Tree-level S-matrix of superstring field theory with homotopy algebra structure

We show that the tree-level S-matrices of the superstring field theories based on the homotopy-algebra structure agree with those obtained in the first-quantized formulation. The proof is given in detail for the heterotic string field theory. The extensions to the type II and open superstring field theories are straightforward.

After the pioneering work by Witten [1], research on the superstring field theory was falling into a long period of stagnation, except for a few important developments [2][3][4][5][6]. Recently, however, several important progress has been made one after another [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], and several complete superstring field theories are now established. Now, there are three complementary formulations, each of which has advantages and disadvantages: the Wess-Zumino-Witten (WZW) -like formulation, the formulation based on the homotopy-algebra structure, and the formulation accompanied with an extra free field. The WZW-like formulation was first proposed for the open superstring by Berkovits in his ingenious paper [4] and afterward extended to the heterotic string field theory [5,6]. Although both of them were originally limited to the NS sector, but have recently been extended to a complete form including the Ramond sector [16,24]. Attempts to construct a WZW-like action for the type II superstring field theory are also being made [22,25]. The homotopy-algebra-based formulation, the open superstring field theory with an A ∞ structure [9], and the heterotic and type II superstring field theories with an L ∞ structure [10], was pioneered by the Munich group. Also in this formulation, all the constructions were initially limited to the NS or NS-NS sector, but soon have been extended to those including the Ramond sector, and now completed [17,24,25].
The formulation accompanied with an extra free field has been developed by Sen for the heterotic and type II superstring field theories [12,15]. In this formulation, a pair of Ramond string fields are introduced, which double the degrees of freedom but half of them is cleverly decoupled from the physical world as a free field. It has been shown that the formulation can also apply to the open superstring field theory [18].
In this paper, we consider the homotopy-algebra-based superstring field theories and show that their tree-level physical S-matrices agree with those calculated by the first-quantized method [26,27]. In the bosonic string theory, the amplitude for each process is given by the integration over the moduli space of punctured Riemann surface. The string field theory provides a triangulation of the moduli space, each region of which is filled with the contribution from a Feynman diagram. Each contribution is necessary to be connected smoothly at the boundaries so that the sum is an integral over the entire moduli space. It is well known that this requirement is essentially equivalent to requiring the action to be gauge invariant [28][29][30][31]. For the superstring field theories, on the other hand, the amplitudes are given by the integral over the super moduli space, so we must also take into account the contribution of the odd moduli integration. It can be incorporated by insertions of the picture changing operator (PCO) [27], which apparently seems to disturb the smooth connection between contributions from Feynman diagrams. However, the gauge-invariant action constructed by utilizing the homotopy algebra structure also includes terms that may fill the gap as contributions from the vertical integration [32]. As the result, the superstring field theories with homotopy algebra structure reproduce the S-matrices obtained in the first-quantized formulation, which the purpose of this paper is to prove.
The study along this direction was previously performed for several tree-level four-string amplitudes: four-NS string amplitude in the open superstring [9], and arbitrary four-string amplitudes in the heterotic string field theory [24], and some typical four-string amplitudes in the type II superstring field theory [25]. 1 In addition to these confirmations by explicit calculation, a general proof for the tree-level S-matrix was also given based on the minimal 2.1 Heterotic string field theory with cyclic L ∞ structure We start by briefly summarizing how the heterotic string field theory based on the L ∞ structure is constructed. The first-quantized heterotic string theory is obtained by combining the leftmoving (holomorphic) superconformal field theory and the right-moving (anti-holomorphic) bosonic conformal field theory. The former consists of a matter sector with c = 15 , the fermionic (conformal) ghosts (b, c) , and the bosonic (superconformal) ghosts (β, γ) . The latter consists of a matter sector with c = 26 , and the fermionic ghosts (b, c). It is useful to 'bosonize' the bosonic ghosts (β, γ) to a pair of fermions (η, ξ) and a chiral boson φ . The Hilbert space of the bosonized ghosts (η, ξ; φ) is called the large Hilbert space H l . In contrast, the Hilbert space of the bosonic ghosts (β, γ) is called the small Hilbert space H s , which can be embedded in H l as states Φ satisfying ηΦ = 0 with a fixed picture number. 2 The subspaces with different picture numbers provide equivalent representations of H s , and can be related to each other by one-to-one mapping using PCO. The heterotic string field takes a value in H s as described below in detail.
