YITP-20-154

Tree-level S-matrix of superstring field theory with homotopy algebra
structure
Hiroshi Kunitomo
Center for Gravitational Physics, Yukawa Institute for Theoretical Physics
Kyoto University, Kyoto 606-8502, Japan
<EMAIL_ADDRESS>

Abstract

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.

###### Contents

  1. 1 Introduction
  2. 2 Tree-level S-matrix of heterotic string field theory
    1. 2.1 Heterotic string field theory with cyclic $L_{\infty}$ structure
    2. 2.2 S-matrix generating function
    3. 2.3 Evaluation of the S-matrix
  3. 3 Extension to the type II superstring field theory
    1. 3.1 Type II superstring field theory with $L_{\infty}$ structure
    2. 3.2 S-matrix generating function and its evaluation
  4. 4 Extension to the open superstring field theory
    1. 4.1 Open superstring field theory with cyclic $A_{\infty}$ structure
    2. 4.2 S-matrix generating function and its evaluation
  5. 5 Summary and discussion
  6. A S-matrix via HPT
  7. B Derivation of (2.54a)
  8. C Proof of (2.69)
  9. D Proof of (2.70)
  10. E Relation to the Erler-Okawa-Takezaki open superstring field theory
  11. F Cyclicity of generalized $A_{\infty}$ structure

##  1 Introduction

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, 14, 13, 12, 15, 18, 16,
17, 22, 19, 20, 21, 23, 11, 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 $A_{\infty}$ structure [9], and the heterotic
and type II superstring field theories with an $L_{\infty}$ 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]. 111The explicit confirmations for four-NS [33] and arbitrary
four- and five-string amplitudes [34] were also performed in the WZW-like
formulation. These can also be considered as a confirmation of those in the
homotopy-algebra-based formulation since the two theories are related to a
simple field redefinition. In addition to these confirmations by explicit
calculation, a general proof for the tree-level S-matrix was also given based
on the minimal model with the homological perturbation theory (HPT) [35]. This
proof is, however, still restricted to the amplitudes with the external
NS(-NS) strings, and the proof for the complete S-matrix is still lacking. We
extend it proof to the general S-matrix and complete the proof in the
homotopy-algebra-based superstring field theories.

The paper is organized as follows. In section 2, we give a general proof for
the heterotic string in detail. We first summarize some basic properties of
the heterotic string field theory with cyclic $L_{\infty}$ structure in
section 2.1, focusing on what is needed for the proof. Then, in section 2.2,
we give the S-matrix generating function at tree-level in a closed form based
on a general argument given in Ref. [36, 37]. The result agrees with that
given by using the argument based on the (almost) minimal model with HPT.
Using the S-matrix generating function, we show that the tree-level physical
S-matrix agrees with that calculated using the first-quantized method in
section 2.3. The proof is given by generalizing the method used in Ref. [35].
It is straightforward to extend the proof to the type II superstring field
theory, which is given in section 3. After summarizing the basic properties in
section 3.1, we give a proof for type II superstring field theory in section
3.2. The extension to the open superstring field theory is also
straightforward if we generalize the construction method in Ref. [17] to the
one applicable more general $A_{\infty}$ structure following the way given in
Ref. [24]. Starting from giving such a generalized open superstring field
theory in section 4.1, we prove the agreement of the S-matrix in section 4.2.
Section 5 is devoted to the summary and discussion. Finally, six appendices
are included. Appendix B is devoted to deriving the final form of the S-matrix
generating functional, which is not given in the text since some technical
details are required. In Appendix A, we explain how the S-matrix is also
obtained as the (almost) minimal model via HPT. Some proofs to confirm the
consistency of the perturbation are also given. Appendices C and D are devoted
to proving some formulas used in the text. In Appendix E, we show that the
generalized open superstring field theory given in Section 4 reduces to that
in Ref. [17] as a special case based on the Witten’s vertex. A simple proof
for the cyclicity of the $A_{\infty}$ structure in the generalized theory is
also given in Appendix F, which was not so simple in the previous
construction.

##  2 Tree-level S-matrix of heterotic string field theory

Let us first take the heterotic string field theory and show that it
reproduces the tree-level S-matrix calculated by the first-quantized method.

###  2.1 Heterotic string field theory with cyclic $L_{\infty}$ structure

We start by briefly summarizing how the heterotic string field theory based on
the $L_{\infty}$ structure is constructed. The first-quantized heterotic
string theory is obtained by combining the left-moving (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 $(\beta,\gamma)$ . The latter consists of a matter sector with $c=26$ ,
and the fermionic ghosts $(\overline{b},\overline{c})$. It is useful to
‘bosonize’ the bosonic ghosts $(\beta,\gamma)$ to a pair of fermions
$(\eta,\xi)$ and a chiral boson $\phi$ . The Hilbert space of the bosonized
ghosts $(\eta,\xi;\phi)$ is called the large Hilbert space $\mathcal{H}_{l}$ .
In contrast, the Hilbert space of the bosonic ghosts $(\beta,\gamma)$ is
called the small Hilbert space $\mathcal{H}_{s}$, which can be embedded in
$\mathcal{H}_{l}$ as states $\Phi$ satisfying $\eta\Phi=0$ with a fixed
picture number.222 For notational simplicity, the zero-mode of $\eta$-ghost
appearing frequently is simply denoted as $\eta$ . The subspaces with
different picture numbers provide equivalent representations of
$\mathcal{H}_{s}$ , and can be related to each other by one-to-one mapping
using PCO. The heterotic string field takes a value in $\mathcal{H}_{s}$ as
described below in detail.

The heterotic string field $\Phi$ is Grassmann even and satisfying the closed
string constraints:

$b_{0}^{-}\Phi\ =\ L_{0}^{-}\Phi\ =\ 0\,.$ (2.1)

It has ghost number 2 and two components

$\Phi\ =\ \Phi_{NS}+\Phi_{R}\ \in\ \mathcal{H}^{res}\ =\
\mathcal{H}_{NS}+\mathcal{H}_{R}^{res}\,,$ (2.2)

where $\mathcal{H}_{NS}$ ($\mathcal{H}_{R}$) is the small Hilbert space of the
NS (Ramond) sector with picture number $-1$ ($-1/2$). The Ramond Hilbert space
$\mathcal{H}_{R}^{res}$ is further restricted by an extra condition:

$\mathcal{H}_{R}^{res}\ =\ \\{\Phi_{R}\in\mathcal{H}_{R}\mid XY\Phi_{R}\ =\
\Phi_{R}\\}\,,$ (2.3)

where $X$ and $Y$ are defined by

$X\ =\ -\delta(\beta_{0})G_{0}+\delta^{\prime}(\beta_{0})b_{0}\,,\qquad Y\ =\
-2c_{0}^{+}\delta^{\prime}(\gamma_{0})\,.$ (2.4)

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 $X=\\{Q,\Xi\\}$ with

$\Xi\ =\
\xi_{0}+(\Theta(\beta_{0})\eta\xi_{0}-\xi_{0})\Pi_{-3/2}+(\xi_{0}\eta\Theta(\beta_{0})-\xi_{0})\Pi_{-1/2}\,.$
(2.5)

Here, $\Pi_{n}$ 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

$XYX\ =\ X\,,\qquad YXY\ =\ Y\,.$ (2.6)

We can show that $XY$ used to define $\mathcal{H}^{res}$ 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

$\mathcal{G}\ =\ \pi^{0}+X\pi^{1}\,,\qquad\mathcal{G}^{-1}\ =\
\pi^{0}+Y\pi^{1}\,,$ (2.7)

with the projection operator $\pi^{0}$ $(\pi^{1})$ onto the NS (Ramond)
component. Then, the restricted Hilbert space $\mathcal{H}^{res}$ can
concisely be written as

$\mathcal{H}^{res}\ =\
\\{\Phi\in\mathcal{H}_{s}=\mathcal{H}_{NS}+\mathcal{H}_{R}\mid\mathcal{G}\mathcal{G}^{-1}\Phi\
=\ \Phi\\}\,.$ (2.8)

The restricted Hilbert space $\mathcal{H}^{res}$ is closed under the action of
the BRST operator: $XYQXY=QXY$ , and any state
$\mathcal{B}\in\mathcal{H}^{res}$ can be expanded in the ghost zero-mode as

$\displaystyle\mathcal{B}=$ $\displaystyle\ \mathcal{B}_{NS}+\mathcal{B}_{R}$
$\displaystyle=$ $\displaystyle\
\left(b_{NS}-c_{0}^{+}B_{NS}\right)+\left(b_{R}-\frac{1}{2}(\gamma_{0}+2c_{0}^{+}G)B_{R}\right)\,,$
(2.9)

where $G=G_{0}+2\gamma_{0}b_{0}$ . In particular, we denote the string field
$\Phi\in\mathcal{H}^{res}$ as

$\Phi\ =\
\left(\phi_{NS}-c_{0}^{+}\psi_{NS}\right)+\left(\phi_{R}-\frac{1}{2}(\gamma_{0}+2c_{0}^{+}G)\psi_{R}\right)\,.$
(2.10)

We can define three symplectic forms for the large, small and restricted
Hilbert space, $\omega_{l}$ , $\omega_{s}$ and $\Omega$ by

$\displaystyle\omega_{l}(\Phi_{1},\Phi_{2})\ =$ $\displaystyle\
(-1)^{|\Phi_{1}|}{}_{l}\langle\Phi_{1}|c_{0}^{-}|\Phi_{2}\rangle_{l}\,,\qquad$
$\displaystyle\Phi_{1},\Phi_{2}\in$ $\displaystyle\ \mathcal{H}_{l}\,,$
(2.11a) $\displaystyle\omega_{s}(\Phi_{1},\Phi_{2})\ =$ $\displaystyle\
(-1)^{|\Phi_{1}|}{}_{s}\langle\Phi_{1}|c_{0}^{-}|\Phi_{2}\rangle_{s}\,,\qquad$
$\displaystyle\Phi_{1},\Phi_{2}\in$ $\displaystyle\ \mathcal{H}_{s}\,,$
(2.11b) $\displaystyle\Omega(\Phi_{1},\Phi_{2})\ =$ $\displaystyle\
(-1)^{|\Phi_{1}|}{}_{s}\langle\Phi_{1}|c_{0}^{-}\mathcal{G}^{-1}|\Phi_{2}\rangle_{s}\,,\qquad$
$\displaystyle\Phi_{1},\Phi_{2}\in$ $\displaystyle\ \mathcal{H}^{res}\,.$
(2.11c)

For $\Phi_{1},\Phi_{2}\in\mathcal{H}^{res}$ , they are related as

$\Omega(\Phi_{1}\,,\Phi_{2})\ =\
\omega_{s}(\Phi_{1}\,,\mathcal{G}^{-1}\Phi_{2})\ =\
\omega_{l}(\xi_{0}\Phi_{1},\mathcal{G}^{-1}\Phi_{2})\,.$ (2.12)

We also use their bilinear map representation defined by

$\displaystyle\langle\omega_{l}|\,:\,\mathcal{H}_{l}$
$\displaystyle\otimes\mathcal{H}_{l}\qquad$ $\displaystyle\ {\longrightarrow}$
$\displaystyle\mathbb{C}$ (2.13) $\displaystyle\ \rotatebox{90.0}{$\in$}$
$\displaystyle\rotatebox{90.0}{$\in$}$ $\displaystyle\Phi_{1}$
$\displaystyle\otimes\Phi_{2}\qquad$ $\displaystyle\longmapsto$
$\displaystyle\qquad\omega_{l}(\Phi$ $\displaystyle{}_{1}\,,\Phi_{2})\,,$

and

$\langle\omega_{s}|\ =\
\langle\omega_{l}|(\xi_{0}\otimes\mathbb{I})\,,\qquad\langle\Omega|\ =\
\langle\omega_{l}|(\xi_{0}\otimes\mathcal{G}^{-1})\,.$ (2.14)

Note that the natural inner product in $\mathcal{H}^{res}$ defined by $\Omega$
has the off-diagonal form

$\Omega(\Phi_{1},\Phi_{2})\ =\
\langle\\!\langle\phi_{NS1}|\psi_{NS2}\rangle\\!\rangle+\langle\\!\langle\psi_{NS1}|\phi_{NS2}\rangle\\!\rangle+\langle\\!\langle\phi_{R1}|\psi_{R2}\rangle\\!\rangle+\langle\\!\langle\psi_{R1}|\phi_{R2}\rangle\\!\rangle\,,$
(2.15)

after integrating out the ghost zero-modes.333 The double bra-ket represents
the BPZ inner-product after integrating out the ghost zero-modes [25].

The action of heterotic string field theory with cyclic $L_{\infty}$ structure
is written as

$I[\Phi]\ =\
\sum_{n=0}^{\infty}\frac{1}{(n+2)!}\,\Omega(\Phi\,,L_{n+1}(\underbrace{\Phi\,,\cdots\,,\Phi}_{n+1}))\,,$
(2.16)

which is invariant under the gauge transformation

$\delta\Phi\ =\
\sum_{n=0}^{\infty}\frac{1}{n!}\,L_{n+1}(\underbrace{\Phi\,,\cdots\,,\Phi}_{n}\,,\Lambda)\,.$
(2.17)

Here, $L_{1}=Q$ and the multi-string products
$L_{n+2}(\Phi_{1},\cdots,\Phi_{n+2})$ are graded commutative, which satisfy
the $L_{\infty}$ relations

$\displaystyle\sum_{\sigma}\sum_{m=1}^{n}(-1)^{\epsilon{(\sigma)}}\frac{1}{m!(n-m)!}L_{n-m+1}(L_{m}(\Phi_{\sigma(1)},\cdots,\Phi_{\sigma(m)}),\Phi_{\sigma(m+1)},\cdots,\Phi_{\sigma(n)})\
=\ 0\,,$ (2.18)

and the cyclicity condition

$\Omega(\Phi_{1}\,,L_{n}(\Phi_{2}\,,\cdots\,,\Phi_{n+1}))\ =\
-(-1)^{|\Phi_{1}|}\,\Omega(L_{n}(\Phi_{1}\,,\cdots\,,\Phi_{n})\,,\Phi_{n+1})\,.$
(2.19)

The symbol $\sigma$ in (2.18) denotes the permutation from $\\{1,\cdots,n\\}$
to $\\{\sigma(1),\cdots,\sigma(n)\\}$ and the factor $\epsilon(\sigma)$ is the
sign factor of permutation of string fields from
$\\{\Phi_{1},\cdots,\Phi_{n}\\}$ to
$\\{\Phi_{\sigma(1)},\cdots,\Phi_{\sigma(n)}\\}$ . The set of the string
products satisfying these conditions is called a cyclic $L_{\infty}$ algebra
$(\mathcal{H}^{res},\Omega,\\{L_{n}\\})$.

For constructing the string products of the cyclic $L_{\infty}$ algebra, we
use the coalgebra representation, which is, for example, summarized in [38,
24]. The heterotic string products are represented by the (Grassmann) odd
coderivation $\bm{L}=\bm{Q}+\bm{L}_{int}$ acting on the symmetrized tensor
algebra $\mathcal{SH}^{res}=\oplus_{n=0}^{\infty}(\mathcal{H}^{res})^{\wedge
n}$ . However, it is not very good idea to directly construct the cyclic
$L_{\infty}$ algebra $(\mathcal{H}^{res},\Omega,\bm{L})$ since $\Omega$ is
asymmetric between the NS and Ramond sectors. Instead, we considered, in [24],
an cyclic $L_{\infty}$ algebra
$(\mathcal{H}_{l},\omega_{l},\bm{Q}-\bm{\eta}+\bm{B})$ first. The coderivation
$\bm{B}$ acting on
$\mathcal{SH}_{l}=\oplus_{n=0}^{\infty}(\mathcal{H}_{l})^{\wedge n}$ is
constructed so that an extended generating function,

$\bm{B}(s,t)\ =\
\sum_{m,n,r=0}^{\infty}s^{m}t^{n}\bm{B}^{(n)}_{m+n+r+1}|^{2r}\,,$ (2.20)

related to $\bm{B}$ as $\bm{B}=\bm{B}(0,1)$ , satisfies two differential
equations

$\displaystyle\partial_{t}\bm{B}(s,t)\ =$ $\displaystyle\
[\bm{Q},\bm{\lambda}(s,t)]+[\bm{B}(s,t),\bm{\lambda}(s,t)]^{1}+s\,[\bm{B}(s,t),\bm{\lambda}(s,t)]^{2}\,,$
(2.21a) $\displaystyle\partial_{s}\bm{B}(s,t)\ =$ $\displaystyle\
[\bm{\eta},\bm{\lambda}(s,t)]-t\,[\bm{B}(s,t),\bm{\lambda}(s,t)]^{2}\,,$
(2.21b)

which simultaneously determine another even coderivation

$\bm{\lambda}(s,t)\ =\
\sum_{m,n,r=0}^{\infty}s^{m}t^{n}\bm{\lambda}^{(n+1)}_{m+n+r+2}|^{2r}\,,$
(2.22)

called the (generating function of) gauge products. Here, the brackets
$[\,\cdot\,,\cdot\,]^{1}$ and $[\,\cdot\,,\,\cdot\,]^{2}$ are defined from the
(graded) commutators by projecting the cyclic Ramond number as

$[\bm{A}|^{2r},\bm{B}|^{2s}]^{1}\ =\
[\bm{A}|^{2r},\bm{B}|^{2s}]|^{2(r+s)}\,,\qquad[\bm{A}|^{2r},\bm{B}|^{2s}]^{2}\
=\ [\bm{A}|^{2r},\bm{B}|^{2s}]|^{2(r+s-1)}\,.$ (2.23)

For the coderivation $\bm{B}(s,t)$ constructed to satisfy (2.21), the
relations

$\displaystyle\
[\bm{Q},\bm{B}(s,t)]+\frac{1}{2}\,[\bm{B}(s,t),\bm{B}(s,t)]^{1}+\frac{s}{2}\,[\bm{B}(s,t),\bm{B}(s,t)]^{2}\
=\ 0\,,$ (2.24a) $\displaystyle\
[\bm{\eta},\bm{B}(s,t)]-\frac{t}{2}\,[\bm{B}(s,t),\bm{B}(s,t)]^{2}\ =\ 0\,,$
(2.24b)

hold, which reduce at $(s,t)=(0,1)$ to the $L_{\infty}$ relation for the
coderivation $\bm{Q}-\bm{\eta}+\bm{B}$ . We have shown that the coderivation
$\bm{L}$ representing the heterotic string products can be obtained from
$\bm{B}$ as

