YITP-21-58

Type II superstring field theory revisited
Hiroshi Kunitomo
Center for Gravitational Physics, Yukawa Institute for Theoretical Physics
Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
<EMAIL_ADDRESS>

Abstract

We reconstruct a complete type II superstring field theory with $L_{\infty}$
structure in a symmetric way concerning the left- and right-moving sectors.
Based on the new construction, we show again that the tree-level S-matrix
agrees with that obtained using the first-quantization method. Not only is
this a simple and elegant reconstruction, but it also enables the action to be
mapped to that in the WZW-like superstring field theory, which has not yet
been constructed and fills the only gap in the WZW-like formulation.

###### Contents

  1. 1 Introduction
  2. 2 Type II superstring field theory with $L_{\infty}$ structure
  3. 3 Symmetric construction of string products
    1. 3.1 $L_{\infty}$ triplet
    2. 3.2 Explicit construction
  4. 4 Tree-level S-matrix
  5. 5 Gauge-invariant action in WZW-like formulation
    1. 5.1 WZW-like action for the NS-NS sector
    2. 5.2 Complete WZW-like action and gauge invariance
  6. 6 Conclusion and discussion
  7. A Projected commutators
  8. B Expansion of $\bm{B}_{3}^{(\bullet,\bullet)}(s,\bar{s})$, $\bm{\lambda}_{3}^{(\bullet,\bullet)}(s,\bar{s})$ and $\bar{\bm{\lambda}}_{3}^{(\bullet,\bullet)}(s,\bar{s})$

##  1 Introduction

Recent significant developments in the superstring field theory [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] have ended long-term
stagnation and made it possible to construct classical gauge-invariant actions
for open [12, 5, 18], heterotic [14, 16, 17], and type II [15, 16, 17]
superstring theories. It is based on three complementary construction methods:
the formulation with the homotopy algebra structure[9, 10, 11, 12, 14, 15],
the Wess-Zumino-Witten (WZW) -like formulation [1, 2, 3, 4, 5, 6, 7, 13, 14,
15], and the formulation accompanied with an extra free field [18, 16, 17].
The only exception is the complete WZW-like action for the type II superstring
field theory, which has not yet been constructed. One of the purposes of this
paper is to give it and fills a gap in the WZW-like formulation.

In the previous papers [14, 15], we have attempted to construct complete WZW-
like actions, including all sectors representing the space-time bosons and
fermions, for the heterotic and the type II superstring field theories by
mapping from those of the formulation with the $L_{\infty}$ algebra method.
This construction worked well for the heterotic string theory but not for the
type II superstring theory. The reason is that we have taken the left-right
asymmetric method for constructing the type II string field theory with an
$L_{\infty}$ structure. In this case, we could only construct an action that
is a hybrid of the WZW-like action and the action with an $L_{\infty}$
structure, which we called the half-WZW-like action. Therefore, we first
revisit the formulation based on $L_{\infty}$ structure and provide an
alternative to the asymmetric method, the symmetric way to construct the
action for the type II superstring field theory. It is another purpose of this
paper.

The paper is organized as follows. In section 2, we summarize how to construct
a gauge-invariant action for the type II superstring field theory based on an
$L_{\infty}$ structure. The string field is suitably constrained and the
gauge-invariant action is written by using the string products satisfying the
$L_{\infty}$ algebra relations. We propose in section 3 a new construction
method of the string products with cyclic $L_{\infty}$ structure symmetric
with respect to the left- and right-moving sectors. After introducing the
coalgebra representation, we suppose $L_{\infty}$ algebra using the string
products with fixed cyclic Ramond numbers. The significant point, which is
different from the other cases, is that their RR output part is including the
picture-changing operator (PCO) explicitly. This breaks the cyclicity of the
algebra but instead suitably decomposes it into three commutative $L_{\infty}$
algebras (rather than four), two constraint $L_{\infty}$ algebras and a
dynamical $L_{\infty}$ algebra. A similarity transformation allows us to
transform dynamical $L_{\infty}$ algebra into the one used for writing down
the action and the other two into the constraints that impose that the first
algebra closes in the small Hilbert space. In order to explicitly construct
such cyclic string products, we give differential equations for their
generating functional. These differential equations recursively determine the
string products with respect to the number of input string fields, with
bosonic products as the initial condition. We show that the new action
correctly derives the same tree-level physical S-matrix as that calculated
using the first-quantization method in section 4. The proof is based on the
homological perturbation theory (HPT), which provides the explicit form of the
tree-level S-matrix in closed form and makes it possible to demonstrate the
agreement. In section 5, we write down a complete WZW-like action for the type
II superstring field theory, which we could not construct previously. After
summarizing how the NS-NS action with $L_{\infty}$ structure was rewritten as
a WZW-like action through the map between string fields in two formulations,
we extend it to all four sectors. It fills the gap in the WZW-like formulation
and should be significant to deepen the understanding of the superstring field
theory. Section 6 is devoted to the conclusion and discussion. Finally, it
contains two appendices. In Appendix A, we define a projected commutator that
plays a significant role in constructing the cyclic products with the
$L_{\infty}$ structure. Appendix B contains the $s$ and $\bar{s}$ expansions
of the generating functionals of cyclic three-string products and
corresponding gauge products to help for understanding the flow how they are
recursively determined.

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

Let us recall how the type II superstring field theory with an $L_{\infty}$
structure has been constructed [10, 15]. The first-quantized Hilbert space,
$\mathcal{H}$ , of type II superstring is composed of four sectors
corresponding to the combination of the NS and Ramond boundary conditions for
the left- and right-moving fermionic coordinates:

$\mathcal{H}\ =\
\mathcal{H}_{NS\textrm{-}NS}+\mathcal{H}_{R\textrm{-}NS}+\mathcal{H}_{NS\textrm{-}R}+\mathcal{H}_{R\textrm{-}R}\,.$
(2.1)

Correspondingly, the type II superstring field $\Phi$ has four components:

$\Phi\ =\
\Phi_{NS\textrm{-}NS}+\Phi_{R\textrm{-}NS}+\Phi_{NS\textrm{-}R}+\Phi_{R\textrm{-}R}\in\mathcal{H}\,,$
(2.2)

which is Grassmann even111Here, for consistent Grassmann property, we should
generally assume that $\Phi$ is GSO projected. and has ghost number 2. We take
the picture number of each component as $(-1,-1)$ , $(-1/2,-1)$ , $(-1,-1/2)$
and $(-1/2,-1/2)$ , respectively. The NS-NS and R-R components,
$\Phi_{NS\textrm{-}NS}$ and $\Phi_{R\textrm{-}R}$ , represent space-time
bosons, and the R-NS and NS-R components, $\Phi_{R\textrm{-}NS}$ and
$\Phi_{NS\textrm{-}R}$ , represent space-time fermions. The string field
$\Phi$ is restricted by the closed string constraints

$b_{0}^{-}\Phi\ =\ L_{0}^{-}\Phi\ =\ 0\,,$ (2.3)

with $b_{0}^{\pm}=b_{0}\pm\bar{b}_{0},\ L_{0}^{\pm}=L_{0}\pm\bar{L}_{0}$ , and
$c_{0}^{\pm}=(c_{0}\pm\bar{c}_{0})/2$ . We call the Hilbert space restricted
by constraints (2.3) the closed string Hilbert space. In addition, it is
necessary to impose an extra constraint by introducing a projection operator
$\mathcal{P}_{\mathcal{G}}=\mathcal{G}\mathcal{G}^{-1}$ with

$\begin{split}\mathcal{G}\ =&\
\pi^{(0,0)}+X\pi^{(1,0)}+\bar{X}\pi^{(0,1)}+X\bar{X}\pi^{(1,1)}\,,\\\
\mathcal{G}^{-1}\ =&\
\pi^{(0,0)}+Y\pi^{(1,0)}+\bar{Y}\pi^{(0,1)}+Y\bar{Y}\pi^{(1,1)}\,.\end{split}$
(2.4)

Here, $\pi^{(a,b)}\ (a,b=0,1)$ is a projection operator onto a component

$\pi^{(0,0)}\Phi\ =\ \Phi_{NS\textrm{-}NS}\,,\ \pi^{(1,0)}\Phi\ =\
\Phi_{R\textrm{-}NS}\,,\ \pi^{(0,1)}\Phi\ =\ \Phi_{NS\textrm{-}R}\,,\
\pi^{(1,1)}\Phi\ =\ \Phi_{R\textrm{-}R}\,,$ (2.5)

and, we take the PCO’s as

$\displaystyle X\ =$ $\displaystyle\
-\delta(\beta_{0})G+\frac{1}{2}(\gamma_{0}\delta(\beta_{0})+\delta(\beta_{0})\gamma_{0})b_{0}^{+}\,,$
$\displaystyle\qquad Y\ =$ $\displaystyle\
-2\frac{G}{L_{0}^{+}}\delta(\gamma_{0})\,,$ (2.6) $\displaystyle\bar{X}\ =$
$\displaystyle\
-\delta(\bar{\beta}_{0})\bar{G}+\frac{1}{2}(\bar{\gamma}_{0}\delta(\bar{\beta}_{0})+\delta(\bar{\beta}_{0})\bar{\gamma}_{0})b_{0}^{+}\,,$
$\displaystyle\qquad\bar{Y}\ =$ $\displaystyle\
-2\frac{\bar{G}}{L_{0}^{+}}\delta(\bar{\gamma}_{0})\,.$ (2.7)

Note that $\mathcal{G}$ satisfies

$\mathcal{G}\mathcal{G}^{-1}\mathcal{G}\ =\
\mathcal{G}\,,\qquad\mathcal{G}^{-1}\mathcal{G}\mathcal{G}^{-1}\ =\
\mathcal{G}^{-1},\qquad[Q,\mathcal{G}]\ =\ 0\,,$ (2.8)

and is almost exact in the large Hilbert space $\mathcal{H}_{l}$:

$\mathcal{G}\ =\ \pi^{(0,0)}+[Q\,,\mathfrak{S}]\,,\qquad\mathfrak{S}\ =\
\Xi\pi^{(1,0)}+\bar{\Xi}\pi^{(0,1)}+\frac{1}{2}(\Xi\bar{X}+X\bar{\Xi})\pi^{(1,1)}\,,$
(2.9)

with222For notational simplicity, we denote the zero modes of fermionic ghosts
$(\xi(z),\eta(z))$ and $(\bar{\xi}(\bar{z}),\bar{\eta}(\bar{z}))$ as
$(\xi,\eta)$ and $(\bar{\xi},\bar{\eta})$ , respectively.

$\displaystyle\Xi\ =$ $\displaystyle\
\xi+(\Theta(\beta_{0})\eta\xi-\xi)P_{-3/2}+(\xi\eta\Theta(\beta_{0})-\xi)P_{-1/2}\,,$
(2.10) $\displaystyle\bar{\Xi}\ =$ $\displaystyle\
\bar{\xi}+(\Theta(\bar{\beta}_{0})\bar{\eta}\bar{\xi}-\bar{\xi})P_{-3/2}+(\bar{\xi}\bar{\eta}\Theta(\beta_{0})-\bar{\xi})P_{-1/2}\,.$
(2.11)

The operator $P_{-3/2}$ and $P_{-1/2}$ are the projector onto the states with
ghost number $-3/2$ and $-1/2$ , respectively. The BRST operator consistently
acts on the string field restricted by (2.12) thanks to the property:
$\mathcal{P}_{\mathcal{G}}Q\mathcal{P}_{\mathcal{G}}=Q\mathcal{P}_{\mathcal{G}}$
. Then, the last constraint

$\mathcal{P}_{\mathcal{G}}\Phi\ =\ \Phi$ (2.12)

restricts the dependence of the ghost zero modes of each component of the
string field as

$\displaystyle\Phi_{NS\textrm{-}NS}\ =$ $\displaystyle\
\phi_{NS\textrm{-}NS}-c_{0}^{+}\psi_{NS\textrm{-}NS}\,,$ (2.13a)
$\displaystyle\Phi_{R\textrm{-}NS}\ =$ $\displaystyle\
\phi_{R\textrm{-}NS}-\frac{1}{2}(\gamma_{0}+2c_{0}^{+}G)\psi_{R\textrm{-}NS}\,,$
(2.13b) $\displaystyle\Phi_{NS\textrm{-}R}\ =$ $\displaystyle\
\phi_{NS\textrm{-}R}-\frac{1}{2}(\bar{\gamma}_{0}+2c_{0}^{+}\bar{G})\psi_{NS\textrm{-}R}\,,$
(2.13c) $\displaystyle\Phi_{R\textrm{-}R}\ =$ $\displaystyle\
\phi_{R\textrm{-}R}-\frac{1}{2}(\gamma_{0}\bar{G}-\bar{\gamma}_{0}G+2c_{0}^{+}G\bar{G})\psi_{R\textrm{-}R}\,.$
(2.13d)

We call the Hilbert space further restricted by (2.12) as the restricted
Hilbert space, $\mathcal{H}^{res}$ . The ghost-zero-mode independent
components $\phi_{i}$ ($\psi_{i}$)
$(i=NS\textrm{-}NS,R\textrm{-}NS,NS\textrm{-}R,R\textrm{-}R)$ are Grassmann
even (odd) and correspond to the fields (anti-fields) in the gauge-fixed basis
when they are quantized by the Batalin-Vilkovisky (BV) formalism. The simplest
and practical gauge is obtained by setting $\psi=0$ , which we call the
Siegel-Ramond (SR) gauge and denote its Hilbert space as $\mathcal{H}_{SR}$ .