The heterotic string field Φ is Grassmann even and satisfying the closed string constraints: It has ghost number 2 and two components where H N S (H R ) is the small Hilbert space of the NS (Ramond) sector with picture number −1 (−1/2). The Ramond Hilbert space H res R is further restricted by an extra condition: where X and Y are defined by The operator X is the PCO on states with picture number −3/2 , and commutative with the BRST operator Q . In the large Hilbert space, it can be written as the BRST exact form Here, Π n is the projection operator onto the states with picture number n . The operator Y acts on states with picture number −1/2 as an inverse of X in the sense that it satisfies For notational simplicity, the zero-mode of η-ghost appearing frequently is simply denoted as η .
We can show that XY used to define H res is a projection operator acting on the sates with picture number −1/2. The ghost number of the string field is equal to the basis states for the classical string field. For later use, it is useful to introduce a notation with the projection operator π 0 (π 1 ) onto the NS (Ramond) component. Then, the restricted Hilbert space H res can concisely be written as The restricted Hilbert space H res is closed under the action of the BRST operator: XY QXY = QXY , and any state B ∈ H res can be expanded in the ghost zero-mode as where G = G 0 + 2γ 0 b 0 . In particular, we denote the string field Φ ∈ H res as We can define three symplectic forms for the large, small and restricted Hilbert space, ω l , ω s and Ω by For Φ 1 , Φ 2 ∈ H res , they are related as We also use their bilinear map representation defined by and (2.14) Note that the natural inner product in H res defined by Ω has the off-diagonal form after integrating out the ghost zero-modes. 3 The action of heterotic string field theory with cyclic L ∞ structure is written as which is invariant under the gauge transformation Here, L 1 = Q and the multi-string products L n+2 (Φ 1 , · · · , Φ n+2 ) are graded commutative, which satisfy the L ∞ relations and the cyclicity condition The symbol σ in (2.18) denotes the permutation from {1, · · · , n} to {σ(1), · · · , σ(n)} and the factor ǫ(σ) is the sign factor of permutation of string fields from {Φ 1 , · · · , Φ n } to {Φ σ(1) , · · · , Φ σ(n) } . The set of the string products satisfying these conditions is called a cyclic L ∞ algebra (H res , Ω, {L n }). For constructing the string products of the cyclic L ∞ algebra, we use the coalgebra representation, which is, for example, summarized in [24,38]. The heterotic string products are represented by the (Grassmann) odd coderivation L = Q + L int acting on the symmetrized tensor algebra SH res = ⊕ ∞ n=0 (H res ) ∧n . However, it is not very good idea to directly construct the cyclic L ∞ algebra (H res , Ω, L) since Ω is asymmetric between the NS and Ramond sectors. Instead, we considered, in [24], an cyclic L ∞ algebra (H l , ω l , Q − η + B) first. The coderivation B acting on SH l = ⊕ ∞ n=0 (H l ) ∧n is constructed so that an extended generating function, hold, which reduce at (s, t) = (0, 1) to the L ∞ relation for the coderivation Q − η + B . We have shown that the coderivation L representing the heterotic string products can be obtained from B as Here, π n is the projection operator π n : SH res → (H res ) ∧n . (2.26) The invertible cohomomorphismF −1 is defined by with π 1 1 = π 1 π 1 (π 0 1 = π 0 π 1 ), and I ∈ SH res is the (multiplicative) identity of the symmetrized tensor algebra. Using a property of cohomomorphismŝ 28) and the relation π 1F = π 1 I + Ξπ 1 1 l , (2.29) obtained by actingF from the right of both sides of (2.27), we can rewrite the second equation in (2.25b) as a self-consistent equation, which we will use later: . (2.30) Finally, we note that the bracket [ , ] 1 or [ , ] 2 can also be defined by projecting the intermediate state onto the NS or Ramond state, respectively. For example, for coderivations A n+1 and B m+1 , we find that with π 1 B n+2 (s, t) = π 1 B(s, t)π n+2 , π 1 λ n+2 (s, t) = π 1 λ(s, t)π n+2 and π(s) = π 0 + sπ 1 . We will use these forms of the relations later.

