Classical Open String Amplitudes and Boundary String Field Theory

Abstract: We show that boundary string field theory realizes the minimal model of open string field theory. More precisely, we observe that the expansion of the (co)homological vector field, $Q$ of boundary string field theory in the cohomology of its linear part reproduces the S-matrices of perturbative string theory. In mathematical terms, boundary string field theory realizes the minimal model map of the cohomological perturbation lemma.


Introduction and Summary
The standard formulations of string theory rely on a choice of background, more precisely, a choice of conformal field theory. Different backgrounds can be realized, for example, by a curved target space manifold. In this case strings move in a curved spacetime. For open strings there is an additional freedom in the choice of a boundary condition, which one usually thinks of as certain brane configurations. Different choices lead to inequivalent theories in general; one says that these theories are background dependent.
It has been known for some time [1,2] that it is possible access nearby backgrounds from a specific. On general grounds, the problem of background deformation covariant string field theory is well posed when the Hilbert space is preserved by the perturbation, [3]. This is the case, in particular for open string field theory on a fixed closed background (ie. bulk CFT) and boundary perturbations generated by operators obtained from the bulk CFT. In this case we then expect that a background independent formulation of string theory should exist.
In [4] Witten proposed a formulation for a background independent classical bosonic open string theory. This theory is formally defined as a Batalin-Vilkovisky (BV) action, S on the space of two dimensional, boundary conformal field theories (BCFT) on the disk. Classical backgrounds correspond to critical point of S. This theory is commonly known boundary string field theory (BSFT).
Witten proposed to parametrize conformal boundary conditions by critical points in the space of local operators, O and corresponding world sheet boundary action V with V = b −1 O. In concrete calculations one is generically forced to treat the V as perturbations around a conformal background. 1 Later, Shatashvili showed [6] that contact terms arising in the perturbative expansion of the boundary interaction spoil the naive proof of gauge invariance and proposed a modification in terms of deformed Virasoro generators (see also [1]) for which he proved gauge invariance to first order in perturbation theory. This, in particular, induces non-linear terms in the cohomological vector field when expanded around a conformal background. On the other hand, since only matter perturbations are considered the BV structure of the theory is no longer explicit.
In section 2, after reviewing the definition of BSFT, we will argue that for background perturbations, O = φ i O i in the physical subspace at a critical point, the cohomological vector field, Q has an expansion in φ i where the expansion coefficients are given by classical open string S-matrix elements, that is, where V (n) denote n-point tree level scattering matrices. Likewise the differential of the action on the physical subspace has a compact expression which can be integrated where the subscript S stands for symmetrization. 2 We can interpret (1.1) and (1.3) as follows: BSFT directly generates the minimal model [7] of open bosonic string field theory (OSFT) [8]. This is different from the usual way of computing homological minimal models, since in the latter is constructed in terms of Feynman diagrams derived from the coefficients (vertices) of the homological vector field. 3 Given that BSFT directly computes the minimal model of OSFT one may ask what the minimal model of BSFT is. In section 3 we will argue on general grounds that there are no vertices in S that connect connect on-shell states to a single off-shell state. Consequently, the only contributions to tree-level S-matrices are given by contact vertices arising in the expansion of Q. Thus, the minimal model of BSFT is again that of OSFT. This also shows that around a conformal background, BSFT and OSFT are perturbatively equivalent where the equivalence is provided by the minimal model map of OSFT.
1 Non-local boundary interaction were argued in [5] to correspond to a shift of the closed string background. 2 The objects V (n) and V (n) S are discussed in the appendix. 3 In mathematical literature this construction is known as the homological perturbation lemma.

Construction of the BSFT action
Boundary string field theory is by definition a BV action. As such its definition depends on the following ingredients: In particular this means ω should be closed.

