Relation between the reducibility structures and between the master actions in the Witten formulation and the Berkovits formulation of open superstring field theory

Developing the analysis in JHEP03 (2014) 044 [arXiv:1312.1677] by the present authors et al., we clarify the relation between the Witten formulation and the Berkovits formulation of open superstring field theory at the level of the master action, namely the solution to the classical master equation in the Batalin-Vilkovisky formalism, which is the key for the path-integral quantization. We first scrutinize the reducibility structure, a detailed gauge structure containing the information about ghost string fields. Then, extending the condition for partial gauge fixing introduced in the above-mentioned paper to the sector of ghost string fields, we investigate the master action. We show that the reducibility structure and the master action under partial gauge fixing of the Berkovits formulation can be regarded as the regularized versions of those in the Witten formulation.


Introduction
Since the decisive formulation of open bosonic string field theory [1], various attempts have been made to construct manifestly covariant open superstring field theory based on the Ramond-Neveu-Schwarz formalism [2][3][4][5][6][7][8][9][10]. They can be classified into two types, according to the way to treat the Hilbert space of the superconformal ghost sector on the worldsheet [11,12]: the approaches based on the small Hilbert space, and those based on the large Hilbert space. The relation between the two types of approach has recently been JHEP10(2015)127 investigated with the technique of partial gauge fixing, especially in the Neveu-Schwarz (NS) sector [13,14]. In the partial gauge fixing, however, ghost string fields, which are necessary for proper gauge fixing, are not taken into consideration; therefore the scope of the analyses is limited to only a certain aspect of the theories. In particular, we can say little about the relation from the viewpoint of the path integral. 1 The aim of the present paper is to acquire a deeper understanding by elucidating the relation at the level of the master action, namely the solution to the classical master equation in the Batalin-Vilkovisky (BV) formalism [16][17][18][19], which is the key for the path-integral quantization of complicated gauge systems such as string field theory.
Historically, the first manifestly covariant open superstring field theory was formulated by Witten [2]. Based on the small Hilbert space, it is a natural extension of open bosonic string field theory. However, it has the problem of divergences caused by the picture-changing operator inserted at the string midpoint [20]. Among the approaches to overcoming this problem, the Berkovits formulation for the NS sector is remarkable [5]. The theory is constructed without using any picture-changing operators, based on the large Hilbert space. Recently, the present authors et al. have manifested the mechanism of how the problem in the Witten formulation is resolved in this formulation [13]. In the paper [13], we have shown that the action and the gauge transformation in the Berkovits formulation can be interpreted as the regularized versions of those in the Witten formulation, using the technique of partial gauge fixing. Imposed on the condition for partial gauge fixing, the Berkovits action can be written in the form including line integrals of the picture-changing operator, rather than its local insertions; therefore the divergences in the Witten formulation are avoided. Inspired by this line-integral regularization mechanism, Erler, Konopka, and Sachs have constructed a new open superstring field theory with the small Hilbert space approach [10], and its relation to the Berkovits formulation, also, has been analyzed by the use of partial gauge fixing [14]. 2 In these studies on the relation between the small Hilbert space approach and the large Hilbert space approach, however, partial gauge fixing is performed without taking into account ghost string fields, and an understanding from the viewpoint of the path integral is missing. In order to deepen our understanding of the relation, in the present paper we perform a detailed analysis on the NS sector in the Berkovits formulation and compare it with the one in the Witten formulation, extending the condition for partial gauge fixing in ref. [13] to the sector of ghost string fields. We first scrutinize the detailed gauge structure called reducibility structure. It contains the information about ghost string fields, and governs the form of the master action, which is the key for the path-integral quantization in the BV formalism. We then investigate the relation between the master actions in the two formulations. Through the analyses, we show that the reducibility structure and the master action under partial gauge fixing of the Berkovits formulation can be regarded as the regularized versions of those in the Witten formulation.

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The present paper is organized as follows. In section 2, we briefly review the Witten formulation and the Berkovits formulation of open superstring field theory, concentrating on the NS sector. We then explain in section 3 partial gauge fixing of the Berkovits formulation introduced in ref. [13]. After that, extending the condition for partial gauge fixing, we investigate the reducibility structures and the master actions in sections 4 and 5, respectively. Finally, section 6 is allocated for summary and discussion. Two appendices are provided to supply details of the analyses.

The Witten formulation and the Berkovits formulation
In the present section, we review two formulations of open superstring field theory, concentrating on the NS sector. One is the Witten formulation [2], and the other is the Berkovits formulation [5]. The action in the former is constructed in the small Hilbert space, and that in the latter is in the large Hilbert space. We first summarize the basics of the two Hilbert spaces in subsection 2.1; then we briefly review the Witten formulation in subsection 2.2 and the Berkovits formulation in subsection 2.3.