A natural symplectic form in the closed string Hilbert space is defined by
using the BPZ-inner product as

$\omega(\Phi_{1}\,,\Phi_{2})\ =\
(-1)^{|\Phi_{1}|}\langle\Phi_{1}|c_{0}^{-}|\Phi_{2}\rangle\,,$ (2.14)

where $\langle\Phi|$ is the BPZ conjugate of $|\Phi\rangle$ . The symbol
$|\Phi|$ denotes the Grassmann property of string field $\Phi$: $|\Phi|=0$ or
1 if $\Phi$ is Grassmann even or odd, respectively. A natural symplectic form
in the restricted Hilbert space $\Omega$ is defined by using $\omega$ as

$\Omega(\Phi_{1}\,,\Phi_{2})\ =\
\omega(\Phi_{1}\,,\mathcal{G}^{-1}\Phi_{2})\,.$ (2.15)

Due to the asymmetry of the inner product among sectors, it is nontrivial to
make the $L_{\infty}$ algebra being cyclic across all the sectors. For later
use, it is also convenient to introduce the symplectic form $\omega_{l}$ in
the large Hilbert space. It is related to $\omega$ as

$\omega_{l}(\xi\bar{\xi}\Phi_{1}\,,\Phi_{2})\ =\ \omega(\Phi_{1}\,,\Phi_{2})$
(2.16)

if we embed $\Phi_{1},\Phi_{2}\in\mathcal{H}$ into $\mathcal{H}_{l}$ as the
fields satisfying the constraint $\eta\Phi_{i}=\bar{\eta}\Phi_{i}=0\ (i=1,2)$
. We also use the bilinear map representation of symplectic forms defined by

$\displaystyle\langle\omega_{l}|\,:\,\mathcal{H}_{l}$
$\displaystyle\otimes\mathcal{H}_{l}\qquad$ $\displaystyle\ {\longrightarrow}$
$\displaystyle\mathbb{C}$ (2.17) $\displaystyle\ \rotatebox{90.0}{$\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

$\langle\omega_{s}|\ =\
\langle\omega_{l}|\xi\bar{\xi}\otimes\mathbb{I}\,,\qquad\langle\Omega|\ =\
\langle\omega_{l}|\xi\bar{\xi}\otimes\mathcal{G}^{-1}\,.$ (2.18)

The action of the type II superstring field theory is defined using the string
products $L_{n}$ mapping $n$ string fields to a string field as

$\displaystyle L_{n}\ :\qquad\qquad$ $\displaystyle\
(\mathcal{H}^{res})^{\wedge n}$ $\displaystyle\ \longrightarrow$
$\displaystyle\ \mathcal{H}^{res}\,,\qquad(n\geq 1)\,,$ $\in$ $\in$ (2.19)
$\displaystyle\ \ \ \Phi_{1}$ $\displaystyle\wedge\ \cdots\wedge\Phi_{n}$
$\displaystyle\ \longmapsto$ $\displaystyle\qquad L_{n}(\Phi_{1}$
$\displaystyle\,,\ \cdots\,,\Phi_{n})\,.$

We identify the one-string product as the BRST operator $L_{1}=Q$ , and each
$(\mathcal{H}^{res})^{\wedge n}$ $(n\geq 2)$ is the symmetrized tensor product
of $\mathcal{H}^{res}$ whose element,
$\Phi_{1}\wedge\cdots\wedge\Phi_{n}\in(\mathcal{H}^{res})^{\wedge n}$ , is
defined by

$\Phi_{1}\wedge\cdots\wedge\Phi_{n}\ =\
\sum_{\sigma}(-1)^{\epsilon(\sigma)}\Phi_{\sigma(1)}\otimes\cdots\otimes\Phi_{\sigma(n)}\,,\qquad\Phi_{i}\in\mathcal{H}^{res}\,,$
(2.20)

where $\sigma$ and $\epsilon(\sigma)$ denote all the permutations of
$\\{1,\cdots,n\\}$ and the sign factor coming from the exchange
$\\{\Phi_{1},\cdots,\Phi_{n}\\}$ to
$\\{\Phi_{\sigma(1)},\cdots,\Phi_{\sigma(n)}\\}$ . Note that the string field
$L_{n}(\Phi_{1},\cdots,\Phi_{n})$ must satisfy the constraints (2.3) and
(2.12):

$\displaystyle\begin{split}&\ b_{0}^{-}L_{n}(\Phi_{1}\,,\cdots\,,\Phi_{n})\ =\
L_{0}^{-}L_{n}(\Phi_{1}\,,\cdots\,,\Phi_{n})\ =\ 0\,,\\\ &\
\mathcal{P}_{\mathcal{G}}L_{n}(\Phi_{1}\,,\cdots\,,\Phi_{n})\ =\
L_{n}(\Phi_{1}\,,\cdots\,,\Phi_{n})\,.\end{split}$ (2.21)

If the string products are equipped with the cyclic $L_{\infty}$ structure,
that is, satisfy the $L_{\infty}$ relations

$\sum_{\sigma}\sum_{m=1}^{n}(-1)^{\epsilon(\sigma)}\frac{1}{m!(n-m)!}L_{n-m+1}(L_{m}(\Phi_{\sigma(1)}\,,\cdots\,,\Phi_{\sigma(m)})\,,\Phi_{\sigma(m+1)}\,,\cdots\,,\Phi_{\sigma(n)})\
=\ 0\,,$ (2.22)

and the cyclicity condition,

$\Omega(\Phi_{1}\,,L_{n}(\Phi_{2}\,,\cdots\,,\Phi_{n+1}))\ =\
-(-1)^{|\Phi_{1}|}\Omega(L_{n}(\Phi_{1}\,,\cdots\,,\Phi_{n})\,,\Phi_{n+1})\,,$
(2.23)

the action of the type II superstring field theory is given by

$I\ =\
\sum_{n=0}^{\infty}\frac{1}{(n+2)!}\Omega(\Phi\,,L_{n+1}(\underbrace{\Phi\,,\cdots\,,\Phi}_{n+1}))\,,$
(2.24)

which is invariant under the gauge transformation

$\delta\Phi\ =\
\sum_{n=0}^{\infty}\frac{1}{n!}L_{n+1}(\underbrace{\Phi\,,\cdots\,,\Phi}_{n}\,,\Lambda)\,.$
(2.25)

The gauge parameter $\Lambda$ has also four components:
$\Lambda=\Lambda_{NS\textrm{-}NS}+\Lambda_{R\textrm{-}NS}+\Lambda_{NS\textrm{-}R}+\Lambda_{R\textrm{-}R}\in\mathcal{H}$
. A set of string products satisfying (2.22) and (2.23) defines a cyclic
$L_{\infty}$ algebra $(\mathcal{H}^{res},\Omega,\\{L_{n}\\})$ . The nontrivial
task is to provide a prescription that gives these string products.

##  3 Symmetric construction of string products

In the previous paper [15], we gave a prescription to construct a set of
string products with an $L_{\infty}$ structure required by complete action of
the type II superstring field theory by extending the asymmetric construction
proposed in [10]. It is sufficient for defining an action and gauge
transformation but is slightly complicated and could not be mapped to those in
the WZW-like formulation. In this paper, we propose another prescription by
extending the symmetric construction in [10].

###  3.1 $L_{\infty}$ triplet

We use the coalgebra representation. The coderivation
$\bm{L}=\sum_{n=0}^{\infty}\bm{L}_{n}$ is defined as an operator acting on
symmetrized tensor algebra

$\mathcal{SH}\ =\ \mathcal{H}^{\wedge
0}\oplus\mathcal{H}\oplus\mathcal{H}^{\wedge 2}\oplus\cdots\,,$ (3.1)

by333The identity operator $\mathbb{I}_{n}$ acting on $\mathcal{H}^{\wedge n}$
is given by
$\mathbb{I}_{n}=\frac{1}{n!}\underbrace{\mathbb{I}\wedge\cdots\wedge\mathbb{I}}_{n}=\underbrace{\mathbb{I}\otimes\cdots\otimes\mathbb{I}}_{n}.$

$\displaystyle L_{n}\Phi_{1}\wedge\cdots\wedge\Phi_{m}\ =$ $\displaystyle\
0\,,\qquad$ $\displaystyle\textrm{for}\ m<n\,,$ (3.2) $\displaystyle
L_{n}\Phi_{1}\wedge\cdots\wedge\Phi_{m}\ =$ $\displaystyle\
L_{n}(\Phi_{1}\,,\cdots\,,\Phi_{n})\,,\qquad$ $\displaystyle\textrm{for}\
m=n\,,$ (3.3) $\displaystyle L_{n}\Phi_{1}\wedge\cdots\wedge\Phi_{m}\ =$
$\displaystyle\
(L_{n}\wedge\mathbb{I}_{m-n})\Phi_{1}\wedge\cdots\wedge\Phi_{m}\,,\qquad$
$\displaystyle\textrm{for}\ m<n\,.$ (3.4)

We denote the projection onto $\mathcal{H}^{\wedge n}$ as $\pi_{n}$ :
$\pi_{n}\mathcal{SH}=\mathcal{H}^{\wedge n}$ . The string products satisfying
$L_{\infty}$ relation (2.22) is given by a nilpotent odd coderivation,
$\bm{L}=\sum_{n=1}^{\infty}\bm{L}_{n}$ satisfying444 In this paper the square
bracket $[\cdot,\cdot]$ denotes the graded commutator. $[\bm{L},\bm{L}]=0$ .
The basic building block of our construction is the odd coderivation

$\bm{B}\ =\
\sum_{p,r=0}^{\infty}\sum_{\bar{p},\bar{r}=0}^{\infty}\bm{B}^{(p,\bar{p})}_{p+r+1,\bar{p}+\bar{r}+1}\mid^{(2r,2\bar{r})}\,,$
(3.5)

acting on the symmetrized tensor algebra, $\mathcal{SH}_{l}$ , generated by
the large Hilbert space. The superscripts $p$ $(\bar{p})$ denotes the left-
(right-) moving picture number of the string product itself and $2r$
$(2\bar{r})$ denotes the left- (right-) moving cyclic Ramond number defined by

Cyclic Ramond No. $=$ No. of Ramond inputs $+$ No. of Ramond outputs.

We used a matrix notation describing the string products as a diagonal matrix
[10]:

$\bm{B}^{(p,\bar{p})}_{p+r+1,\bar{p}+\bar{r}+1}\mid^{(2r,2\bar{r})}\ =\
\delta_{p+r,\bar{p}+\bar{r}}\bm{B}^{(p,\bar{p})}_{p+r+1}\mid^{(2r,2\bar{r})}\,.$
(3.6)

The coderivation $\bm{B}$ can be decomposed into four components according to
the sector of outputs as:

$\displaystyle\pi^{(0,0)}_{1}\bm{B}\ =$ $\displaystyle\
\sum_{p,r=0}^{\infty}\sum_{\bar{p},\bar{r}=0}^{\infty}\pi_{1}\bm{B}^{(p,\bar{p})}_{p+r+1,\bar{p}+\bar{r}+1}\mid^{(2r,2\bar{r})}_{(2r,2\bar{r})}\,,$
(3.7a) $\displaystyle\pi^{(1,0)}_{1}\bm{B}\ =$ $\displaystyle\
\sum_{p,r=0}^{\infty}\sum_{\bar{p},\bar{r}=0}^{\infty}\pi_{1}\bm{B}^{(p,\bar{p})}_{p+r+1,\bar{p}+\bar{r}+1}\mid^{(2r,2\bar{r})}_{(2r-2,2\bar{r})}\,,$
(3.7b) $\displaystyle\pi^{(0,1)}_{1}\bm{B}\ =$ $\displaystyle\
\sum_{p,r=0}^{\infty}\sum_{\bar{p},\bar{r}=0}^{\infty}\pi_{1}\bm{B}^{(p,\bar{p})}_{p+r+1,\bar{p}+\bar{r}+1}\mid^{(2r,2\bar{r})}_{(2r,2\bar{r}-2)}\,,$
(3.7c) $\displaystyle\pi^{(1,1)}_{1}\bm{B}\ =$ $\displaystyle\
\sum_{p,r=0}^{\infty}\sum_{\bar{p},\bar{r}=0}^{\infty}\pi_{1}\bm{B}^{(p,\bar{p})}_{p+r+1,\bar{p}+\bar{r}+1}\mid^{(2r,2\bar{r})}_{(2r-2,2\bar{r}-2)},$
(3.7d)

where $\pi_{1}^{(a,b)}=\pi^{(a,b)}\pi_{1}$ . The subscripts $2r$ $(2\bar{r})$
denotes the left- (right-) moving Ramond number defined by

Ramond No. $=$ No. of Ramond inputs $-$ No. of Ramond outputs.

Although it is not difficult to construct the products $\bm{B}$ to be cyclic
(with respect to $\omega_{l}$), they cannot be used as $\bm{L}$ in the action
(2.24) since they have wrong picture numbers except for the first component
(3.7a). The picture number deficit of the components (3.7b), (3.7c), and
(3.7d) from those of $\bm{L}$ are $[1,0]$ , $[0,1]$ and $[1,1]$ ,
respectively: the picture number of their outputs are $(-3/2,-1)$ ,
$(-1,-2/3)$ , and $(-3/2,-3/2)$ , respectively.