S-matrix generating function
In the quantum field theory, a perturbative amplitude is conventionally calculated utilizing the Feynman's method: drawing the possible (connected and amputated) Feynman diagrams for a specific process, combining the propagators and vertices, and evaluating them with integrating the loop momentum (momenta) if it is the one at the loop level. Also for the open [9], heterotic [24], and type II [25] superstring field theories, we have already calculated various tree-level fourpoint amplitudes based on the homotopy-algebra-based formulation. 4 In this section, we discuss all the (tree-level) amplitudes collectively by considering the S-matrix generating functional at the tree level. We first quantize the theory following the BV formalism. From the off-diagonal form of the inner product (2.15) in H res , we can identify φ = φ N S + φ R and ψ = ψ N S + ψ R in (2.10) to the field and anti-field of the BV formalism in the gauge-fixed basis [41]. In this basis, the classical BV master action has the same form as the classical action (2.16), but Φ is now the quantum string field including the states in H res with all the ghost numbers. 5 The Siegel-Ramond (SR) gauge is obtained by simply setting ψ = 0 . We call the Hilbert space of the quantum field in this gauge H SR : Φ = φ ∈ H SR . Then the gauge fixed action and the n-point vertices for the Feynman rules to calculate amplitudes. Here, however, instead of considering each amplitude independently, we consider the generating functional of S-matrix elements which 4 Several four-and five-point amplitudes have also been calculated by the WZW-like open superstring field theory [33,34]. 5 We use the same symbol for the classical and quantum string fields and their Hilbert spaces, but it would not be confused. The ghost number of the quantum field remains two, which is defined by the sum of two kinds of ghost numbers: the ghost number of basis states and the ghost number of coefficient space-time fields (space-time ghost number). 6 This is a schematic expression. See Ref. [31] for precise treatment.
can be obtained by evaluating the effective action at its stationary configuration [36]. At the tree-level, in particular, it reduces to the classical action evaluated at the classical solution and is calculated as follows.
Let us first define the on-shell subspace, 39) and the BRST invariant projection operator, Then, we introduce an operator which defines a Hodge-Kodaira decomposition of H res , This Q + is called a (contracting) homotopy operator of Q , which is compatible with Ω : Under the SR gauge condition ψ = 0 , H t = ∅ and the quantum field φ is decomposed to the on-shell and off-shell subspaces: with The classical equation of motion in the SR gauge can now be written in the form of an integral equation: At the tree level, the S-matrix generating functional is given by evaluating the classical action at the solution of the equation (2.48). Its explicit expression can be found using the coalgebra representation as follows. Define the homotopy, projection and identity operators, H ,P , andÎ by which act on the symmetric tensor algebra SH SR and satisfy the relationŝ Using Q + φ = 0 and the general property of the coderivation, we can show that Eq.
The solution φ cl of this equation can be solved for φ 0 as 7 and thus The S-matrix generating functional is a functional of φ 0 obtained by evaluating the gauge fixed action (2.36) at the classical solution (2.53), and is given by as is derived in Appendix B. This can also be represented by using the multi-linear map 8 7 The relation (2.52) shows that the map i ′ ≡ (Î − HL int ) −1P (deformed inclusion map in the language of HPT) is a cohomomorphism. This was first shown in Ref. [39] for the first couple of orders in L int , and subsequently proven in Ref. [40]. We also provide an independent proof in Appendix A. 8 Note that the quadratic term in S[φ 0 ] vanishes due to the on-shell property of φ 0 .
The amplitude for (n + 3)-string scattering can be obtained as where φ i ∈ H 0 are the wave functions (the vertex operators in the first-quantized formulation) of external string states . We can discuss various (n + 3)-string scattering amplitudes together using S n+3 | (or all the scattering amplitudes using S| ). As an example, the various four-string scattering amplitudes are represented by Taking into account the fact that the output of the product π 1 L satisfies the closed string constraints in Eq. (2.1) and has off-shell momentum for inputs with generic momenta, we can rearrange it as from which we can see that the specific amplitude for each process agrees with that calculated by using the Feynman rules in Ref. [24]. The S-matrix considered here is the total S-matrix in the covariant BRST formalism. The physical S-matrix is obtained by projecting it (or equivalently the external states) onto the physical subspace H Q ⊂ H 0 defined by the (relative) BRST cohomology [42][43][44][45]: (2.60) Decoupling of unphysical states from the physical S-matrix is guaranteed by The last equality follows from the fact that the internal states are generically off-shell.
Here, we have derived the S-matrix generating functional (2.54) following the physical consideration given in Ref. [36]. However, it can also be obtained as an (almost) minimal model of the the cyclic L ∞ algebra (H res , Ω, L) by means of HPT [46,47]. This alternative derivation is given in Appendix A.