The small Hilbert space and the large Hilbert space
The small Hilbert space and the large Hilbert space are basic concepts in the Ramond-Neveu-Schwarz formalism. In order to see the difference between the two spaces, we fermionize the superconformal ghosts β and γ as in refs. [11,12]: β = e −φ ∂ξ , γ = η e φ . (2.1) The fields ξ and η are fermionic, whereas φ is bosonic.
Here and in what follows, we omit the normal-ordering symbol with respect to the SL(2, R)-invariant vacuum for simplicity, and use the convention in which appropriate cocycle factors are implicitly included, so that e lφ (l ∈ odd) anticommute with fermionic operators. The fundamental operator product expansions (OPEs) of ξ, η, and φ are given by with "∼" denoting the equality up to non-singular terms. In (2.1), the operator ξ, whose conformal weight is zero, is accompanied by the derivative symbol ∂. In fact, we can describe superstring theory, not using the bare ξ. In other words, we can describe it without the zero mode ξ 0 . 3 The superstring Hilbert space containing only the states which can be constructed without ξ 0 is called the small Hilbert space, and the one including also JHEP10(2015)127 the states involving ξ 0 is called the large Hilbert space. If a state A is in the small Hilbert space H small , it is annihilated by the zero mode of η, and vice versa: A ∈ H small ⇐⇒ η 0 A = 0 . (2.4) It follows from the OPEs of ξ and η that the zero modes ξ 0 and η 0 satisfy Therefore any state ϕ in the large Hilbert space can be written in terms of two states A and B in the small Hilbert space as with Thus we could say that the large Hilbert space is twice as large as the small one.
In each of the spaces, there are two important quantum numbers: the world-sheet ghost number g and the picture number p. They are defined by the charges Q g and Q p below: Here we have used the doubling trick, 4 and have denoted by C the counterclockwise unit circle centered at the origin. The ghost number and the picture number of the BRST operator Q, for example, are one and zero, respectively. In the fermionized description (2.1), we have with the BRST current j B given by 5 where T m is the matter energy-momentum tensor and G m is the matter supercurrent. The action of Q upon ξ gives the picture-changing operator [11,12] X :  [24] of states A and B in the small Hilbert space and that in the large Hilbert space, respectively. The inner product in the small Hilbert space, A, B , vanishes unless the sum 4 For the doubling trick, see refs. [22,23], for example. 5 The total derivative term 3 4 ∂ 2 c makes jB primary. of the ghost number of A and of B is equal to three, and the sum of the picture number of A and of B is equal to minus two:

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Here g(A) and g(B) are the ghost number of A and of B, respectively; p(A) and p(B) are the picture number of A and of B, respectively. By contrast, the inner product in the large Hilbert space, A, B , vanishes unless the sum of the ghost numbers is equal to two and the sum of the picture numbers is equal to minus one: The two inner products are related as 6

The Witten formulation
The first formulation of manifestly covariant open superstring field theory was proposed by Witten [2], based on the small Hilbert space approach. It is a natural extension of the cubic open bosonic string field theory [1], with the action composed of the string fields in the natural picture: an NS string field of picture number minus one and a Ramond string field of picture number minus a half. However, it has the problem of divergences caused by the picture-changing operator inserted at the string midpoint [20]. In the present subsection, we review this open superstring field theory and its problem, focusing on the NS sector. The NS-sector action in the Witten formulation, S W , is given by 6 See appendix B of ref. [25] for the reason why the imaginary unit is necessary.

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Here g denotes the open string coupling constant, X mid denotes the picture-changing operator (2.11) inserted at the string midpoint, Ψ W is a Grassmann-odd NS open superstring field of even parity under the Gliozzi-Scherk-Olive (GSO) projection, and the symbol " * " represents the multiplication in the space of string fields [1]. (All the superstring fields to appear in the present paper are GSO even.) For later convenience, we have appended the superscript "W" to the string field in the Witten formulation. The world-sheet ghost number g and the picture number p of Ψ W are +1 and −1, respectively. In what follows, the quantum number (g, p) of a string field will often be indicated by its subscript. For example, the Ψ W will be written also as Ψ W (1,−1) . As is mentioned in subsection 2.1, inner products of the form A, B vanish unless p(A) + p(B) = −2. Therefore, without the insertion of X, which raises picture number by one, the cubic term in the action (2.17) would identically be zero, and the interacting theory could not be described. On the other hand, the very midpoint insertion of X causes the two serious problems: scattering amplitudes are divergent even at the tree level, and gauge transformation is not well-defined [20]. Here we focus on the latter problem, which is related to the main subject of the present paper. The gauge transformation in the Witten formulation is given by where Λ W (0,−1) is a Grassmann-even gauge parameter of ghost number zero and picture number minus one. (Note that in the Witten formulation, all the NS string fields are in the −1 picture.) In the variation of the action under the above transformation, the terms of order g 0 or g 1 vanish, and that of order g 2 takes the form This would be zero if X mid X mid were finite, but the fact is that the OPE of the picturechanging operator with itself is singular: Thus the product X mid X mid , in which two X's collide at the string midpoint, is divergent, and the gauge transformation (2.18) is not well-defined.

The Berkovits formulation
In order to remedy the problems in the Witten formulation, Berkovits has formulated open superstring field theory without using any picture-changing operators [5]. This theory, unlike the Witten one, is constructed in the large Hilbert space. The NS-sector action in the Berkovits formulation, S B , takes the following Wess-Zumino-Witten form: 7 A factor of the imaginary unit in each term is necessary in order for the action to be real. See appendix B of ref. [25].

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Here Φ (0,0) is a GSO-even NS string field whose Grassmann parity is even. It carries no ghost number and no picture number, as is indicated by the subscript (0, 0). In the above equations and in what follows, we omit the multiplication symbol " * " for simplicity, but products of string fields are always defined by Witten's star product.
The operator η 0 , as well as the BRST operator Q, acts as the derivation upon string fields, satisfying In virtue of this, the action (2.21) is invariant under the transformation of the form [26] Note that there are not one but two gauge parameters ǫ (−1,0) and ǫ (−1,1) . As the result of the extension of the superstring Hilbert space, we have larger gauge symmetry in the Berkovits formulation than in the Witten one. Furthermore, the two parameters are in different pictures from each other. In the Witten formulation, all the NS string fields are in the same picture, and the picture-changing operation is realized by X, whereas in the Berkovits formulation, picture numbers of string fields are not fixed to the same value. Nevertheless, as shown in ref. [13], the two formulations are related to each other: if we perform partial gauge fixing in the Berkovits formulation, the resultant action and the residual gauge transformation can be regarded as the regularized version of the action and of the gauge transformation in the Witten formulation.