Using the coderivation $\bm{B}$ , we can construct three independent
$L_{\infty}$ algebras, $(\mathcal{H}_{l},\bm{D})$ , $(\mathcal{H}_{l},\bm{C})$
and $(\mathcal{H}_{l},\bar{\bm{C}})$ , which are called an $L_{\infty}$
triplet in [13], with

$\displaystyle\pi_{1}\bm{D}\ =$ $\displaystyle\
\pi_{1}\bm{Q}+\pi_{1}^{(0,0)}\bm{B}\,,$ (3.8a) $\displaystyle\pi_{1}\bm{C}\ =$
$\displaystyle\
\pi_{1}\bm{\eta}-\pi_{1}^{(1,0)}\bm{B}-\frac{1}{2}\bar{X}\pi_{1}^{(1,1)}\bm{B}\,,$
(3.8b) $\displaystyle\pi_{1}\bar{\bm{C}}\ =$ $\displaystyle\
\pi_{1}\bar{\bm{\eta}}-\pi_{1}^{(0,1)}\bm{B}-\frac{1}{2}X\pi_{1}^{(1,1)}\bm{B}\,.$
(3.8c)

We can merge them into an $L_{\infty}$ algebra,
$(\mathcal{H}_{l},\bm{D}-\bm{C}-\bar{\bm{C}})$ , as

$\pi_{1}(\bm{D}-\bm{C}-\bar{\bm{C}})\ =\
\pi_{1}\bm{Q}-\pi_{1}\bm{\eta}-\pi_{1}\bar{\bm{\eta}}+(\pi_{1}-\pi_{1}^{(1,1)})\bm{B}+\frac{1}{2}(X+\bar{X})\pi_{1}^{(1,1)}\bm{B}\,,$
(3.9)

because the $L_{\infty}$ relation

$[\,\bm{D}-\bm{C}-\bar{\bm{C}}\,,\bm{D}-\bm{C}-\bar{\bm{C}}\,]\ =\ 0\,,$
(3.10)

holds for each picture number deficit and can decompose as

$\displaystyle\begin{split}&\ [\,\bm{D}\,,\bm{D}\,]\ =\ [\,\bm{C}\,,\bm{C}\,]\
=\ [\,\bar{\bm{C}}\,,\bar{\bm{C}}\,]\ =\ 0\,,\\\ &\ [\,\bm{D}\,,\bm{C}\,]\ =\
[\,\bm{D}\,,\bar{\bm{C}}\,]\ =\ [\,\bm{C}\,,\bar{\bm{C}}\,]\ =\
0\,.\end{split}$ (3.11)

Note that it is difficult to make this merged $L_{\infty}$ algebra
$(\mathcal{H}_{l},\bm{D}-\bm{C}-\bar{\bm{C}})$ cyclic, unlike the heterotic
string field theory case, due to the last (R-R) component. Once this
$L_{\infty}$ triplet is constructed, we can transform it into the triplet
$(\bm{\eta},\bar{\bm{\eta}};\bm{L})$ closed in $\mathcal{H}^{res}$ as

$\displaystyle\ \pi_{1}\hat{\bm{F}}^{-1}\bm{C}\hat{\bm{F}}\ =\
\pi_{1}\bm{\eta}\,,\qquad\pi_{1}\hat{\bm{F}}^{-1}\bar{\bm{C}}\hat{\bm{F}}\ =\
\pi_{1}\bar{\bm{\eta}}\,,$ (3.12a) $\displaystyle\pi_{1}\bm{L}\,\equiv$
$\displaystyle\
\pi_{1}\hat{\bm{F}}^{-1}\bm{D}\hat{\bm{F}}=\pi_{1}\bm{Q}+\mathcal{G}\pi_{1}\bm{b}\,,\qquad\pi_{1}\bm{b}\
=\ \pi_{1}\bm{B}\hat{\bm{F}}\,.$ (3.12b)

by using the cohomomorphism

$\pi_{1}\hat{\bm{F}}^{-1}\ =\
\pi_{1}\mathbb{I}-\mathfrak{S}\pi_{1}\bm{B}\,,\qquad\mathfrak{S}\ =\
\Xi\pi^{(1,0)}+\bar{\Xi}\pi^{(0,1)}+\frac{1}{2}(\Xi\bar{X}+X\bar{\Xi})\pi^{(1,1)}\,.$
(3.13)

As was shown in [14], if $\bm{B}$ is cyclic with respect to $\omega_{l}$ ,
$\bm{L}$ is cyclic with respect to $\Omega$ , and thus gives a cyclic
$L_{\infty}$ algebra $(\mathcal{H}^{res},\Omega,\bm{L})$ used for the action.

###  3.2 Explicit construction

Now, the task is to construct odd coderivation $\bm{B}$ satisfying the
$L_{\infty}$ relations (3.11) written for $\bm{B}$ as

$\displaystyle[\,\bm{Q}\,,\bm{B}\,]+\frac{1}{2}[\,\bm{B}\,,\bm{B}\,]^{11}\ =$
$\displaystyle\ 0\,,$ (3.14a)
$\displaystyle[\,\bm{\eta}\,,\bm{B}\,]-\frac{1}{2}[\,\bm{B}\,,\bm{B}\,]^{21}-\frac{1}{4}[\,\bm{B}\,,\bm{B}\,]^{22}_{\bar{X}}\
=$ $\displaystyle\ 0\,,$ (3.14b)
$\displaystyle[\,\bar{\bm{\eta}}\,,\bm{B}\,]-\frac{1}{2}[\,\bm{B}\,,\bm{B}\,]^{12}-\frac{1}{4}[\,\bm{B}\,,\bm{B}\,]^{22}_{X}\
=$ $\displaystyle\ 0\,.$ (3.14c)

Here, the square bracket $[\cdot,\cdot]^{ab}$ with $a,b=1$ or $2$ denotes the
projected commutator defined by projecting onto specific cyclic Ramond
numbers, an extension of that introduced in [14] for the heterotic string
field theory. In addition, similar bracket with subscript $\mathcal{X}=X$ or
$\bar{X}$ is defined by inserting $\mathcal{X}$ at the intermediate state555
This bracket cannot always be well-defined since $X$ and $\bar{X}$ are the
PCO’s acting on the states with picture number $-3/2$ .. We give an explicit
definition of these brackets in Appendix A, in which we also summarize the
Jacobi identities they satisfy.

Assume that the cyclic $L_{\infty}$ algebra
$(\mathcal{H}_{l},\omega_{l},\bm{L}^{(0,0)}=\sum_{n=0}^{\infty}\bm{L}^{(0,0)}_{n+1})$
without picture number, which is straightforwardly constructed similar to that
of the bosonic string field theory [19, 20, 21], is known666 This is actually
a cyclic $L_{\infty}$ algebra $(\mathcal{H},\omega_{s},\bm{L}^{(0,0)})$ closed
in the small Hilbert space. . We define a generating functional

$\bm{L}^{(0,0)}(s,\bar{s})\ =\
\bm{Q}+\sum_{m,\bar{m},r,\bar{r}}s^{m}\bar{s}^{\bar{m}}\bm{L}^{(0,0)}_{m+r+1,\bar{m}+\bar{r}+1}\mid^{(2r,2\bar{r})}\
\equiv\ \bm{Q}+\bm{L}^{B}(s,\bar{s})\,,$ (3.15)

with

$\qquad\bm{L}^{(0,0)}_{m+r+1,\bar{m}+\bar{r}+1}\mid^{(2r,2\bar{r})}\ =\
\delta_{m+r,\bar{m}+\bar{r}}\bm{L}^{(0,0)}_{m+r+1}\mid^{(2r,2\bar{r})}\,,$
(3.16)

which reduces to $\bm{L}^{(0,0)}$ at $(s,\bar{s})=(1,1)$ . The parameter $s$
or $\bar{s}$ is counting the left- or right-moving picture number deficit from
$\bm{B}$ , respectively. We can show that $\bm{L}^{B}(s,\bar{s})$ satisfies

$\displaystyle[\,\bm{Q}\,,\bm{L}^{B}(s,\bar{s})\,]$ $\displaystyle\
+\frac{1}{2}[\,\bm{L}^{B}(s,\bar{s})\,,\bm{L}^{B}(s,\bar{s})\,]^{11}+\frac{s}{2}[\,\bm{L}^{B}(s,\bar{s})\,,\bm{L}^{B}(s,\bar{s})\,]^{21}$
$\displaystyle\
+\frac{\bar{s}}{2}[\,\bm{L}^{B}(s,\bar{s})\,,\bm{L}^{B}(s,\bar{s})\,]^{12}+\frac{s\bar{s}}{2}[\,\bm{L}^{B}(s,\bar{s})\,,\bm{L}^{B}(s,\bar{s})\,]^{22}\
=\ 0\,,$ (3.17a) derived from the $L_{\infty}$ relation
$[\bm{L}^{(0,0)},\bm{L}^{(0,0)}]=0$ . It is also closed in the small Hilbert
space, $[\,\bm{\eta}\,,\bm{L}^{B}(s,\bar{s})\,]\ \ =\
0\,,\qquad[\,\bar{\bm{\eta}}\,,\bm{L}^{B}(s,\bar{s})\,]\ \ =\ 0\,,$ (3.17b)

since it can be constructed without using $\xi$ or $\bar{\xi}$. Then we extend
$\bm{B}$ to include those with non-zero picture number deficit from $\bm{B}$
and define a generating functional

$\displaystyle\begin{split}\bm{B}(s,\bar{s},t)\ =&\
\sum_{p,m,r=0}^{\infty}\sum_{\bar{p},\bar{r},\bar{m}=0}^{\infty}t^{p+\bar{p}}s^{m}\bar{s}^{\bar{m}}\bm{B}^{(p,\bar{p})}_{p+m+r+1,\bar{p}+\bar{m}+\bar{r}+1}\mid^{(2r,2\bar{r})}\\\
\equiv&\
\sum_{p,\bar{p}=0}^{\infty}t^{p+\bar{p}}\bm{B}^{(p,\bar{p})}(s,\bar{s})\,,\end{split}$
(3.18)

by introducing another parameter $t$ counting the (total) picture number. The
$L_{\infty}$ relations (3.14) are extended for $\bm{B}(s,\bar{s},t)$ to

$\displaystyle\bm{I}(s,\bar{s},t)\ \equiv$ $\displaystyle\
[\bm{Q}\,,\bm{B}(s,\bar{s},t)]+\frac{1}{2}[\bm{B}(s,\bar{s},t),\bm{B}(s,\bar{s},t)]^{11}\
$ $\displaystyle\
+\frac{s}{2}\Big{(}[\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{21}+t[\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{\bar{X}}\Big{)}$
$\displaystyle\
+\frac{\bar{s}}{2}\Big{(}[\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{12}+t[\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{X}\Big{)}$
$\displaystyle\
+\frac{s\bar{s}}{2}[\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}\ =\ 0\,,$
(3.19a) $\displaystyle\bm{J}(s,\bar{s},t)\ \equiv$ $\displaystyle\
[\bm{\eta}\,,\bm{B}(s,\bar{s},t)]$ $\displaystyle\
-\frac{t}{2}\left([\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{21}+\frac{t}{2}[\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{\bar{X}}\right)\
=\ 0\,,$ (3.19b) $\displaystyle\bar{\bm{J}}(s,\bar{s},t)\ \equiv$
$\displaystyle\ [\bar{\eta},\bm{B}(s,\bar{s},t)]$ $\displaystyle\
-\frac{t}{2}\left([\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{12}+\frac{t}{2}[\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{X}\right)\
=\ 0\,.$ (3.19c)

The string product $\bm{B}$ is obtained as $\bm{B}=\bm{B}(0,0,1)$ , and the
relations (3.19) reduce to (3.14) for $\bm{B}$ . We can show that if
$\bm{B}(s,\bar{s},t)$ satisfies the differential equations,

$\displaystyle\partial_{t}\bm{B}(s,\bar{s},t)\ =$ $\displaystyle\
[\bm{Q}\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]+[\bm{B}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{11}$
$\displaystyle\
+s\Big{(}[\bm{B}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{21}+t[\bm{B}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{\bar{X}}\Big{)}$
$\displaystyle\
+\bar{s}\Big{(}[\bm{B}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{12}+t[\bm{B}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{X}\Big{)}$
$\displaystyle\
+s\bar{s}[\bm{B}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}$
$\displaystyle\
+\frac{s}{2}[\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{\bar{\Xi}}+\frac{\bar{s}}{2}[\bm{B}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{\Xi}\,,$
(3.20a) $\displaystyle\partial_{s}\bm{B}(s,\bar{s},t)\ =$ $\displaystyle\
[\bm{\eta}\,,\bm{\lambda}(s,\bar{s},t)]\ $ $\displaystyle\
-t\Big{(}[\bm{B}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{21}+\frac{t}{2}[\bm{B}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{\bar{X}}\Big{)}\,,$
(3.20b) $\displaystyle\partial_{\bar{s}}\bm{B}(s,\bar{s},t)\ =$
$\displaystyle\ [\bar{\bm{\eta}}\,,\bar{\bm{\lambda}}(s,\bar{s},t)]$
$\displaystyle\
-t\Big{(}[\bm{B}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{12}+\frac{t}{2}[\bm{B}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{X}\Big{)}\,,$
(3.20c)

the $L_{\infty}$ relations (3.19) hold. Here, we introduced two degree-even
coderivations $\bm{\lambda}(s,\bar{s},t)$ and
$\bar{\bm{\lambda}}(s,\bar{s},t)$ satisfying