2.
A degree 1 vector field V which squares to zero. 3. The vector field V should generate a symmetry of ω. Property 1 and 3 imply that i V ω is closed: This allows us to define locally an action S via dS = i V ω. Property 2 then implies that S satisfies the classical master equation {S, S} = 0 which is equivalent to S being gauge invariant under the transformations generated by V .
In the context of BSFT [4] these objects were identified as follows: First of all the space of fields F the space of all two dimensional sigma models defined on the disc with fixed bulk CFT action I 0 . Points in F are represented by boundary operators O. The complete world sheet action is then where b −1 is defined in terms of the closed string anti-ghost field. We will come to its proper definition later. Tangent  The symplectic form at a point O is the expectation value of two tangents taken with respect to I: The measure on the boundary is normalized to one. We use the same convention as [6] for denoting expectation values: · is taken with respect to I, while · corresponds to I 0 . In [4] it was already shown that this form is closed.
The We defer the precise definition of Q to the next subsection. However, we already state that there are some issues when we take its definition too naively. Indeed, in [6] it was shown that contact terms generate corrections to V . Also, it is not obvious that V squares to zero or that it generates a symmetry of ω when one includes these contact terms. Nevertheless it was argued in [6] that the latter property holds to first order in the perturbation of the background. We already gave a criterion for V generating a symmetry of ω, namely closedness of i V ω. However, one may be more ambitious by showing that one can integrate i V ω, meaning that we can find an explicit formula for S. As explained in the introduction this is actually the case for the action we found. Furthermore, we can infer the nilpotency of V because it generates the minimal model of open string theory which, by construction, forms an A ∞ algebra. The limitation of our derivation is, however, that up to now these formulas only work for on-shell perturbations.
On the other hand, since we work on-shell in perturbation theory, we can make two important simplifications. First of all instead of working on the disc we work on the upper half-plane. All of the following calculations can also be done on the disc, but the formulas of certain conformal transformations are more involved (for example scaling transformations translated to the disc). Conformal symmetry also allows us to the positions of the operator insertions in the symplectic structure (2.3) to 0 and ∞:

The b-ghost and the cohomological vector field
In this subsection we want to find a consistent definition of the operator b −1 and the cohomological vector field V = {Q, O}. In [4] the following definitions were proposed The contour C α is the unit circle of radius 1 − α, α 1 and v is the vector field generating rotations. A natural question is then in which sense these operator equations hold when these objects are accompanied by other operator insertions. We expect that in

They should not depend on the other insertions
In the following we will find expressions for b −1 and Q satisfying the three properties if the perturbation is taken to be on-shell with respect to the unperturbed background. We start with b −1 . Here the argument is in fact somewhat circular, because this object already enters into the definition of the boundary action. However, we can make a guess and see whether it works. In a conformal background we know the action of b −1 . It is given by b(v)O i , where the contour runs only around the operator insertion O i . In conformal field theory we know that this just removes a c-ghost insertion from the operator O i . Let us check that this is compatible with (2.7). To a fixed order in the perturbation we have where in this case b −1 is taken with respect to the unperturbed background. Since b 2 −1 = 0 the second term in the above expression drops. We find that the boundary interaction does not contribute to the contour integral Cα b(v). Therefore as expected. We use the same trick for {Q, O}. We expand (2.7) to a particular order in the perturbation of the background, Here the BRST charge will now give non-vanishing contributions from both, the O i and b −1 O.
Since we work in a conformal background, we know the action of the BRST charge. We denote it by {Q 0 , ·}. In particular, for an on-shell perturbation O we have Similarly to perturbative open string theory these produce boundary terms in moduli space, parametrized by the positions of the operators O. We distinguish between the following two cases: 1. Only operators O are close to each other.
2. Some of the operators O are close to one of the O i as in figure 1.
The first case appears even without operator insertions, so considering Figure 1. A contribution to the boundary of the moduli space giving a correction to the BRST operator. In the limit a → 0 the disc is equivalent to two discs connected by a narrow neck. Through this a new local operator is defined inserted in the left disc. Its coefficients with respect to the basis is the expectation value of the right disc.
shows that these should not contribute. The partition function is defined so that the above gives zero. They are so to say part of the definition of · and do not contribute to {Q, O}.
The second case arises for each new operator inserted. We will see that they also produce a finite contributions described by a new local operator inserted at the position of the operator O i , which can be interpreted as the correction to {Q, O i } with respect to the conformal background. So we should find {Q, O i } = {Q 0 , O i } + δO i where δO i is some local operator that we will determine in the next subsection.