Partial gauge fixing in the Berkovits formulation
The two formulations of open superstring field theory introduced in the preceding section may look completely different. They are, however, related to each other through partial gauge fixing [13]. By fixing part of the gauge in the Berkovits formulation, we can show that the free theories are equivalent; moreover, in the interacting case, the Berkovits formulation can be interpreted as the regularized version of the Witten one. In the present section, we review this relation. We first explain the basic idea of partial gauge fixing and demonstrate the equivalence of the two formulations for the case of free theory in subsection 3.1. Then we introduce in subsection 3.2 a one-parameter family of conditions for partial gauge fixing, which is useful for the analysis of interacting theory. After that, in subsection 3.3, we investigate the residual gauge symmetry under partial gauge fixing and explain its relation to the Witten gauge transformation (2.18) in more detail and in a more sophisticated manner than in ref. [13].

The basic idea of partial gauge fixing
Let us begin by reviewing the basic idea of partial gauge fixing. For this purpose, we consider the free theories, showing their equivalence. The equation of motion in the free Witten theory is given by

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and that in the free Berkovits theory is given by Because the string field Ψ W (1,−1) in (3.1) is in the small Hilbert space, it satisfies Using this equation and the identity we can rewrite (3.1) as Therefore, for any solution Ψ W (−1,1) to the equation of motion (3.1) in the Witten formulation, we have a solution ξ 0 Ψ W (−1,1) to the equation (3.2) in the Berkovits formulation. In fact, by the use of the gauge transformation, the string field Φ (0,0) in the Berkovits formulation can always be brought to the form where Ψ (1,−1) is some string field depending on Φ (0,0) . Indeed, in the free theory, 8 the gauge transformation (2.24) reduces to and therefore if we consider the transformation specified by the resultant gauge transform takes the form In this manner, setting Φ (0,0) in the form (3.6) corresponds to fixing part of the gauge. The condition for this partial gauge fixing is given by Under this condition, we can show also that in free theory the action in the Berkovits formulation reduces to the gauge-invariant action in the Witten formulation. To see this, let us start with the free Berkovits action 8 For the case of interacting theory, see appendix A.

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When Φ (0,0) is written in the form (3.6), we obtain (In the last equality, we have used the relation (2.14).) Thus (3.12) coincides with the free Witten action under the identification (3.14) In the above argument, we have only used the following properties of ξ 0 : Therefore we may replace ξ 0 with a Grassmann-odd operator Ξ satisfying and may consider the condition Ξ Φ (0,0) = 0 . The relation (2.14) is then generalized to and we can show the equivalence of the two free theories under the partial gauge fixing (3.17) in the same manner as before, identifying Ψ (1,−1) = η 0 Φ (0,0) in the Berkovits formulation with Ψ W (1,−1) in the Witten formulation.

A one-parameter family of conditions for partial gauge fixing
In the preceding subsection, we examined only the free theories. To show their equivalence, we did not have to specify the form of Ξ in (3.17). In the interacting case, however, the choice of Ξ becomes important. A particular type of Ξ helps us to manifest the relation between the two formulations. In the present subsection, we review such useful gauge choices proposed in ref. [13]. We consider a one-parameter family of conditions for partial gauge fixing of the form Here Ξ λ are operators defined by integrals along the counterclockwise unit circle C centered at the origin: (3.21) As is explained in ref. [13], in the limit λ → 0 we have

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From the viewpoint of the state-operator correspondence in the conformal frame on the upper half-plane, the open string lies on the unit upper half-circle centered at the origin, with its midpoint at z = i. Therefore, the operator Ξ λ approaches ξ mid , the midpoint insertion of ξ, as the parameter λ tends to zero: Furthermore, from (3.22) we obtain where X λ is the BRST transform of Ξ λ : Hence X λ becomes X mid , the midpoint insertion of the picture-changing operator, when λ goes to zero: We also note that Ξ λ are BPZ even:

Residual gauge symmetry under partial gauge fixing
The condition (3.19) considered in the preceding subsection eliminates the gauge degrees of freedom in the Berkovits formulation only partially. Therefore, there remains residual gauge symmetry even after the condition is imposed. In the present subsection, we investigate the residual gauge transformation which preserves condition (3.19), which can be regarded as the regularized version of the Witten gauge transformation (2.18), in more detail and in a more sophisticated manner than in ref. [13]. For this purpose, it is convenient to express (2.24) in terms of Φ = Φ (0,0) rather than G. In order to perform this rewriting, we introduce the adjoint operator ad gΦ = g ad Φ , whose action upon a string field A is defined by This operator satisfies From [Φ, G] = 0, for an arbitrary variation of Φ we obtain Therefore, the gauge variation of Φ is given by
Because the residual transformation satisfies (3.32), we have This equation can be solved recursively in η 0 ǫ (−1,1) , and we obtain In the last equality, we have used the fact that the sum in (3.34) converges for small g because the operator is O(g). Substituting (3.34) into (3.31), we find that the explicit form of the residual transformation is given by Thus the residual gauge transformation in the Berkovits formulation is parameterized by a single gauge parameter as the gauge transformation in the Witten formulation is. In fact, the former can be regarded as the regularized version of the latter. To see this, we express Φ as and consider the residual transformation in terms of the string field Ψ = η 0 Φ, which will correspond to Ψ W (1,−1) in the Witten formulation. Expanding (3.36) in g, we obtain Unlike the gauge parameter Λ W (0,−1) in the Witten formulation, which is in the small Hilbert space, the parameter ǫ (−1,0) in the Berkovits formulation moves around the whole of the large Hilbert space. Nevertheless, as shown in appendix B, without loss of generality we may assume that ǫ (−1,0) satisfies the condition (3.39)

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Therefore, ǫ (−1,0) can be written as for some Λ (0,−1) ∈ H small , and the Λ (0,−1) is naturally related to the gauge parameter in the Witten formulation. This can be seen as follows. First, we transform the term Here we have used the relations Then, eq. (3.38) becomes The point is that in the singular limit λ → 0, the operators X λ and Ξ λ in (3.43) effectively become the midpoint insertions X mid and ξ mid , respectively, and we have In the last line, we have used for any pair of string fields A and B. It should be noted that the O(g 2 ) term in (3.44) is divergent. Apart from this O(g 2 ) term, however, the last line of (3.44) precisely coincides with the Witten gauge transformation (2.18) under the identification Thus we find that in this singular limit, the residual transformation (3.38) is naturally related to the Witten gauge transformation. We could say that the O(g 2 ) term plays the role of the counterterm. In fact, the Berkovits theory under the gauge (3.19) can be regarded as the regularized version of the Witten one [13].