$\displaystyle\bm{L}\ =$ $\displaystyle\
\hat{\bm{F}}^{-1}(\bm{Q}+\pi^{0}\bm{B})\hat{\bm{F}}\,,$ (2.25a)
$\displaystyle\pi_{1}\bm{L}_{int}\ =$ $\displaystyle\
\mathcal{G}\pi_{1}\bm{l}\,,\qquad\pi_{1}\bm{l}\ =\
\pi_{1}\bm{B}\hat{\bm{F}}\,.$ (2.25b)

Here, $\pi_{n}$ is the projection operator

$\pi_{n}\,:\,\mathcal{SH}^{res}\rightarrow(\mathcal{H}^{res})^{\wedge n}\,.$
(2.26)

The invertible cohomomorphism $\hat{\bm{F}}^{-1}$ is defined by

$\pi_{1}\hat{\bm{F}}^{-1}\ =\ \pi_{1}\mathbb{I}-\Xi\pi_{1}^{1}\bm{B}\,,$
(2.27)

with $\pi_{1}^{1}=\pi^{1}\pi_{1}$ ($\pi^{0}_{1}=\pi^{0}\pi_{1}$), and
$\mathbb{I}\in\mathcal{SH}^{res}$ is the (multiplicative) identity of the
symmetrized tensor algebra. Using a property of cohomomorphisms

$\hat{\bm{F}}\ =\
\pi_{0}+\sum_{n=1}^{\infty}\frac{1}{n!}\left(\pi_{1}\hat{\bm{F}}\right)^{\wedge
n}\,,$ (2.28)

and the relation

$\pi_{1}\hat{\bm{F}}\ =\ \pi_{1}\mathbb{I}+\Xi\pi_{1}^{1}\bm{l}\,,$ (2.29)

obtained by acting $\hat{\bm{F}}$ 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:

$\pi_{1}\bm{l}\ =\
\sum_{n=0}^{\infty}\pi_{1}\bm{B}_{n+2}\left(\frac{1}{(n+2)!}\Big{(}\pi_{1}\mathbb{I}+\Xi\pi_{1}^{1}\bm{l}\Big{)}^{\wedge(n+2)}\right)\,.$
(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 $\bm{A}_{n+1}$ and $\bm{B}_{m+1}$
, we find that

$\displaystyle[\bm{A}_{n+1},\bm{B}_{m+1}]^{1}\ =$ $\displaystyle\
\bm{A}_{n+1}\big{(}\pi^{0}_{1}\bm{B}_{m+1}\wedge\mathbb{I}_{n}\big{)}-(-1)^{AB}\bm{B}_{m+1}\big{(}\pi^{0}_{1}\bm{A}_{n+1}\wedge\mathbb{I}_{m}\big{)}\,,$
(2.31) $\displaystyle[\bm{A}_{n+1},\bm{B}_{m+1}]^{2}\ =$ $\displaystyle\
\bm{A}_{n+1}\big{(}\pi^{1}_{1}\bm{B}_{m+1}\wedge\mathbb{I}_{n}\big{)}-(-1)^{AB}\bm{B}_{m+1}\big{(}\pi^{1}_{1}\bm{A}_{n+1}\wedge\mathbb{I}_{m}\big{)}\,.$
(2.32)

In this definition, it is easy to see that $[\,,]=[\,,]^{1}+[\,,]^{2}$ holds.
We can rewrite the relations in Eqs. (2.21) and (2.24) using this
representation as

$\displaystyle\partial_{t}\bm{B}_{n+2}(s,t)\ =$ $\displaystyle\
[\bm{Q},\bm{\lambda}_{n+2}(s,t)]+\sum_{m=0}^{n-1}\Big{(}\bm{B}_{m+2}(s,t)\big{(}\pi(s)\pi_{1}\bm{\lambda}_{n-m+1}(s,t)\wedge\mathbb{I}_{m+1}\big{)}$
$\displaystyle\hskip
128.0374pt-\bm{\lambda}_{m+2}(s,t)\big{(}\pi(s)\pi_{1}\bm{B}_{n-m+1}(s,t)\wedge\mathbb{I}_{m+1}\big{)}\Big{)}\,,$
(2.33a) $\displaystyle\partial_{s}\bm{B}_{n+2}(s,t)\ =$ $\displaystyle\
[\bm{\eta},\bm{\lambda}_{n+2}(s,t)]-\sum_{m=0}^{n-1}\Big{(}\bm{B}_{m+2}(s,t)\big{(}t\pi^{1}_{1}\bm{\lambda}_{n-m+1}(s,t)\wedge\mathbb{I}_{m+1}\big{)}$
$\displaystyle\hskip
128.0374pt-\bm{\lambda}_{m+2}(s,t)\big{(}t\pi^{1}_{1}\bm{B}_{n-m+1}(s,t)\wedge\mathbb{I}_{m+1}\big{)}\Big{)}\,,$
(2.33b)

and

$\displaystyle[\bm{Q},\bm{B}_{n+2}(s,t)]\ =$ $\displaystyle\
-\sum_{m=0}^{n-1}\bm{B}_{m+2}(s,t)\big{(}\pi(s)\pi_{1}\bm{B}_{n-m+1}(s,t)\wedge\mathbb{I}_{m+1}\big{)}\,,$
(2.34a) $\displaystyle[\bm{\eta},\bm{B}_{n+2}(s,t)]\ =$ $\displaystyle\
\sum_{m=0}^{n-1}\bm{B}_{m+2}(s,t)\big{(}t\pi^{1}_{1}\bm{B}_{n-m+1}(s,t)\wedge\mathbb{I}_{m+1}\big{)}\,,$
(2.34b)

with $\pi_{1}\bm{B}_{n+2}(s,t)=\pi_{1}\bm{B}(s,t)\pi_{n+2}$ ,
$\pi_{1}\bm{\lambda}_{n+2}(s,t)=\pi_{1}\bm{\lambda}(s,t)\pi_{n+2}$ and
$\pi(s)=\pi^{0}+s\pi^{1}$ . We will use these forms of the relations later.

###  2.2 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 four-point amplitudes based on the homotopy-algebra-based
formulation.444Several four- and five-point amplitudes have also been
calculated by the WZW-like open superstring field theory [33, 34].  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 $\mathcal{H}^{res}$ , we can identify
$\phi=\phi_{NS}+\phi_{R}$ and $\psi=\psi_{NS}+\psi_{R}$ 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),

$I[\Phi]\ =\
\sum_{n=0}^{\infty}\frac{1}{(n+2)!}\,\Omega(\Phi\,,L_{n+1}(\underbrace{\Phi\,,\cdots\,,\Phi}_{n+1}))\,,$
(2.35)

but $\Phi$ is now the quantum string field including the states in
$\mathcal{H}^{res}$ with all the ghost numbers.555We 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). The Siegel-Ramond (SR) gauge is obtained by simply setting $\psi=0$ .
We call the Hilbert space of the quantum field in this gauge
$\mathcal{H}_{SR}$ : $\Phi=\phi\in\mathcal{H}_{SR}$ . Then the gauge fixed
action

$I[\phi]=\
\frac{1}{2}\langle\phi|c_{0}^{-}\mathcal{G}^{-1}Q|\phi\rangle+\sum_{n=3}^{\infty}\frac{1}{n!}\langle\phi|c_{0}^{-}|l_{n-1}\left(\phi^{n-1}\right)\rangle\,,$
(2.36)

is invariant under the BRST transformation666 This is a schematic expression.
See Ref. [31] for precise treatment.

$\delta_{BRST}\phi\ =\ \left.\frac{\partial
I[\Phi]}{\partial\psi}\right|_{\psi=0}\,.$ (2.37)

From (2.36), we can read off the propagator

$\Pi\ \equiv\
-\frac{b_{0}^{+}b_{0}^{-}\mathcal{G}}{L_{0}^{+}}\int^{2\pi}_{0}\frac{d\theta}{2\pi}e^{-i\theta
L_{0}^{-}}\,,$ (2.38a) and the $n$-point vertices $c_{0}^{-}l_{n-1}\,,$
(2.38b)

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 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,

$\mathcal{H}_{0}\ =\ \\{\Phi\in\mathcal{H}^{res}\,\mid
L_{0}^{+}\Phi_{NS}=G_{0}\Phi_{R}=0\\}\,,$ (2.39)

and the BRST invariant projection operator,

$\qquad P_{0}\,:\,\mathcal{H}^{res}\longrightarrow\mathcal{H}_{0}\,,\qquad
P_{0}^{2}\ =\ P_{0}\,,\quad[Q,P_{0}]\ =\ 0\,.$ (2.40)

Then, we introduce an operator

$Q^{+}\ =\ \frac{1}{L_{0}^{+}}b_{0}^{+}(1-P_{0})\,,$ (2.41)

satisfying

$\displaystyle\ Q^{+}Q+QQ^{+}+P_{0}\ =\ \mathbb{I}\,,$ (2.42a) $\displaystyle\
Q^{+}P_{0}\ =\ P_{0}Q^{+}\ =\ Q^{+}Q^{+}\ =\ 0\,,$ (2.42b)

which defines a Hodge-Kodaira decomposition of $\mathcal{H}^{res}$ ,

$\mathcal{H}^{res}\ =\ \mathcal{H}^{p}+\mathcal{H}^{t}+\mathcal{H}^{u}\,,\\\ $
(2.43)

with

$\mathcal{H}^{p}\ =\ P_{0}\mathcal{H}^{res}\,,\qquad\mathcal{H}^{t}\ =\
QQ^{+}\mathcal{H}^{res}\,,\qquad\mathcal{H}^{u}\ =\
Q^{+}Q\mathcal{H}^{res}\,.$ (2.44)

This $Q^{+}$ is called a (contracting) homotopy operator of $Q$ , which is
compatible with $\Omega$ :

$\Omega(\mathcal{H}_{0},\mathcal{H}^{u})\ =\
\Omega(\mathcal{H}^{u},\mathcal{H}^{u})\ =\ 0\,.$ (2.45)

Under the SR gauge condition $\psi=0$ , $\mathcal{H}^{t}=\emptyset$ and the
quantum field $\phi$ is decomposed to the on-shell and off-shell subspaces:

$\phi\ \in\ \mathcal{H}_{0}+\overline{\mathcal{H}}_{0}\,,$ (2.46)

with

$\mathcal{H}_{0}\ =\ \mathcal{H}^{p}\cap\mathcal{H}_{SR}\ =\
P_{0}\mathcal{H}_{SR}\,,\qquad\overline{\mathcal{H}}_{0}\ =\
\mathcal{H}^{u}\cap\mathcal{H}_{SR}\ =\ (1-P_{0})\mathcal{H}_{SR}\,.$ (2.47)

The classical equation of motion in the SR gauge can now be written in the
form of an integral equation:

$\phi\ =\
\phi_{0}-Q^{+}\pi_{1}\bm{L}_{int}(e^{\wedge\phi})\,,\qquad\phi_{0}\in\mathcal{H}_{0}\,.$
(2.48)

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, $\bm{H}$ ,
$\hat{\bm{P}}$ , and $\hat{\bm{I}}$ by

$\displaystyle\bm{H}\ =$ $\displaystyle\
\sum_{r,s=0}^{\infty}\frac{1}{(r+s+1)!}\,(-Q^{+})\wedge\underbrace{\mathbb{I}_{1}\wedge\cdots\wedge\mathbb{I}_{1}}_{r}\wedge\underbrace{P_{0}\wedge\cdots\wedge
P_{0}}_{s}\,,$ (2.49a) $\displaystyle\hat{\bm{P}}\ =$ $\displaystyle\
\sum_{n=0}^{\infty}P_{n}\ =\
\sum_{n=0}^{\infty}\frac{1}{n!}\underbrace{P_{0}\wedge\cdots\wedge
P_{0}}_{n}\,,$ (2.49b) $\displaystyle\hat{\bm{I}}\ =$ $\displaystyle\
\sum_{n=0}^{\infty}\mathbb{I}_{n}\ =\
\sum_{n=0}^{\infty}\frac{1}{n!}\underbrace{\mathbb{I}_{1}\wedge\cdots\wedge\mathbb{I}_{1}}_{n}\,,$
(2.49c)

which act on the symmetric tensor algebra $\mathcal{SH}_{SR}$ and satisfy the
relations

$\displaystyle\ \hat{\bm{P}}\ =\ \hat{\bm{I}}+\bm{H}\bm{Q}+\bm{Q}\bm{H}\,,$
(2.50a) $\displaystyle\ \hat{\bm{P}}^{2}\ =\
\hat{\bm{P}}\,,\qquad[\bm{Q},\hat{\bm{P}}]\ =\ 0\,,\qquad$ (2.50b)
$\displaystyle\ \bm{H}\hat{\bm{P}}\ =\ \hat{\bm{P}}\bm{H}\ =\ \bm{H}\bm{H}\ =\
0\,.$ (2.50c)

Using $Q^{+}\phi=0$ and the general property of the coderivation, we can show
that Eq. (2.48) is equivalent to

$\bm{H}\bm{L}(e^{\wedge\phi})\ =\ 0\,.$ (2.51)

The solution $\phi_{cl}$ of this equation can be solved for $\phi_{0}$
as777The relation (2.52) shows that the map
$\bm{i}^{\prime}\equiv(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}$
(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 $\bm{L}_{int}$ ,
and subsequently proven in Ref. [40]. We also provide an independent proof in
Appendix A.

$e^{\wedge\phi_{cl}}\ =\
(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}(e^{\wedge\phi_{0}})\,,$
(2.52)

and thus

$\phi_{cl}\ =\
\pi_{1}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}(e^{\wedge\phi_{0}})\,.$
(2.53)

The S-matrix generating functional is a functional of $\phi_{0}$ obtained by
evaluating the gauge fixed action (2.36) at the classical solution (2.53), and
is given by

$\displaystyle S[\phi_{0}]\ \equiv$ $\displaystyle\ I[\phi_{cl}]$
$\displaystyle=$ $\displaystyle\
\sum_{n=0}^{\infty}\frac{1}{(n+2)!}\Omega(\phi_{0},S_{n+1}(\phi_{0}\,,\cdots\,,\phi_{0}))\,,$
(2.54a) with $\bm{S}\ =\ \bm{Q}\hat{\bm{P}}+\bm{S}_{int}\,,\qquad\bm{S}_{int}\
=\
\hat{\bm{P}}\bm{L}_{int}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}\,,$
(2.54b)

as is derived in Appendix B. This can also be represented by using the multi-
linear map888 Note that the quadratic term in $S[\phi_{0}]$ vanishes due to
the on-shell property of $\phi_{0}$ .  $\langle S|=\sum_{n=0}^{\infty}\langle
S_{(n+3)}|$ with

$\displaystyle\langle S|\ =$ $\displaystyle\
\langle\Omega|P_{0}\otimes\pi_{1}\bm{S}_{int}\,:\quad$
$\displaystyle\mathcal{H}_{0}\otimes\mathcal{SH}_{0}\ \longrightarrow\
\mathbb{C}\,,$ (2.55a) $\displaystyle\langle S_{n+3}|\ =$ $\displaystyle\
\langle\Omega|P_{0}\otimes\pi_{1}\bm{S}_{int}\pi_{n+2}\,:\quad$
$\displaystyle\mathcal{H}_{0}\otimes(\mathcal{H}_{0})^{\wedge(n+2)}\
\longrightarrow\ \mathbb{C}\,.$ (2.55b)

The amplitude for $(n+3)$-string scattering can be obtained as

$S_{n+3}(\phi_{1},\cdots,\phi_{n+3})\ =\ \langle
S_{n+3}|(\phi_{1}\,\otimes\,\phi_{2}\wedge\cdots\wedge\phi_{n+3})\,,$ (2.56)

where $\phi_{i}\in\mathcal{H}_{0}$ 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
$\langle S_{n+3}|$ (or all the scattering amplitudes using $\langle S|$ ). As
an example, the various four-string scattering amplitudes are represented by

$\displaystyle\langle S_{4}|\ =$ $\displaystyle\ \langle\Omega|P_{0}\otimes
P_{0}\pi_{1}(\bm{L}_{3}-\bm{L}_{2}(\mathbb{I}\wedge
Q^{+}\pi_{1}\bm{L}_{2}))P_{3}\pi_{3}\,.$ (2.57)

Taking into account the fact that the output of the product $\pi_{1}\bm{L}$
satisfies the closed string constraints in Eq. (2.1) and has off-shell
momentum for inputs with generic momenta, we can rearrange it as

$\langle S_{4}|\ =\ \langle\omega_{s}|P_{0}\otimes
P_{0}\pi_{1}\bigg{(}\bm{l}_{3}+\bm{l}_{2}\Big{(}\mathbb{I}\wedge\left(\frac{b_{0}^{+}\mathcal{G}}{L_{0}^{+}}\right)\pi_{1}\bm{l}_{2}\Big{)}\bigg{)}P_{3}\pi_{3}\,,$
(2.58)

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
$\mathcal{H}_{Q}\subset\mathcal{H}_{0}$ defined by the (relative) BRST
cohomology [42, 43, 44, 45]:

$\langle S^{phys}|\ =\ \langle S|\hat{\bm{\mathcal{P}}}\,,$ (2.59)

with

$\mathcal{P}\,:\,\mathcal{H}_{0}\ \longrightarrow\
\mathcal{H}_{Q}\,,\qquad\hat{\bm{\mathcal{P}}}\ =\
\sum_{n=0}^{\infty}\frac{1}{n!}\underbrace{\mathcal{P}\wedge\cdots\wedge\mathcal{P}}_{n}\,.$
(2.60)

Decoupling of unphysical states from the physical S-matrix is guaranteed by

$\displaystyle[\bm{Q}\,,\,\bm{S}_{int}]\ =$ $\displaystyle\
\hat{\bm{P}}\Big{(}[\bm{Q}\,,\,\bm{L}_{int}]-\bm{L}_{int}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}[\bm{Q}\,,\,\bm{H}\bm{L}_{int}]\Big{)}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}$
$\displaystyle=$ $\displaystyle\
\hat{\bm{P}}\bm{L}_{int}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}\bm{L}_{int}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}\
=\ 0\,.$ (2.61)

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 $L_{\infty}$ algebra
$(\mathcal{H}^{res},\Omega,\bm{L})$ by means of HPT [46, 47]. This alternative
derivation is given in Appendix A.