$[\bm{\eta}\,,\bar{\bm{\lambda}}(s,\bar{s},t)]\ =\
0\,,\qquad[\bar{\bm{\eta}}\,,\bm{\lambda}(s,\bar{s},t)]\ =\ 0\,,$ (3.21)

and used an abbreviated notation

$\bm{\lambda}(s,\bar{s},t)+\bar{\bm{\lambda}}(s,\bar{s},t)\ =\
(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)\,.$ (3.22)

These degree-even coderivations, $\bm{\lambda}(s,\bar{s},t)$ and
$\bar{\bm{\lambda}}(s,\bar{s},t)$ , are called (generating functionals of)
gauge products and can be expanded in the parameters as

$\displaystyle\bm{\lambda}(s,\bar{s},t)\ =$ $\displaystyle\
\sum_{p,m,r=0}^{\infty}\sum_{\bar{p},\bar{r},\bar{m}=0}^{\infty}t^{p+\bar{p}}s^{m}\bar{s}^{\bar{m}}\bm{\lambda}^{(p+1,\bar{p})}_{p+m+r+2,\bar{p}+\bar{m}+\bar{r}+1}\mid^{(2r,2\bar{r})}$
$\displaystyle\equiv$ $\displaystyle\
\sum_{p,\bar{p}=0}^{\infty}t^{p+\bar{p}}\bm{\lambda}^{(p+1,\bar{p})}(s,\bar{s})\,,$
(3.23) $\displaystyle\bar{\bm{\lambda}}(s,\bar{s},t)\ =$ $\displaystyle\
\sum_{p,m,r=0}^{\infty}\sum_{\bar{p},\bar{r},\bar{m}=0}^{\infty}t^{p+\bar{p}}s^{m}\bar{s}^{\bar{m}}\bar{\bm{\lambda}}^{(p,\bar{p}+1)}_{p+m+r+1,\bar{p}+\bar{m}+\bar{r}+2}\mid^{(2r,2\bar{r})}$
$\displaystyle\equiv$ $\displaystyle\
\sum_{p,\bar{p}=0}^{\infty}t^{p+\bar{p}}\bar{\bm{\lambda}}^{(p,\bar{p}+1)}(s,\bar{s})\,.$
(3.24)

The bracket with subscript, $[\cdot,\cdot]^{22}_{\mathfrak{X}}$ with
$\mathfrak{X}=\Xi$ or $\bar{\Xi}$ , is the (graded) commutator with
$\mathfrak{X}$ inserted at the intermediate state, whose explicit definition
is given in Appendix A. By differentiating (3.19a) by $t$ and using the
differential equation (3.20a), we find that

$\displaystyle\partial_{t}\bm{I}(s,\bar{s},t)\ =$ $\displaystyle\
[\bm{I}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{11}$
$\displaystyle\
+s\Big{(}[\bm{I}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{21}+t[\bm{I}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{\bar{X}}\Big{)}$
$\displaystyle\
+\bar{s}\Big{(}[\bm{I}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{12}+t[\bm{I}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{X}\Big{)}$
$\displaystyle\
+s\bar{s}[\bm{I}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}$
$\displaystyle\
+s[\bm{I}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{\bar{\Xi}}+\bar{s}[\bm{I}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{\Xi}\,,$
(3.25)

which implies if $\bm{I}(s,\bar{s},0)=0$ , then $\bm{I}(s,\bar{s},t)=0$ for
${}^{\forall}t$ . Similarly, by differentiating (3.19b) and (3.19c) by $t$ and
using the differential equations (3.20) we obtain

$\displaystyle\partial_{t}\bm{J}(s,\bar{s},t)\ =$ $\displaystyle\
-\partial_{s}\bm{I}(s,\bar{s},t)+[\bm{J}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{11}$
$\displaystyle\
+s\Big{(}[\bm{J}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{21}+t[\bm{J}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{\bar{X}}\Big{)}$
$\displaystyle\
+\bar{s}\Big{(}[\bm{J}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{12}+t[\bm{J}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{X}\Big{)}$
$\displaystyle\
+s\bar{s}[\bm{J}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}$
$\displaystyle\
-t\Big{(}[\bm{I}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{21}+\frac{t}{2}[\bm{I}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{\bar{X}}\Big{)}$
$\displaystyle\
+s[\bm{J}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{\bar{\Xi}}+\bar{s}[\bm{J}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{\Xi}\,,$
(3.26a) $\displaystyle\partial_{t}\bar{\bm{J}}(s,\bar{s},t)\ =$
$\displaystyle\
-\partial_{\bar{s}}\bm{I}(s,\bar{s},t)+[\bar{\bm{J}}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{11}$
$\displaystyle\
+s\Big{(}[\bar{\bm{J}}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{21}+t[\bar{\bm{J}}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{\bar{X}}\Big{)}$
$\displaystyle\
+\bar{s}\Big{(}[\bar{\bm{J}}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{12}+t[\bar{\bm{J}}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{X}\Big{)}$
$\displaystyle\
+s\bar{s}[\bar{\bm{J}}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}$
$\displaystyle\
-t\Big{(}[\bm{I}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{12}+\frac{t}{2}[\bar{\bm{J}}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})(s,\bar{s},t)]^{22}_{X}\Big{)}$
$\displaystyle\
+s[\bar{\bm{J}}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{\bar{\Xi}}+\bar{s}[\bar{\bm{J}}(s,\bar{s},t)\,,\bm{B}(s,\bar{s},t)]^{22}_{\Xi}\,,$
(3.26b)

which imply if $\bm{I}(s,\bar{s},t)=0$ and
$\bm{J}(s,\bar{s},0)=\bar{\bm{J}}(s,\bar{s},0)=0$ , then
$\bm{J}(s,\bar{s},t)=\bar{\bm{J}}(s,\bar{s},t)=0$ for ${}^{\forall}t$ . On the
other hand, (3.19) reduce to

$\displaystyle\bm{I}(s,\bar{s},0)\ =$ $\displaystyle\
[\bm{Q}\,,\bm{B}(s,\bar{s},0)]+\frac{1}{2}[\bm{B}(s,\bar{s},0),\bm{B}(s,\bar{s},0)]^{11}+\frac{s}{2}[\bm{B}(s,\bar{s},0)\,,\bm{B}(s,\bar{s},0)]^{21}$
$\displaystyle\hskip
51.21495pt+\frac{\bar{s}}{2}[\bm{B}(s,\bar{s},0)\,,\bm{B}(s,\bar{s},0)]^{12}+\frac{s\bar{s}}{2}[\bm{B}(s,\bar{s},0)\,,\bm{B}(s,\bar{s},0)]^{22}\
=\ 0\,,$ (3.27a) $\bm{J}(s,\bar{s},0)\ =\ [\bm{\eta}\,,\bm{B}(s,\bar{s},0)]\
=\ 0\,,\qquad\bar{\bm{J}}(s,\bar{s},0)\ =\
[\bar{\bm{\eta}}\,,\bm{B}(s,\bar{s},0)]\ =\ 0\,,$ (3.27b)

at $t=0$ , which hold if we set

$\bm{B}(s,\bar{s},0)=\ \bm{B}^{(0,0)}(s,\bar{s})\ =\ \bm{L}^{B}(s,\bar{s})$
(3.28)

as was seen in (3.17). Thus, if $\bm{B}(s,\bar{s},t)$ satisfies the
differential equations (3.20) with the initial condition (3.28), the relations
(3.19) hold. We obtain the string product $\bm{B}$ satisfying the relations
(3.2) as $\bm{B}=\bm{B}(0,0,1)$ .

Under the initial condition (3.28), we can explicitly solve the differential
equations (3.20) and find
$\bm{B}(s,\bar{s},t)=\sum_{n=2}^{\infty}\bm{B}_{n}(s,\bar{s},t)$ in ascending
order of $n$. First, the 2-string products $\bm{B}_{2}(s,\bar{s},t)$ ,
$\bm{\lambda}_{2}(s,\bar{s},t)$ , and $\bar{\bm{\lambda}}_{2}(s,\bar{s},t)$
are expanded in $t$ as

$\displaystyle\bm{B}_{2}(s,\bar{s},t)\ =$ $\displaystyle\
\bm{B}_{2}^{(0,0)}(s,\bar{s})+t\left(\bm{B}_{2}^{(1,0)}(s,\bar{s})+\bm{B}_{2}^{(0,1)}(s,\bar{s})\right)+t^{2}\bm{B}_{2}^{(1,1)}(s,\bar{s})\,,$
(3.29a) $\displaystyle\bm{\lambda}_{2}(s,\bar{s},t)\ =$ $\displaystyle\
\bm{\lambda}_{2}^{(1,0)}(s,\bar{s})+t\bm{\lambda}_{2}^{(1,1)}(s,\bar{s})\,,$
(3.29b) $\displaystyle\bar{\bm{\lambda}}_{2}(s,\bar{s},t)\ =$ $\displaystyle\
\bar{\bm{\lambda}}_{2}^{(0,1)}(s,\bar{s})+t\bar{\bm{\lambda}}_{2}^{(1,1)}(s,\bar{s})\,.$
(3.29c)

We can decompose the equations (3.20) to

$\displaystyle\bm{B}_{2}^{(1,0)}(s,\bar{s})\ =$ $\displaystyle\
[\,\bm{Q}\,,\bm{\lambda}^{(1,0)}(s,\bar{s})\,]\,,\qquad$
$\displaystyle\bm{B}_{2}^{(0,1)}(s,\bar{s})\ =$ $\displaystyle\
[\,\bm{Q}\,,\bar{\bm{\lambda}}^{(1,0)}(s,\bar{s})\,]\,,$ (3.30a)
$\displaystyle\partial_{s}\bm{B}_{2}^{(0,0)}(s,\bar{s})\ =$ $\displaystyle\
[\,\bm{\eta}\,,\bm{\lambda}_{2}^{(1,0)}(s,\bar{s})\,]\,,\qquad$
$\displaystyle\partial_{\bar{s}}\bm{B}_{2}^{(0,0)}(s,\bar{s})\ =$
$\displaystyle\
[\,\bar{\eta}\,,\bar{\bm{\lambda}}_{2}^{(0,1)}(s,\bar{s})\,]\,.$ (3.30b)
$\displaystyle 2\bm{B}_{2}^{(1,1)}(s,\bar{s})\ =$ $\displaystyle\
[\,\bm{Q}\,,\bm{\lambda}_{2}^{(1,1)}(s,\bar{s})+\bar{\bm{\lambda}}_{2}^{(1,1)}(s,\bar{s})\,]\,,$
(3.30c) $\displaystyle\partial_{s}\bm{B}_{2}^{(0,1)}(s,\bar{s})\ =$
$\displaystyle\
[\,\bm{\eta}\,,\bm{\lambda}_{2}^{(1,1)}(s,\bar{s})\,]\,,\qquad$
$\displaystyle\partial_{\bar{s}}\bm{B}_{2}^{(1,0)}(s,\bar{s})\ =$
$\displaystyle\
[\,\bar{\eta}\,,\bar{\bm{\lambda}}_{2}^{(1,1)}(s,\bar{s})\,]\,,$ (3.30d)

for each order of $t$ and each picture number. Note that there is no nonlinear
term in these equations (3.30) for the 2-string products. First, eqs. (3.30b)
can be solved for $\bm{\lambda}_{2}^{(1,0)}(s,\bar{s})$ and
$\bar{\bm{\lambda}}_{2}^{(0,1)}(s,\bar{s})$ as

$\displaystyle\pi_{1}\bm{\lambda}_{2}^{(1,0)}(s,\bar{s})\ =$ $\displaystyle\
\frac{1}{3}\left(\xi\partial_{s}L^{B}_{2}(s,\bar{s})-\partial_{s}L^{B}_{2}(s,\bar{s})(\xi\pi_{1}\wedge\mathbb{I})\right)\
\equiv\ \xi\circ\pi_{1}\partial_{s}\bm{L}^{B}_{2}(s,\bar{s})\,,$ (3.31a)
$\displaystyle\pi_{1}\bar{\bm{\lambda}}_{2}^{(0,1)}(s,\bar{s})\ =$
$\displaystyle\
\frac{1}{3}\left(\bar{\xi}\partial_{\bar{s}}L^{B}_{2}(s,\bar{s})-\partial_{\bar{s}}L^{B}_{2}(s,\bar{s})(\bar{\xi}\pi_{1}\wedge\mathbb{I})\right)\
\equiv\ \bar{\xi}\circ\pi_{1}\partial_{\bar{s}}\bm{L}^{B}_{2}(s,\bar{s})\,,$
(3.31b)

under the initial condition
$\bm{B}_{2}^{(0,0)}(s,\bar{s})=\bm{L}_{2}^{B}(s,\bar{s})$ . Then, eqs. (3.30a)
determine $\bm{B}_{2}^{(1,0)}(s,\bar{s})$ and $\bm{B}_{2}^{(0,1)}(s,\bar{s})$
as