Evaluation of the S-matrix
Using the multilinear representation (2.55a), we can show that all the tree-level amplitudes of the heterotic string field theory agree with those calculated in the first-quantized formulation.
Let us define two maps on the symmetrized tensor algebra SH res : Here, as shown in Appendix A, the first mapî ′ is a cohomomorphism appeared as a deformed chain map in the HPT, and thus determined by its component π 1î ′ : SH res → H res . Another map Σ is related with S int in (2.54b) and S| in (2.55a) as We can deduce a relation, between two maps. Then, from the recursive relation (2.30), we have the classical Dyson-Schwinger equation We extend the Σ with two parameters t and s counting the picture number and picture number deficit, respectively, so as to satisfy the generalized Dyson-Schwinger equation with G(s, t) = π 0 + (tX + s)π 1 is determined to satisfy ∆(0, 1) = ∆ and Then, the derivatives of the extended map Σ(s, t) satisfy We can find from the definition that the extended S-matrix S(s, t)| defined by can be expanded as 9 The superscripts with (·) and [ · ] indicate the picture number and the picture number deficit, respectively. Thus, S n+3 | = S n+3 (0, 1)| is the sum of the amplitudes of several processes as For example, S 4 | is the sum of three amplitudes (2.77) The first and third terms are four-R and four-NS string scattering amplitudes, respectively. The second term is further the sum of two expressions of the two-R-two-NS scattering amplitude, depending on whether the output (of Σ) is the NS or R state: Each amplitude can be calculated by solving the Dyson-Schwinger equation (2.65). We find which agree with those calculated in Ref. [24], and an alternative expression of the two-NS-two-R amplitude, It is not difficult to prove that the physical S-matrix agrees with that calculated in the firstquantized formulation. From (2.71) we find that the extended S-matrix satisfies the relation Then, using the expansion in (2.74b), we can find the relation between components with different picture numbers: The explicit form of the right hand side is not important since it does not contribute to the physical S-matrix thanks to the fact that QP = ηP = 0 . By using this relation repeatedly, we can find the physical amplitude can eventually be written as In this final form, the PCOs in an amplitude are acting on the external states in a way that is common to all the Feynman diagrams, so each amplitude is written as integral of a smooth function (section) over the whole moduli space including external states accompanied by PCOs.
If we further note that the differences in PCOs we use and in the states they act can be written in the form of [Q, [η, * ]] , 10 we can replace X 0 s with the local ones X(z)s so that all the external NS and R states have the picture number either −1 or 0 and −1/2 or −3/2 , respectively. Then, we can conclude that the heterotic string field theory reproduces the tree-level S-matrix calculated in the covariant first-quantized formulation [26].

Extension to the type II superstring field theory
In the previous section, we show that the tree-level physical S-matrix of the heterotic string field theory agrees with that calculated in the conventional first-quantized formulation. This proof is easily extended to the type II superstring field theory since it was constructed by repeating the prescription developed for the heterotic string field theory [25].