JHEP10(2015)127 4 Reducibility structure
In subsection 3.3, we have considered the residual gauge transformation under partial gauge fixing of the Berkovits theory, manifesting its relation to the Witten gauge transformation. Developing our analysis further, in the present section, we investigate more detailed gauge structure called reducibility structure. The structure is important in that it governs quantization procedure. For example, it determines whether the quantization requires not only ordinary ghosts but also additional ghosts such as ghosts for ghosts. We first explain the concept of reducibility in subsection 4.1, and illustrate reducibility structure with open bosonic string field theory in subsection 4.2. We then review the reducibility structure of the Witten formulation in subsection 4.3, and explain that of the Berkovits formulation in subsection 4.4. After that, in subsection 4.5, we examine the relation between the reducibility structures of the two formulations. We find that the reducibility structure of the Berkovits formulation under the partial gauge fixing includes, as a sub-structure, the regularized version of the reducibility structure of the Witten formulation.

The concept of reducibility
In order to explain the concept of reducibility, let us consider a gauge system described by a classical action S = S[ϕ], which depends on a classical field ϕ, and assume that the action is invariant under gauge transformation of the form 9 In the above equation, we have appended the subscript "ǫ 0 " to the variation symbol δ in order to indicate that the variation is parameterized by ǫ 0 . We use a similar notation in what follows in the present subsection (and only in the present subsection). In general, if we change the parameter ǫ 0 as with ǫ 1 parameterizing the variation of ǫ 0 , the form of δ ǫ 0 ϕ changes accordingly. In some theories, however, for some particular choice of δ ǫ 1 ǫ 0 , the gauge variation δ ǫ 0 ϕ is invariant under (4.2) if we use the equation of motion where the symbol "≃" denotes the equality which holds if (but not necessarily only if) we use the equation of motion. Furthermore, in complicated gauge systems, there exists also a variation

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which keeps δ ǫ 1 ǫ 0 invariant under the use of the equation of motion: In general, a gauge system is said to be N -th stage reducible if there exist a series of parameters ǫ k (0 ≤ k ≤ N ) and a series of variations As we see in the following subsections, string field theory is infinitely reducible, with the above N infinity.

Reducibility structure of open bosonic string field theory
To illustrate reducibility structure, let us consider open bosonic string field theory [1], for example. Its action takes the form where Ψ 1 is a Grassmann-odd string field of world-sheet ghost number one. In what follows, the world-sheet ghost number g of a bosonic string field is indicated by a subscript as in Ψ g . The BRST operator and the BPZ inner product in the Hilbert space of the bosonic theory are denoted by the same symbols Q and , , respectively, as in the Berkovits formulation, for simplicity. Let us first examine the free theory. The gauge transformation is given by For convenience, we have appended the subscript "0" to the variation symbol δ. The point is that because Q is nilpotent, δ 0 Ψ 1 does not change under the following variation of the gauge parameter: Moreover, eq. (4.10) is invariant under the variation of the form In fact, there exists a series of variations of parameters. The n-th variation is not affected by the (n + 1)-st variation of the form In other words, we have δ n+1 δ n Λ −(n−1) = 0 (n ≥ 0) . (4.14)

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The Grassmann parity of Λ −n is even (resp. odd) if n is even (resp. odd). Note that the above equation holds without using the equation of motion. This is a feature of the free string field theory. In the interacting theory, however, this is not the case. Let us next see this.
By the presence of the interaction, the gauge transformation (4.9) is replaced with Unlike the free theory, the interacting theory does not have a variation δ 1 Λ 0 which keeps δ 0 Ψ 1 strictly invariant. However, we find that the variation does not change (4.15) if we use the equation of motion Indeed we have which vanishes under the use of the equation of motion. Moreover, eq. (4.16) is invariant under the variation if we use (4.17). In fact, if the equation of motion holds, the n-th variation in the interacting theory with [ , } denoting the graded commutator is not affected by the (n + 1)-st variation of the form We thus find that the open bosonic string field theory is infinitely reducible. It should be noted that it is because g is zero that in the free theory δ n+1 δ n Λ −(n−1) exactly vanish.

Reducibility structure of the Witten formulation
As we have seen in subsection 2.2, open superstring field theory in the Witten formulation has the problem of divergences caused by the collision of picture-changing operators. Apart from such divergences, however, its reducibility structure is quite similar to that of open bosonic string field theory. In particular, the structures of the free theories are exactly the same. Indeed, the gauge variation JHEP10 (2015)127 in the free open superstring field theory does not change under the variation of the form 24) and the n-th variation is not affected by the (n + 1)-st variation where the Grassmann parity of Λ W (−n,−1) is even (resp. odd) if n is even (resp. odd). Let us next consider the interacting NS-sector theory. In this case, there arise divergences caused by the picture-changing operator, but we examine formal gauge structure, neglecting such divergences. In the interacting theory, the gauge transformation is given by As in the case of the bosonic theory, the interacting NS-sector theory does not have a variation δ 1 Λ W (0,−1) which keeps δ 0 Ψ W (1,−1) strictly invariant. However, the variation does not change (4.28) formally if we use the equation of motion Indeed we have which vanishes under the use of the equation of motion if we neglect the divergence caused by the X mid in front of the commutator and that in δS W δΨ W . Similarly, if we use (4.30), the n-th variation is not affected formally by the (n + 1)-st variation JHEP10(2015)127