$\displaystyle\pi_{1}\bm{B}_{2}^{(1,0)}(s,\bar{s})\ =$ $\displaystyle\
\frac{1}{3}\left(X_{0}\partial_{s}L^{B}_{2}(s,\bar{s})+\partial_{s}L^{B}_{2}(s,\bar{s})(X_{0}\pi\wedge\mathbb{I}_{2})\right)\
\equiv\ X\circ\pi_{1}\partial_{s}\bm{L}_{2}^{B}(s,\bar{s})\,,$ (3.32a)
$\displaystyle\pi_{1}\bm{B}_{2}^{(0,1)}(s,\bar{s})\ =$ $\displaystyle\
\frac{1}{3}\left(\bar{X}_{0}\partial_{s}L^{B}_{2}(s,\bar{s})+\partial_{s}L^{B}_{2}(s,\bar{s})(\bar{X}_{0}\pi\wedge\mathbb{I}_{2})\right)\
\equiv\ \bar{X}\circ\pi_{1}\partial_{\bar{s}}\bm{L}_{2}^{B}(s,\bar{s})\,.$
(3.32b)

Using these results, the equations (3.30d) are solved for
$\pi_{1}\bm{\lambda}^{(1,1)}(s,\bar{s})$ and
$\pi_{1}\bar{\bm{\lambda}}^{(1,1)}(s,\bar{s})$ as

$\displaystyle\pi_{1}\bm{\lambda}^{(1,1)}(s,\bar{s})\ =$ $\displaystyle\
\xi\circ\pi_{1}\partial_{s}\bm{B}_{2}^{(0,1)}(s,\bar{s})\
=\xi\circ\bar{X}\circ\pi_{1}\partial_{s}\partial_{\bar{s}}\bm{L}_{2}^{B}(s,\bar{s})\,,$
(3.33a) $\displaystyle\pi_{1}\bar{\bm{\lambda}}^{(1,1)}(s,\bar{s})\ =$
$\displaystyle\
\bar{\xi}\circ\pi_{1}\partial_{\bar{s}}\bm{B}_{2}^{(1,0)}(s,\bar{s})\
=\bar{\xi}\circ
X\circ\pi_{1}\partial_{s}\partial_{\bar{s}}\bm{L}_{2}^{B}(s,\bar{s})\,,$
(3.33b)

which determine $\bm{B}_{2}^{(1,1)}(s,\bar{s})$ as

$\pi_{1}\bm{B}_{2}^{(1,1)}(s,\bar{s})\ =\
X\circ\bar{X}\circ\pi_{1}\partial_{s}\partial_{\bar{s}}\bm{L}_{2}^{B}(s,\bar{s})\,,$
(3.34)

from (3.30c). We illustrate in Fig. 1 the flow of how the 2-string (gauge)
products are determined.

Figure 1: The flow of how the 2-string (gauge) products are determined.

The string and gauge products with specific cyclic Ramond numbers are obtained
as coefficients by expanding in $s$ and $\bar{s}$ :

$\displaystyle\bm{B}_{2}^{(0,0)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{2}^{(0,0)}\mid^{(2,2)}+s\bm{B}_{2}^{(0,0)}\mid^{(0,2)}+\bar{s}\bm{B}_{2}^{(0,0)}\mid^{(2,0)}+s\bar{s}\bm{B}_{2}^{(0,0)}\mid^{(0,0)}\,,$
(3.35) $\displaystyle\bm{B}_{2}^{(1,0)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{2}^{(1,0)}\mid^{(0,2)}+\bar{s}\bm{B}_{2}^{(1,0)}\mid^{(0,0)}\,,$
(3.36) $\displaystyle\bm{B}_{2}^{(0,1)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{2}^{(0,1)}\mid^{(2,0)}+s\bm{B}_{2}^{(0,1)}\mid^{(0,0)}\,,$ (3.37)
$\displaystyle\bm{B}_{2}^{(1,1)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{2}^{(1,1)}\mid^{(0,0)}\,,$ (3.38)
$\displaystyle\bm{\lambda}_{2}^{(1,0)}(s,\bar{s})\ =$ $\displaystyle\
\bm{\lambda}_{2}^{(1,0)}\mid^{(0,2)}+\bar{s}\bm{\lambda}_{2}^{(1,0)}\mid^{(0,0)}\,,\qquad\bm{\lambda}_{2}^{(1,1)}(s,\bar{s})\
=\ \bm{\lambda}_{2}^{(1,1)}\mid^{(0,0)}\,,$ (3.39)
$\displaystyle\bar{\bm{\lambda}}_{2}^{(0,1)}(s,\bar{s})\ =$ $\displaystyle\
\bm{\lambda}_{2}^{(0,1)}\mid^{(2,0)}+s\bar{\bm{\lambda}}_{2}^{(0,1)}\mid^{(0,0)}\,,\qquad\bar{\bm{\lambda}}_{2}^{(1,1)}(s,\bar{s})\
=\ \bar{\bm{\lambda}}_{2}^{(1,1)}\mid^{(0,0)}\,.$ (3.40)

The explicit form of each product is found by expanding the solutions
(3.31)-(3.34) in $s$ and $\bar{s}$ . In particular, the 2-string product
satisfying (3.14) is given by

$\displaystyle\pi_{1}\bm{B}_{2}\ =$ $\displaystyle\
\pi_{1}\bm{B}_{2}^{(0,0)}\mid^{(2,2)}+\pi_{1}\bm{B}_{2}^{(1,0)}\mid^{(0,2)}+\pi_{1}\bm{B}_{2}^{(0,1)}\mid^{(2,0)}+\pi_{1}\bm{B}_{2}^{(1,1)}\mid^{(0,0)}$
$\displaystyle=$ $\displaystyle\
\pi_{1}\bm{L}_{2}^{B}\mid^{(2,2)}+X\circ\pi_{1}\bm{L}_{2}^{B}\mid^{(0,2)}+\bar{X}\circ\pi_{1}\bm{L}_{2}^{B}\mid^{(2,0)}+X\circ\bar{X}\circ\pi_{1}\bm{L}_{2}^{B}\mid^{(0,0)}\,.$
(3.41)

The flow of determining each component is given in Fig. 2.

Figure 2: The flow of how each component of the 2-string (gauge) products is
determined.

Similarly, expanding the 3-string products as

$\displaystyle\bm{B}_{3}(s,\bar{s},t)\ =$ $\displaystyle\
\bm{B}_{3}^{(0,0)}(s,\bar{s})+t\left(\bm{B}_{3}^{(1,0)}(s,\bar{s})+\bm{B}_{3}^{(0,1)}(s,\bar{s})\right)$
$\displaystyle\
+t^{2}\left(\bm{B}_{3}^{(2,0)}(s,\bar{s})+\bm{B}_{3}^{(1,1)}(s,\bar{s})+\bm{B}_{3}^{(0,2)}(s,\bar{s})\right)$
$\displaystyle\
+t^{3}\left(\bm{B}_{3}^{(2,1)}(s,\bar{s})+\bm{B}_{3}^{(1,2)}(s,\bar{s})\right)+t^{4}\bm{B}_{3}^{(2,2)}(s,\bar{s})\,,$
(3.42) $\displaystyle\bm{\lambda}_{3}(s,\bar{s},t)\ =$ $\displaystyle\
\bm{\lambda}_{3}^{(1,0)}(s,\bar{s})+t\left(\bm{\lambda}_{3}^{(2,0)}(s,\bar{s})+\bm{\lambda}_{3}^{(1,1)}(s,\bar{s})\right)$
$\displaystyle\
+t^{2}\left(\bm{\lambda}_{3}^{(2,1)}(s,\bar{s})+\bm{\lambda}_{3}^{(1,2)}(s,\bar{s})\right)+t^{3}\bm{\lambda}_{3}^{(2,2)}(s,\bar{s})\,,$
(3.43) $\displaystyle\bar{\bm{\lambda}}_{3}(s,\bar{s},t)\ =$ $\displaystyle\
\bar{\bm{\lambda}}_{3}^{(0,1)}(s,\bar{s})+t\left(\bar{\bm{\lambda}}_{3}^{(1,1)}(s,\bar{s})+\bar{\bm{\lambda}}_{3}^{(0,2)}(s,\bar{s})\right)$
$\displaystyle\
+t^{2}\left(\bar{\bm{\lambda}}_{3}^{(2,1)}(s,\bar{s})+\bar{\bm{\lambda}}_{3}^{(1,2)}(s,\bar{s})\right)+t^{3}\bar{\bm{\lambda}}_{3}^{(2,2)}(s,\bar{s})\,,$
(3.44)

the differential equations (3.20) for 3-string products at
$\mathcal{O}(t^{0})$ become

$\displaystyle\bm{B}_{3}^{(1,0)}(s,\bar{s})\ =$ $\displaystyle\
[\,\bm{Q}\,,\bm{\lambda}_{3}^{(1,0)}(s,\bar{s})\,]+[\,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,,\bm{\lambda}_{2}^{(1,0)}(s,\bar{s})\,]^{11}$
$\displaystyle\
+s[\,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,,\bm{\lambda}_{2}^{(1,0)}(s,\bar{s})\,]^{21}+\bar{s}[\,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,,\bm{\lambda}_{2}^{(1,0)}(s,\bar{s})\,]^{12}$
$\displaystyle\
+s\bar{s}[\,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,,\bm{\lambda}_{2}^{(1,0)}(s,\bar{s})\,]^{22}+\frac{\bar{s}}{2}[\,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,]^{22}_{\Xi}\,,$
(3.45a) $\displaystyle\bm{B}_{3}^{(0,1)}(s,\bar{s})\ =$ $\displaystyle\
[\,\bm{Q}\,,\bar{\bm{\lambda}}_{3}^{(0,1)}(s,\bar{s})\,]+[\,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,,\bar{\bm{\lambda}}_{2}^{(0,1)}(s,\bar{s})\,]^{11}$
$\displaystyle\
+s[\,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,,\bar{\bm{\lambda}}_{2}^{(0,1)}(s,\bar{s})\,]^{21}+\bar{s}[\,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,,\bar{\bm{\lambda}}_{2}^{(0,1)}(s,\bar{s})\,]^{12}$
$\displaystyle\
+s\bar{s}[\,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,,\bar{\bm{\lambda}}_{2}^{(0,1)}(s,\bar{s})\,]^{22}+\frac{\bar{s}}{2}[\,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,,\bm{B}_{2}^{(0,0)}(s,\bar{s})\,]^{22}_{\bar{\Xi}}\,,$
(3.45b) $\displaystyle\partial_{s}\bm{B}_{3}^{(0,0)}(s,\bar{s})\ =$
$\displaystyle\
[\,\bm{\eta}\,,\bm{\lambda}_{3}^{(1,0)}(s,\bar{s})\,]\,,\qquad\partial_{\bar{s}}\bm{B}_{3}^{(0,0)}(s,\bar{s})\
=\ [\,\bar{\bm{\eta}}\,,\bar{\bm{\lambda}}_{3}(s,\bar{s})\,]\,.$ (3.45c)

Under the initial condition
$\bm{B}_{3}^{(0,0)}(s,\bar{s})=\bm{L}_{3}^{B}(s,\bar{s})$ , eqs. (3.45c) are
solved for $\bm{\lambda}_{3}^{(1,0)}(s,\bar{s})$ and
$\bar{\bm{\lambda}}_{3}^{(0,1)}(s,\bar{s})$ as

$\displaystyle\pi_{1}\bm{\lambda}_{3}^{(1,0)}(s,\bar{s})\ =$ $\displaystyle\
\xi\circ\partial_{s}L_{3}^{B}(s,\bar{s})\ =\
\frac{1}{4}\left(\xi\partial_{s}L^{B}_{3}(s,\bar{s})-\partial_{s}L_{3}^{B}(s,\bar{s})(\xi\pi_{1}\wedge\mathbb{I}_{2})\right)\,,$
(3.46) $\displaystyle\pi_{1}\bar{\bm{\lambda}}_{3}^{(0,1)}(s,\bar{s})\ =$
$\displaystyle\ \bar{\xi}\circ\partial_{\bar{s}}L_{3}^{B}(s,\bar{s})\ =\
\frac{1}{4}\left(\bar{\xi}\partial_{\bar{s}}L^{B}_{3}(s,\bar{s})-\partial_{\bar{s}}L_{3}^{B}(s,\bar{s})(\bar{\xi}\pi_{1}\wedge\mathbb{I}_{2})\right)\,.$
(3.47)

Then, all the terms on the right-hand sides of eqs. (3.45a) and (3.45b) are
given, and thus, they determine $\bm{B}_{3}^{(1,0)}(s,\bar{s})$ and
$\bm{B}_{3}^{(0,1)}(s,\bar{s})$ . We can repeat similar procedures at each
order of $t$ and determine all the products
$\bm{B}_{3}^{(\bullet,\bullet)}(s,\bar{s})$,
$\bm{\lambda}_{3}^{(\bullet,\bullet)}(s,\bar{s})$ , and
$\bar{\bm{\lambda}}_{3}^{(\bullet,\bullet)}(s,\bar{s})$ . In general, the
equations (3.20) at each order of $t$ have the form777 Depending on the
picture number, $\bm{\lambda}_{3}^{(\bullet,\bullet)}(s,\bar{s})$ or
$\bar{\bm{\lambda}}_{3}^{(\bullet,\bullet)}(s,\bar{s})$ is missing in eq.
(3.48a).