Type II superstring field theory with L ∞ structure
The type II superstring field Φ is Grassmann even and has four components with ghost number 2: G = X X , X = π 0 + Xπ 1 , X = π 0 + Xπ 1 , (3.85) 10 For example, if we map the worldsheet to the complex z plane so that the string 1 and 2 are on the points Z 1 and Z 2 , respectively, the differences can be written as where (X 0 ) 1 and (X 0 ) 2 are X 0 s acting on the string 1 and 2, respectively.
where π 0 and π 1 (π 0 and π 1 ) are the projection operators onto the NS and R states in the holomorphic (anti-holomorphic) sectors, respectively. We take the non-local operators satisfying the same relations as those satisfied by Y and Y , as inverse picture changing operators, which is necessary to consistently impose all the constraints. 11 The constraint (2.8) can now be split into two conditions which we use later. Any sate B ∈ H res can be expanded in the ghost zero-mode as so we denote the string field Φ ∈ H res as The on-shell subspace H 0 ⊂ H res is defined by The natural symplectic forms Ω , ω s , and ω l in the H res = H res L ⊗ H res R , H s = (H s ) L ⊗ (H s ) R , and H l = (H l ) L ⊗ (H l ) R , respectively, are defined similarly to those in Eqs. (2.11), and are related for Φ 1 , Φ 2 ∈ H res as and with Ω as The action and the gauge transformation have the same form as those of the heterotic string field theory, except that the symplectic form Ω and the string products L n+1 are now those for the type II superstring field theory. The string products are constructed by repeating twice the prescription developed for the heterotic string products. We first apply it to the holomorphic sector and obtain the heterotic All the quantities and relations in this first step have the same form as those of the heterotic string field theory in appearance. Next, in the second step, we repeat the prescription for the anti-holomorphic sector and construct the type II superstring product as The product B is obtained from the generating function B(s, t) = ∞ m,n,r=0 by solving the differential equation with the initial condition The cohomomorphismF is given byF which reduce to the L ∞ relation of the coderivation Q − η + B at (s, t) = (0, 1) . If you note that the initial heterotic product has the form of L int H = X l H and look carefully the differential equation (3.102), it is found that the final form of B has the form of and thus π 1 L int is written as the form in which it is manifest that the products is closed in H res :

S-matrix generating function and its evaluation
Since the equation of motion has the same form as that of the heterotic string field theory, the S-matrix generating function can also be written in the same form as (2.55a) with where L int is the type II superstring product given by (3.107). By repeating the proof for the heterotic string amplitudes, we can again show that the physical amplitudes of type II superstring are obtained as integrals of the smooth functions (sections) over the whole moduli space with appropriate picture changed external states. Let us briefly summarize the procedure for the type II superstring. If we define the mapŝ the S-matrix can be represented by Σ similar to that of the heterotic string field theory as The Σ is recursively determined by the equation hold similar to the case of the heterotic string field theory. The explicit form of ρ(s, t) is not relevant for the physical S-matrix, but obtained similarly to that of the heterotic string given in Appendix D. The relations (3.115) provides the key equation with T (s, t) = ξ 0 • ρ(s, t) . Then, the extended S-matrix    Hence, for the physical S-matrix, we have in a similar way to the case of the heterotic string field theory. Here,P is the projection operator onto the physical Hilbert space satisfying QP = ηP = ηP = 0 . The amplitudes with no anti-holomorphic picture number S (0) n+3 | is nothing but those of the heterotic string field theory, S So, now it is easy to see that without explicitly repeating the procedure again. Substituting this into the first step result (3.120), the physical S-matrix of the type II superstring field theory can be written as From the similar consideration to that in the heterotic string field theory, this agrees with the tree-level physical (n + 3)-string scattering amplitudes obtained in the covariant first-quantized formulation.

Extension to the open superstring field theory
The proof given for closed superstring theories in the previous sections can also be applied to the case of the open superstring field theory. In this section, we first extend our construction method to the open superstring field theory, and then, prove that it reproduces the well-known tree-level S-matrix.