Corrections to the BRST operator
According to (2.6) the BRST operator acts on the boundary insertions in the following way: The integral of the total derivative will give a contribution whenever it collides with another operator. As explained before collisions with operators b −1 O do not appear explicitly. The only relevant ones are those with the O i . We focus on one of these collision in (2.7), since by linearity, we just get the sum of every single one. We denote the distance of O to one of the O i by a, a regulator which eventually will be set to zero and is chosen so that O is closer to O i than to the rest of the insertions. Then (2.13) contributes as where we denoted all other operator insertions by Ω. There is a relative plus sign since open string operators are fermionic objects. In the first expression we want to separate the operators O i (x i ) and O(x i +a) from the rest of the operator insertions. We do this by splitting the upper half-plane along a circle of diameter slightly larger than a centered at x i + a 2 , so that the two operators are close to the circle. This amounts to inserting a complete set of states (see figure 2) By translation invariance the choice of x does not matter. Of course, we cannot separate the integrated operators whenever they are inserted between x i and x i + a. Hence we split each integration range We then find We can now use a scale transformation by a factor 1 a around x i in the first correlator: (2.18) We see that for certain h l we get divergences in the limit a → 0. These divergences can be subtracted by suitable counter terms. The finite contribution comes from h l − h i = 0. This implies, in particular, that for on-shell insertions (h i = 0), O l c has zero conformal weight as well. We denote this contribution for fixed k by since, as is clear from (2.18), it agrees precisely with the perturbative (k + 2)-point string amplitude, symmetrized in all but two entries. Concerning the divergent terms arising in the above limit, it is well known that in string theory divergences also arise when intermediate states have negative dimension, eg. [9]. This stems from the fact that the Schwinger parametrization of the propagator is only valid for L 0 > 0. So we can think of these divergences as artifacts of representing the amplitude as an integral over positions. Figure 2. The insertion of a complete set of states at x + a 2 .
In the second correlator we have the integral over the region E a , which for a → 0 will cover the whole integration range. Hence in this limit the correlator becomes lim a→0 n≥0 Combining this with the first correlator we see that the contribution to δO i coming from this limit is There is one more contribution coming from O inserted at x i − a giving the same contribution but with different ordering, this showing that the expansion of the cohomological vector field Q at order k in the cohomology of Q 0 is indeed given by the perturbative S-matrix of order k. This the proves that BSFT with Q defined as above, indeed realized the minimal model map. From the identification with the perturbative S-matrix we may furthermore, conclude that exact states decouple in (2.22). where we have used translation invariance on the boundary of the disk. The action (2.27) is actually independent of a by conformal symmetry and this allows us to fix a = 1 without restricting the generality. Note that so far we did not assume any concrete renormalization prescription, in analogy to the definition of string amplitudes in perturbative string theory. The action can then be integrated as was explained in section 1. It is just a sum of perturbative S-matrices. One can then deal with divergences as explained in e.g. [9].

A simple off-shell example
The key property that allowed us to express the corrections to the BRST operator Q 0 induced by the background is identity (2.13). For generic off-shell deformations (2.13) receives corrections which, in turn, are not given by contact terms. In this case we are not able to determine the corrections Q 0 to all orders 4 . An exception to this is the tachyon at zero momentum around a conformal theory with Neumann boundary conditions. The vertex operator for this background shift is given by O(θ) = T c(θ). The corresponding anti-field is described bỹ O(θ) =T c∂c(θ). This example is somewhat trivial to compute because b −1 O = T and so the background adds no terms beyond linear order to the cohomological vector field: (2.28) The differential of the action is then Normalizing the ghost 3-point function as For completeness we also give the integrated action This is tachyon action first obtained in [10].

Superstring
The extension to the cohomology of the NS-sector of the open superstring is straight forward. 5 The background field is naturally integrated over supermoduli space which means that V is in the 0-picture. At the same time it is natural to take O(0) and O(∞) in the −1 picture since they are not integrated in (2.25). 6 With this the key identity equation (2.13) still holds for the superstring. Furthermore, since O and dO are in the −1 picture (dO is contained in Ω in (2.2) ) this requires that O j (x i ) and O j c (∞) in (2.15) are in the −1 picture as well. Consequently, (2.22) and (2.23) reproduce the superstring S-matrix with the correct picture assignments. Recall, also that on-shell, the precise location of the picture −1 operators is irrelevant as long as the global picture number is correct.
The extension to the Ramond sector is less clear. Indeed this requires two Ramond fields in the − 1 2 picture and one NS field in −1 picture, or four of the Ramond fields in the − 1 2 picture. There does not seem to be a natural way to incorporate this into the definition (2.25). Of course, since Ramond fields represent space-time fermions it is not natural to have non-vanishing Ramond fields in the background. On the other hand, perturbative string amplitudes involving 2n fermions are generically non-vanishing. Thus, if BSFT is to realize the minimal model for super string theory the BSFT action should be non-vanishing for an arbitrary even number of Ramond fields. An alternative possibility is that BSFT is minimal only with respect to the NS sector whereas the NS-fields coupling to fermions are not integrated out. This is not in contradiction with what we found before since the NS-sector is a closed subsector of the theory. In that case we would expect vertices with at most two Ramond fields. Then one might postulate that the three unintegrated fields appearing in (2.27) carry picture −1, − 1 2 , − 1 2 respectively. While this is probably consistent on shell this construction is nevertheless not very natural.