$\displaystyle\bm{B}_{3}^{(\bullet,\bullet)}(s,\bar{s})\ =$ $\displaystyle\
[\,\bm{Q}\,,\bm{\lambda}_{3}^{(\bullet,\bullet)}(s,\bar{s})+\bar{\bm{\lambda}}_{3}^{(\bullet,\bullet)}(s,\bar{s})\,]+\cdots\,,$
(3.48a)
$\displaystyle[\,\bm{\eta}\,,\bm{\lambda}^{(\bullet,\bullet)}_{3}(s,\bar{s})\,]\
=$ $\displaystyle\
\cdots\,,\qquad[\,\bar{\bm{\eta}}\,,\bar{\bm{\lambda}}^{(\bullet,\bullet)}_{3}(s,\bar{s})\,]\
=\ \cdots\,,$ (3.48b)

where $\cdots$ denotes the terms including only the products already given in
the lower order. We can solve eqs. (3.48b) for
$\bm{\lambda}_{3}^{(\bullet,\bullet)}(s,\bar{s})$ or
$\bar{\bm{\lambda}}_{3}^{(\bullet,\bullet)}(s,\bar{s})$ , and then eq. (3.48a)
determine $\bm{B}_{3}^{(\bullet,\bullet)}(s,\bar{s})$ . We give the flow of
how the 3-string (gauge) products are determined in Fig. 3. Their expansion in
$s$ and $\bar{s}$ and the flow of how each component, (gauge) products with
specific cyclic Ramond number, are determined are given in Appendix B.

Figure 3: The flow of how 3-string (gauge) products are determined.

In principle, we can repeat a similar procedure for higher-string products
$\bm{B}_{n}^{(\bullet,\bullet)}(s,\bar{s})$ ,
$\bm{\lambda}_{n}^{(\bullet,\bullet)}(s,\bar{s})$ , and
$\bar{\bm{\lambda}}_{n}^{(\bullet,\bullet)}(s,\bar{s})$ in ascending order of
$n$ and determine all the string and gauge products $\bm{B}(s,\bar{s},t)$ ,
$\bm{\lambda}(s,\bar{s},t)$ and $\bar{\bm{\lambda}}(s,\bar{s},t)$ .

Before closing this section, we rewrite the (extended) $L_{\infty}$-relations
(3.19) and the differential equations (3.20) as

$\displaystyle\
[\bm{Q}\,,\bm{B}_{n+2}(s,\bar{s},t)]+\sum_{m=0}^{n-1}\bm{B}_{m+2}(s,\bar{s},t)\Big{(}\pi(s,\bar{s},t)\pi_{1}\bm{B}_{n-m+1}(s,\bar{s},t)\wedge\mathbb{I}_{m+1}\Big{)}\
=\ 0\,,$ (3.49a) $\displaystyle\
[\bm{\eta}\,,\bm{B}_{n+2}(s,\bar{s},t)]+\sum_{m=0}^{n-1}\bm{B}_{m+2}(s,\bar{s},t)\Big{(}\pi(t)\pi_{1}\bm{B}_{n-m+1}(s,\bar{s},t)\wedge\mathbb{I}_{m+1}\Big{)}\
=\ 0\,,$ (3.49b) $\displaystyle\
[\bar{\bm{\eta}}\,,\bm{B}_{n+2}(s,\bar{s},t)]+\sum_{m=0}^{n-1}\bm{B}_{m+2}(s,\bar{s},t)\Big{(}\bar{\pi}(t)\pi_{1}\bm{B}_{n-m+1}(s,\bar{s},t)\wedge\mathbb{I}_{m+1}\Big{)}\
=\ 0\,,$ (3.49c)

and

$\displaystyle\partial_{t}\bm{B}_{n+2}(s,\bar{s},t)\ =$ $\displaystyle\
[\bm{Q}\,,(\bm{\lambda}+\bar{\bm{\lambda}})_{n+2}(s,\bar{s},t)]$
$\displaystyle\
+\sum_{m=0}^{n-1}\Big{(}\bm{B}_{m+2}(s,\bar{s},t)\left(\pi(s,\bar{s},t)\pi_{1}(\bm{\lambda}+\bar{\bm{\lambda}})_{n-m+1}(s,\bar{s},t)\wedge\mathbb{I}_{m+1}\right)$
$\displaystyle\hskip
42.67912pt-(\bm{\lambda}+\bar{\bm{\lambda}})_{m+2}(s,\bar{s},t)\left(\pi(s,\bar{s},t)\pi_{1}\bm{B}_{n-m+1}(s,\bar{s},t)\wedge\mathbb{I}_{m+1}\right)\Big{)}$
$\displaystyle\
+\sum_{m=0}^{n-1}\bm{B}_{m+2}(s,\bar{s},t)\left((s\bar{\Xi}+\bar{s}\Xi)\pi_{1}^{(1,1)}\bm{B}_{n-m+1}(s,\bar{s},t)\wedge\mathbb{I}_{m+1}\right)\,,$
(3.50a) $\displaystyle\partial_{s}\bm{B}_{n+2}(s,\bar{s},t)\ =$
$\displaystyle\ [\bm{\eta}\,,\bm{\lambda}_{n+2}(s,\bar{s},t)]$
$\displaystyle+\sum_{m=0}^{n-1}\Big{(}\bm{B}_{m+2}(s,\bar{s},t)\left(\pi(t)\pi_{1}(\bm{\lambda}+\bar{\bm{\lambda}})_{n-m+1}(s,\bar{s},t)\wedge\mathbb{I}_{m+1}\right)$
$\displaystyle\hskip
42.67912pt-(\bm{\lambda}+\bar{\bm{\lambda}})_{m+2}(s,\bar{s},t)\left(\pi(t)\pi_{1}\bm{B}_{n-m+1}(s,\bar{s},t)\wedge\mathbb{I}_{m+1}\right)\Big{)}\,,$
(3.50b) $\displaystyle\partial_{\bar{s}}\bm{B}_{n+2}(s,\bar{s},t)\ =$
$\displaystyle\ [\bar{\bm{\eta}}\,,\bar{\bm{\lambda}}_{n+2}(s,\bar{s},t)]$
$\displaystyle+\sum_{m=0}^{n-1}\Big{(}\bm{B}_{m+2}(s,\bar{s},t)\left(\bar{\pi}(t)\pi_{1}(\bm{\lambda}+\bar{\bm{\lambda}})_{n-m+1}(s,\bar{s},t)\wedge\mathbb{I}_{m+1}\right)$
$\displaystyle\hskip
42.67912pt-(\bm{\lambda}+\bar{\bm{\lambda}})_{m+2}(s,\bar{s},t)\left(\bar{\pi}(t)\pi_{1}\bm{B}_{n-m+1}(s,\bar{s},t)\wedge\mathbb{I}_{m+1}\right)\Big{)}\,,$
(3.50c)

where

$\begin{split}&\ \pi(s,\bar{s},t)\ =\
\pi^{(0,0)}+s\pi^{(1,0)}+\bar{s}\pi^{(0,1)}+\big{(}t\left(s\bar{X}+\bar{s}X\right)+s\bar{s}\big{)}\pi^{(1,1)}\,,\\\
&\ \pi(t)\ =\
-t\left(\pi^{(1,0)}+\frac{t}{2}\bar{X}\pi^{(1,1)}\right)\,,\qquad\bar{\pi}(t)\
=\ -t\left(\pi^{(0,1)}+\frac{t}{2}X\pi^{(1,1)}\right)\,.\end{split}$ (3.51)

These alternative forms are convenient for use in the next section [24].

##  4 Tree-level S-matrix

Using HPT, we can show that the tree-level S-matrix derived from (super)string
field theory agrees with that calculated using the first-quantization method
[22, 23, 24]. We show in this section that the new prescription proposed in
the previous section simplifies this proof for the type II superstring field
theory.

As a preparation, let us define the on-shell subspace,

$\mathcal{H}^{p}\ =\ \\{\Phi\in\mathcal{H}^{res}\mid
L_{0}^{+}\Phi_{NS\textrm{-}NS}=G_{0}\Phi_{R\textrm{-}NS}=\bar{G}_{0}\Phi_{NS\textrm{-}R}=G_{0}\Phi_{R\textrm{-}R}=\bar{G}_{0}\Phi_{R\textrm{-}R}=0\\}\,,$
(4.1)

and the BRST invariant projection operator,

$P_{0}\ :\ \mathcal{H}^{res}\longrightarrow\mathcal{H}^{p}\,,\qquad
P_{0}^{2}=P_{0}\,,\qquad[Q,P_{0}]=0\,.$ (4.2)

The homotopy operator $Q^{+}$ of $Q$ satisfying

$\displaystyle Q^{+}Q+QQ^{+}+P_{0}\ =\ \mathbb{I}\,,$ (4.3) $\displaystyle
Q^{+}P_{0}=P_{0}Q^{+}=Q^{+}Q^{+}=0\,,$ (4.4)

is defined by

$Q^{+}\ =\ \frac{1}{L_{0}^{+}}b_{0}^{+}(1-P_{0})\,,$ (4.5)

and provides a Hodge-Kodaira decomposition of $\mathcal{H}^{res}$ ,

$\mathcal{H}^{res}\ =\ \mathcal{H}^{p}+\mathcal{H}^{t}+\mathcal{H}^{u}\,,\\\ $
(4.6)

with

$\mathcal{H}^{p}=P_{0}\mathcal{H}^{res}\,,\qquad\mathcal{H}^{t}=QQ^{+}\mathcal{H}^{res}\,,\qquad\mathcal{H}^{u}=Q^{+}Q\mathcal{H}^{res}\,.$
(4.7)

It is compatible with $\Omega$ :

$\Omega(\mathcal{H}^{p}\,,\mathcal{H}^{u})\ =\
\Omega(\mathcal{H}^{u}\,,\mathcal{H}^{u})\ =\ 0\,.$ (4.8)

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

$p\ =\ P_{0}\,:\,\mathcal{H}^{res}\ \twoheadrightarrow\
\mathcal{H}^{p}\,,\qquad i\ =\ P_{0}\,:\,\mathcal{H}^{p}\ \hookrightarrow\
\mathcal{H}^{res}\,,$ (4.9)

satisfying the relations

$pi\ =\ P_{0}\,,\qquad ip\ =\ \mathbb{I}-Q^{+}Q-QQ^{+}\,.$ (4.10)

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

$\phi\ =\ P_{0}\phi+(1-P_{0})\phi\ \in\
\mathcal{H}_{0}+\overline{\mathcal{H}}_{0}\,,$ (4.11)

where888 To be precise, we must remove the ghost number constraint for the
quantum string field, but components with non-zero space-time ghost number do
not contribute the tree-level S-matrix.

$\mathcal{H}_{0}=\mathcal{H}^{p}\cap\mathcal{H}_{SR}=P_{0}\mathcal{H}_{SR}\,,\qquad\overline{\mathcal{H}}_{0}=\mathcal{H}^{u}\cap\mathcal{H}_{SR}=(1-P_{0})\mathcal{H}_{SR}\,.$
(4.12)

The chain complex $(\mathcal{H}^{p},QP_{0})$ , with the SR gauge condition,
defines the relative BRST cohomology. Lift these (equivalence) data to the
chain complexes $(\mathcal{SH}^{res},\bm{Q})$ and $(\mathcal{SH}^{p},\bm{QP})$
with the chain maps

$\hat{\bm{p}}\ =\ \hat{\bm{P}}\,:\,\mathcal{SH}^{res}\ \twoheadrightarrow\
\mathcal{SH}^{p}\,,\qquad\hat{\bm{i}}\ =\ \hat{\bm{P}}\,:\,\mathcal{SH}^{p}\
\hookrightarrow\ \mathcal{SH}^{res}\,,$ (4.13)

satisfying

$\hat{\bm{p}}\hat{\bm{i}}\ =\ \hat{\bm{P}}\,,\qquad\hat{\bm{i}}\hat{\bm{p}}\
=\ \hat{\bm{I}}+\bm{H}\bm{Q}+\bm{Q}\bm{H}\,,$ (4.14)

where

$\displaystyle\hat{\bm{P}}\ =$ $\displaystyle\ \sum_{n=0}^{\infty}P_{n}\ =\
\sum_{n=0}^{\infty}\frac{1}{n!}\underbrace{P_{0}\wedge\cdots\wedge
P_{0}}_{n}\,,$ (4.15) $\displaystyle\hat{\bm{I}}\ =$ $\displaystyle\
\sum_{n=0}^{\infty}\ =\
\sum_{n=0}^{\infty}\frac{1}{n!}\underbrace{\mathbb{I}_{1}\wedge\cdots\mathbb{I}_{1}}_{n}\,,$
(4.16) $\displaystyle\bm{H}\ =$ $\displaystyle\
\sum_{r,s=0}^{\infty}\frac{1}{(r+s+1)!}(-Q^{+})\wedge\underbrace{\mathbb{I}_{1}\wedge\cdots\wedge\mathbb{I}_{1}}_{r}\wedge\underbrace{P_{0}\wedge\cdots\wedge
P_{0}}_{s}\,.$ (4.17)