Reducibility structure of the Berkovits formulation
Now that we have examined the reducibility structure of the Witten formulation, let us next consider that of the Berkovits formulation. Here we will review the structure before partial gauge fixing, following refs. [15,27]. As we will see, the Berkovits theory, also, is infinitely reducible. Let us begin by analyzing the free theory. The gauge transformation in the free theory is given by In the matrix notation, this can be written as In virtue of the relation eq. (4.36) does not change under the variation of the form Moreover, eq. (4.38) is invariant under the variation In fact, there exists a series of variations of parameters. The n-th variation is not affected by the (n + 1)-st variation of the form
As in the case of the free Witten theory, the above equation holds without the use of the equation of motion. Next let us consider the reducibility structure of the interacting theory. For this purpose, it is convenient to introduce the deformed BRST operator [27,28] Q := e −ad gΦ Q e ad gΦ . (4.44) This operator is nilpotent as Q is: However, it does not anticommute with η 0 : for an arbitrary string field A, we have The right-hand side is proportional to the derivative of the action 10 and therefore we obtain Q, η 0 ≃ 0 . In order to investigate the reducibility structure, we start from the following form of the gauge transformation (see (3.31)): Unlike the free theory, the interacting theory does not have a variation of the gauge parameters which keeps δ 0 Φ strictly invariant. Eq. (4.48) tells us, however, that the variation does not change (4.49) if we use the equation of motion

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Indeed, noting we have Furthermore, eq. (4.50) is invariant under the variation if we use (4.51). In fact, if the equation of motion is satisfied, the n-th variation in the interacting theory is not affected by the (n + 1)-st variation of the form Note that in the first matrix on the right-hand side of (4.55), the first row contains Q (not Q) and e ad gΦ η 0 , whereas the others contain Q and η 0 .

Relation between the reducibility structures of the two formulations
In the preceding subsection, we have explained the reducibility structure of the Berkovits formulation before partial gauge fixing. There, starting from (4.49) with the two gauge parameters independent, we have obtained a series of the variations (4.55), which satisfy
Hence the reducibility structure is altered. In the present subsection, we will investigate how it is altered by the partial gauge fixing, and show that it includes, as a sub-structure, the regularized version of the reducibility structure of the Witten formulation. Before beginning the analysis, let us summarize below the results to be obtained, in order to clarify the direction in which we are going.

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Now let us confirm the above. We first investigate how the reducibility structure is altered by the partial gauge fixing (3.19). For this purpose, it is convenient to start from the following form of the residual gauge transformation: with η 0 ǫ (−1,1) depending on Qǫ (−1,0) as in (3.34).
First, let us investigate the consequence of the condition Ξ λ ǫ (−1,0) = 0 . This equation can be solved as follows, in the same manner that we used to obtain (3.34). From (4.72), we have Solving this equation recursively, we obtain

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In the last equality, we have used the fact that the sum in (4.74) converges for small g because the operator − η 0 Ξ λ e ad gΦ − 1 (4.75) is O(g). We thus find that η 0 ǫ (−2,1) can be expressed in terms of Qǫ (−2,0) , and that the explicit form of (4.65) under the constraint (4.72) is given by Here and in what follows, the "sub" on the variation symbol indicates that we consider the reducibility sub-structure specified by (4.61). In order to find the variation δ sub 2 ǫ (−2,0) which satisfies the reducibility relation The proof of (4.80) goes along the same lines as that of (4.66). It follows from (4.80) that we have and therefore if the parameter ǫ (−n,n−1) (n ≥ 2) vanishes from the reducibility structure, so does ǫ (−(n+1), n) . Hence, as a consequence of condition (4.71), all the parameters of the form ǫ (−n,n−1) (∀n ≥ 2) disappear from the reducibility structure.

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Next let us consider the condition Ξ λ ǫ (−2,0) = 0 , (4.83) which can be realized by the use of the degree of freedom of the transformation (4.78). Under this condition, the parameters ǫ (−3,0) and ǫ (−3,1) in (4.78) are not independent any longer. Because the structure of (4.78) is exactly the same as that of (4.65), we can express η 0 ǫ (−3,1) in terms of Qǫ (−3,0) , following the same argument as before. This time, through the condition (4.83), the parameters ǫ (−3,1) , ǫ (−4,2) , ǫ (−5, 3) , and so forth are eliminated from the set of independent parameters. We can proceed our argument in this manner, and in the end we obtain the reducibility sub-structure described only by the parameters ǫ (−n,0) (n ≥ 1) with Ξ λ ǫ (−n,0) = 0 (∀n ≥ 1) . (4.84) The variations of these parameters take the same form as (4.76):  Because the reducibility sub-structure (4.85) of the Berkovits formulation has no singularity for λ = 0, we conclude that it can be regarded as the regularized version of the reducibility structure of the Witten formulation.