###  2.3 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 $\mathcal{SH}^{res}$
:

$\hat{\bm{i}}^{\prime}\ =\
(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}\,,\qquad\bm{\Sigma}\ =\
\bm{l}\hat{\bm{i}}^{\prime}\,.$ (2.62)

Here, as shown in Appendix A, the first map $\hat{\bm{i}}^{\prime}$ is a
cohomomorphism appeared as a deformed chain map in the HPT, and thus
determined by its component $\pi_{1}\hat{\bm{i}}^{\prime}$ :
$\mathcal{SH}^{res}\rightarrow\mathcal{H}^{res}$ . Another map $\bm{\Sigma}$
is related with $\bm{S}_{int}$ in (2.54b) and $\langle S|$ in (2.55a) as

$\pi_{1}\bm{S}_{int}\ =\ \mathcal{G}P_{0}\pi_{1}\bm{\Sigma}\,,\qquad\langle
S|\ =\ \langle\omega_{l}|\xi_{0}P_{0}\otimes P_{0}\pi_{1}\bm{\Sigma}\,.$
(2.63)

We can deduce a relation,

$\pi_{1}\hat{\bm{i}}^{\prime}\ =\
P_{0}\pi_{1}-Q^{+}\mathcal{G}\pi_{1}\bm{\Sigma}\,,$ (2.64)

between two maps. Then, from the recursive relation (2.30), we have the
classical Dyson-Schwinger equation

$\displaystyle\pi_{1}\bm{\Sigma}\ =$ $\displaystyle\
\sum_{n=0}^{\infty}\pi_{1}\bm{B}_{n+2}\left(\frac{1}{(n+2)!}(\pi_{1}\hat{\bm{i}}^{\prime}+\Xi\pi_{1}^{1}\bm{\Sigma})^{\wedge(n+2)}\right)$
$\displaystyle=$ $\displaystyle\
\sum_{n=0}^{\infty}\pi_{1}\bm{B}_{n+2}\left(\frac{1}{(n+2)!}\left(P_{0}\pi_{1}-\Delta\pi_{1}\bm{\Sigma}\right)^{\wedge(n+2)}\right)\,,$
(2.65)

where

$\Delta\ =\ Q^{+}\mathcal{G}-\Xi\pi^{1}\,.$ (2.66)

We extend the $\bm{\Sigma}$ with two parameters $t$ and $s$ counting the
picture number and picture number deficit, respectively, so as to satisfy the
generalized Dyson-Schwinger equation

$\pi_{1}\bm{\Sigma}(s,t)\ =\
\sum_{n=0}^{\infty}\pi_{1}\bm{B}_{n+2}(s,t)\left(\frac{1}{(n+2)!}\left(P_{0}\pi_{1}-\Delta(s,t)\pi_{1}\bm{\Sigma}(s,t)\right)^{\wedge(n+2)}\right)\,,$
(2.67)

where

$\Delta(s,t)\ =\ Q^{+}\mathcal{G}(s,t)-t\,\Xi\pi^{1}\,,$ (2.68)

with $\mathcal{G}(s,t)=\pi^{0}+(tX+s)\pi^{1}$ is determined to satisfy
$\Delta(0,1)=\Delta$ and

$[\bm{Q},\pi_{1}\bm{\Sigma}(s,t)]\ =\ [\bm{\eta},\pi_{1}\bm{\Sigma}(s,t)]\ =\
0\,.$ (2.69)

Then, the derivatives of the extended map $\bm{\Sigma}(s,t)$ satisfy

$\pi_{1}\partial_{t}\bm{\Sigma}(s,t)\ =\
[\bm{Q},\pi_{1}\bm{\rho}(s,t)]\,,\qquad\pi_{1}\partial_{s}\bm{\Sigma}(s,t)\ =\
[\bm{\eta},\pi_{1}\bm{\rho}(s,t)]\,,$ (2.70)

where concrete form of $\bm{\rho}(s,t)$ is not relevant for the physical
S-matrix. The derivations of Eqs. (2.69) and (2.70) are given in Appendices C
and D, respectively. Combining the two equations (2.70), we can derive the key
relation

$\pi_{1}\partial_{t}\bm{\Sigma}(s,t)-X_{0}\circ\pi_{1}\partial_{s}\bm{\Sigma}(s,t)\
=\ [\bm{Q},[\bm{\eta},\pi_{1}\bm{T}(s,t)]]\,,$ (2.71)

with $\pi_{1}\bm{T}(s,t)=\xi_{0}\circ\pi_{1}\bm{\rho}(s,t)$. Here, for the
operator $\mathcal{O}=\xi_{0}$ or $X_{0}$ , the operation $\mathcal{O}\,\circ$
on $\pi_{1}\bm{D}_{n}$ is defined by

$\mathcal{O}\circ\pi_{1}\bm{D}_{n}\ =\
\frac{1}{n+1}\Big{(}\mathcal{O}D_{n}+(-1)^{|\mathcal{O}||D|}D_{n}(\mathcal{O}\pi_{1}\wedge\mathbb{I}_{n-1})\Big{)}\,.$
(2.72)

We can find from the definition that the extended S-matrix $\langle S(s,t)|$
defined by

$\langle S(s,t)|\ =\ \langle\omega_{l}|\xi_{0}P_{0}\otimes
P_{0}\pi_{1}\bm{\Sigma}(s,t)$ (2.73)

can be expanded as999 Note that the lower bound of the summation with respect
to $p$ is actually restricted by the constraint that the cyclic Ramond number
($=$ No. of R-inputs $+$ No. of R-outputs) has to be less than the number of
external lines ($=$ No. of inputs $+$ No. of outputs).

$\displaystyle\langle S(s,t)|\ =$ $\displaystyle\ \sum_{n=0}^{\infty}\langle
S_{n+3}(s,t)|\,,$ (2.74a) $\displaystyle\langle S_{n+3}(s,t)|\ =$
$\displaystyle\ \sum_{m=0}^{n+1}s^{m}\langle S_{n+3}^{[m]}(t)|\,,\quad\langle
S_{n+3}^{[m]}(t)|\ =\ \sum_{p=0}^{n-m+1}t^{p}\langle
S_{n+3}^{(p)}|^{2(n-m-p+1)}\,,$ (2.74b)

induced by the expansion

$\displaystyle\bm{\Sigma}(s,t)\ =$ $\displaystyle\
\sum_{n=0}^{\infty}\bm{\Sigma}_{n+2}(s,t)\,,$ (2.75a)
$\displaystyle\bm{\Sigma}_{n+2}(s,t)\ =$ $\displaystyle\
\sum_{m=0}^{n+1}s^{m}\bm{\Sigma}_{n+2}^{[m]}(t)\,,\quad\bm{\Sigma}_{n+2}^{[m]}(t)\,,\quad=\
\sum_{p=0}^{n-m+1}t^{p}\bm{\Sigma}_{n+2}^{(p)}|^{2(n-m-p+1)}\,.$ (2.75b)

The superscripts with $(\cdot)$ and $[\,\cdot\,]$ indicate the picture number
and the picture number deficit, respectively. Thus, $\langle S_{n+3}|=\langle
S_{n+3}(0,1)|$ is the sum of the amplitudes of several processes as

$\langle S_{n+3}|\ =\ \sum_{p=0}^{n+1}\langle S^{(p)}_{n+3}|^{2(n-p+1)}\,.$
(2.76)

For example, $\langle S_{4}|$ is the sum of three amplitudes

$\displaystyle\langle S_{4}|\ =$ $\displaystyle\ \langle
S_{4}^{(0)}|^{4}+\langle S_{4}^{(1)}|^{2}+\langle S_{4}^{(2)}|^{0}\,.$ (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
$\bm{\Sigma}$) is the NS or R state:

$\langle S_{4}^{(1)}|^{2}\ =\ \langle S_{4}^{(1)}|^{2}_{2}+\langle
S_{4}^{(1)}|^{2}_{0}\,.$ (2.78)

Each amplitude can be calculated by solving the Dyson-Schwinger equation
(2.65). We find

$\displaystyle\langle S_{4}^{(0)}|^{4}_{2}\ =$ $\displaystyle\
\langle\omega_{l}|\xi_{0}P_{0}\otimes
P_{0}\pi_{1}^{1}\left(\bm{B}_{3}^{(0)}|^{4}_{2}-\bm{B}_{2}^{(0)}|^{2}_{0}\left(\mathbb{I}\wedge\frac{b_{0}^{+}}{L_{0}^{+}}\pi_{1}^{0}\bm{B}_{2}^{(0)}|^{2}_{2})\right)\right)P_{3}\pi_{3}\,,$
(2.79a) $\displaystyle\langle S_{4}^{(1)}|^{2}_{0}\ =$ $\displaystyle\
\langle\omega_{l}|\xi_{0}P_{0}\otimes
P_{0}\pi_{1}^{1}\bigg{(}\bm{B}_{3}^{(1)}|^{2}_{0}-\bm{B}_{2}^{(0)}|^{2}_{0}\left(\mathbb{I}\wedge\frac{b_{0}^{+}}{L_{0}^{+}}\pi_{1}^{0}\bm{B}_{2}^{(1)}|^{0}_{0})\right)$
$\displaystyle\hskip
128.0374pt-\bm{B}_{2}^{(0)}|^{2}_{0}\left(\mathbb{I}\wedge\left(\frac{b_{0}^{+}X}{L_{0}^{+}}-\Xi\right)\pi_{1}^{0}\bm{B}_{2}^{(0)}|^{2}_{0})\right)\bigg{)}P_{3}\pi_{3}\,,$
(2.79b) $\displaystyle\langle S_{4}^{(2)}|^{0}_{0}\ =$ $\displaystyle\
\langle\omega_{l}|\xi_{0}P_{0}\otimes
P_{0}\pi_{1}^{0}\left(\bm{B}_{3}^{(2)}|^{0}_{0}-\bm{B}_{2}^{(1)}|^{0}_{0}\left(\mathbb{I}\wedge\frac{b_{0}^{+}}{L_{0}^{+}}\pi_{1}^{0}\bm{B}_{2}^{(1)}|^{0}_{0})\right)\right)P_{3}\pi_{3}\,,$
(2.79c) which agree with those calculated in Ref. [24], and an alternative
expression of the two-NS-two-R amplitude, $\displaystyle\langle
S_{4}^{(1)}|^{2}_{2}\ =$ $\displaystyle\ \langle\omega_{l}|\xi_{0}P_{0}\otimes
P_{0}\pi_{1}^{0}\bigg{(}\bm{B}_{3}^{(1)}|^{2}_{2}-\bm{B}_{2}^{(1)}|^{0}_{0}\left(\mathbb{I}\wedge\frac{b_{0}^{+}}{L_{0}^{+}}\pi_{1}^{0}\bm{B}_{2}^{(0)}|^{2}_{2})\right)$
$\displaystyle\hskip
128.0374pt-\bm{B}_{2}^{(0)}|^{2}_{2}\left(\mathbb{I}\wedge\left(\frac{b_{0}^{+}X}{L_{0}^{+}}-\Xi\right)\pi_{1}^{1}\bm{B}_{2}^{(0)}|^{2}_{0})\right)\bigg{)}P_{3}\pi_{3}\,.$
(2.79d)

It is not difficult to prove that the physical S-matrix agrees with that
calculated in the first-quantized formulation. From (2.71) we find that the
extended S-matrix satisfies the relation

$\partial_{t}\langle S(s,t)|\ -\partial_{s}\langle S(s,t)|(X_{0})_{cyc}=\
\langle\omega_{l}|\xi_{0}P_{0}\otimes
P_{0}\pi_{1}[\bm{Q},[\bm{\eta},\bm{T}(s,t)]]\,,$ (2.80)

where $\langle S(s,t)|(X_{0})_{cyc}$ is defined by

$\langle S(s,t)|(X_{0})_{cyc}\ =\ \sum_{n=0}^{\infty}\langle
S_{n+3}(s,t)|(X_{0}\otimes\mathbb{I}_{n+2}+\mathbb{I}\otimes\mathbb{I}_{n+1}\wedge
X_{0})\,.$ (2.81)

Then, using the expansion in (2.74b), we can find the relation between
components with different picture numbers:

$(p+1)\langle S^{(p+1)}_{n+3}|^{2(n-m-p)}\ -\ (m+1)\langle
S^{(p)}_{n+3}|^{2(n-m-p)}(X_{0})_{cyc}\ =\
\langle\omega_{l}|\xi_{0}P_{0}\otimes
P_{0}\pi_{1}[\bm{Q},[\bm{\eta},\ast\,]]\,.$ (2.82)

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
$\bm{Q}\hat{\bm{\mathcal{P}}}=\bm{\eta}\hat{\bm{\mathcal{P}}}=0$ . By using
this relation repeatedly, we can find the physical amplitude can eventually be
written as

$\langle S^{phys}_{n+3}|\ =\ \sum_{p=0}^{n+1}\langle
S^{(0)}_{n+3}|^{2(n-p+1)}((X_{0})_{cyc})^{p}\hat{\bm{\mathcal{P}}}\,.$ (2.83)

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 $[Q,[\eta,\ *\ ]]$ ,101010 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
$\displaystyle(X_{0})_{1}-(X_{0})_{2}\ =$ $\displaystyle\
[Q\,,\,[\eta\,,\,\xi_{0}\oint_{Z_{1}}\frac{dz_{1}}{2\pi
iz_{1}}\oint_{Z_{2}}\frac{dz_{2}}{2\pi
iz_{2}}\int^{z_{1}}_{z_{2}}dw\partial\xi(w)]]\,,$
$\displaystyle(X_{0})_{1}-X(Z_{1})\ =$ $\displaystyle\
[Q\,,\,[\eta\,,\,\xi_{0}\oint_{Z_{1}}\frac{dz_{1}}{2\pi
iz_{1}}\int^{z_{1}}_{Z_{1}}dz\partial\xi(z)]]\,,$ where $(X_{0})_{1}$ and
$(X_{0})_{2}$ are $X_{0}$s acting on the string 1 and 2, respectively.  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].

##  3 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].

###  3.1 Type II superstring field theory with $L_{\infty}$ structure

The type II superstring field $\Phi$ is Grassmann even and has four components
with ghost number 2:

$\Phi\ =\
\Phi_{NS\textrm{-}NS}+\Phi_{R\textrm{-}NS}+\Phi_{NS\textrm{-}R}+\Phi_{R\textrm{-}R}\in\
\mathcal{H}^{res}=\mathcal{H}_{NS\textrm{-}NS}+\mathcal{H}_{R\textrm{-}NS}^{res}++\mathcal{H}_{NS\textrm{-}R}^{res}+\mathcal{H}_{R\textrm{-}R}^{res}\,.$
(3.84)

The NS-NS component $\Phi_{NS\textrm{-}NS}$ , R-NS component
$\Phi_{R\textrm{-}NS}$ , NS-R component $\Phi_{NS\textrm{-}R}$ , and R-R
component $\Phi_{R\textrm{-}R}$ have picture numbers $(-1,-1)$ , $(-1/2,-1)$ ,
$(-1,-1/2)$ , and $(-1/2,-1/2)$ , respectively. The Hilbert space
$\mathcal{H}^{res}$ is restricted by the closed string constraints (2.1) and
an extra condition (2.8) with

$\displaystyle\mathcal{G}\ =$ $\displaystyle\
\mathscr{X}\overline{\mathscr{X}}\,,\qquad$ $\displaystyle\mathscr{X}\ =$
$\displaystyle\ \pi^{0}+X\pi^{1}\,,\qquad$
$\displaystyle\overline{\mathscr{X}}\ =$ $\displaystyle\
\overline{\pi}^{0}+\overline{X}\overline{\pi}^{1}\,,$ (3.85)
$\displaystyle\mathcal{G}^{-1}\ =$ $\displaystyle\
\mathscr{Y}\overline{\mathscr{Y}}\,,\qquad$ $\displaystyle\mathscr{Y}\ =$
$\displaystyle\ \pi^{0}+\mathcal{Y}\pi^{1}\,,\qquad$
$\displaystyle\overline{\mathscr{Y}}\ =$ $\displaystyle\
\overline{\pi}^{0}+\overline{\mathcal{Y}}\overline{\pi}^{1}\,,$ (3.86)

where $\pi^{0}$ and $\pi^{1}$ ($\overline{\pi}^{0}$ and $\overline{\pi}^{1}$)
are the projection operators onto the NS and R states in the holomorphic
(anti-holomorphic) sectors, respectively. We take the non-local operators

$\mathcal{Y}\ =\
-2\frac{G}{L_{0}^{+}}\delta(\gamma_{0})\,,\qquad\overline{\mathcal{Y}}\
-2\frac{\overline{G}}{L_{0}^{+}}\delta(\overline{\gamma}_{0})\,,$ (3.87)

satisfying the same relations as those satisfied by $Y$ and $\overline{Y}$ ,

$X\mathcal{Y}X\ =\ X\,,\qquad\mathcal{Y}X\mathcal{Y}\ =\
\mathcal{Y}\,,\qquad\overline{X}\,\overline{\mathcal{Y}}\,\overline{X}\ =\
\overline{X}\,,\qquad\overline{\mathcal{Y}}\,\overline{X}\,\overline{\mathcal{Y}}\
=\ \overline{\mathcal{Y}}\,,$ (3.88)

as inverse picture changing operators, which is necessary to consistently
impose all the constraints.111111 In Ref. [25], we took these non-local
inverse changing operators only for the R-R sector, which is enough for
consistency. The result is the same in both cases if we expand the string
field in the ghost zero-modes as in Eq. (3.90). The constraint (2.8) can now
be split into two conditions

$\mathscr{X}\mathscr{Y}\Phi\ =\
\overline{\mathscr{X}}\,\overline{\mathscr{Y}}\Phi=\Phi\,,$ (3.89)

which we use later. Any sate $\mathcal{B}\in\mathcal{H}^{res}$ can be expanded
in the ghost zero-mode as

$\displaystyle\mathcal{B}=$ $\displaystyle\
\mathcal{B}_{NS\textrm{-}NS}+\mathcal{B}_{R\textrm{-}NS}+\mathcal{B}_{NS\textrm{-}R}+\mathcal{B}_{R\textrm{-}R}$
$\displaystyle=$ $\displaystyle\
\left(b_{NS}-c_{0}^{+}B_{NS\textrm{-}NS}\right)+\left(b_{R\textrm{-}NS}-\frac{1}{2}(\gamma_{0}+2c_{0}^{+}G)B_{R\textrm{-}NS}\right)$
$\displaystyle\
+\left(b_{NS\textrm{-}R}-\frac{1}{2}(\overline{\gamma}_{0}+2c_{0}^{+}\overline{G})B_{NS\textrm{-}R}\right)+\left(b_{R\textrm{-}R}-\frac{1}{2}(\gamma_{0}\overline{G}-\overline{\gamma}_{0}G+2c_{0}^{+}G\overline{G})B_{R\textrm{-}R}\right),$
(3.90)

so we denote the string field $\Phi\in\mathcal{H}^{res}$ as

$\displaystyle\Phi=$ $\displaystyle\
\Phi_{NS\textrm{-}NS}+\Phi_{R\textrm{-}NS}+\Phi_{NS\textrm{-}R}+\Phi_{R\textrm{-}R}$
$\displaystyle=$ $\displaystyle\
\left(\phi_{NS}-c_{0}^{+}\psi_{NS\textrm{-}NS}\right)+\left(\phi_{R\textrm{-}NS}-\frac{1}{2}(\gamma_{0}+2c_{0}^{+}G)\psi_{R\textrm{-}NS}\right)$
$\displaystyle\
+\left(\phi_{NS\textrm{-}R}-\frac{1}{2}(\overline{\gamma}_{0}+2c_{0}^{+}\overline{G})\psi_{NS\textrm{-}R}\right)+\left(\phi_{R\textrm{-}R}-\frac{1}{2}(\gamma_{0}\overline{G}-\overline{\gamma}_{0}G+2c_{0}^{+}G\overline{G})\psi_{R\textrm{-}R}\right).$
(3.91)