The projection, identity, and homotopy operators, $\hat{\bm{P}}$ ,
$\hat{\bm{I}}$ , and $\bm{H}$ , satisfy the relations

$\displaystyle\ \hat{\bm{P}}\ =\ \hat{\bm{I}}+\bm{H}\bm{Q}+\bm{Q}\bm{H}\,,$
(4.18) $\displaystyle\ \hat{\bm{P}}^{2}\ =\
\hat{\bm{P}}\,,\qquad[\,\bm{Q}\,,\hat{\bm{P}}\,]\ =\ 0\,,$ (4.19)
$\displaystyle\ \bm{H}\hat{\bm{P}}\ =\ \hat{\bm{P}}\bm{H}\ =\ \bm{H}\bm{H}\ =\
0\,.$ (4.20)

If we perturb $\bm{Q}$ by $\bm{L}_{int}=\sum_{n=0}^{\infty}\bm{L}_{n+2}$ so
that $(\bm{Q}+\bm{L}_{int})^{2}=0$ , the homological perturbation lemma tells
us that chain complexes and chain maps are deformed as

$\displaystyle(\mathcal{SH}^{res},\bm{L}=\bm{Q}+\bm{L}_{int})\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
9.68907pt\raise 10.0486pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.38196pt\hbox{$\scriptstyle{\bm{p}^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 4.30554pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
10.24422pt\raise-10.16943pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.86389pt\hbox{$\scriptstyle{\bm{i}^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 3.0pt\raise-4.30554pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces(\mathcal{SH}^{p},\bm{S}=\bm{QP}+\bm{S}_{int})\,,$
(4.23)

with

$\displaystyle\hat{\bm{p}}^{\prime}\ =$ $\displaystyle\
\hat{\bm{P}}(\hat{\bm{I}}-\bm{L}_{int}\bm{H})^{-1}\,,$ (4.24)
$\displaystyle\hat{\bm{i}}^{\prime}\ =$ $\displaystyle\
(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}\,,$ (4.25)
$\displaystyle\bm{S}_{int}\ =$ $\displaystyle\
\hat{\bm{P}}\bm{L}_{int}(\hat{\bm{I}}-\bm{H}\bm{L}_{int})^{-1}\hat{\bm{P}}\,.$
(4.26)

Here, we defined $(\hat{\bm{I}}-\bm{\mathcal{O}})^{-1}$ by the formal series:

$(\hat{\bm{I}}-\bm{\mathcal{O}})^{-1}\ =\
\sum_{n=0}^{\infty}\mathcal{O}^{n}\,.$ (4.27)

For convenience, introduce $\bm{\Sigma}$ with

$\pi_{1}\bm{S}_{int}\ =\
\mathcal{G}P_{0}\pi_{1}\bm{\Sigma}\,,\qquad\bm{\Sigma}\ =\
\bm{B}\hat{\bm{F}}\hat{\bm{i}}^{\prime}\,.$ (4.28)

The multi-linear representation,

$\langle S|\ =\ \langle\Omega|P_{0}\otimes\pi_{1}\bm{S}_{int}\ =\
\langle\omega_{l}|\xi\bar{\xi}P_{0}\otimes P_{0}\pi_{1}\bm{\Sigma}\,,$ (4.29)

defines a map from $\mathcal{H}^{p}\otimes\mathcal{SH}^{p}$ to $\mathbb{C}$ .
The total S-matrix (at the tree level) is given by $\langle S|$ by restricting
the external states onto the gauge-fixed states, that is,
$\mathcal{H}^{p}\otimes\mathcal{SH}^{p}$ to
$\mathcal{H}_{0}\otimes\mathcal{SH}_{0}$ . If we expand $\bm{\Sigma}$ in the
number of inputs, $\bm{\Sigma}=\sum_{n=0}^{\infty}\bm{\Sigma}_{n+2}$ , it
induces the expansion of the S-matrix in the number of external states:

$\langle S|\ =\ \sum_{n=0}^{\infty}\langle S_{n+3}|\ \,,\qquad\langle
S_{n+3}|\ =\ \langle\omega_{l}|\xi\bar{\xi}P_{0}\otimes
P_{0}\pi_{1}\bm{\Sigma}_{n+2}\,.$ (4.30)

Each term with specific number of external states is further classified by the
number of external Ramond states:

$\displaystyle\langle S_{n+3}|\ =$ $\displaystyle\ \sum_{0\leq
r,\bar{r}\leq(n+3)/2}\langle S_{n+3}|^{(2r,2\bar{r})}$ $\displaystyle=$
$\displaystyle\ \sum_{0\leq
r,\bar{r}\leq(n+3)/2}\langle\omega_{l}|\xi\bar{\xi}P_{0}\otimes
P_{0}\pi_{1}\bm{\Sigma}_{n+2}|^{(2r,2\bar{r})}\,.$ (4.31)

For example, $\langle S_{3}|$ contains four terms:

$\langle S_{3}|\ =\ \langle S_{3}|^{(0,0)}+\langle S_{3}|^{(0,2)}+\langle
S_{3}|^{(2,0)}+\langle S_{3}|^{(2,2)}\,.$ (4.32)

The first, second, and third terms give $(\textrm{NS-NS})^{3}$ , $(\textrm{NS-
NS})\textrm{-}(\textrm{R-NS})^{2}$ , and $(\textrm{NS-
NS})\textrm{-}(\textrm{NS-R})^{2}$ amplitudes, respectively. The fourth term,
on the other hand, includes two amplitudes $(\textrm{NS-
NS})$-$(\textrm{R-R})^{2}$ and
$(\textrm{R-NS})\textrm{-}(\textrm{NS-R})\textrm{-}(\textrm{R-R})$ , which
cannot be distinguished by the Ramond number of the external states999 To be
precise, the second, third, or fourth term contains two different
representations for an amplitude due to the asymmetry that distinguishes one
external state (output) from the others (inputs)..

Since the Hilbert space $\mathcal{H}_{0}$ still contains unphysical states in
general, the physical S-matrix is defined by projecting it onto the physical
subspace, that is, by taking physical states defined by the relative BRST
cohomology,
$\mathcal{H}_{Q}=\textrm{Ker}\,Q/\textrm{Im}\,Q\subset\mathcal{H}_{0}$ , as
external states:

$\langle S^{phys}|\ =\ \langle S|\hat{\bm{\mathcal{P}}}\,,$ (4.33)

with

$\hat{\bm{\mathcal{P}}}\ =\
\sum_{n=0}^{\infty}\frac{1}{n!}\underbrace{\mathcal{P}\wedge\cdots\wedge\mathcal{P}}_{n}\,,\qquad\mathcal{P}\
:\ \mathcal{H}_{0}\ \longrightarrow\ \mathcal{H}_{Q}\,.$ (4.34)

The unitarity of the physical S-matrix is guaranteed by the BRST invariance,
$[\bm{Q},\bm{S}_{int}]=0$ .

By using the relations of the cohomomorphisms

$\pi_{1}\hat{\bm{F}}\ =\
\pi_{1}\mathbb{I}+\mathfrak{S}\pi_{1}\bm{B}\hat{\bm{F}}\,,\qquad\pi_{1}\hat{\bm{i}}^{\prime}\
=\
P_{0}\pi_{1}-Q^{+}\mathcal{G}\pi_{1}\bm{B}\hat{\bm{F}}\hat{\bm{i}}^{\prime}$
(4.35)

following from the definitions (3.13) of $\hat{\bm{F}}^{-1}$ and (4.25) of
$\hat{\bm{i}}^{\prime}$, respectively, we can show that the Dyson-Schwinger
(DS) equation for $\bm{\Sigma}$ ,

$\pi_{1}\bm{\Sigma}\ =\
\sum_{n=0}^{\infty}\pi_{1}\bm{B}_{n+2}\left(\frac{1}{(n+2)!}\big{(}P_{0}\pi_{1}-(Q^{+}\mathcal{G}-\mathfrak{S})\pi_{1}\bm{\Sigma}\big{)}^{\wedge(n+2)}\right)\,,$
(4.36)

holds. By expanding $\bm{\Sigma}$ in the number of inputs, it becomes the
recurrence relation

$\pi_{1}\bm{\Sigma}_{n+2}\ =\
\sum_{m=0}^{n}\pi_{1}\bm{B}_{m+2}\left(\frac{1}{(m+2)!}\big{(}P_{0}\pi_{1}-(Q^{+}\mathcal{G}-\mathfrak{S})\pi_{1}\sum_{l=0}^{n-m-1}\bm{\Sigma}_{l+2}\big{)}^{\wedge(m+2)}\right)\pi_{n+2}\,,$
(4.37)

which determine $\bm{\Sigma}_{n+2}$ recursively. We extend it with three
parameters $s$ , $\bar{s}$ , and $t$ to the extended DS equation,

$\displaystyle\pi_{1}\bm{\Sigma}(s,\bar{s},t)_{n+2}\ $ $\displaystyle\hskip
14.22636pt=\
\sum_{m=0}^{n}\pi_{1}\bm{B}_{m+2}(s,\bar{s},t)\left(\frac{1}{(m+2)!}\big{(}P_{0}\pi_{1}-\Delta(s,\bar{s},t)\pi_{1}\sum_{l=0}^{n-m-1}\bm{\Sigma}_{l+2}(s,\bar{s},t)\big{)}^{\wedge(m+2)}\right)\,,$
(4.38)

which reduces to the DS equation (4.37) at $(s,\bar{s},t)=(0,0,1)$ . Here,
$\Delta(s,\bar{s},t)$ defined by

$\Delta(s,\bar{s},t)=Q^{+}\mathcal{G}(s,\bar{s},t)-\mathfrak{S}(t)\,,$ (4.39)

with

$\displaystyle\mathcal{G}(s,\bar{s},t)\,=$
$\displaystyle\,\pi^{(0,0)}+(tX+s)\pi^{(1,0)}+(t\bar{X}+\bar{s})\pi^{(0,1)}+\left(t^{2}X\bar{X}+t(s\bar{X}+\bar{s}X)+s\bar{s}\right)\pi^{(1,1)}\,,$
(4.40) $\displaystyle\mathfrak{S}(t)\,=$
$\displaystyle\,t\,\Xi\pi^{(1,0)}+t\,\bar{\Xi}\pi^{(0,1)}+\frac{t^{2}}{2}(\Xi\bar{X}+X\bar{\Xi})\pi^{(1,1)}\,,$
(4.41)

was determined to satisfy the relations

$\displaystyle\ [Q\,,\Delta(s,\bar{s},t)]\ =\
\pi(s,\bar{s},t)-P_{0}\mathcal{G}(s,\bar{s},t)\,,$ (4.42a) $\displaystyle\
[\eta\,,\Delta(s,\bar{s},t)]\ =\
\pi(t)\,,\qquad[\bar{\eta}\,,\Delta(s,\bar{s},t)]\ =\ \bar{\pi}(t)\,,$ (4.42b)
and $\displaystyle\partial_{t}\Delta(s,\bar{s},t)\ =$ $\displaystyle\
-\left[Q,Q^{+}\left(\partial_{t}\mathfrak{S}(t)+(s\bar{\Xi}+\bar{s}\Xi)\pi^{(1,1)}\right)\right]+(s\bar{\Xi}+\bar{s}\Xi)\pi^{(1,1)}$
$\displaystyle\
+P_{0}\left(\partial_{t}\mathfrak{S}(t)+(s\bar{\Xi}+\bar{s}\Xi)\pi^{(1,1)}\right)\,,$
(4.42c) $\displaystyle\partial_{s}\Delta(s,\bar{s},t)\ =$ $\displaystyle\
-\left[\eta\,,Q^{+}\left(\partial_{t}\mathfrak{S}(t)+(s\bar{\Xi}+\bar{s}\Xi)\pi^{(1,1)}\right)\right]\,,$
(4.42d) $\displaystyle\partial_{\bar{s}}\Delta(s,\bar{s},t)\ =$
$\displaystyle\
-\left[\bar{\eta}\,,Q^{+}\left(\partial_{t}\mathfrak{S}(t)+(s\bar{\Xi}+\bar{s}\Xi)\pi^{(1,1)}\right)\right]\,.$
(4.42e)

These relations (4.42) make the $\bm{\Sigma}(s,\bar{s},t)$ satisfying (4.38)
is BRST invariant and is in the small Hilbert space:

$[\,\bm{Q}\,,\bm{\Sigma}(s,\bar{s},t)\,]\ =\ 0\,,$ (4.43a)
$[\,\bm{\eta}\,,\bm{\Sigma}(s,\bar{s},t)\,]\ =\
0\,,\qquad[\,\bar{\bm{\eta}}\,,\bm{\Sigma}(s,\bar{s},t)\,]\ =\ 0\,,$ (4.43b)

which can be shown, similar to the case in [24], using the alternative form of
the (extended) $L_{\infty}$-relations (3.49) and the fact that generic
intermediate states are off-shell. Next, differentiating eq. (4.38) by
parameters, we obtain the relations