Master action in the Batalin-Vilkovisky formalism
Quantization of complicated gauge systems such as string field theory is often performed with the Batalin-Vilkovisky (BV) formalism [16][17][18][19], which is an extension of the BRST formalism. In this formalism, the most important process for gauge fixing is construction of the master action, or the solution to the classical master equation. The equation is

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an extension of the Ward-Takahashi identity, and the point is that given a reducibility structure, we can in principle construct its solution. Taking this into account, we expect from the result in subsection 4.5 that the master action in the Berkovits formulation will be related to that in the Witten formulation. In the present section, we will show that it is indeed the case: the former reduces to the regularized version of the latter after partial gauge fixing. Because superstring field theory in the Witten formulation has the same structure as bosonic string field theory [1], apart from the problem of the picture-changing operator, we can easily obtain its master action S W formally as in the bosonic case [29]. It is given by Here the Ψ W (g,−1) with g ≤ 0 are ghost fields, and those with g ≥ 2 are antighost fields. 11 All of these Ψ's are Grassmann odd. Note that if we neglect the divergences caused by the picture-changing operator, the above action (5.1) is indeed a solution to the classical master equation of the form where δ R and δ L denote the right and the left variation, respectively. We will demonstrate that the master action in the Berkovits formulation reduces to the regularized version of (5.1) after we perform partial gauge fixing and integrate out auxiliary components.

Relation between the master actions in the free theories
Let us begin by considering the free theories, in which the coupling g is equal to zero. In this case, the master action (5.1) in the Witten formulation becomes Noting eq. (2.12), we can rewrite this as with 11 In the present paper, we will not distinguish antighosts and antifields in the BV formalism for simplicity.
In the language of the BV formalism, we will consider only a gauge-fixing fermion such that antifields of minimal-sector fields are identified with antighosts.

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The free master action S B quad in the Berkovits formulation is given in ref. [15]: , Furthermore, as explained in ref. [25], it is invariant under the transformations bellow: where Λ's are gauge parameters. In the rest of the present subsection, we are going to show that S B quad reduces to S W quad under partial gauge fixing for the above symmetry, which is an extension of the original gauge symmetry (4.35). 12 In fact, each S B n (n ≥ 0) reduces to S W n . For showing this, it is convenient to decompose Φ (g,p) as follows: The string fields Φ − (g,p) and Φ Ξ (g,p) are in the small Hilbert space. Note that the subscript (g, p) on them is simply carried over from Φ (g,p) and does not indicate the ghost numbers and the picture numbers of Φ − (g,p) and Φ Ξ (g,p) . We impose the following conditions for partial gauge fixing: It should be noted that this set of conditions coincides with a subset of the conditions for complete gauge fixing of the free master action considered in subsection 2.2 of ref. [15], merely by replacing Ξ λ in (5.13) with ξ 0 . We have already learned in subsection 3.1 that S B 0 reduces to S W 0 under the condition Therefore, what we have to show in the present subsection is the reduction of S B n to S W n for n ≥ 1. Let us first consider the action 1) . (5.16) As can be seen from (5.13), the field Φ (2,−1) does not submit to any constraints, whereas Φ (−1,0) and Φ (−1,1) are subject to the conditions Noting (2.16), we find that the second term on the rightmost side of (5.16) becomes Because Φ Ξ (−1,1) appears only in this term, it acts as a Lagrange multiplier field which imposes Φ Ξ (2,−1) = 0 . After integrating out Φ Ξ (−1,1) , the action S B 1 therefore reduces to In the third equality, we have used (2.16): both Φ − (2,−1) and X λ Φ Ξ (−1,0) are in the small Hilbert space (note (3.42c)), and therefore Φ − Next we consider the action S B 2 . It takes the form

Relation between the master actions in the interacting theories
Now that we have confirmed the correspondence of the master actions in the free theories, let us next manifest the relation between the interacting theories. Unlike the Witten theory, the Berkovits theory remains regular even if the interaction is present, being free from the midpoint insertion of the picture-changing operator. In fact, the master action in the Berkovits formulation, S B , can be interpreted as the regularized version of that in the Witten formulation. In particular, in the singular limit λ → 0 of the partial gauge fixing (5.13), the action S B , which is non-polynomial, reproduces the formal cubic master action ( and Φ − (n+1,−1) survive out of the ghosts Φ (−n,m) (1 ≤ n, 0 ≤ m ≤ n) and the antighosts Φ (n+1,−m) (1 ≤ m ≤ n), with the survivors corresponding to Ψ W (−n+1,−1) and −Ψ W (n+1,−1) in the Witten formulation. As we will see, the interacting theory succeeds to this structure, with the right-hand side of (5.33) receiving O(g) correction: In the process of the reduction of the master action S B quad + S B cubic , these O(g) corrections include terms involving fields which do not appear in the completely reduced action in the free theory. However, substituting the relations (5.34) themselves back for these fields, we will find that such extra terms are of order g 2 , and therefore they can be neglected in our analysis. We will obtain in the end the completely reduced action described only by the fields Φ Ξ (0,0) , Φ Ξ (−n,0) and Φ − (n+1,−1) (n ≥ 1) as in the free theory. In the rest of the present subsection, we confirm what we have summarized above.

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The complete form of the cubic terms of the master action in the Berkovits formulation, S B cubic , was first shown in ref. [28], but there are many other expressions which are related to one another through canonical transformations in the BV formalism. 14 They can in general be divided into four types of term as follows according to the numbers of Q and η 0 [27]: where S B Qη , S B Q , S B η , and S B N are the terms with one Q and one η 0 , with one Q and no η 0 , with no Q and one η 0 , and with no Q and no η 0 , respectively. In fact, S B Qη is exactly the cubic term in the original action S B : Furthermore, the form of S B N is uniquely determined as (For the proof of the uniqueness, see ref. [27].) Here and in what follows, we append the superscript " * " to antighosts for convenience, and consequently Φ Ξ (n+1,−m) and Φ − (n+1,−m) (1 ≤ m ≤ n) will be written as Φ * Ξ (n+1,−m) and Φ * − (n+1,−m) , respectively. The degrees of freedom of canonical transformations in the BV formalism are reflected in S B Q and S B η . Among many different expressions of S B Q and S B η , in the present paper we will use the following one [27]: The advantage of this choice is that S B cubic does not include terms quadratic in an auxiliary field Φ Ξ (−n,m) (1 ≤ m ≤ n). In fact, at any step of the reduction process, a descendant of 14 The sum of S B quad and S B cubic satisfies the master equation (5.9) in the following sense:

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S B quad + S B cubic does not include such terms as long as we neglect O(g 2 ) terms. Therefore we can treat the auxiliary fields simply as Lagrange multipliers as in the free theory analysis.
Our strategy for investigating the relation between S B and S W under the conditions (5.13) (or equivalently (5.14)) is as follows. First, we pick up from S B quad + S B cubic the terms including a first Lagrange multiplier field Φ Ξ (−n,n) (n ≥ 1), which imposes a constraint on Φ * Ξ (n+1,−n) , and then integrate out all of these multipliers. Next, from the resultant action, we pick up the terms including a second Lagrange multiplier field Φ Ξ (−n,n−1) (n ≥ 2), which imposes a constraint on Φ * Ξ (n+1,−(n−1)) , and then integrate out these multipliers. Continuing this process, we lastly obtain the completely reduced action of S B , and compare it with S W .
Following the above strategy, let us begin by examining the terms including a first Lagrange multiplier field Φ Ξ (−n,n) (n ≥ 1): where (5.40a), (5.40b), and (5.40c) are the contributions from S B quad , S B Q , and S B η , respectively. There are no contributions from S B Qη and S B N . Using the decomposition (5.11), the conditions (5.13), and the BPZ evenness of Ξ λ , we can express the sum of the above terms as Thus we find that Φ Ξ (−n,n) imposes the constraint where the factor η 0 Ξ λ acts as a projector into the small Hilbert space. Applying this constraint to the fields Φ * (2+n+b, −1−n−b) and Φ * (n+1, −n) , which are on the right-hand side of (5.42), we have

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Here and in what follows, the symbol δ n,m denotes the Kronecker delta: δ n,m := 1 for n = m 0 for n = m . (5.44) (In the first equality of (5.43b), we have used (5.14b).) Substituting these back into (5.42), we obtain Note that the range of the summation for a got narrower: the index runs from one, not zero. As can be seen from (5.45), after the Lagrange multiplier Φ Ξ (−n,n) is integrated out, cubic terms including Φ * Ξ (n+1,−n) become O(g 2 ) and can be neglected in the present analysis. Instead, part of the quadratic term i Φ * (n+1,−n) , QΦ (−n,n−1) in S B quad contributes to terms of order g. Indeed we have , QΦ (−n,n−1) (5.46) with eqs. (5.14) and (5.45) imposed, and the second term on the right-hand side is of order g.
Next let us examine the terms including a second Lagrange multiplier field Φ (−n,n−1) (n ≥ 2). Among these terms, those in S B Q do not contribute in the present analysis because they are of order g 2 owing to the constraint (5.45). Below is what can be relevant: out Φ Ξ (−n,n−1) (n ≥ 2), we obtain the constraint The crucial point here is that all the terms in (5.47) are linear in Φ (−n,n−1) , as was mentioned below (5.39). If we did not have (5.43a) and could not neglect the a = 0 term in (5.42), eq. (5.47d) would include the term which is quadratic in Φ (−n,n−1) , in which case we could not treat Φ Ξ (−n,n−1) as a Lagrange multiplier.

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In the second equality, we performed the deformation similar to that we did in obtaining (5.51) from (5.48). Note the ranges of the summations for a, k, and l. The constraint (5.55) tells us that after the Lagrange multiplier Φ Ξ (−n,n−2) is integrated out, cubic terms including Φ * Ξ (n+1,−(n−2)) become O(g 2 ) and can be neglected, but the quadratic term i Φ * (n+1,−(n−2)) , QΦ (−n,n−3) in S B quad includes an O(g) part: i Φ * (n+1,−(n−2)) , QΦ (−n,n−3) = i δ n,3 Φ * − (4,−1) , QΦ (−3,0) with (5.14) and (5.55) imposed. Furthermore, if we take account of (5.55), eqs. (5.53) and (5.51) become as follows: In this manner, as we integrate out Lagrange multipliers, cubic terms including Φ * Ξ (n+1,−m) (1 ≤ m ≤ n) drop away, and S B quad generates O(g) terms; moreover, the ranges of the infinite series in the constraints become narrower step by step. The ultimate forms of the constraints are given by Thus, among the fields Φ * Ξ (n+1,−m) , only Φ * Ξ (n+1,−1) are relevant in our analysis. We finally find that the completely reduced action S B red takes the form with the conditions (5.14) and the constraints (5.59) imposed. The first two terms on the right-hand side are nothing but the quadratic term and the cubic term in the original action S B . The relation between the original actions S B and S W has already been manifested in the previous paper [13]: under the partial gauge fixing (5.15), the action S B can be regarded as the regularized version of S W , and we have Therefore, in what follows, we concentrate on the third term on the right-hand side of (5.61). It can be written as After the same calculation as performed in (5.20), the first term on the right-hand side reduces to the following form: Eq. (5.64) is equal to the sum of the quadratic terms (5.6b) under the identification which is equivalent to (5.32). Furthermore, the second term on the right-hand side of (5.63) can be deformed as

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In the second equality, we have used (5.59). Rewriting the cubic term in (5.1) as we find that in the singular limit λ → 0, eq. (5.67) coincides with the cubic term in the master action S W minus the one in the original action S W up to O(g 2 ) terms under the identification (5.66), namely (5.32). We thus have Because the Berkovits formulation is regular regardless of the presence of the interaction, from the above result we can conclude that S B is the regularized version of S W , with the terms higher order in g in S B playing the role of the counterterms for canceling the divergences in the Witten formulation.