The on-shell subspace $\mathcal{H}_{0}\subset\mathcal{H}^{res}$ is defined by

$\displaystyle\mathcal{H}_{0}\ =\
\\{\Phi\in\mathcal{H}^{res}\,|\,L_{0}^{+}\Phi_{NS\textrm{-}NS}=G_{0}\Phi_{R\textrm{-}NS}=\overline{G}_{0}\Phi_{NS\textrm{-}R}=G_{0}\Phi_{R\textrm{-}R}=\overline{G}_{0}\Phi_{R\textrm{-}R}=0\\}\,.$
(3.92)

The natural symplectic forms $\Omega$ , $\omega_{s}$ , and $\omega_{l}$ in the
$\mathcal{H}^{res}=\mathcal{H}^{res}_{L}\otimes\mathcal{H}^{res}_{R}$ ,
$\mathcal{H}_{s}=(\mathcal{H}_{s})_{L}\otimes(\mathcal{H}_{s})_{R}$ , and
$\mathcal{H}_{l}=(\mathcal{H}_{l})_{L}\otimes(\mathcal{H}_{l})_{R}$ ,
respectively, are defined similarly to those in Eqs.(2.11), and are related
for $\Phi_{1},\Phi_{2}\in\mathcal{H}^{res}$ as

$\Omega(\Phi_{1}\,,\Phi_{2})\ =\
\omega_{s}(\Phi_{1}\,,\mathcal{G}^{-1}\Phi_{2})\ =\
\omega_{l}(\xi_{0}\overline{\xi}_{0}\Phi_{1},\mathcal{G}^{-1}\Phi_{2})\,.$
(3.93)

It is also useful to introduce $\Omega_{m}$ in the restricted medium Hilbert
space
$\mathcal{H}_{m}^{res}=(\mathcal{H}_{l})_{L}\otimes\mathcal{H}^{res}_{R}$ and
$\omega_{m}$ in the medium Hilbert space
$\mathcal{H}_{m}=(\mathcal{H}_{l})_{L}\otimes(\mathcal{H}_{s})_{R}$ , and
their counterparts the holomorphic and anti-holomorphic sectors exchanged
$\Omega_{\overline{m}}$ in
$\mathcal{H}_{\overline{m}}^{res}=\mathcal{H}^{res}_{L}\otimes(\mathcal{H}_{l})_{R}$
and $\omega_{\overline{m}}$ in
$\mathcal{H}_{\overline{m}}=(\mathcal{H}_{s})_{L}\otimes(\mathcal{H}_{l})_{R}$
, which are defined in an obvious way. They are related with $\omega_{l}$ as

$\displaystyle\Omega_{m}(\Phi_{1}\,,\Phi_{2})\ =$ $\displaystyle\
\omega_{m}(\Phi_{1}\,,\overline{\mathscr{Y}}\Phi_{2})\ =$ $\displaystyle\
\omega_{l}(\overline{\xi_{0}}\Phi_{1}\,,\overline{\mathscr{Y}}\Phi_{2})\,,\qquad\Phi_{1},\Phi_{2}\in$
$\displaystyle\mathcal{H}_{m}^{res}\,,$ (3.94)
$\displaystyle\Omega_{\overline{m}}(\Phi_{1}\,,\Phi_{2})\ =$ $\displaystyle\
\omega_{\overline{m}}(\Phi_{1}\,,\mathscr{Y}\Phi_{2})\ =$ $\displaystyle\
\omega_{l}(\xi_{0}\Phi_{1}\,,\mathscr{Y}\Phi_{2})\,,\qquad\Phi_{1},\Phi_{2}\in$
$\displaystyle\mathcal{H}_{\overline{m}}^{res}\,,$ (3.95)

and with $\Omega$ as

$\displaystyle\Omega_{m}(\xi_{0}\Phi_{1}\,,\mathscr{Y}\Phi_{2})\ =\
\Omega_{\overline{m}}(\overline{\xi_{0}}\Phi_{1}\,,\overline{\mathscr{Y}}\Phi_{2})\
=\
\Omega(\Phi_{1}\,,\Phi_{2})\,,\qquad\Phi_{1},\Phi_{2}\in\mathcal{H}^{res}\,.$
(3.96)

The action and the gauge transformation have the same form as those of the
heterotic string field theory,

$\displaystyle I[\Phi]\ =$ $\displaystyle\
\sum_{n=0}^{\infty}\frac{1}{(n+2)!}\,\Omega(\Phi\,,L_{n+1}(\underbrace{\Phi\,,\cdots\,,\Phi}_{n+1}))\,,$
(3.97a) $\displaystyle\delta\Phi\ =$ $\displaystyle\
\sum_{n=0}^{\infty}\frac{1}{n!}\,L_{n+1}(\underbrace{\Phi\,,\cdots\,,\Phi}_{n}\,,\Lambda)\,,$
(3.97b)

except that the symplectic form $\Omega$ 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 product
$\bm{L}_{H}=\bm{Q}+\bm{L}_{H}^{\,int}$ with

$\displaystyle\bm{L}_{H}\ =$ $\displaystyle\
\hat{\bm{F}}^{-1}(\bm{Q}+\pi^{0}\bm{B})\hat{\bm{F}}\,,$ (3.98a) $\displaystyle
n\pi_{1}\bm{L}_{H}^{int}\ =$ $\displaystyle\
\mathscr{X}\pi_{1}\bm{l}_{H}\,,\qquad\pi_{1}\bm{l}_{H}\ =\
\pi_{1}\bm{B}\hat{\bm{F}}\,.$ (3.98b)

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

$\displaystyle\bm{L}\ =$ $\displaystyle\
\hat{\overline{\bm{F}}}^{-1}(\bm{Q}+\overline{\pi}^{0}\overline{\bm{B}})\hat{\overline{\bm{F}}}\,,$
(3.99a) $\displaystyle\pi_{1}\bm{L}_{int}\ =$ $\displaystyle\
\overline{\mathscr{X}}\pi_{1}\overline{\bm{l}}\,,\qquad\pi_{1}\overline{\bm{l}}\
=\ \pi_{1}\overline{\bm{B}}\,\hat{\overline{\bm{F}}}\,.$ (3.99b)

The product $\overline{\bm{B}}$ is obtained from the generating function

$\overline{\bm{B}}(s,t)\ =\
\sum_{\overline{m},\overline{n},\overline{r}=0}^{\infty}s^{\overline{m}}t^{\overline{n}}\,\overline{\bm{B}}^{(\overline{n})}_{\overline{m}+\overline{n}+\overline{r}+1}|^{2\overline{r}}\
\equiv\sum_{\overline{n}=0}^{\infty}t^{\overline{n}}\,\overline{\bm{B}}^{(\overline{n})}(s)\,,$
(3.100)

with the gauge product

$\overline{\bm{\lambda}}(s,t)\ =\
\sum_{\overline{m},\overline{n},\overline{r}=0}^{\infty}s^{\overline{m}}t^{\overline{n}}\,\overline{\bm{\lambda}}^{(\overline{n}+1)}_{\overline{m}+\overline{n}+\overline{r}+2}|^{2\overline{r}}\,,$
(3.101)

by solving the differential equation

$\displaystyle\partial_{t}\overline{\bm{B}}(s,t)\ =$ $\displaystyle\
[\bm{Q},\overline{\bm{\lambda}}(s,t)]+[\overline{\bm{B}}(s,t),\overline{\bm{\lambda}}(s,t)]^{\overline{1}}+s\,[\overline{\bm{B}}(s,t),\overline{\bm{\lambda}}(s,t)]^{\overline{2}}\,,$
(3.102a) $\displaystyle\partial_{s}\overline{\bm{B}}(s,t)\ =$ $\displaystyle\
[\overline{\bm{\eta}},\overline{\bm{\lambda}}(s,t)]-t\,[\overline{\bm{B}}(s,t),\overline{\bm{\lambda}}(s,t)]^{\overline{2}}\,,$
(3.102b)

with the initial condition

$\overline{\bm{B}}^{(\overline{0})}(s)\ =\ \bm{L}_{H}^{\,int}(s)\ =\
\sum_{\overline{m},\overline{r}=0}^{\infty}s^{\overline{m}}(\bm{L}_{H})_{\overline{m}+\overline{r}+1}|^{2\overline{r}}\,.$
(3.103)

The cohomomorphism $\hat{\overline{\bm{F}}}$ is given by

$\hat{\overline{\bm{F}}}^{-1}\ =\
\pi_{1}\mathbb{I}-\overline{\Xi}\overline{\pi}^{1}_{1}\overline{\bm{B}}\,,$
(3.104)

where $\overline{\pi}_{1}^{1}=\pi_{1}\overline{\pi}^{1}$ . Here, the picture
and cyclic Ramond numbers are those of the anti-holomorphic sector, which are
made clear by letters with bar. The commutators
$[\cdot\,,\cdot]^{\overline{1},\overline{2}}$ are the analogues of
$[\cdot\,,\cdot]^{1,2}$ projected by the cyclic Ramond number of the anti-
holomorphic sector. It would be obvious that $\overline{\bm{B}}(s,t)$
satisfies the similar relations to Eqs. (2.24) as

$\displaystyle\
[\bm{Q},\overline{\bm{B}}(s,t)]+\frac{1}{2}\,[\overline{\bm{B}}(s,t),\overline{\bm{B}}(s,t)]^{\overline{1}}+\frac{s}{2}\,[\overline{\bm{B}}(s,t),\overline{\bm{B}}(s,t)]^{\overline{2}}\
=\ 0\,,$ (3.105a) $\displaystyle\
[\overline{\bm{\eta}},\overline{\bm{B}}(s,t)]-\frac{t}{2}\,[\overline{\bm{B}}(s,t),\overline{\bm{B}}(s,t)]^{2}\
=\ 0\,,$ (3.105b)

which reduce to the $L_{\infty}$ relation of the coderivation
$\bm{Q}-\overline{\bm{\eta}}+\overline{\bm{B}}$ at $(s,t)=(0,1)$ . If you note
that the initial heterotic product has the form of
$\bm{L}_{H}^{\,int}=\mathscr{X}\bm{l}_{H}$ and look carefully the differential
equation (3.102), it is found that the final form of $\overline{\bm{B}}$ has
the form of

$\pi_{1}\overline{\bm{B}}\ =\ \mathscr{X}\pi_{1}\overline{\bm{b}}\,,$ (3.106)

and thus $\pi_{1}\bm{L}_{int}$ is written as the form in which it is manifest
that the products is closed in $\mathcal{H}^{res}$ :

$\pi_{1}\bm{L}_{int}\ =\ \mathcal{G}\pi_{1}\bm{l}\,,\qquad\pi_{1}\bm{l}\ =\
\pi_{1}\overline{\bm{b}}\,\hat{\overline{\bm{F}}}\,.$ (3.107)

###  3.2 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

$\bm{S}_{int}\ =\
\hat{\bm{P}}\bm{L}_{int}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}\,,$
(3.108)

where $\bm{L}_{int}$ 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 maps

$\hat{\bm{i}}^{\prime}\ =\
(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}\,,\qquad\pi_{1}\overline{\bm{\Sigma}}\
=\ \pi_{1}\overline{\bm{l}}\hat{\bm{i}}^{\prime}\,,$ (3.109)

the S-matrix can be represented by $\overline{\bm{\Sigma}}$ similar to that of
the heterotic string field theory as

$\displaystyle\pi_{1}\bm{S}_{int}\ =$ $\displaystyle\
\overline{\mathscr{X}}P_{0}\pi_{1}\overline{\bm{\Sigma}}\,,$ (3.110)
$\displaystyle\langle S|\ =$ $\displaystyle\
\langle\Omega|P_{0}\otimes\pi_{1}\bm{S}_{int}\ =\
\langle\Omega_{\overline{m}}|\overline{\xi}_{0}P_{0}\otimes
P_{0}\pi_{1}\overline{\bm{\Sigma}}\,.$ (3.111)

The $\overline{\bm{\Sigma}}$ is recursively determined by the equation

$\pi_{1}\overline{\Sigma}\ =\
\sum_{n=0}^{\infty}\pi_{1}\overline{\bm{B}}_{n+2}\bigg{(}\frac{1}{(n+2)!}(P_{0}\pi_{1}-\overline{\Delta}\pi_{1}\bm{\Sigma})^{\wedge(n+2)}\bigg{)}\,,$
(3.112)

with

$\overline{\Delta}=Q^{+}\overline{\mathscr{X}}-\overline{\Xi}\overline{\pi}^{1}\,.$
(3.113)

If we define two parameter extension $\overline{\bm{\Sigma}}(s,t)$ using
$\overline{\bm{B}}(s,t)$ and

$\overline{\Delta}(s,t)\ =\
Q^{+}\overline{\mathscr{X}}(s,t)-t\,\overline{\Xi}\overline{\pi}^{1}\,,\qquad\overline{\mathscr{X}}(s,t)\
=\ \overline{\pi}^{0}+(t\overline{X}+s)\overline{\pi}^{1}\,,$ (3.114)

it is easy to see that the relations

$\partial_{t}\overline{\bm{\Sigma}}(s,t)\ =\
[\bm{Q},\overline{\bm{\rho}}(s,t)]\,,\qquad\partial_{s}\overline{\bm{\Sigma}}(s,t)\
=\ [\overline{\bm{\eta}},\overline{\bm{\rho}}(s,t)]\,,$ (3.115)

hold similar to the case of the heterotic string field theory. The explicit
form of $\overline{\bm{\rho}}(s,t)$ 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

$\partial_{t}\overline{\bm{\Sigma}}(s,t)-\overline{X}_{0}\circ\partial_{s}\overline{\bm{\Sigma}}(s,t)\
=\ [\bm{Q},[\overline{\bm{\eta}},\overline{\bm{T}}(s,t)]]\,,$ (3.116)

with $\overline{\bm{T}}(s,t)=\overline{\xi}_{0}\circ\overline{\bm{\rho}}(s,t)$
. Then, the extended S-matrix

$\displaystyle\langle S_{n+3}(s,t)|\ =$ $\displaystyle\
\sum_{\overline{m}=0}^{n+1}s^{\overline{m}}\langle
S_{n+3}^{[\overline{m}]}(t)|\,,\quad\langle S_{n+3}^{[\overline{m}]}(t)|\ =\
\sum_{\overline{p}=0}^{n-\overline{m}+1}t^{\overline{p}}\langle
S_{n+3}^{(\overline{p})}|^{2(n-\overline{m}-\overline{p}+1)}\,,$ (3.117)

induced by the expansion

$\overline{\bm{\Sigma}}_{n+2}(s,t)\ =\
\sum_{\overline{m}=0}^{n+1}s^{\overline{m}}\,\overline{\bm{\Sigma}}_{n+2}^{[\overline{m}]}(t)\,,\quad\overline{\bm{\Sigma}}_{n+2}^{[\overline{m}]}(t)\,,\quad=\
\sum_{\overline{p}=0}^{n-\overline{m}+1}t^{\overline{p}}\,\overline{\bm{\Sigma}}_{n+2}^{(\overline{p})}|^{2(n-\overline{m}-\overline{p}+1)}\,,$
(3.118)

satisfies the relation

$\partial_{t}\langle S(s,t)|\ -\partial_{s}\langle S(s,t)|\overline{X}_{0}=\
\langle\Omega_{m}|\overline{\xi}_{0}P_{0}\otimes
P_{0}\pi_{1}[\bm{Q},[\overline{\bm{\eta}},\overline{\bm{T}}(s,t)]]\,.$ (3.119)

Hence, for the physical S-matrix, we have

$\langle S^{phys}_{n+3}|\ =\ \sum_{\overline{p}=0}^{n+1}\langle
S_{n+3}^{(\overline{0})}|^{2(n-\overline{p}+1)}(\overline{X}_{0})^{\overline{p}}\hat{\bm{\mathcal{P}}}\,,$
(3.120)

in a similar way to the case of the heterotic string field theory. Here,
$\hat{\bm{\mathcal{P}}}$ is the projection operator onto the physical Hilbert
space satisfying
$\bm{Q}\hat{\bm{\mathcal{P}}}=\bm{\eta}\hat{\bm{\mathcal{P}}}=\overline{\bm{\eta}}\hat{\bm{\mathcal{P}}}=0$
. The amplitudes with no anti-holomorphic picture number $\langle
S_{n+3}^{(\overline{0})}|$ is nothing but those of the heterotic string field
theory,

$\langle S_{n+3}^{(\overline{0})}|\ =\ \langle(S_{H})_{n+3}|\,,$ (3.121)

by construction. It is embedded in the type II superstring Hilbert space as

$\langle S_{H}|\ =\
\langle\Omega_{m}|\xi_{0}P_{0}\otimes\mathscr{Y}\pi_{1}\bm{S}_{H}\ =\
\langle\Omega_{m}|\xi_{0}P_{0}\otimes P_{0}\pi_{1}\bm{\Sigma}_{H}\,,$ (3.122)

with

$\bm{S}_{H}\ =\
\hat{\bm{P}}\bm{L}_{H}^{int}(\hat{\bm{I}}-\bm{H}\bm{L}_{H}^{int})^{-1}\hat{\bm{P}}\,.$
(3.123)

So, now it is easy to see that

$\langle(S_{H})_{n+3}|\ =\ \sum_{p=0}^{n+1}\langle
S^{(0)}_{n+3}|^{2(r-p+1)}(X_{0})^{p}\,,$ (3.124)

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

$\displaystyle\langle S_{n+3}^{phys}|\ =\
\sum_{p=0}^{n+1}\sum_{\overline{p}=0}^{n+1}\langle
S_{n+3}^{(0,\overline{0})}|^{\left(2(n-p+1),2(n-\overline{p}+1)\right)}(X_{0})^{p}(\overline{X}_{0})^{\overline{p}}\hat{\bm{\mathcal{P}}}\,.$
(3.125)

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.

