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 JHEP 03 (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
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.
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 normalordering 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 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 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: 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 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 := Q · ξ = e φ G m + c∂ξ + b∂η e 2φ + ∂ bη e 2φ , (2.11) which raises picture number by one. We list the ghost number g and the picture number p, together with the conformal weight h, of various operators in table 1. 3 In the present paper, an operator O of conformal weight h is expanded in the coordinate z on the upper half-plane as O(z) = n On z n+h .

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 6 See appendix B of ref. [25] for the reason why the imaginary unit is necessary.
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 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 picture-changing 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 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 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].
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 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 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 : (g, p) = (−1, 1) , ξ 2 0 = 0 , {ξ 0 , η 0 } = 1 . 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: As is explained in ref. [13], in the limit λ → 0 we have 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 Now suppose that Φ obeys condition (3.19). In order for the gauge transform of Φ to keep the condition, the variation (3.31) has to satisfy Ξ λ δΦ = 0 . Under this constraint, the gauge parameters ǫ (−1,0) and ǫ (−1,1) are not independent any longer. As a matter of fact, we can express η 0 ǫ (−1,1) in terms of Φ and Qǫ (−1,0) as follows. 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 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 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 (1 + η 0 Ξ λ )Qǫ (−1,0) in (3.38) as 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 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].

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 For such δ ǫ 1 ǫ 0 , we have 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 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 As we see in the following subsections, string field theory is infinitely reducible, with the above N infinity. 9 For simplicity, we consider a gauge system described by a single field ϕ and a single gauge parameter ǫ0.

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 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 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 in the free open superstring field theory does not change under the variation of the form and the n-th variation is not affected by the (n + 1)-st variation Thus we have 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 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 is not affected formally by the (n + 1)-st variation

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 In other words, we have The Grassmann parity of ǫ (−n,m) (0 ≤ m ≤ n) is even (resp. odd) if n is even (resp. odd). 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 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 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 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 For example, the variations δ 1 ǫ (−1,0) and δ 1 ǫ (−1,1) are given by (see (4.50)) If once the partial gauge fixing is performed, however, the parameters ǫ (−1,0) and ǫ (−1,1) are not independent any longer: η 0 ǫ (−1,1) can be expressed in terms of Qǫ (−1,0) as in (3.34). 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.
Here and in what follows, we append a hat to a variation symbol when we consider a variation under the partial gauge fixing. In order to find the variation δ 1 ǫ (−1,0) which satisfies the reducibility relation let us try performing on ǫ (−1,0) the transformation as in (4.58a). Then, through the relation (3.34), η 0 ǫ (−1,1) is transformed accordingly, with its variation δ 1 η 0 ǫ (−1,1) different from (4.52b); hence δ 1 δ 0 Φ does not coincide with (4.53). Nevertheless, eq. (4.65) does lead to the desired relation (4.64). The point is that the variation δ 1 η 0 ǫ (−1,1) induced by (4.65) and the δ 1 η 0 ǫ (−1,1) in (4.52b) is effectively the same: In virtue of this relation, we obtain We can indeed confirm (4.66) as follows, using eqs. (3.34), (4.65), (4.52b), and (4.48): We thus find that the variation (4.65) is of an appropriate form. Because the parameters ǫ (−2,0) and ǫ (−2,1) in (4.65) are independent, we can continue the analysis on the reducibility structure in exactly the same manner as in subsection 4.4. The result is that the expression of the n-th variation is the same as (4.55) except that the parameters ǫ (−n,n) and ǫ (−(n+1),n+1) do not appear: Thus all the parameters of the form ǫ (−n,n) (n ≥ 1) has disappeared from the reducibility structure, and we have  Solving this equation recursively, we obtain 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 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.

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 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 where 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 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 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 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 The field Φ (3,−1) does not obey any conditions, but the others do: Noting (2.16), we realize that the last term on the rightmost side of (5.22) becomes 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.
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 sum of S B quad and S B cubic satisfies the master equation (5.9) in the following sense: 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 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 Here and in what follows, the symbol δ n,m denotes the Kronecker delta: (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 i Φ * (n+1,−n) , QΦ (−n,n−1) = i δ n,1 Φ * − (2,−1) , QΦ (−1,0) + i Ξ λ Φ * Ξ (n+1,−n) , 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: where the terms (5.47a), (5.47b), (5.47c), and (5.47d) are the contributions from S B quad , S B N , S B η , and (5.46), respectively. Note that S B N contributes only for n = 2. After integrating 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. Applying the constraint (5.48) to the fields Φ * (n+1, −(n−1)) , Φ * (3+n+b, −1−n−b) , and Φ * (n+2, −n) , which are on the right-hand side of (5.48), we have Substituting these back into (5.48) and using the relation Ξ λ QΞ λ = Ξ λ X λ , we obtain Now the ranges of the summation for a and that for l became narrower. Because of the relation (5.51), cubic terms including Φ * Ξ (n+1,−(n−1)) become O(g 2 ) after we integrate out the Lagrange multiplier Φ Ξ (−n,n−1) , and can therefore be neglected in our analysis. Instead, part of the quadratic term i Φ * (n+1,−(n−1)) , QΦ (−n,n−2) in S B quad contributes to terms of order g: i Φ * (n+1,−(n−1)) , QΦ (−n,n−2) = i δ n,2 Φ * − (3,−1) , QΦ (−2,0) + i Ξ λ Φ * Ξ (n+1,−(n−1)) , QΦ (−n,n−2) (5.52) with eqs. (5.14) and (5.51) imposed. It should be added that after (5.51) is taken into account, eq. (5.45) becomes  Let us go one more step further. We examine the terms including a third Lagrange multiplier field Φ (−n,n−2) (n ≥ 3). Owing to (5.53) and (5.51), those in S B Q do not contribute to terms of order g, and S B N contributes only for n = 3. Therefore, what can be relevant is where the terms (5.54a), (5.54b), (5.54c), and (5.54d) are the contributions from S B quad , S B N , S B η , and (5.52), respectively. Integrating out Φ Ξ (−n,n−2) (n ≥ 3), we obtain the constraint In the second equality, we performed the deformation similar to that we did to obtain (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) + i Ξ λ Φ * Ξ (n+1,−(n−2)) , QΦ (−n,n−3) (5.56) 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: where we have defined ψ n (n ∈ Z) by 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 In the second equality, we have used (5.59). Rewriting the cubic term in (5.1) as

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 15 For some approaches to this problem, see refs. [27,30].
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.
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 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 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 is O(g). Thus we have completed the proof of the properness.
B Proof of the properness of condition (3.39) Because the gauge parameter ǫ (−1,0) in the Berkovits formulation resides in the large Hilbert space, it can be decomposed as 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 2. Then we show that the gauge transformation specified by the parameters ǫ (−1,0) and ǫ (−1,1) and the one specified by their transforms ǫ (−1,0) + δ 1 ǫ (−1,0) and ǫ (−1,1) + δ 1 ǫ (−1,1) differ merely by a trivial gauge transformation, and therefore the two gauge transformations are equivalent.
The second part ensures that all the genuine gauge transformations can be generated by ǫ (−1,0) + δ 1 ǫ (−1,0) and ǫ (−1,1) + δ 1 ǫ (−1,1) . Therefore, together with the first part, we can conclude that all the possible gauge transformations of the form (2.24) can be covered by the ǫ (−1,0) satisfying (B.2) and the other parameter ǫ (−1,1) . It should be noted that in the following analysis, the string field Φ is not subject to any constraints, such as the condition (3.19) for partial gauge fixing.