Summary and discussion
For the purpose of acquiring a deeper understanding of the relation between the small Hilbert space approach and the large Hilbert space approach to open superstring field theory, in the present paper we have investigated the Berkovits formulation in detail with the technique of partial gauge fixing, and have clarified its relation to the Witten formulation at the level of the reducibility structure and the master action. In particular, the master action in the Berkovits formulation has turned out to reduce to the regularized version of that in the Witten formulation after partial gauge fixing. As shown in subsection 4.5, behind this relation is the correspondence between a reducibility sub-structure of the Berkovits formulation and the reducibility structure of the Witten formulation. In general, the form of a reducibility structure governs that of a master action. For our analysis of the master action in the Berkovits formulation, it was sufficient to investigate its quadratic terms and cubic terms. In fact, its higher-order terms have not been completely obtained yet. 15 We expect, however, that our result will be useful also for solving this problem. In order to see the point in which the difficulty lies, let us review the way to construct the master action in general. In principle, one can construct a master action S, namely a solution to the classical master equation in the BV formalism systematically in the following manner. First, one expands S in what is called the antifield number: 1) 15 For some approaches to this problem, see refs. [27,30].

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where S (n) (n ≥ 0) denotes the sum of all the terms of antifield number n, with S (0) coinciding with the original action S. Consequently, the master equation is decomposed into its subequations. Then, using the subequations, one can determine S (N +1) (N ≥ 0) if one knows the form of the reducibility structure and the actions S (0) , S (1) , . . . , S (N ) . Thus, given an S (= S (0) ) and a reducibility structure, all the S (n) 's can be obtained one by one. In some theories such as Yang-Mills theories, only a finite number of S (n) 's are nonzero: In this case, one can solve the master equation completely, merely by carrying out the above-mentioned procedure. There are, however, other theories in which (6.2) does not hold: no matter how large n 0 ≥ 0 one may take, there exists some n (≥ n 0 ) satisfying S (n) = 0, and hence the procedure cannot terminate at any finite step. In fact, string field theory is exactly one of such complicated gauge theories. A strategy to find a solution in this class of theories is as follows.
2. Second, from the form of N n=0 S (n) , one infers the complete solution S.

Third, one confirms that the inferred S satisfies the master equation.
In open bosonic string field theory [1], the second step is readily performed, and the above strategy works successfully. In fact, the solution S is of the same form as the original action S: one can obtain S merely by eliminating the world-sheet ghost number constraint on the string field in the original action [31][32][33][34][35]. It should be noted that behind this result is the mathematical structure called A ∞ [36][37][38][39][40][41]. 16 In the Berkovits formulation of open superstring field theory, however, this kind of structure is obscure; 17 as shown in refs. [15,25,27,28], the solution cannot take the same form as the original action, and it is difficult to infer the complete solution. In this situation, crucial it can be how one performs the first step of the above strategy. One may think that whatever calculation procedure one may adopt, the resultant S (n) 's are the same, and that the more S (n) 's one obtains, the easier it will become to infer the complete solution. The fact is, however, that the result does depend on the manner in which one performs the calculation: there are degrees of freedom of canonical transformations in the BV formalism, which are essentially the degrees of freedom of field redefinition, and therefore the solution to the master equation is not unique. Thus, what form of a solution one obtains depends on how one calculates. In particular, what expression of the reducibility structure one starts from is very important. The point is that only certain forms of partial solutions may be appropriate to infer the 16 Closed bosonic string field theory has another mathematical structure, L∞, and its master action can also be readily obtained in virtue of this structure [42][43][44]. 17 Recently, Erler, Konopka, and Sachs have formulated a new open superstring field theory which possesses a manifest A∞ structure [10]. In this theory, the master action can be constructed trivially, but is difficult to treat because of the existence of complicated non-associative multi-string products. By contrast, the master action in the Berkovits formulation, if it is constructed, must be easy to treat, formulated only in terms of Witten's star product.

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complete solution. We expect, however, our result in the present paper will be useful for approaching this problem. Starting from the reducibility structure of the expression (4.55) and adopting the calculation procedure which keeps the relation to the Witten formulation manifest, we will be able to fix most of the degrees of freedom of canonical transformations.

Acknowledgments
The work of S. T. was supported in part by the Special Postdoctoral Researcher Program at RIKEN.
A Properness of the condition (3.17) for partial gauge fixing In the present appendix, we prove the properness of the condition (3.17) for partial gauge fixing: we show that for any string field Φ = Φ (0,0) , there exists a gauge transform Φ + δΦ which satisfies Ξ Φ + δΦ = 0 . that is, Then, we can solve (A.5) recursively as In the last equality, we have used the fact that the sum in (A.6) converges for small g because the operator is O(g). Thus we have completed the proof of the properness. and thus it contains two components: η 0 Ξ λ ǫ (−1,0) in H small and Ξ λ η 0 ǫ (−1,0) in Ξ λ H small . Nevertheless, all the possible gauge transformations of the form (2.24) can be covered only by the latter component of ǫ (−1,0) and the other gauge parameter ǫ (−1,1) , which moves around the whole of the large Hilbert space. Hence, as far as the gauge transformation is concerned, without loss of generality we may assume ǫ (−1,0) ∈ Ξ λ H small , that is, In the present appendix, we prove the above claim. We will show that if we take into account what is called trivial gauge transformations, we can indeed impose condition (B.2), namely (3.39).

B.1 Trivial gauge transformations
Before proving the properness of (B.2), in the present subsection we will explain the concept of trivial gauge transformations because it plays a crucial role in our proof. In order to explain the concept, let us consider a system described by a classical action S = S[ϕ], which depends on classical bosonic fields ϕ i , with the index i distinguishing different fields. 18 The action S is invariant under transformations of the form variations δ trv ϕ i become identically zero when we use the equations of motion. We would like to remark that the degrees of freedom of trivial gauge transformations exist even in non-gauge theories.
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