Open superstring field theory with cyclic A ∞ structure
The open superstring field Φ is Grassmann odd and has two components with ghost number 1: The NS component Φ N S and R component Φ R have picture numbers −1 and −1/2 , respectively. The Hilbert space H res is restricted by a constraint (2.8) with The string field Φ ∈ H res is expanded in the ghost zero-modes as The on-shell subspace H 0 ⊂ H res and the projector P 0 are introduced as The symplectic forms are defined by using the BPZ inner product. The degree of generic states Φ 1,2 are defined by deg(Φ 1,2 ) = (|Φ 1,2 | + 1) mod 2 , and thus, in particular, the degree of the open superstring field Φ is even. The action and gauge transformation are now given by where ǫ(i, j) = j k=i deg(Φ k ) . Our construction method of the string products of the heterotic string field theory is also applicable to those of the open superstring field theory. It is achieved by simply replacing the coderivation B(s, t) acting on the symmetrized tensor algebra SH l with the coderivation acting on the tensor algebra T H res = ⊕ ∞ n=0 (H res ) ⊗n . If A(s, t) satisfies the differential equations At (s, t) = (0, 1) , they reduce to the A ∞ relations of the coderivation Q − η + A(0, 1) . All the open superstring products, and simultaneously the gauge products, are determined in the same way as those of the heterotic string field theory by solving the differential equations in (4.136) starting from the initial condition Here, M n+2 are the open string products without PCO-insertions, which we call the open bosonic string products. The cyclic A ∞ algebra M = Q+M int is constructed from A = A(0, 1) as using the cohomomorphism This construction is an extension of that in Ref. [9] proposed for the products in the NS sector. If we restrict our construction to the NS sector, the cohomomorphismF becomes trivial, π 0 1F = π 0 1 I , and the differential equations (4.136) reduce to those in Ref. [9] for the string product It is also a generalization of that in Ref. [17], in which a complete action of the open superstring field theory based on the A ∞ algebra was first constructed. The string products with A ∞ structure (4.140) completely agree with those constructed in Ref. [17] if we take the associative bosonic product, M B (s) = m 2 | 0 + s m 2 | 2 , as an initial condition, 12 which is explicitly in Appendix E. Our construction method, however, allows us to take more general initial bosonic products, and provides more general superstring products which were not be able to be constructed before. This degree of freedom allows for the various open string field theories realizing different triangulations of the moduli space, which will be useful in analyzing specific problems. The proof of cyclicity becomes easier in our construction, which we give in Appendix F.

S-matrix generating function and its evaluation
The tree-level S-matrix generating functional is obtained in a similar way to the heterotic string field theory: Here, all the quantities are those acting on T H res , which are similarly defined for the open superstring, using the open string homotopy operator and satisfy the same relation as (2.50). If we introduce two mapŝ the map Σ is related to S int as and satisfies the Dyson-Schwinger equation where ∆ = Q + G − Ξπ 1 . These basic relations and the differential equations (4.136) have the same form as those of the heterotic string field theory, and therefore it is easy to show that if we suppose that the two-parameter extension Σ(s, t) follows the extended Dyson-Schwinger equation with certain ρ(s, t) determined in a manner similar to that for the heterotic string given in Appendix D. The relations (4.152b) lead to the same equation as (2.71), from which we find that S(s, t)| = ω l |ξ 0 P 0 ⊗ P 0 π 1 Σ(s, t) satisfies the equation except for the terms which vanish when acting on physical states. Consequently, the physical S-matrix at the tree level can be rewritten as and agrees with that in the first-quantized formulation since it has the form of integral of smooth functions (sections) with picture changed external states over the whole open-string moduli spaces.