##  4 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.

###  4.1 Open superstring field theory with cyclic $A_{\infty}$ structure

The open superstring field $\Phi$ is Grassmann odd and has two components with
ghost number 1:

$\Phi\ =\ \Phi_{NS}+\Phi_{R}\ \in\mathcal{H}^{res}\ =\
\mathcal{H}_{NS}+\mathcal{H}^{res}_{R}\,.$ (4.126)

The NS component $\Phi_{NS}$ and R component $\Phi_{R}$ have picture numbers
$-1$ and $-1/2$ , respectively. The Hilbert space $\mathcal{H}^{res}$ is
restricted by a constraint (2.8) with

$\displaystyle\mathcal{G}\ =$ $\displaystyle\ \pi^{0}+X\pi^{1}\,,\qquad$
$\displaystyle X\ =$ $\displaystyle\
-\delta(\beta_{0})G_{0}+\delta^{\prime}(\beta_{0})b_{0}\,,$ (4.127a)
$\displaystyle\mathcal{G}^{-1}\ =$ $\displaystyle\ \pi^{0}+Y\pi^{1}\,,\qquad$
$\displaystyle Y\ =$ $\displaystyle\ -c_{0}\delta^{\prime}(\gamma_{0})\,.$
(4.127b)

The string field $\Phi\in\mathcal{H}^{res}$ is expanded in the ghost zero-
modes as

$\displaystyle\Phi\ =\
(\phi_{NS}-c_{0}\psi_{NS})+\left(\phi_{R}-\frac{1}{2}(\gamma_{0}+c_{0}G)\psi_{R}\right)\,.$
(4.128)

The on-shell subspace $\mathcal{H}_{0}\subset\mathcal{H}^{res}$ and the
projector $P_{0}$ are introduced as

$P_{0}\,:\,\mathcal{H}^{res}\longrightarrow\mathcal{H}_{0}\ =\
\\{\Phi\in{\mathcal{H}^{res}\mid L_{0}\Phi_{NS}\ =\ G_{0}\Phi_{R}\ =\
0\,}\\}\,.$ (4.129)

The symplectic forms are defined by

$\displaystyle\omega_{l}(\Phi_{1},\Phi_{2})\ =$ $\displaystyle\
(-1)^{\textrm{deg}(\Phi_{1})}{}_{l}\langle\Phi_{1}|\Phi_{2}\rangle_{l}\,,\qquad$
$\displaystyle\Phi_{1},\Phi_{2}\in$ $\displaystyle\ \mathcal{H}_{l}\,,$
(4.130a) $\displaystyle\omega_{s}(\Phi_{1},\Phi_{2})\ =$ $\displaystyle\
(-1)^{\textrm{deg}(\Phi_{1})}{}_{s}\langle\Phi_{1}|\Phi_{2}\rangle_{s}\,,\qquad$
$\displaystyle\Phi_{1},\Phi_{2}\in$ $\displaystyle\ \mathcal{H}_{s}\,,$
(4.130b) $\displaystyle\Omega(\Phi_{1},\Phi_{2})\ =$ $\displaystyle\
(-1)^{\textrm{deg}(\Phi_{1})}{}_{s}\langle\Phi_{1}|\mathcal{G}^{-1}|\Phi_{2}\rangle_{s}\,,\qquad$
$\displaystyle\Phi_{1},\Phi_{2}\in$ $\displaystyle\ \mathcal{H}^{res}\,,$
(4.130c)

using the BPZ inner product. The degree of generic states $\Phi_{1,2}$ are
defined by $\textrm{deg}(\Phi_{1,2})=(|\Phi_{1,2}|+1)\ \textrm{mod}\ 2$ , and
thus, in particular, the degree of the open superstring field $\Phi$ is even.

The action and gauge transformation are now given by

$I[\Phi]\ =\
\sum_{n=0}^{\infty}\frac{1}{n+2}\Omega(\Phi,M_{n+1}(\underbrace{\Phi,\cdots,\Phi}_{n+1}))\,,$
(4.131)

and

$\delta\Phi\ =\
\sum_{n=0}^{\infty}\sum_{k=0}^{n}M_{n+1}(\underbrace{\Phi,\cdots,\Phi}_{n-k},\Lambda,\underbrace{\Phi,\cdots,\Phi}_{k})\,,$
(4.132)

by using the non-commutative open-superstring products, $M_{1}=Q$ and
$M_{n+2}\ (n\geq 0)$ , which satisfy the $A_{\infty}$ relations

$\displaystyle\begin{split}&\
\sum_{m=0}^{n}\sum_{j=0}^{n-m}(-1)^{\epsilon(1,j)}\ \\\ &\hskip
28.45274pt\times\
M_{n-m+1}(\Phi_{1}\,,\cdots,\Phi_{j}\,,M_{m+1}(\Phi_{j+1}\,,\cdots,\Phi_{j+m+1})\,,\Phi_{j+m+2}\,,\cdots,\Phi_{n+1})\
=\ 0\,,\end{split}$ (4.133)

and the cyclicity condition

$\Omega(M_{n}(\Phi_{1}\,,\cdots\,,\Phi_{n}),\Phi_{n+1})\ =\
(-1)^{\textrm{deg}(\Phi_{1})\epsilon(2,n+1)}\Omega(M_{n}(\Phi_{2}\,,\cdots\,,\Phi_{n+1}),\Phi_{1})\,,$
(4.134)

where $\epsilon(i,j)=\sum_{k=i}^{j}\textrm{deg}(\Phi_{k})$ .

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 $\bm{B}(s,t)$ acting on the
symmetrized tensor algebra $\mathcal{SH}_{l}$ with the coderivation

$\bm{A}(s,t)\ =\
\sum_{n,m,r=0}^{\infty}s^{m}t^{n}\bm{A}^{(n)}_{m+n+r+1}|^{2r}\,,$ (4.135)

acting on the tensor algebra
$\mathcal{TH}^{res}=\oplus_{n=0}^{\infty}(\mathcal{H}^{res})^{\otimes n}$ . If
$\bm{A}(s,t)$ satisfies the differential equations

$\displaystyle\partial_{t}\bm{A}(s,t)\ =$ $\displaystyle\
[\bm{Q},\bm{\mu}(s,t)]+[\bm{A}(s,t),\bm{\mu}(s,t)]^{1}+s\,[\bm{A}(s,t),\bm{\mu}(s,t)]^{2}\,,$
(4.136a) $\displaystyle\partial_{s}\bm{A}(s,t)\ =$ $\displaystyle\
[\bm{\eta},\bm{\mu}(s,t)]-t\,[\bm{A}(s,t),\bm{\mu}(s,t)]^{2}\,,$ (4.136b)

with the (even) coderivation of the open superstring gauge products

$\bm{\mu}(s,t)\ =\
\sum_{m,n,r=0}^{\infty}s^{m}t^{n}\bm{\mu}^{(n+1)}_{m+n+r+2}|^{2r}\,,$ (4.137)

it satisfies the similar relations to Eqs. (2.24),

$\displaystyle\
[\bm{Q},\bm{A}(s,t)]+\frac{1}{2}\,[\bm{A}(s,t),\bm{A}(s,t)]^{1}+\frac{s}{2}\,[\bm{A}(s,t),\bm{A}(s,t)]^{2}\
=\ 0\,,$ (4.138a) $\displaystyle\
[\bm{\eta},\bm{A}(s,t)]-\frac{t}{2}\,[\bm{A}(s,t),\bm{A}(s,t)]^{2}\ =\ 0\,.$
(4.138b)

At $(s,t)=(0,1)$ , they reduce to the $A_{\infty}$ relations of the
coderivation $\bm{Q}-\bm{\eta}+\bm{A}(0,1)$ . 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

$\bm{A}(s,0)\ =\ \bm{M}_{B}^{(0)}(s)\ =\
\sum_{m,r=0}^{\infty}s^{m}\bm{M}^{(0)}_{m+r+1}|^{2r}\,.$ (4.139)

Here, $\bm{M}^{(0)}_{n+2}$ are the open string products without PCO-
insertions, which we call the open bosonic string products. The cyclic
$A_{\infty}$ algebra $\bm{M}=\bm{Q}+\bm{M}_{int}$ is constructed from
$\bm{A}=\bm{A}(0,1)$ as

$\displaystyle\bm{M}\ =$ $\displaystyle\
\hat{\bm{F}}^{-1}(\bm{Q}+\pi^{0}\bm{A})\hat{\bm{F}}\,,$ (4.140a)
$\displaystyle\pi_{1}\bm{M}_{int}\ =$ $\displaystyle\
\mathcal{G}\bm{a}\,,\qquad\pi_{1}\bm{a}\ =\ \pi_{1}\bm{A}\hat{\bm{F}}\,,$
(4.140b)

using the cohomomorphism

$\pi_{1}\hat{\bm{F}}^{-1}\ =\ \pi_{1}\mathbb{I}-\Xi\pi_{1}^{1}\bm{A}\,.$
(4.141)

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 cohomomorphism $\hat{\bm{F}}$ becomes trivial,
$\pi_{1}^{0}\hat{\bm{F}}=\pi_{1}^{0}\mathbb{I}$ , and the differential
equations (4.136) reduce to those in Ref. [9] for the string product

$\bm{A}|^{0}(s,t)\ =\
\sum_{n,m=0}^{\infty}s^{m}t^{n}\bm{A}^{(n)}_{n+m+1}|^{0}\,.$ (4.142)

It is also a generalization of that in Ref. [17], in which a complete action
of the open superstring field theory based on the $A_{\infty}$ algebra was
first constructed. The string products with $A_{\infty}$ structure (4.140)
completely agree with those constructed in Ref. [17] if we take the
associative bosonic product,
$\bm{M}_{B}^{(0)}(s)=\bm{m}_{2}|^{0}+s\,\bm{m}_{2}|^{2}\,,$ as an initial
condition,121212 The associative product $m_{2}$ can, for example, be defined
by using the Witten’s star product as
$m_{2}(\Phi_{1},\Phi_{2})=(-1)^{\textrm{deg}(\Phi_{1})}\Phi_{1}*\Phi_{2}$ .
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.

###  4.2 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:

$\langle S|\ =\
\langle\Omega|P_{0}\otimes\pi_{1}\bm{S}_{int}\,:\,\mathcal{H}_{0}\otimes\mathcal{TH}_{0}\longrightarrow\mathbb{C}\,,$
(4.143)

with

$\bm{S}_{int}\ =\
\hat{\bm{P}}\bm{M}_{int}(\bm{I}-\bm{H}\bm{M}_{int})^{-1}\hat{\bm{P}}\,.$
(4.144)

Here, all the quantities are those acting on $\mathcal{TH}^{res}$ , which are
similarly defined for the open superstring,

$\bm{H}\ =\ \sum_{r,s=0}^{\infty}\mathbb{I}^{\otimes r}\otimes(-Q^{+})\otimes
P_{0}^{\otimes s}\,,\qquad\hat{\bm{P}}\ =\ \sum_{n=0}^{\infty}P_{0}^{\otimes
n}\,,\qquad\hat{\bm{I}}\ =\ \sum_{n=0}^{\infty}\mathbb{I}^{\otimes n}\,,$
(4.145)

using the open string homotopy operator

$Q^{+}\ =\ \frac{1}{L_{0}}b_{0}(1-P_{0})\,,$ (4.146)

and satisfy the same relation as (2.50). If we introduce two maps

$\hat{\bm{i}}^{\prime}\ =\
(\hat{\bm{I}}-\bm{H}\bm{M}_{int})^{-1}\hat{\bm{P}}\,,\qquad\bm{\Sigma}\ =\
\bm{a}\hat{\bm{i}}^{\prime}\,,$ (4.147)

the map $\bm{\Sigma}$ is related to $\bm{S}_{int}$ as

$\pi_{1}\bm{S}_{int}\ =\ \mathcal{G}P_{0}\pi_{1}\bm{\Sigma}\,,\quad\langle S|\
=\ \langle\omega_{l}|\xi_{0}P_{0}\otimes P_{0}\pi_{1}\bm{\Sigma}\,,$ (4.148)

and satisfies the Dyson-Schwinger equation

$\pi_{1}\bm{\Sigma}\ =\
\sum_{n=0}^{\infty}\pi_{1}\bm{A}_{n+2}\Big{(}(P_{0}\pi_{1}-\Delta\pi_{1}\bm{\Sigma})^{\otimes(n+2)}\Big{)}\,,$
(4.149)

where $\Delta=Q^{+}\mathcal{G}-\Xi\pi^{1}$ . 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 $\bm{\Sigma}(s,t)$ follows the extended Dyson-
Schwinger equation

$\pi_{1}\bm{\Sigma}(s,t)\ =\
\sum_{n=0}^{\infty}\pi_{1}\bm{A}_{n+2}(s,t)\Big{(}(P_{0}\pi_{1}-\Delta(s,t)\pi_{1}\bm{\Sigma}(s,t))^{\otimes(n+2)}\Big{)}\,,$
(4.150)

with

$\Delta(s,t)=Q^{+}\mathcal{G}(s,t)-t\Xi\pi^{1}\,,\qquad\mathcal{G}(s,t)=\pi^{0}+(tX+s)\pi^{1}\,,$
(4.151)

$\bm{\Sigma}(s,t)$ commutes with $\bm{Q}$ and $\bm{\eta}$ ,

$[\bm{Q}\,,\bm{\Sigma}(s,t)]\ =\ 0\,,\qquad[\bm{\eta}\,,\bm{\Sigma}(s,t)]\ =\
0\,,$ (4.152a) and satisfies $\partial_{t}\bm{\Sigma}(s,t)\ =\
[\bm{Q},\bm{\rho}(s,t)]\,,\qquad\partial_{s}\bm{\Sigma}(s,t)\ =\
[\bm{\eta},\bm{\rho}(s,t)]\,,$ (4.152b)

with certain $\bm{\rho}(s,t)$ 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 $\langle
S(s,t)|=\langle\omega_{l}|\xi_{0}P_{0}\otimes P_{0}\pi_{1}\bm{\Sigma}(s,t)$
satisfies the equation

$\partial_{t}\langle S(s,t)|-\partial_{s}\langle S(s,t)|X_{0}\ =\ 0\,,$
(4.153)

with

$\langle S(s,t)|X_{0}\ =\ \sum_{n=0}^{\infty}\langle
S_{n+3}(s,t)|\Big{(}X_{0}\otimes\mathbb{I}_{n+2}+\mathbb{I}\otimes\sum_{r=0}^{n+1}\mathbb{I}_{r}\otimes
X_{0}\otimes\mathbb{I}_{n-r+1}\Big{)}\,,$ (4.154)

except for the terms which vanish when acting on physical states.
Consequently, the physical S-matrix at the tree level can be rewritten as

$\langle S^{phys}|\ =\ \sum_{n=0}^{\infty}\sum_{p=0}^{n+1}\langle
S^{(0)}|^{2(n-p+1)}(X_{0})^{p}\hat{\bm{\mathcal{P}}}\,,$ (4.155)

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.

##  5 Summary and discussion

In this paper, we have shown that the tree-level physical S-matrices derived
from the homotopy-algebra-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 $\bm{B}(s,t)$ 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 [50, 32]. 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.

Acknowledgments

The author would like to thank Tatsuya Sugimoto for collaboration in the early
stage of this work. He also would like to thank Jakub Vošmera for drawing his
attention to Refs. [39, 40]. He also would like to express his appreciation to
organizers of “Fundamental Aspects of String Theory” ICTP-SAIFR, Sao Paulo
1-12 June 2020, in particular Nathan Berkovits, for providing exciting
atmosphere through the online workshop. This work was supported in part by
JSPS Grant-in-Aid for Scientific Research (C) Grant Number JP18K03645.

##  Appendix 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 $L_{\infty}$ algebra
($\mathcal{H}^{res},\Omega,\bm{L}$) by means of HPT.131313 The minimal model
usually considered [46, 47] is the S-matrix naturally appeared in the light-
cone gauge, in which all the unphysical model is gauged away [58].

Consider two chain complexes $(\mathcal{H}^{res},d=Q)$ and
$(\mathcal{H}_{0},D=QP_{0})$ with chain maps

$p\ =\ P_{0}\,:\,\mathcal{H}^{res}\twoheadrightarrow\mathcal{H}_{0}\,,\qquad
i\ =\ P_{0}\,:\,\mathcal{H}_{0}\hookrightarrow\mathcal{H}^{res}\,,$ (A.156)

satisfying

$pi\ =\ \mathbb{I}_{\mathcal{H}_{0}}\ =\ P_{0}\,,\qquad ip\ =\
\mathbb{I}_{\mathcal{H}^{res}}+hd+dh\ =\ \mathbb{I}-Q^{+}Q-QQ^{+}\,.$ (A.157)

The chain complex $(\mathcal{H}_{0},QP_{0})$ , 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. (2.49):
$(\mathcal{SH}^{res},\bm{d}=\bm{Q})$ and $(\mathcal{SH}_{0},\bm{D}=\bm{QP})$
with

$\displaystyle\hat{\bm{p}}\ =\
\hat{\bm{P}}\,:\,\mathcal{SH}^{res}\twoheadrightarrow\mathcal{SH}_{0}\,,\qquad\hat{\bm{i}}\
=\ \hat{\bm{P}}\,:\,\mathcal{SH}_{0}\hookrightarrow\mathcal{SH}^{res}\,,$
(A.158)

satisfying

$\hat{\bm{p}}\hat{\bm{i}}\ =\ \hat{\bm{P}}\,,\qquad\hat{\bm{i}}\hat{\bm{p}}\
=\ \hat{\bm{I}}+\bm{HQ}+\bm{QH}\,.$ (A.159)

Then, if we perturb $\bm{Q}$ by $\bm{L}_{int}$ so that
$(\bm{Q}+\bm{L}_{int})^{2}=0$ , the homological perturbation lemma tells us
that the equivalence data are deformed as

$\displaystyle\bm{d}^{\prime}\ =$ $\displaystyle\ \bm{Q}+\bm{L}_{int}\ =\
\bm{L}\,,$ (A.160) $\displaystyle\hat{\bm{i}}^{\prime}\ =$ $\displaystyle\
(\hat{\bm{I}}-\bm{HL}_{int})^{-1}\hat{\bm{P}}\,,$ (A.161)
$\displaystyle\hat{\bm{p}}^{\prime}\ =$ $\displaystyle\
\hat{\bm{P}}(\hat{\bm{I}}-\bm{L}_{int}\bm{H})^{-1}\,,$ (A.162)
$\displaystyle\bm{D}^{\prime}\ =$ $\displaystyle\
\bm{QP}+\hat{\bm{P}}\bm{L}_{int}(\hat{\bm{I}}-\bm{HL}_{int})^{-1}\hat{\bm{P}}\
=\ \bm{QP}+\bm{S}_{int}\,,$ (A.163) $\displaystyle\bm{h}^{\prime}\ =$
$\displaystyle\ \bm{H}(\hat{\bm{I}}-\bm{L}_{int}\bm{H})^{-1}\ =\
(\hat{\bm{I}}-\bm{HL}_{int}^{-1})\bm{H}\,.$ (A.164)

We confirm below that the almost minimal model
$(\mathcal{SH}_{0},\bm{D}^{\prime})$ is actually $L_{\infty}$-quasi-isomorphic
to $(\mathcal{SH}^{res},\bm{d}^{\prime}=\bm{L})$ by showing that the deformed
maps $\hat{\bm{i}}^{\prime}$ and $\hat{\bm{p}}^{\prime}$ are cohomomorphisms,
and $\bm{D}^{\prime}$ is a coderivation.