Alternative definition of the cohomological vector field
For generic off-shell perturbations the construction described above does not generalize directly because the correction to the BRST charge Q will not be given by boundary terms since the bulk BRST current does not generate total derivatives when acting on generic off-shell perturbations. In view of this we explore an alternative definition of Q through where we take C a (0) to be the circle around 0 with radius a. The limit a → 0 then ensures that {Q, O(0)} will again be a local operator inserted at 0. This definition seems more natural from the traditional point of view of defining the action of charge as a small contour integral around the operator it acts on. We will show that this equivalent to our previous one in the case of on-shell operators.

On-shell perturbations
We argue on the level of the action. As before Q will act on operators coming from the background, this time only on those which are inside the circle of radius a. Therefore we split the integral along the boundary according to R = E a ∪ [−a, a]: as before.

General off-shell perturbations
For arbitrary off-shell perturbations we could not find an explicit expression for the cohomological vector field. However we will argue that the coefficients are restricted by the resonance condition given in [6]. On-shell we found that the cutoff dependence of the expansion coeffi- of the cohomological vector field is determined by the scaling dimension of the basis vectors O i . We expect this to hold also in the off-shell case. So The cutoff-independent terms are then exactly those satisfying the resonance condition all others should be subtracted via counter-terms. The resonance condition has an important consequence for the equation of motion If φ i is off-shell (h i = 0), then at least one of the φ i k is also off-shell due to (3.6). Thus an off-shell field cannot be sourced by on-shell fields only. A consequence of this is that there are no vertices in BSFT that contain a single off-shell state. Since there are no tree-level Feynman diagrams containing only vertices with two or more internal lines the S-matrices computed with these vertices are the vertices themselves. Thus, the minimal model of BSFT is again that of OSFT. Moreover, we see that classical OSFT is equivalent to BSFT perturbatively, because they predict the same scattering amplitudes.

Conclusions
We were able to show that for deformations of the background in the cohomology of Q 0 , the open string BRST operator without background, the expansion of the cohomological vector field V , of BSFT are given by the S-matrices of perturbative open string theory which, in turn, is the minimal model of open string field theory. Thus, BSFT, when restricted to the cohomology of OSFT realizes the minimal model of OSFT. This clarifies the relation between OSFT and BSFT: in the cohomology of Q 0 they are related by the minimal model map. This also explains why BSFT has no propagator, a fact that was often considered an odd property of BSFT. In addition this equivalence also implies that V is nilpotent in some neighborhood a vanishing background, which is a non-trivial result. The extension of this equivalence can easily be extended to the NS sector of the open superstring.
The inclusion of the Ramond sector is, however, problematic since there is no natural way to assign picture to the background in this sector. It would be interesting to find a consistent generalization of (2.25) to supermoduli space. Another limitation of our approach is that it does not easily allow to give a detailed description of the cohomological vector field off-shell. In particular, we can not guarantee that V squares to zero off-shell and thus whether BSFT does indeed define a consistent off-shell BV action, in some neighborhood of a vanishing background. Progress in this direction would certainly be helpful in order to decide if BSFT can resolve some of the infrared issues in string perturbation theory.

A Cyclic symmetry
In this section we want to explain the well known but important cyclic symmetry property of open string amplitudes.
We begin with the first non-trivial case, the four-point scattering amplitude: Suppose we are given four string states represented by conformally invariant operators O i , where i ∈ {1, ..., 4}. Those can have arbitrary ghost number and hence arbitrary statistics, but we assume that they are multiplied by a string field φ i so that the total degree is equal to one, just in case of ordinary open string field theory. We assume that the punctures, where the operators are inserted, are in a given order on the boundary of the disc. The moduli space of this punctured disc is then one dimensional.
The usual representation of the four-point S-matrix with given cyclic order (1234) is The overall sign of this transformation is +1, because O 1 has to commute with one even and two odd objects. This shows that the four-point S-matrix has indeed cyclic symmetry. We also introduce the symmetric correlator V S , which is just the sum over all cyclically inequivalent correlators (meaning that they cannot mapped to each other by a cyclic permutation). There is a nice and compact way to represent this symmetric correlator. Notice that we have |S 4 /C 4 | = |S 3 | = 6 inequivalent cyclic orderings. Half of them can be written as The integration range splits into three parts: (−∞, ∞) = (−∞, 0] ∪ [0, 1] ∪ [1, ∞). This produces (1,2,3,4), (1,2,4,3) and (1,4,3,2). We can get the other half by adding the same amplitude with index 2 and 3 swapped. Therefore We can again split the integration ranges according to the ordering of the operators (with reference point at infinity). It is then clear that the first half produces all orderings of the tuple (1, 2, ..., n − 1, n) with 1 fixed and 2 always to the left of 3. Hence we can again generate all cyclically inequivalent orderings by adding the amplitude with just 2 and 3 swapped. There is also no overcounting because we fix the position of 1 to be leftmost.