Summary and discussion
In this paper, we have shown that the tree-level physical S-matrices derived from the homotopyalgebra-based superstring field theories reproduce those obtained in the first-quantized formulation. For the heterotic string field theory, the differential equations (2.21) for the (basic building blocks of) the string products B(s, t) play a key role. Utilizing these differential equations we can eventually derive a sequence of decent equations (2.82) for the amplitudes. They allow us to rewrite the physical S-matrix in such a way that the equivalence to the one obtained in the first-quantized formulation is transparent. The extension to the type II and open superstring field theories has been straightforward since the key differential equations have essentially the same form. The open superstring field theory considered in this paper is a generalization of the previously constructed one and provides various theories realizing different triangulations of the moduli space.
It is interesting to extend the proof to the S-matrix at the loop level. In order to consider it, however, we need to extend the discussion to the one based on the quantum (or loop) homotopy algebras [31,48,49]. For closed superstring field theories, it is related to the problem of how to avoid the unphysical spurious singularities [32,50]. A consistent superstring field theory needs to be constructed to provide an algorithm that calculates the scattering amplitude as an integral over the moduli space that smoothly connects the contributions from various Feynman diagrams without hitting spurious singularities [51]. Some recent achievements [52][53][54][55][56] will help further progress.
There are many other interesting issues that can only be performed by means of the string field theories [57], and further studies are expected to be done.

A S-matrix via HPT
In this appendix, we derive the tree-level S-matrix of the heterotic string field theory (2.54) as the almost minimal model of the L ∞ algebra (H res , Ω, L) by means of HPT. 13 Consider two chain complexes (H res , d = Q) and (H 0 , D = QP 0 ) with chain maps The chain complex (H 0 , QP 0 ) , with the gauge conditions, defines the relative BRST cohomology [59][60][61]. In order to obtain the S-matrix (2.54b) we first lift these equivalence data to those acting on the corresponding symmetrized tensor algebra defined by Eqs. Then, if we perturb Q by L int so that (Q + L int ) 2 = 0 , the homological perturbation lemma tells us that the equivalence data are deformed as We confirm below that the almost minimal model (SH 0 , D ′ ) is actually L ∞ -quasi-isomorphic to (SH res , d ′ = L) by showing that the deformed mapsî ′ andp ′ are cohomomorphisms, and D ′ is a coderivation.
However, the complicated factor in front of the ∆ in the right hand side becomes simple if it acts on (P ⊗ A + A ⊗P ) or (H ⊗Î −Î ⊗ H) with any map A on SH res , and we find that ∞ r,s,t,u=0 by using the relation for 1 ≤ β ≤ u 0 . This shows that (A.176) holds for u = u 0 + 1 , and hence the formula (A.176) is proven by mathematical induction. Using this formula (A.176), the left hand side of the relation (A.174) can be calculated by noting that 1 t! I ∧t is the identity on (H res ) ∧t : (A.179) The proof of the formula (A.175) is the following. We assume that s ≥ u , which is possible without loss of generality since the formula is symmetric with respect to s and u , and use the mathematical induction with respect to u . First, rewrite the formula as (A.180) For u = 0 , the left hand side is calculated as by using the relation (A.178) repeatedly. Then, suppose that it is true for u = u 0 . The left hand side of (A.180) for u = u 0 + 1 can be calculated as The last equality follows from the assumption of the induction. Hence, the formula (A.175) is true for ∀ u by mathematical induction. Using these properties of H , we can show, order by order in the power of H , that the linear map on SH 0 ,î respectively, obtained by reversing the order of the factors. We omit the explicit proof since it is parallel to that forî ′ .

(C.205)
It was also used the fact that the internal states are generically off-shell. Hence, from the principle of mathematical induction, we can conclude that [Q, π 1 Σ(s, t)] = 0 . Similarly, we find from Eq. (C.199) that using the relation (2.34b) and [η , ∆(s, t)] = −tπ 1 , (C.207) with the off-shell-ness of the internal states. Hence, it can also be concluded that [η, π 1 Σ(s, t)] = 0 from the principle of mathematical induction.

E Relation to the Erler-Okawa-Takezaki open superstring field theory
When starting from the cubic theory, M  satisfying the initial conditions A(0) = m 2 | 2 and A(0) = m 2 | 0 . These equations are slight modification of those proposed in Ref. [17]. In the previous method, almost the same equations that respect the Ramond number, instead of the cyclic Ramond number, directly provide (the generating function of) the products with A ∞ structure. In our method, on the other hand, Eqs. (E.227) provide an intermediate products, which have to be transformed to the final form by the cohomomorphism. It can be seen, however, that the final products are the same.