Note that coderivation $\bm{D}$ and cohomomorphism $\hat{\bm{F}}$ satisfy

$\displaystyle\Delta\bm{D}\ =$ $\displaystyle\
(\bm{D}\otimes\mathbb{I}+\mathbb{I}\otimes\bm{D})\Delta\,,$ (A.165)
$\displaystyle\Delta\hat{\bm{F}}=$ $\displaystyle\
(\hat{\bm{F}}\otimes\hat{\bm{F}})\Delta\,,$ (A.166)

with respect to the coproduct $\Delta$ . On the other hand, the homotopy
operator $\bm{H}$ defined by (2.49a) itself does not have any good property
under the coproduct $\Delta$ . We have

$\displaystyle\Delta\bm{H}\ =$ $\displaystyle\
\sum_{r,s,t,u=0}^{\infty}\begin{pmatrix}r+t\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ u\end{pmatrix}$ $\displaystyle\hskip
14.22636pt\times\Big{\\{}(-Q^{+})\wedge\mathbb{I}^{\wedge r}\wedge
P_{0}^{\wedge s}\otimes\mathbb{I}^{\wedge t}\wedge P_{0}^{\wedge
u}+\mathbb{I}^{\wedge r}\wedge P_{0}^{\wedge
s}\otimes(-Q^{+})\wedge\mathbb{I}^{\wedge t}\wedge P_{0}^{\wedge
u}\Big{\\}}\Delta\,.$ (A.167)

However, the complicated factor in front of the $\Delta$ in the right hand
side becomes simple if it acts on
$(\hat{\bm{P}}\otimes\bm{A}+\bm{A}\otimes\hat{\bm{P}})$ or
$(\bm{H}\otimes\hat{\bm{I}}-\hat{\bm{I}}\otimes\bm{H})$ with any map $\bm{A}$
on $\mathcal{SH}^{res}$ , and we find that

$\displaystyle\ \sum_{r,s,t,u=0}^{\infty}\begin{pmatrix}r+t\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ u\end{pmatrix}$ $\displaystyle\hskip
28.45274pt\times\Big{(}(-Q^{+})\wedge\mathbb{I}^{\wedge r}\wedge P_{0}^{\wedge
s}\otimes\mathbb{I}^{\wedge t}\wedge P_{0}^{\wedge u}+\mathbb{I}^{\wedge
r}\wedge P_{0}^{\wedge s}\otimes(-Q^{+})\wedge\mathbb{I}^{\wedge t}\wedge
P_{0}^{\wedge
u}\Big{)}\Big{(}\hat{\bm{P}}\otimes\bm{A}+\bm{A}\otimes\hat{\bm{P}}\Big{)}$
$\displaystyle\ =\ \hat{\bm{P}}\otimes\bm{HA}+\bm{HA}\otimes\hat{\bm{P}}\,,$
(A.168)

or

$\displaystyle\ \sum_{r,s,t,u=0}^{\infty}\begin{pmatrix}r+t\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ u\end{pmatrix}$ $\displaystyle\hskip
28.45274pt\times\Big{(}(-Q^{+})\wedge\mathbb{I}^{\wedge r}\wedge P_{0}^{\wedge
s}\otimes\mathbb{I}^{\wedge t}\wedge P_{0}^{\wedge u}+\mathbb{I}^{\wedge
r}\wedge P_{0}^{\wedge s}\otimes(-Q^{+})\wedge\mathbb{I}^{\wedge t}\wedge
P_{0}^{\wedge
u}\Big{)}\Big{(}\bm{H}\otimes\hat{\bm{I}}-\hat{\bm{I}}\otimes\bm{H}\Big{)}$
$\displaystyle\ =\ -\bm{H}\otimes\bm{H}\,.$ (A.169)

The first relation (A.168) is shown by using the formula

$\sum_{u=0}^{t}\begin{pmatrix}r+t-u\\\ r\end{pmatrix}\begin{pmatrix}s+u\\\
s\end{pmatrix}\ =\ \begin{pmatrix}r+s+t+1\\\ t\end{pmatrix}\,,$ (A.170)

which can be proven by mathematical induction with respect to $t$ as follows.
For $t=0$ (and $t=1$) , Eq. (A.170) is clearly true. Suppose that it holds for
$t=t_{0}$ , then we have

$\displaystyle\sum_{u=0}^{t_{0}+1}\begin{pmatrix}r+t_{0}+1-u\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ s\end{pmatrix}\ =$ $\displaystyle\
\begin{pmatrix}r+t_{0}+1\\\
r\end{pmatrix}+\sum_{u=1}^{t_{0}}\begin{pmatrix}r+t_{0}+1-u\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ s\end{pmatrix}\
+\begin{pmatrix}s+t_{0}+1\\\ s\end{pmatrix}$ $\displaystyle=$ $\displaystyle\
\sum_{u=0}^{t_{0}}\frac{r+t_{0}+1-u}{t_{0}+1}\begin{pmatrix}r+t_{0}-u\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ s\end{pmatrix}$ $\displaystyle\
+\sum_{u=1}^{t_{0}+1}\frac{s+u}{t_{0}+1}\begin{pmatrix}r+t_{0}+1-u\\\
r\end{pmatrix}\begin{pmatrix}s+u-1\\\ s\end{pmatrix}$ $\displaystyle=$
$\displaystyle\
\frac{r+s+t_{0}+2}{t_{0}+1}\sum_{u=0}^{t_{0}}\begin{pmatrix}r+t_{0}-u\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ s\end{pmatrix}$ $\displaystyle=$
$\displaystyle\ \begin{pmatrix}r+s+t_{0}+2\\\ t_{0}+1\end{pmatrix}\,,$ (A.171)

using the assumption of the induction in the last equality. We used simple
relations of the binomial coefficients

$t\begin{pmatrix}t-1\\\ r\end{pmatrix}\ =\ (t-r)\begin{pmatrix}t\\\
r\end{pmatrix}\ =\ (r+1)\begin{pmatrix}t\\\ r+1\end{pmatrix}\,,$ (A.172)

for $0\leq r<t\in\mathbb{Z}$ , and the relation

$\displaystyle\begin{pmatrix}r+t_{0}+1-u\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ s\end{pmatrix}\ =$ $\displaystyle\
\frac{r+t_{0}+1-u}{t_{0}+1}\begin{pmatrix}r+t_{0}-u\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ s\end{pmatrix}$ $\displaystyle\
+\frac{s+u}{t_{0}+1}\begin{pmatrix}r+t_{0}+1-u\\\
r\end{pmatrix}\begin{pmatrix}s+u-1\\\ s\end{pmatrix}\,,$ (A.173)

for $1\leq u\leq t_{0}$ , which can easily be confirmed using (A.172). Since
Eq. (A.171) shows that Eq. (A.170) is true for $t=t_{0}+1$ , Eq. (A.170) is
true by mathematical induction.

The second relation (A.169) follows from the relation

$\frac{1}{t!}\Big{(}\mathbb{I}^{\wedge t}\wedge P_{0}^{\wedge
u}\Big{)}\frac{1}{(t+u+1)!}\sum_{\alpha=0}^{t+u}\Big{(}\mathbb{I}^{\wedge(t+u-\alpha)}\wedge
P_{0}^{\wedge\alpha}\Big{)}\ =\
\frac{1}{(t+1)!}\sum_{\alpha=0}^{t}\Big{(}\mathbb{I}^{\wedge(t-\alpha)}\wedge
P_{0}^{\wedge(u+\alpha)}\Big{)}\,,$ (A.174)

and the formula

$\displaystyle\sum_{\alpha=0}^{u}\begin{pmatrix}r+t-s-\alpha+1\\\
r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\
s\end{pmatrix}+\sum_{\alpha=0}^{s}\begin{pmatrix}r+t-u-\alpha+1\\\
t-u\end{pmatrix}\begin{pmatrix}u+\alpha\\\ u\end{pmatrix}\ =\
\begin{pmatrix}r+t+2\\\ r+1\end{pmatrix}\,,$ (A.175)

for $r\geq s$ and $t\geq u$ . In order to show the relation (A.174), we first
prove the formula

$\sum_{\beta=0}^{u}\begin{pmatrix}u\\\
\beta\end{pmatrix}\frac{(t+u-\alpha-\beta)!}{(t-\alpha)!}\frac{(\alpha+\beta)!}{\alpha!}\
=\ \frac{(t+u+1)!}{(t+1)!}\,,$ (A.176)

by mathematical induction with respect to $u$ as follows. For $u=0$ , it
clearly holds. Suppose that it holds for $u=u_{0}$ , then

$\displaystyle\ \sum_{\beta=0}^{u_{0}+1}\begin{pmatrix}u_{0}+1\\\
\beta\end{pmatrix}\frac{(t+u_{o}+1-\alpha-\beta)!}{(t-\alpha)!}\frac{(\alpha+\beta)!}{\alpha!}$
$\displaystyle\ =\
\frac{(t+u_{0}+1-\alpha)!}{(t-\alpha)!}+\sum_{\beta=1}^{u_{0}}\begin{pmatrix}u_{0}+1\\\
\beta\end{pmatrix}\frac{(t+u_{0}+1-\alpha-\beta)!}{(t-\alpha)!}\frac{(\alpha+\beta)!}{\alpha!}+\frac{(\alpha+u_{0}+1)!}{\alpha!}$
$\displaystyle\ =\ \sum_{\beta=0}^{u_{0}}\begin{pmatrix}u_{0}\\\
\beta\end{pmatrix}\frac{(t+u_{0}+1-\alpha-\beta)}{(t-\alpha)!}\frac{(\alpha+\beta)!}{\alpha!}+\sum_{\beta=1}^{u_{0}+1}\begin{pmatrix}u_{0}\\\
\beta-1\end{pmatrix}\frac{(t+u_{0}+1-\alpha-\beta)!}{(t-\alpha)!}\frac{(\alpha+\beta)!}{\alpha!}$
$\displaystyle\ =\ (t+u_{0}+2)\sum_{\beta=0}^{u_{0}}\begin{pmatrix}u_{0}\\\
\beta\end{pmatrix}\frac{(t+u_{0}-\alpha-\beta)!}{(t-\alpha)!}\frac{(\alpha+\beta)!}{\alpha!}$
$\displaystyle\ =\ \frac{(t+u_{0}+2)!}{(t+1)!}\,,$ (A.177)

by using the relation

$\begin{pmatrix}u_{0}+1\\\ \beta\end{pmatrix}\ =\ \begin{pmatrix}u_{0}\\\
\beta\end{pmatrix}+\begin{pmatrix}u_{0}\\\ \beta-1\end{pmatrix}\,,$ (A.178)

for $1\leq\beta\leq u_{0}$ . 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 $\frac{1}{t!}\mathbb{I}^{\wedge t}$ is the identity on
$(\mathcal{H}^{res})^{\wedge t}$ :

$\displaystyle\ \frac{1}{t!}\Big{(}\mathbb{I}^{\wedge t}\wedge P_{0}^{\wedge
u}\Big{)}\frac{1}{(t+u+1)!}\sum_{\alpha=0}^{t+u}\Big{(}\mathbb{I}^{\wedge(t+u-\alpha)}\wedge
P_{0}^{\wedge\alpha}\Big{)}\ $ $\displaystyle\ =\
\frac{1}{(t+u+1)!}\sum_{\beta=0}^{u}\sum_{\alpha=\beta}^{t+\beta}\begin{pmatrix}u\\\
\beta\end{pmatrix}\frac{(t+u-\alpha)!}{(t+\beta-\alpha)!}\frac{\alpha!}{(\alpha-\beta)!}\,\mathbb{I}^{\wedge(t+\beta-\alpha)}\wedge
P_{0}^{\wedge(u+\alpha-\beta)}$ $\displaystyle\ =\
\frac{1}{(t+u+1)!}\sum_{\alpha=0}^{t}\Bigg{(}\sum_{\beta=0}^{u}\begin{pmatrix}u\\\
\beta\end{pmatrix}\frac{(t+u-\alpha-\beta)!}{(t-\alpha)}\frac{(\alpha+\beta)!}{\alpha!}\Bigg{)}\mathbb{I}^{\wedge(t-\alpha)}\wedge
P_{0}^{\wedge(u+\alpha)}$ $\displaystyle\ =\
\frac{1}{(t+1)!}\sum_{\alpha=0}^{t}\mathbb{I}^{\wedge(t-\alpha)}\wedge
P_{0}^{\wedge(u+\alpha)}\,.$ (A.179)

The proof of the formula (A.175) is the following. We assume that $s\geq u$ ,
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

$\sum_{\alpha=0}^{u}\begin{pmatrix}r+t-s-\alpha+1\\\
r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\
s\end{pmatrix}+\sum_{\alpha=0}^{s}\begin{pmatrix}r+t-s-u+\alpha+1\\\
r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u-\alpha\\\ s-\alpha\end{pmatrix}\
=\ \begin{pmatrix}r+t+2\\\ r+1\end{pmatrix}\,.$ (A.180)

For $u=0$ , the left hand side is calculated as

$\displaystyle\ \begin{pmatrix}r+t-s+1\\\
r-s\end{pmatrix}+\sum_{\alpha=0}^{s}\begin{pmatrix}r+t-s+\alpha+1\\\
r-s+\alpha+1\end{pmatrix}$ $\displaystyle\ =\ \begin{pmatrix}r+t-s+1\\\
r-s\end{pmatrix}+\begin{pmatrix}r+t-s+1\\\
r-s+1\end{pmatrix}+\sum_{\alpha=1}^{s}\begin{pmatrix}r+t-s+\alpha+1\\\
r-s+\alpha+1\end{pmatrix}$ $\displaystyle\ =\ \cdots$ $\displaystyle\ =\
\begin{pmatrix}r+t-s+k+1\\\ r-s+k\end{pmatrix}+\begin{pmatrix}r+t-s+k+1\\\
r-s+k+1\end{pmatrix}+\sum_{\alpha=k+1}^{s}\begin{pmatrix}r+t-s+\alpha+1\\\
r-s+\alpha+1\end{pmatrix}$ $\displaystyle\ =\ \cdots$ $\displaystyle\ =\
\begin{pmatrix}r+t+1\\\ r\end{pmatrix}+\begin{pmatrix}r+t+1\\\
r+1\end{pmatrix}\ =\ \begin{pmatrix}r+t+2\\\ r+1\end{pmatrix}\,,$ (A.181)

