlation functions in CFT ′′ in terms of those in CFT.
Since |s(k) is a state built by the α µ −n ,ᾱ ν −n and the ghost oscillators on the state |k , it is clear from eqs.(A.6) that the only states Φ 2,α2 | for which the right hand side of eq.(A.5) does not vanish trivially are those built by the α µ n ,ᾱ ν n and the ghost oscillators on ϕ| ⊗ −k|. A simple analysis of BRST cohomology shows that the number of unphysical states built on ϕ| ⊗ −k| does have a smooth limit as k → 0 since no new physical state appears at this value of the momentum. ⋆ Thus it is possible to choose the basis of unphysical states Φ 2,α2 | for momentum k in such a way that as k → 0, these states reduce to the unphysical states in the k = 0 ⋆ Note that we are taking the limit keeping k 2 = 0 all the time.
sector. In this case the k → 0 limit of the right hand side of eq.(A.5) agrees with its k = 0 value. This, in turn, shows that the components Φ 2,α2 |c − 0Q B K|s(k) ′′ has smooth limit as k → 0, and this limit agrees with the corresponding expression with |s(k) replaced by |s(0) .
Let us now analyze the components Φ 3,k3 |c − 0 K|s(k) ′′ . Using eq.(3.60) of ref. [13], and eq.(A.3) we see that, The only physical state Φ 3,k3 | for which eq.(A.7) is non-trivial is −k|⊗ ϕ|c −1c−1 c + 0 . But the right hand side of eq.(A.7) vanishes for such a state. Using the equation We shall now evaluate λ Φ 3,k3 |c − 0 K|s(0) ′′ and compare with the right hand side of eq.(A.8) in the k → 0 limit. Using eq.(3.60) of ref. [13] we get, Using eqs.(A.2), (A.8), and (A.9) we see that the We now note that the matrix element appearing on the left hand side of eq.(A. 10) has not been determined previously. Hence we can use eq.(A.10) to define this matrix element. This, in turn, guarantees that in the k → 0 limit K|s(k) reduces to the state K|s(0) , and hence proves that eq.(3.16) is satisfied even when one or more of the external states correspond to the zero momentum dilaton state The result of this appendix may also be interpreted in the following way. Note that the pure gauge state, reduces to a physical state |Φ 2,m (0) 2 in the k → 0 limit. Our result shows that, thereby showing that eq.(3.42) can be made compatible with eq.(3.60) of ref. [13].

APPENDIX B
In this appendix we shall analyze the left hand side of eq.(4.14) and show that it can be brought into the form of eq.(4.15). Using eq.(2.37) and (2.33) and the fact [12] [13] that when CFT and CFT ′′ differ by a marginal perturbation, then S has the form: rs is defined through the relation, Let us define,∆ The left hand side of eq.(4.14) may now be written as, rτ2σ2 ) + Other pairings of µ 2 , ν 2 , τ 2 , σ 2 + O(λ 2 ) (B.6) We assume that the only physical states that can appear in the µ 2 ν 2 → τ 2 σ 2 channel are tachyonic. This, in turn, shows that if k is the momentum flowing in this channel, then the corresponding vertex operator e ik.X has dimension h > 0.
As a result, the only possibleL + 0 = 0 states propagating in this channel are of the form c 1c1 e ik.X(0) |Ṽ , where |Ṽ is a dimension (1 − h, 1 − h) primary state in the internal conformal field theory. In other words, the only possibleL 0 + = 0 states propagating in this channel are the tachyonic physical states.
This, in turn, implies that ifÃ τ2σ2k2 vanish for all k 2 , then the only possible intermediate states |Φ 2,t , |Φ 2,s appearing in eq.(B.6) are theL + 0 = 0 states. (We have already assumed that the momenta flowing in the u and t channels are such that they do not allowL + 0 = 0 states to appear as intermediate states in these channels). We shall now show that the right hand side of eq.(B.6) vanishes in this case. In theL + 0 = 0 sector, a basis of unphysical states can be chosen all of which are annihilated by b + 0 (see, for example, ref. [22]). This gives rise to the standard expression for the propagator∆ rs : Computation of the terms inside the first square bracket in eq.(B.6) then reduces to the standard computation of a four point function in string field theory defined by the actionS(Ψ) with physical external states. Standard manipulations given in ref. [13] then shows that this amplitude is identical to the corresponding amplitude calculated with the actionŜ(Ψ). In the analysis of ref. [13] the kinetic term ofŜ(Ψ) was split into two pieces: The inverse of the first term on the right hand side of eq.(B.8) in the b + 0 = 0 gauge, i.e., was taken as the propagator, whereas the second term on the right hand side of eq.(B.8) was taken as the interaction term. Here { Φ ′c 2,r |} is the basis of states conjugate to {|Φ 2,r } with respect to the inner product | , i.e., Φ ′c 2,r |Φ 2,s = δ rs (B.10) Thus, by repeating the analysis of ref. [13] one can show that the contribution from the set of terms inside the first square bracket on the right hand side of eq.(B.6) takes the form: where, We now note that, where, Finally, since b − 0 b + 0 (L + 0 ) −1 is hermitian with respect to the inner product | , using eq.(B.17) the last term on the right hand side of eq.(B.15) may be written as, Eq.(B.15) may now be written as, We now note that in eq.(B.14) the sum over r and s run over unphysical states only, sinceÂ µ2ν2r etc. are taken to be non-vanishing only for such states. ⋆ Taking ⋆ Actually, it isÃ (3) µ2ν2r etc. which vanish for r = α 2 , but this implies thatÂ (3) µ2ν2r is of order λ for r = α 2 .
Summing over a complete set of states, the last two terms in the curly bracket may be expressed as, This shows that ifÃ (3) µ2ν2k2 andÃ (3) σ2τ2k2 vanish, then the left hand side of eq.(4.14) vanishes. This, in turn, implies that the left hand side of eq.(4.14) must be of the form given in eq.(4.15). Although the quantities Σ (1) k2µ2ν2 and hence S (1) k2µ2ν2 may be determined by careful analysis (similar to the one in ref. [13] which determined Φ c 2,k2 |K|Φ 2,l2 ), we shall not carry out that analysis here.
Let us now compute W [2] β2k2 . From eqs.(2.8) and (C.2) we see that, |Φ n,r = |Φ n,r + λW We now note that in the generic case, the basis of unphysical states can be taken to be the same in CFT and CFT ′′ ; these are the states that are not annihilated by