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

$\displaystyle\sum_{\alpha=0}^{u_{0}+1}$ $\displaystyle\
\begin{pmatrix}r+t-s-\alpha+1\\\ r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\
s\end{pmatrix}+\sum_{\alpha=0}^{s}\begin{pmatrix}r+t-s-u_{0}+\alpha\\\
r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u_{0}-\alpha+1\\\
s-\alpha\end{pmatrix}$ $\displaystyle=$ $\displaystyle\
\sum_{\alpha=0}^{u_{0}}\begin{pmatrix}r+t-s-\alpha+1\\\
r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\
s\end{pmatrix}+\begin{pmatrix}r+t-s-u_{0}\\\
r-s\end{pmatrix}\begin{pmatrix}s+u_{0}+1\\\ s\end{pmatrix}$ $\displaystyle\
+\begin{pmatrix}r+t-s-u_{0}\\\ r-s+1\end{pmatrix}\begin{pmatrix}s+u_{0}+1\\\
s\end{pmatrix}+\sum_{\alpha=1}^{s}\begin{pmatrix}r+t-s-u_{0}+\alpha\\\
r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u_{0}-\alpha+1\\\
s-\alpha\end{pmatrix}$ $\displaystyle=$ $\displaystyle\
\sum_{\alpha=0}^{u_{0}}\begin{pmatrix}r+t-s-\alpha+1\\\
r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\ s\end{pmatrix}\ $ $\displaystyle\
+\begin{pmatrix}r+t-s-u_{0}+1\\\ r-s+1\end{pmatrix}\begin{pmatrix}s+u_{0}+1\\\
s\end{pmatrix}+\sum_{\alpha=1}^{s}\begin{pmatrix}r+t-s-u_{0}+\alpha\\\
r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u_{0}-\alpha+1\\\
s-\alpha\end{pmatrix}$ $\displaystyle=$ $\displaystyle\
\sum_{\alpha=0}^{u_{0}}\begin{pmatrix}r+t-s-\alpha+1\\\
r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\ s\end{pmatrix}$ $\displaystyle\
+\begin{pmatrix}r+t-s-u_{0}+1\\\ r-s+1\end{pmatrix}\begin{pmatrix}s+u_{0}\\\
s\end{pmatrix}+\begin{pmatrix}r+t-s-u_{0}+1\\\
r-s+1\end{pmatrix}\begin{pmatrix}s+u_{0}\\\ s-1\end{pmatrix}$ $\displaystyle\
+\begin{pmatrix}r+t-s-u_{0}+1\\\ r-s+2\end{pmatrix}\begin{pmatrix}s+u_{0}\\\
s-1\end{pmatrix}+\sum_{\alpha=2}^{s}\begin{pmatrix}r+t-s-u_{0}+\alpha\\\
r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u_{0}-\alpha+1\\\
s-\alpha\end{pmatrix}$ $\displaystyle=$ $\displaystyle\
\sum_{\alpha=0}^{u_{0}}\begin{pmatrix}r+t-s-\alpha+1\\\
r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\
s\end{pmatrix}+\begin{pmatrix}r+t-s-u_{0}+1\\\
r-s+1\end{pmatrix}\begin{pmatrix}s+u_{0}\\\ s\end{pmatrix}$ $\displaystyle\
+\begin{pmatrix}r+t-s-u_{0}+2\\\ r-s+2\end{pmatrix}\begin{pmatrix}s+u_{0}\\\
s-1\end{pmatrix}$ $\displaystyle\ +\sum_{\alpha=2}^{s}\begin{pmatrix}r+t-s-
u_{0}+\alpha\\\ r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u_{0}-\alpha+1\\\
s-\alpha\end{pmatrix}$ $\displaystyle=$ $\displaystyle\
\sum_{\alpha=0}^{u_{0}}\begin{pmatrix}r+t-s-\alpha+1\\\
r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\
s\end{pmatrix}+\begin{pmatrix}r+t-s-u_{0}+1\\\
r-s+1\end{pmatrix}\begin{pmatrix}s+u_{0}\\\ s\end{pmatrix}$ $\displaystyle\
+\begin{pmatrix}r+t-s-u_{0}+2\\\ r-s+2\end{pmatrix}\begin{pmatrix}s+u_{0}-1\\\
s-1\end{pmatrix}+\begin{pmatrix}r+t-s-u_{0}+2\\\
r-s+2\end{pmatrix}\begin{pmatrix}s+u_{0}-1\\\ s-2\end{pmatrix}$
$\displaystyle\ +\begin{pmatrix}r+t-s-u_{0}+2\\\
r-s+3\end{pmatrix}\begin{pmatrix}s+u_{0}-1\\\
s-2\end{pmatrix}+\sum_{\alpha=3}^{s}\begin{pmatrix}r+t-s-u_{0}+\alpha\\\
r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u_{0}-\alpha+1\\\
s-\alpha\end{pmatrix}$ $\displaystyle=$ $\displaystyle\ \cdots$
$\displaystyle=$ $\displaystyle\
\sum_{\alpha=0}^{u_{0}}\begin{pmatrix}r+t-s-\alpha+1\\\
r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\
s\end{pmatrix}+\sum_{\alpha=0}^{k-1}\begin{pmatrix}r+t-s-u_{0}+\alpha+1\\\
r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u_{0}-\alpha\\\
s-\alpha\end{pmatrix}$ $\displaystyle\ +\begin{pmatrix}r+t-s-u_{0}+k\\\
r-s+k\end{pmatrix}\begin{pmatrix}s+u_{0}-k+2\\\ s-k+1\end{pmatrix}$
$\displaystyle\ +\sum_{\alpha=k}^{s}\begin{pmatrix}r+t-s-u_{0}+\alpha\\\
r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u_{0}-\alpha+1\\\
s-\alpha\end{pmatrix}$ $\displaystyle=$ $\displaystyle\ \cdots$
$\displaystyle=$ $\displaystyle\
\sum_{\alpha=0}^{u_{0}}\begin{pmatrix}r+t-s-\alpha+1\\\
r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\
s\end{pmatrix}+\sum_{\alpha=0}^{s-1}\begin{pmatrix}r+t-s-u_{0}+\alpha+1\\\
r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u_{0}-\alpha\\\
s-\alpha\end{pmatrix}$ $\displaystyle\ +\begin{pmatrix}r+t-u_{0}\\\
r\end{pmatrix}+\begin{pmatrix}r+t-u_{0}\\\ r+1\end{pmatrix}$ $\displaystyle=$
$\displaystyle\ \sum_{\alpha=0}^{u_{0}}\begin{pmatrix}r+t-s-\alpha+1\\\
r-s\end{pmatrix}\begin{pmatrix}s+\alpha\\\
s\end{pmatrix}+\sum_{\alpha=0}^{s}\begin{pmatrix}r+t-s-u_{0}+\alpha+1\\\
r-s+\alpha+1\end{pmatrix}\begin{pmatrix}s+u_{0}-\alpha\\\
s-\alpha\end{pmatrix}$ $\displaystyle=$ $\displaystyle\
\begin{pmatrix}r+t+2\\\ r+1\end{pmatrix}\,.$ (A.182)

The last equality follows from the assumption of the induction. Hence, the
formula (A.175) is true for ${}^{\forall}u$ by mathematical induction.

Using these properties of $\bm{H}$ , we can show, order by order in the power
of $\bm{H}$ , that the linear map on $\mathcal{SH}_{0}$ ,

$\displaystyle\hat{\bm{i}}^{\prime}\ =$ $\displaystyle\
(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}\ =\
\sum_{n=0}^{\infty}(\bm{H}\bm{L}_{int})^{n}\hat{\bm{P}}\,,$ (A.183)

satisfies the property of the cohomomorphism

$\Delta\hat{\bm{i}}^{\prime}\ =\
(\hat{\bm{i}}^{\prime}\otimes\hat{\bm{i}}^{\prime})\Delta\,.$ (A.184)

At $\mathcal{O}((\bf{H})^{0})$ , Eq. (A.184) reduces to
$\Delta\hat{\bm{P}}=(\hat{\bm{P}}\otimes\hat{\bm{P}})\Delta$ , which follows
from the definition (2.49b) of $\hat{\bm{P}}$ . For
$\mathcal{O}((\bm{H})^{n})$ with $n\geq 1$ , we use the mathematical
induction. If we note that

$\Delta\bm{L}_{int}\hat{\bm{P}}\ =\
(\hat{\bm{P}}\otimes\bm{L}_{int}\hat{\bm{P}}+\bm{L}_{int}\hat{\bm{P}}\otimes\hat{\bm{P}})\Delta\,,$
(A.185)

and the factor in the right hand side has the form mentioned above, we can see
that $\mathcal{O}(\bm{H})$ of Eq. (A.184),

$\Delta(\bm{H}\bm{L}_{int}\hat{\bm{P}})\ =\
(\hat{\bm{P}}\otimes\bm{H}\bm{L}_{int}\hat{\bm{P}}+\bm{H}\bm{L}_{int}\hat{\bm{P}}\otimes\hat{\bm{P}})\Delta\,,$
(A.186)

also holds thanks to the relation (A.168). If we suppose that
$\mathcal{O}((\bm{H})^{n_{0}})$ of (A.184),

$\Delta\big{(}(\bm{H}\bm{L}_{int})^{n_{0}}\hat{\bm{P}}\big{)}\ =\
\Big{(}\sum_{m=0}^{n_{0}}\big{(}(\bm{H}\bm{L}_{int})^{n_{0}-m}\hat{\bm{P}}\otimes(\bm{H}\bm{L}_{int})^{m}\hat{\bm{P}}\big{)}\Big{)}\Delta\,,$
(A.187)

holds, then from

$\displaystyle\Delta\bm{L}_{int}$
$\displaystyle(\bm{H}\bm{L}_{int})^{n_{0}}\hat{\bm{P}}\ $ $\displaystyle=$
$\displaystyle\
\Big{(}\hat{\bm{P}}\otimes\bm{L}_{int}(\bm{H}\bm{L}_{int})^{n_{0}}\hat{\bm{P}}+\bm{L}_{int}(\bm{H}\bm{L}_{int})^{n_{0}}\hat{\bm{P}}\otimes\hat{\bm{P}}$
$\displaystyle\
+(\bm{H}\otimes\hat{\bm{I}}-\hat{\bm{I}}\otimes\bm{H})\sum_{m=0}^{n_{0}-1}\big{(}\bm{L}_{int}(\bm{H}\bm{L}_{int})^{n_{0}-m-1}\hat{\bm{P}}\otimes\bm{L}_{int}(\bm{H}\bm{L}_{int})^{m}\hat{\bm{P}}\big{)}\Big{)}\Delta\,,$
(A.188)

the relation at $\mathcal{O}((\bm{H})^{n_{0}+1})$ ,

$\Delta\big{(}(\bm{H}\bm{L}_{int})^{n_{0}+1}\hat{\bm{P}}\big{)}\ =\
\Big{(}\sum_{m=0}^{n_{0}+1}\big{(}(\bm{H}\bm{L}_{int})^{n_{0}-m+1}\hat{\bm{P}}\otimes(\bm{H}\bm{L}_{int})^{m}\hat{\bm{P}}\big{)}\Big{)}\Delta\,,$
(A.189)

holds again thanks to the relations (A.168) and (A.169). Hence, Eq. (A.184)
holds at any order, $\mathcal{O}((\bm{H})^{n})$ with ${}^{\forall}n$ , due to
the mathematical induction, and thus, the map $\hat{\bm{i}}^{\prime}$ is a
cohomomorphism. Note that the cohomomorphism $\hat{\bm{i}}^{\prime}$ is cyclic
with respect to $\Omega$ in the sense that it satisfies

$\langle\Omega|\pi_{1}\hat{\bm{i}}^{\prime}\otimes\pi_{1}\hat{\bm{i}}^{\prime}\
=\ \langle\Omega|P_{0}\otimes P_{0}\,.$ (A.190)

This follows from Eqs. (2.42b), and the fact that the symplectic form
$\langle\Omega|$ compatible with the decomposition (2.43) satisfies

$\langle\Omega|\mathbb{I}\otimes Q^{+}\ =\
\langle\Omega|Q^{+}\otimes\mathbb{I}\,,\qquad\langle\Omega|\mathbb{I}\otimes
P_{0}\ =\ \langle\Omega|P_{0}\otimes\mathbb{I}\,.$ (A.191)

We can also show that $\hat{\bm{p}}^{\prime}$ is a cohomomorphism using the
similar relations to (A.168) and (A.169),

$\displaystyle\Big{(}\hat{\bm{P}}\otimes\bm{A}+\bm{A}\otimes\hat{\bm{P}}\Big{)}$
$\displaystyle\ \sum_{r,s,t,u=0}^{\infty}\begin{pmatrix}r+t\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ u\end{pmatrix}$ $\displaystyle\hskip
28.45274pt\times\Big{(}(-Q^{+})\wedge\mathbb{I}^{\wedge r}\wedge P_{0}^{\wedge
s}\otimes\mathbb{I}^{\wedge t}\wedge P_{0}^{\wedge u}+\mathbb{I}^{\wedge
r}\wedge P_{0}^{\wedge s}\otimes(-Q^{+})\wedge\mathbb{I}^{\wedge t}\wedge
P_{0}^{\wedge u}\Big{)}$ $\displaystyle\ =\
\hat{\bm{P}}\otimes\bm{AH}+\bm{AH}\otimes\hat{\bm{P}}\,,$ (A.192)

and

$\displaystyle\Big{(}\bm{H}\otimes\hat{\bm{I}}-\hat{\bm{I}}\otimes\bm{H}\Big{)}$
$\displaystyle\ \sum_{r,s,t,u=0}^{\infty}\begin{pmatrix}r+t\\\
r\end{pmatrix}\begin{pmatrix}s+u\\\ u\end{pmatrix}$ $\displaystyle\hskip
28.45274pt\times\Big{(}(-Q^{+})\wedge\mathbb{I}^{\wedge r}\wedge P_{0}^{\wedge
s}\otimes\mathbb{I}^{\wedge t}\wedge P_{0}^{\wedge u}+\mathbb{I}^{\wedge
r}\wedge P_{0}^{\wedge s}\otimes(-Q^{+})\wedge\mathbb{I}^{\wedge t}\wedge
P_{0}^{\wedge u}\Big{)}$ $\displaystyle\ =\ \bm{H}\otimes\bm{H}\,,$ (A.193)

respectively, obtained by reversing the order of the factors. We omit the
explicit proof since it is parallel to that for $\hat{\bm{i}}^{\prime}$ .

##  Appendix B Derivation of (2.54a)

Note that the action (2.35) can also be written as a WZW-like form

$I[\Phi]\ =\
\int_{0}^{1}dt\,\Omega(\partial_{t}\Phi(t),\pi_{1}\bm{L}(e^{\wedge\Phi(t)}))\,,$
(B.194)

where $\Phi(t)$ is a one-parameter, $t\in[0,1]$ , extension of the string
field $\Phi$ satisfying $\Phi(1)=\Phi$ and $\Phi(0)=0$ . This can easily be
seen as follows. First, note that the action (B.194) agrees with the action
(2.35) if we take $\Phi(t)=t\Phi$ . On the other hand, it is easy to show that
an arbitrary variation of the integrand becomes total $t$-derivative,

$\delta\Omega(\partial_{t}\Phi(t),\pi_{1}\bm{L}(e^{\wedge\Phi(t)}))\ =\
\partial_{t}\Omega(\delta\Phi(t),\pi_{1}\bm{L}(e^{\wedge\Phi(t)}))\,.$ (B.195)

Thus, the action (2.35) is not changed by deforming the $t$-dependence of
$\Phi(t)$ as long as keeping the boundary condition:
$\delta\Phi(0)=\delta\Phi(1)=0$ . Then, we extend the classical solution
(2.52) as

$\phi_{cl}(t)\ =\
\pi_{1}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}(e^{\wedge
t\phi_{0}})\,,$ (B.196)

and evaluate the action as

$\displaystyle I[\phi_{cl}]\ =$ $\displaystyle\
\int_{0}^{1}dt\,\Omega(\pi_{1}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}(\phi_{0}\wedge
e^{\wedge
t\phi_{0}}),\pi_{1}\bm{L}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}(e^{\wedge
t\phi_{0}}))$ $\displaystyle=$ $\displaystyle\
\int_{0}^{1}dt\,\langle\Omega|\,\phi_{0}\otimes\pi_{1}\bm{S}(e^{\wedge
t\phi_{0}})\,,$ (B.197)

using the bilinear map representation and the relation

$(\hat{\bm{I}}-\bm{H}\bm{L}_{int})\bm{L}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\
=\ \bm{Q}+\hat{\bm{P}}\bm{L}_{int}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})\,,$
(B.198)

that follows from (2.50a), (A.190), and
$\bm{L}^{2}=(\bm{Q}+\bm{L}_{int})^{2}=0$ . The S-matrix generating functional
(2.54a) is obtained by explicitly carrying out the $t$-integration.

##  Appendix C Proof of (2.69)

In order to explicitly investigate $\bm{\Sigma}(s,t)$ , it is necessary to
expand it in the number of inputs as (2.75a). The generalized Dyson-Schwinger
equation (2.67) is then written as

$\displaystyle\pi_{1}\bm{\Sigma}_{n+2}(s,t)\ =$ $\displaystyle\
\sum_{m=0}^{\infty}\pi_{1}\bm{B}_{m+2}(s,t)\bigg{(}\frac{1}{(m+2)!}\Big{(}P_{0}\pi_{1}-\Delta(s,t)\sum_{l=0}^{n-m-1}\pi_{1}\bm{\Sigma}_{l+2}(s,t)\Big{)}^{\wedge(m+2)}\bigg{)}\pi_{n+2}\,,$
$\displaystyle\equiv$ $\displaystyle\
\sum_{m=0}^{n}\pi_{1}\bm{B}_{m+2}(s,t)\Big{(}D_{m+2}(s,t)\Big{)}\pi_{n+2}\,,$
(C.199)

where we defined

$D_{M}(s,t)\ \equiv\
\frac{1}{M!}\Big{(}P_{0}\pi_{1}-\Delta(s,t)\pi_{1}\bm{\Sigma}(s,t)\Big{)}^{\wedge
M}\,,$ (C.200)

for notational simplicity. Note that Eq. (C.199) is a recurrence relation
whose right hand side includes only $\bm{\Sigma}_{l+2}(s,t)$ with $l<n$ . It
determines all the $\bm{\Sigma}_{n+2}(s,t)$ $(n\geq 0)$ recursively from the
relation for $n=0$ : $\pi_{1}\bm{\Sigma}_{2}(s,t)=\pi_{1}\bm{B}_{2}(s,t)P_{2}$
. Then, Eq. (2.69) can be proved by complete induction with respect to the
number of external strings. For $n=0$ , we have

$\displaystyle[\bm{Q},\pi_{1}\bm{\Sigma}_{2}(s,t)]\ =$ $\displaystyle\
[\bm{Q},\pi_{1}\bm{B}_{2}(s,t)P_{2}]\ =$ $\displaystyle\ 0\,,$ (C.201)
$\displaystyle[\bm{\eta},\pi_{1}\bm{\Sigma}_{2}(s,t)]\ =$ $\displaystyle\
[\bm{\eta},\pi_{1}\bm{B}_{2}(s,t)P_{2}]\ =$ $\displaystyle\ 0\,,$ (C.202)

which follow from the relations for $\bm{B}_{2}(s,t)$ in Eqs. (2.34). Next,
suppose that

$[\bm{Q},\pi_{1}\bm{\Sigma}_{l+2}(s,t)]\ =\
[\bm{\eta},\pi_{1}\bm{\Sigma}_{l+2}(s,t)]\ =\ 0\,,$ (C.203)

for $0\leq l\leq n-1$ . Then, we find from Eq. (C.199) that

$\displaystyle\ [\bm{Q},\pi_{1}\bm{\Sigma}_{n+2}(s,t)]$ $\displaystyle\ =\
-\sum_{l=0}^{n-1}\pi_{1}\bm{B}_{l+2}(s,t)\bigg{(}D_{l+1}(s,t)\wedge\pi(s)\pi_{1}\sum_{m=0}^{n-l-1}\bm{B}_{m+2}(s,t)\Big{(}D_{m+2}(s,t)\Big{)}\bigg{)}\pi_{n+2}$
$\displaystyle\hskip
28.45274pt+\sum_{m=0}^{n-1}\pi_{1}\bm{B}_{m+2}(s,t)\bigg{(}D_{m+1}(s,t)\wedge\pi(s)\pi_{1}\bm{\Sigma}(s,t)\bigg{)}\pi_{n+2}$
$\displaystyle\ =\ 0\,,$ (C.204)

using the relation (2.34a) and

$[Q\,,\Delta(s,t)]\ =\ \pi(s)-P_{0}\mathcal{G}(s,t)\,.$ (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
$[\bm{Q},\pi_{1}\bm{\Sigma}(s,t)]=0$ .

Similarly, we find from Eq. (C.199) that

$\displaystyle\ [\bm{\eta},\pi_{1}\bm{\Sigma}_{n+2}(s,t)]$ $\displaystyle\ =\
-\sum_{l=0}^{n-1}\pi_{1}\bm{B}_{l+2}(s,t)\bigg{(}D_{l+1}(s,t)\wedge
t\pi^{1}_{1}\sum_{m=0}^{n-l-1}\bm{B}_{m+2}(s,t)\Big{(}D_{m+2}(s,t)\Big{)}\bigg{)}\pi_{n+2}$
$\displaystyle\hskip
28.45274pt+\sum_{m=0}^{n-1}\pi_{1}\bm{B}_{m+2}(s,t)\bigg{(}D_{m+1}(s,t)\wedge
t\pi^{1}_{1}\bm{\Sigma}(s,t)\bigg{)}\pi_{n+2}$ $\displaystyle\ =\ 0\,,$
(C.206)

using the relation (2.34b) and

$[\eta\,,\Delta(s,t)]\ =\ -t\pi^{1}\,,$ (C.207)

with the off-shell-ness of the internal states. Hence, it can also be
concluded that $[\bm{\eta},\pi_{1}\bm{\Sigma}(s,t)]=0$ from the principle of
mathematical induction.

##  Appendix D Proof of (2.70)

The proof of (2.70) is given by complete induction with respect to the number
of inputs. For $\bm{\Sigma}_{2}(s,t)=\bm{B}_{2}(s,t)P_{2}$ , we have

$\pi_{1}\partial_{t}\bm{\Sigma}_{2}(s,t)\ =\
[\bm{Q},\pi_{1}\bm{\rho}_{2}(s,t)]\,,$ (D.208)

with $\pi_{1}\bm{\rho}_{2}(s,t)=\pi_{1}\bm{\lambda}_{2}(s,t)P_{2}$ using
(2.33a) with $n=0$ :

$\partial_{t}\bm{B}_{2}(s,t)\ =\ [\bm{Q}\,,\bm{\lambda}_{2}(s,t)]\,.$ (D.209)

Next, for

$\displaystyle\bm{\Sigma}_{3}(s,t)\ =$ $\displaystyle\
\bm{B}_{3}(s,t)P_{3}-\bm{B}_{2}(s,t)\left(\Delta(s,t)\pi_{1}\bm{\Sigma}_{2}(s,t)\wedge
P_{0}\pi_{1}\right)\,,$ (D.210)

we can find that

$\pi_{1}\partial_{t}\bm{\Sigma}_{3}(s,t)\ =\
[\bm{Q},\pi_{1}\bm{\rho}_{3}(s,t)]\,,$ (D.211)

with

$\displaystyle\pi_{1}\bm{\rho}_{3}(s,t)\ =$ $\displaystyle\
\pi_{1}\bm{\lambda}_{3}(s,t)P_{3}-\pi_{1}\bm{\lambda}_{2}(s,t)\Big{(}\Delta(s,t)\pi_{1}\bm{\Sigma}_{2}(s,t)\wedge
P_{0}\pi_{1}\Big{)}$ $\displaystyle\hskip
28.45274pt-\pi_{1}\bm{B}_{2}(s,t)\Big{(}\big{[}\Delta(s,t)\pi_{1}\bm{\rho}_{2}(s,t)+Q^{+}\Xi\pi_{1}^{1}\bm{\Sigma}_{2}(s,t)\big{]}\wedge
P_{0}\pi_{1}\Big{)}\,,$ (D.212)

using (2.33a) with $n=1$ ,

$\displaystyle\partial_{t}\bm{B}_{3}(s,t)\ =$ $\displaystyle\
[\bm{Q}\,,\bm{\lambda}_{3}(s,t)]$ $\displaystyle\
+\bm{B}_{2}(s,t)\Big{(}\pi(s)\pi_{1}\bm{\lambda}_{2}(s,t)\wedge\mathbb{I}_{1}\Big{)}-\bm{\lambda}_{2}(s,t)\Big{(}\pi(s)\pi_{1}\bm{B}_{2}(s,t)\wedge\mathbb{I}_{1}\Big{)}\,,$
(D.213)

and

$\partial_{t}\Delta(s,t)\ =\ -[Q\,,Q^{+}\Xi\pi^{1}]-\Xi P_{0}\pi^{1}\,.$
(D.214)

It was also used the off-shell-ness of the internal state. Next, suppose that

$\displaystyle\pi_{1}\partial_{t}\bm{\Sigma}_{l+2}(s,t)\ =$ $\displaystyle\
[\bm{Q},\pi_{1}\bm{\rho}_{l+2}(s,t)]\,,$ (D.215)
$\displaystyle\pi_{1}\bm{\rho}_{l+2}(s,t)\ =$ $\displaystyle\
\sum_{m=0}^{l}\pi_{1}\bm{\lambda}_{m+2}(s,t)\big{(}D_{m+2}(s,t)\big{)}\pi_{l+2}$
$\displaystyle\
-\sum_{m=0}^{l-1}\pi_{1}\bm{B}_{m+2}(s,t)\Big{(}D_{m+1}(s,t)\wedge\pi_{1}E(s,t)\Big{)}\pi_{l+2}\,,$
(D.216)

with

$E(s,t)\ =\ \Delta(s,t)\bm{\rho}(s,t)+Q^{+}\Xi\pi^{1}\bm{\Sigma}(s,t)\,,$
(D.217)

for $0\leq l\leq n-1$ .141414Note that the right hand side of (D.216) includes
only $\bm{\Sigma}_{k+2}(s,t)$ and $\bm{\rho}_{k+2}(s,t)$ with $0\leq k<l$ .
Then, we can find that

$\displaystyle\pi_{1}\partial_{t}\bm{\Sigma}_{n+2}(s,t)\ =$ $\displaystyle\
[\bm{Q}\,,\pi_{1}\bm{\rho}_{n+2}(s,t)]\,,$ (D.218)

with $\bm{\rho}_{n+2}(s,t)$ obtained by setting $l=n$ in Eq. (D.216). Hence,
from the principle of mathematical induction we conclude that
$\partial_{t}\Sigma(s,t)=[\bm{Q},\bm{\rho}(s,t)]$ with $\bm{\rho}(s,t)$
recursively determined by the equation

$\displaystyle\ \pi_{1}\bm{\rho}(s,t)=$ $\displaystyle\
\sum_{m=0}^{\infty}\pi_{1}\bm{\lambda}_{m+2}(s,t)\big{(}D_{m+2}(s,t)\big{)}P_{n+2}\pi_{n+2}$
$\displaystyle\
-\sum_{m=0}^{\infty}\pi_{1}\bm{B}_{m+2}(s,t)\Big{(}D_{m+1}(s,t)\wedge\pi_{1}E(s,t)\Big{)}P_{n+2}\pi_{n+2}\,.$
(D.219)

Similarly, for $\bm{\Sigma}_{2}(s,t)$ and $\bm{\Sigma}_{3}(s,t)$ , we have

$\pi_{1}\partial_{s}\bm{\Sigma}_{2}(s,t)\ =\
[\bm{\eta},\pi_{1}\bm{\rho}_{2}(s,t)]\,,\qquad\pi_{1}\partial_{s}\bm{\Sigma}_{3}(s,t)\
=\ [\bm{\eta},\pi_{1}\bm{\rho}_{3}(s,t)]\,.$ (D.220)

If we assume that

$\displaystyle\ \pi_{1}\partial_{s}\bm{\Sigma}_{l+2}(s,t)\ =\
[\bm{\eta},\pi_{1}\bm{\rho}_{l+2}(s,t)]\,,\quad\textrm{for}\quad 0\leq l\leq
n-1\,,$ (D.221)

then, we can show that

$\displaystyle\pi_{1}\partial_{s}\bm{\Sigma}_{n+2}(s,t)\ =\
[\bm{\eta},\pi_{1}\bm{\rho}_{n+2}(s,t)]\,.$ (D.222)

Hence, it is proven from the principle of mathematical induction that
$\pi_{1}\partial_{s}\bm{\Sigma}(s,t)=[\bm{\eta},\pi_{1}\bm{\rho}(s,t)]$ .

##  Appendix E Relation to the Erler-Okawa-Takezaki open superstring field
theory

When starting from the cubic theory,
$\bm{M}_{B}^{(0)}(s)=\bm{m}_{2}|^{0}+s\,\bm{m}_{2}|^{2}$ , it is easy to see
that that the generating functions $\bm{A}(s,t)$ and $\bm{\mu}(s,t)$ of the
open superstring product and gauge product have the restricted form

$\displaystyle\bm{A}(s,t)\ =$ $\displaystyle\ \bm{A}(t)+s\bm{\mathcal{A}}(t)$
$\displaystyle=$ $\displaystyle\
\bm{A}|^{0}(t)+\bm{A}|^{2}(t)+s\,\bm{\mathcal{A}}|^{0}(t)\,,$ (E.223)
$\displaystyle\bm{\mu}(s,t)\ =$ $\displaystyle\ \bm{\mu}|^{0}(t)\,,$ (E.224)

where

$\displaystyle\bm{A}|^{0}(t)\ =$ $\displaystyle\
\sum_{n=0}^{\infty}t^{n+1}\bm{A}^{(n+1)}_{n+2}|^{0}\,,\quad\bm{A}|^{2}(t)\ =\
\sum_{n=0}^{\infty}t^{n}\bm{A}^{(n)}_{n+2}|^{2}\,,$
$\displaystyle\bm{\mathcal{A}}|^{0}(t)\ =$ $\displaystyle\
\sum_{n=0}^{\infty}t^{n}\bm{\mathcal{A}}^{(n)}_{n+2}|^{0}\,,$ (E.225)
$\displaystyle\bm{\mu}|^{0}(t)\ =$ $\displaystyle\
\sum_{n=0}^{\infty}t^{n}\bm{\mu}^{(n+1)}_{n+2}|^{0}\,.$ (E.226)

Then, $[\bm{A}(s,t),\bm{\mu}(s,t)]^{2}\equiv 0$ since the gauge product
$\bm{\mu}(s,t)$ is only nonvanishing in the NS sector. The differential
equations (4.136) are now simplified and decomposed to the four equations

$\displaystyle\partial_{t}\bm{A}|^{0}(t)\ =$ $\displaystyle\
[\bm{Q},\bm{\mu}|^{0}(t)]+[\bm{A}|^{0}(t),\bm{\mu}|^{0}(t)]\,,$ (E.227a)
$\displaystyle\partial_{t}\bm{A}|^{2}(t)\ =$ $\displaystyle\
[\bm{A}|^{2}(t),\bm{\mu}|^{0}(t)]\,,$ (E.227b)
$\displaystyle\partial_{t}\bm{\mathcal{A}}|^{0}(t)\ =$ $\displaystyle\
[\bm{\mathcal{A}}|^{0}(t),\bm{\mu}|^{0}(t)]\,,$ (E.227c)
$\displaystyle[\bm{\eta},\bm{\mu}|^{0}(t)]\ =$ $\displaystyle\
\bm{\mathcal{A}}|^{0}(t)\,,$ (E.227d)

satisfying the initial conditions $\bm{A}(0)=\bm{m}_{2}|^{2}$ and
$\bm{\mathcal{A}}(0)=\bm{m}_{2}|^{0}$ . 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
$A_{\infty}$ 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.

In the present construction, we have an intermediate cyclic $A_{\infty}$
structure $\bm{Q}-\bm{\eta}+\bm{A}(t)$ which split into two (anti-)commutative
$A_{\infty}$ structures

$\displaystyle\bm{D}(t)\ =$ $\displaystyle\
\bm{Q}+\bm{A}|^{0}_{0}(t)+\bm{A}|^{2}_{2}(t)\,,$ (E.228)
$\displaystyle\bm{C}(t)\ =$ $\displaystyle\ \bm{\eta}-\bm{A}|^{2}_{0}(t)\,.$
(E.229)

Using the fact that $\bm{A}(t)$ satisfies the differential equations (E.227),
they can be rewritten as

$\displaystyle\bm{D}(t)\ =$ $\displaystyle\
\hat{\bm{g}}(t)^{-1}(\bm{Q}+\bm{m}_{2}|_{2})\hat{\bm{g}}(t)\,,$ (E.230)
$\displaystyle\bm{C}(t)\ =$ $\displaystyle\
\hat{\bm{g}}(t)^{-1}(\bm{\eta}-\bm{m}_{2}|_{0})\hat{\bm{g}}(t)\,,$ (E.231)

where the cohomomorphism $\hat{\bm{g}}(t)$ is given by the path-ordered
exponential of $\bm{\mu}|^{0}(t)$ :

$\hat{\bm{g}}(t)\ =\
\vec{\mathcal{P}}\exp\left[\int^{t}_{0}dt^{\prime}\bm{\mu}|^{0}(t^{\prime})\right]\,.$
(E.232)

A relation

$\hat{\bm{g}}(t)\bm{\eta}\hat{\bm{g}}(t)^{-1}\ =\
\bm{\eta}-\bm{m}_{2}|^{0}\,,$ (E.233)

follows from the Eq. (E.227d) are also used. The cohomomorphism
$\pi_{1}\hat{\bm{F}}(t)^{-1}=\pi_{1}\mathbb{I}-\Xi\pi_{1}^{1}\bm{A}(t)$
transforming $\bm{D}(t)$ to the final $A_{\infty}$ structure $\bm{M}(t)$ , is
similarly written as

$\hat{\bm{F}}(t)^{-1}\ =\
\hat{\bm{g}}(t)^{-1}(\hat{\bm{F}}^{-1})\hat{\bm{g}}(t)\,,\quad\pi_{1}\hat{\bm{F}}^{-1}\
=\ \pi_{1}\mathbb{I}-\Xi\pi_{1}^{1}\bm{m}_{2}|^{2}_{0}\,,$ (E.234)

and thus, we find that

$\displaystyle\bm{M}(t)\ =$ $\displaystyle\
\hat{\bm{F}}(t)^{-1}\bm{D}(t)\hat{\bm{F}}(t)\ =$ $\displaystyle\
(\hat{\bm{F}}\hat{\bm{g}}(t))^{-1}(\bm{Q}+\bm{m}_{2}|_{2})\hat{\bm{F}}\hat{\bm{g}}(t)\,,$
(E.235) $\displaystyle\bm{\eta}\ =$ $\displaystyle\
\hat{\bm{F}}(t)^{-1}\bm{C}(t)\hat{\bm{F}}(t)\ =$ $\displaystyle\
(\hat{\bm{F}}\hat{\bm{g}}(t))^{-1}(\bm{\eta}-\bm{m}_{2}|_{0})\hat{\bm{F}}\hat{\bm{g}}(t)\,.$
(E.236)

It was proven in Ref. [23] that the $A_{\infty}$ structure $\bm{M}=\bm{M}(1)$
agrees with that given in Ref. [17].

##  Appendix F Cyclicity of generalized $A_{\infty}$ structure

In this appendix, we show that the $A_{\infty}$ algebra (4.140) is cyclic with
respect to $\Omega$ . First, we note that $\bm{A}$ is cyclic with respect to
$\omega_{l}$ by construction [9]:

$\langle\omega_{l}|(\pi_{1}\bm{A}\otimes\pi_{1}+\pi_{1}\otimes\pi_{1}\bm{A})\
=\ 0\,.$ (F.237)

Then, we can show that $\bm{a}$ is also cyclic with respect to $\omega_{l}$ as

$\displaystyle\langle\omega_{l}|(\pi_{1}\bm{a}\otimes\pi_{1}+\pi_{1}\otimes\pi_{1}\bm{a})\
=$ $\displaystyle\
\langle\omega_{l}|(\pi_{1}\bm{A}\hat{\bm{F}}\otimes\pi_{1}+\pi_{1}\otimes\pi_{1}\bm{A}\hat{\bm{F}})$
$\displaystyle=$ $\displaystyle\
\langle\omega_{l}|\big{(}\pi_{1}\bm{A}\hat{\bm{F}}\otimes(\pi_{1}\hat{\bm{F}}-\Xi\pi_{1}^{1}\bm{A}\hat{\bm{F}})+(\pi_{1}\hat{\bm{F}}-\Xi\pi_{1}^{1}\bm{A}\hat{\bm{F}})\otimes\pi_{1}\bm{A}\hat{\bm{F}}\big{)}$
$\displaystyle=$ $\displaystyle\
\langle\omega_{l}|\big{(}(\pi_{1}\bm{A}\otimes\pi_{1}+\pi_{1}\otimes\pi_{1}\bm{A})(\hat{\bm{F}}\otimes\hat{\bm{F}})$
$\displaystyle\hskip
28.45274pt+(\mathbb{I}\otimes\Xi-\Xi\otimes\mathbb{I})(\pi_{1}\bm{A}\hat{\bm{F}}\otimes\pi_{1}\bm{A}\hat{\bm{F}})\big{)}$
$\displaystyle=$ $\displaystyle\ 0\,,$ (F.238)

where we used the relation

$\pi_{1}\mathbb{I}\ =\
\pi_{1}\hat{\bm{F}}-\Xi\pi_{1}^{1}\bm{A}\hat{\bm{F}}\,,$ (F.239)

following from the definition of $\hat{\bm{F}}^{-1}$ (4.141), and the fact
that $\Xi$ is a BPZ even:

$\langle\omega_{l}|(\mathbb{I}\otimes\Xi-\Xi\otimes\mathbb{I})\ =\ 0\,.$
(F.240)

If we note that $[\bm{\eta},\bm{a}]=0$ it is easy to see that $\bm{a}$ is also
cyclic with respect to $\omega_{s}$ :

$\displaystyle\langle\omega_{s}|(\pi_{1}\bm{a}\otimes\mathbb{I}+\mathbb{I}\otimes\pi_{1}\bm{a})\
=$ $\displaystyle\
\langle\omega_{l}|(\xi_{0}\otimes\mathbb{I})(\pi_{1}\bm{a}\otimes\mathbb{I}+\mathbb{I}\otimes\pi_{1}\bm{a})$
$\displaystyle=$ $\displaystyle\
\sum_{n=0}^{\infty}\langle\omega_{l}|(\xi_{0}\otimes\mathbb{I})(\pi_{1}\bm{a}\otimes\mathbb{I}+\mathbb{I}\otimes\pi_{1}\bm{a})\pi_{n+3}([\eta\,,\xi_{0}]\otimes\mathbb{I}_{n+2})$
$\displaystyle=$ $\displaystyle\
-\sum_{n=0}^{\infty}\langle\omega_{l}|(\pi_{1}\bm{a}\otimes\mathbb{I}+\mathbb{I}\otimes\pi_{1}\bm{a})\pi_{n+3}(\xi_{0}\otimes\mathbb{I}_{n+2})\
$ $\displaystyle=$ $\displaystyle\ 0\,,$ (F.241)

on $\mathcal{H}_{s}$ . Since the BRST operator satisfies

$\langle\Omega|(Q\otimes\mathbb{I}+\mathbb{I}\otimes Q)\ =\ 0\,,$ (F.242)

on $\mathcal{H}^{res}$ , we find that

$\displaystyle\langle\Omega|(\pi_{1}\bm{M}\otimes\mathbb{I}+\mathbb{I}\otimes\pi_{1}\bm{M})\
=$ $\displaystyle\
\langle\omega_{s}|(\pi_{1}\bm{a}\otimes\mathbb{I}+\mathbb{I}\otimes\pi_{1}\bm{a})\
=\ 0\,,$ (F.243)

on $\mathcal{H}^{res}$ .

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