$\pi_{1}\partial_{t}\bm{\Sigma}(s,\bar{s},t)\ =\
\pi_{1}[\,\bm{Q}\,,\bm{\rho}(s,\bar{s},t)\,]\,,$ (4.44a)
$\pi_{1}\partial_{s}\bm{\Sigma}(s,\bar{s},t)\ =\
\pi_{1}[\,\bm{\eta}\,,\bm{\rho}(s,\bar{s},t)\,]\,,\qquad\pi_{1}\partial_{\bar{s}}\bm{\Sigma}(s,\bar{s},t)\
=\ \pi_{1}[\,\bar{\bm{\eta}}\,,\bm{\rho}(s,\bar{s},t)\,]\,,$ (4.44b)

using the (alternative form of) the differential equations (3.50), where
$\bm{\rho}(s,\bar{s},t)$ is the degree even map determined by the recurrence
relation

$\displaystyle\pi_{1}\bm{\rho}_{n+2}(s,\bar{s},t)\ =$ $\displaystyle\
\sum_{m=0}^{n}(\bm{\lambda}+\bar{\bm{\lambda}})_{m+2}(s,\bar{s},t)\Big{(}D_{m+2}(s,\bar{s},t)\Big{)}P_{n+2}\pi_{n+2}$
$\displaystyle\
-\sum_{m=0}^{n}\pi_{1}\bm{B}_{m+2}(s,\bar{s},t)\Big{(}D_{m+1}(s,\bar{s},t)\wedge\pi_{1}\bm{E}(s,\bar{s},t)\Big{)}P_{n+2}\pi_{n+2}\,,$
(4.45)

with

$D_{M}(s,\bar{s},t)\ =\
\frac{1}{M!}\big{(}P_{0}\pi_{1}-\Delta(s,\bar{s},t)\pi_{1}\bm{\Sigma}(s,\bar{s},t)\big{)}^{\wedge
M}\,,$ (4.46) $\pi_{1}\bm{E}(s,\bar{s},t)\ =\
\Delta(s,\bar{s},t)\pi_{1}\bm{\rho}+Q^{+}(\partial_{t}\mathfrak{S}+(s\bar{\Xi}+\bar{s}\Xi)\pi^{(1,1)})\pi_{1}\bm{\Sigma}(s,\bar{s},t)\,.$
(4.47)

Combining the relations (4.44), we can derive the key relation

$\pi_{1}\partial_{t}\bm{\Sigma}(s,\bar{s},t)-X_{0}\circ\pi_{1}\partial_{s}\bm{\Sigma}(s,\bar{s},t)-\bar{X}_{0}\circ\pi_{1}\partial_{\bar{s}}\bm{\Sigma}(s,\bar{s},t)\
=\
[\,\bm{Q}\,,[\,\bm{\eta}\,,[\,\bar{\bm{\eta}}\,,\bm{T}(s,\bar{s},t)\,]\,]\,]\,,$
(4.48)

with
$\pi_{1}\bm{T}(s,\bar{s},t)=\bar{\xi}\circ\xi\circ\pi_{1}\bm{\rho}(s,\bar{s},t)$
.

We can also extend the S-matrix using
$\bm{\Sigma}(s,\bar{s},t)=\sum_{n=0}^{\infty}\bm{\Sigma}_{n+2}(s,\bar{s},t)$
as

$\langle S(s,\bar{s},t)|\ =\ \sum_{n=0}^{\infty}\langle S_{n+3}(s,\bar{s},t)|\
=\ \sum_{n=0}^{\infty}\langle\omega_{l}|\xi\bar{\xi}P_{0}\otimes
P_{0}\pi_{1}\bm{\Sigma}_{n+2}(s,\bar{s},t)\,,$ (4.49)

which reduces the S-matrix (4.30) at $(s,\bar{s},t)=(0,0,1)$ . The extended
S-matrix element $\langle S_{n+3}(s,\bar{s},t)|$ is expanded in the parameters
as101010 We consider that $\langle S_{n+3}^{(p,\bar{p})}|^{(2r,2\bar{r})}$
with the parameters outside the range
$(0,0)\leq(p,\bar{p})\leq(n+1-r,n+1-\bar{r})$ , $0\leq 2r,2\bar{r}\leq n+3$ is
equal to zero.

$\displaystyle\langle S_{n+3}(s,\bar{s},t)|\ =$ $\displaystyle\
\sum_{m=0}s^{m}\bar{s}^{\bar{m}}\langle S_{n+3}^{[m,\bar{m}]}(t)|\,,$ (4.50a)
$\displaystyle\langle S_{n+3}^{[m,\bar{m}]}(t)|\ =$ $\displaystyle\
\sum_{p=0}^{n-m+1}\sum_{\bar{p}=0}^{n-\bar{m}+1}t^{p+\bar{p}}\langle
S_{n+3}^{(p,\bar{p})}|^{(2(n-m-p+1),2(n-\bar{m}-\bar{p}+1))}\,.$ (4.50b)

The key relation (4.48) induces the equation for the extended S-matrix,

$\displaystyle\partial_{t}\langle S(s,\bar{s},t)|-\partial_{s}\langle
S(s,\bar{s},t)|(X_{0})_{cyc}$ $\displaystyle-\partial_{\bar{s}}\langle
S(s,\bar{s},t)|(\bar{X}_{0})_{cyc}\ $ $\displaystyle=\
\langle\omega_{l}|\xi\bar{\xi}P_{0}\otimes
P_{0}\pi_{1}[\bm{Q},[\bm{\eta},\bar{\bm{\eta}},\bm{T}(s,\bar{s},t)]]\,,$
(4.51)

where

$\langle S(s,\bar{s},t)|(\mathcal{X}_{0})_{cyc}\ =\ \sum_{n=0}^{\infty}\langle
S_{n+3}(s,\bar{s},t)|(\mathcal{X}_{0}\otimes\mathbb{I}_{n+2}+\mathbb{I}\otimes\mathbb{I}_{n+1}\wedge\mathcal{X}_{0}\pi_{1})\,,$
(4.52)

for $\mathcal{X}_{0}=X_{0}$ or $\bar{X}_{0}$ . Note that the right hand side
of (4.51) does not contribute to the physical S-matrix. For each amplitude, we
have

$\displaystyle(p+\bar{p})\langle S_{n+3}^{(p,\bar{p})}|^{(2r,2\bar{r})}\ =$
$\displaystyle\ (n-p+1-r)\langle
S_{n+3}^{(p-1,\bar{p})}|^{(2r,2\bar{r})}(X_{0})_{cyc}$ $\displaystyle\hskip
42.67912pt+(n-\bar{p}+1-\bar{r})\langle
S_{n+3}^{(p,\bar{p}-1)}|^{(2r,2\bar{r})}(\bar{X}_{0})_{cyc}+\cdots\,,$ (4.53)

where dots on the right-hand side represent the terms vanishing on the
physical subspace $\mathcal{H}_{Q}$ . Let us apply this relation on
$(p+\bar{p})!\langle S_{n+3}^{(p,\bar{p})}|^{(2r,2\bar{r})}$ , repeatedly. By
using the relation once, we can reduce $p$ or $\bar{p}$ by 1 with the factor
$(n-r-p+2)$ or $(n-\bar{r}-\bar{p}+2)$ , respectively. By repeatedly using the
relation $(p+\bar{p})$ times, we eventually reach $\langle
S_{n+3}^{(0,0)}|^{(2r,2\bar{r})}$ through ${}_{p+\bar{p}}C_{p}$ paths. The
number of paths can be obtained by counting the places to reduce $p$ in the
$(p+\bar{p})$ steps. The coefficients coming from the relation are the same
for all the paths, and equal to

$\frac{(n-r+1)!}{(n-r-p+1)!}\frac{(n-\bar{r}+1)!}{(n-\bar{r}-\bar{p}+1)!}\,.$
(4.54)

Thus, we eventually find that

$\langle S_{n+3}^{(p,\bar{p})}|^{(2r,2\bar{r})}\ =\ \begin{pmatrix}n-r+1\\\
p\end{pmatrix}\begin{pmatrix}n-\bar{r}+1\\\ \bar{p}\end{pmatrix}\langle
S_{n+3}^{(0,0)}|^{(2r,2\bar{r})}(X_{0})_{cyc}^{p}(\bar{X}_{0})_{cyc}^{\bar{p}}\,,$
(4.55)

except for the terms vanishing on $\mathcal{H}_{Q}$ . Then, the physical
amplitudes obtained by projecting $\langle
S_{n+3}^{(p,\bar{p})}|^{(2r,2\bar{r})}$ with $(p,\bar{p})=(n-r+1,n-\bar{r}+1)$
onto $\mathcal{H}_{Q}$ is written as

$\langle S_{n+3}^{phys}|^{(2r,2\bar{r})}\ =\
\langle(S_{B})_{n+3}^{phys}|^{(2r,2\bar{r})}(X_{0})_{cyc}^{n-r+1}(\bar{X}_{0})_{cyc}^{n-\bar{r}+1}\,,$
(4.56)

where $\langle(S_{B})^{phys}_{n+3}|^{(2r,2\bar{r})}$ in the right-hand side is
the bosonic physical amplitude defined by

$\langle(S_{B})^{phys}_{n+3}|^{(2r,2\bar{r})}\ =\ \langle
S_{n+3}^{(0,0)}|^{(2r,2\bar{r})}\hat{\bm{\mathcal{P}}}\,.$ (4.57)

Since the physical amplitude is independent of how we insert PCO’s, we can
appropriately deform and move them to agree with that obtained using the
first-quantization method.

##  5 Gauge-invariant action in WZW-like formulation

Using an alternative method, symmetric construction, we have constructed the
string products (interactions) with $L_{\infty}$ structure for the type II
superstring. The new gauge-invariant action looks different from the previous
one but is the same and nothing essentially new. However, a difference appears
when we try to map it to the WZW-like action through a field redefinition.
Previously, we could only find a map to the half-WZW-like action [15], but the
new construction enables us to construct a map to the complete WZW-like
action.

###  5.1 WZW-like action for the NS-NS sector

We first summarize the results obtained in [13] on the construction of the
WZW-like action for the NS-NS sector.

For the NS-NS sector, generating functional of string products with the
$L_{\infty}$ structure, $\bm{L}^{NS\textrm{-}NS}(s,\bar{s},t)$ , has been
constructed by imposing the differential equations [10],

$\displaystyle\ \partial_{t}\bm{L}^{NS\textrm{-}NS}(s,\bar{s},t)\ =\
[\bm{L}^{NS\textrm{-}NS}(s,\bar{s},t)\,,(\bm{\lambda}^{NS\textrm{-}NS}+\bar{\bm{\lambda}}^{NS\textrm{-}NS})(s,\bar{s},t)]\,,$
(5.1a) $\displaystyle\ \partial_{s}\bm{L}^{NS\textrm{-}NS}(s,\bar{s},t)\ =\
[\bm{\eta}\,,\bm{\lambda}^{NS\textrm{-}NS}(s,\bar{s},t)]\,,\qquad[\bm{\eta}\,,\bar{\bm{\lambda}}^{NS\textrm{-}NS}(s,\bar{s},t)]\
=\ 0\,,$ (5.1b) $\displaystyle\
\partial_{\bar{s}}\bm{L}^{NS\textrm{-}NS}(s,\bar{s},t)\ =\
[\bar{\bm{\eta}}\,,\bar{\bm{\lambda}}^{NS\textrm{-}NS}(s,\bar{s},t)]\,,\qquad[\bar{\bm{\eta}}\,,\bm{\lambda}^{NS\textrm{-}NS}(s,\bar{s},t)]\
=\ 0\,,$ (5.1c)

introducing the (generating functional of) gauge products
$\bm{\lambda}^{NS\textrm{-}NS}(s,\bar{s},t)$ and
$\bar{\bm{\lambda}}^{NS\textrm{-}NS}(s,\bar{s},t)$ . The type II superstring
field theory with the $L_{\infty}$ structure is characterized by the
$L_{\infty}$ triplet $(\bm{\eta},\bar{\bm{\eta}};\bm{L}^{NS\textrm{-}NS})$
through the equations

$\displaystyle\eta\Phi_{NS\textrm{-}NS}\ =$ $\displaystyle\ 0\,,$ (5.2a)
$\displaystyle\bar{\eta}\Phi_{NS\textrm{-}NS}\ =$ $\displaystyle\ 0\,,$ (5.2b)
$\displaystyle\pi_{1}\bm{L}^{NS\textrm{-}NS}(e^{\wedge\Phi_{NS\textrm{-}NS}})\
=$ $\displaystyle\ 0\,.$ (5.2c)

The first two (5.2a) and (5.2b) are the constraints imposing
$\Phi_{NS\textrm{-}NS}$ is in the small Hilbert space, and the last one (5.2c)
is the equation of motion. Using the cohomomorphism

$\hat{\bm{g}}\ =\
\vec{\mathcal{P}}\exp\left(\int_{0}^{1}dt(\bm{\lambda}^{NS\textrm{-}NS}+\bar{\bm{\lambda}}^{NS\textrm{-}NS})(0,0,t)\right)\,,$
(5.3)

where $\vec{\mathcal{P}}$ denotes the path-ordered product from left to right,
we can transform the original triplet
$(\bm{\eta},\bar{\bm{\eta}};\bm{L}^{NS\textrm{-}NS})$ and equations (5.2) to
the dual $L_{\infty}$ triplet
$(\bm{L}^{\eta}=\hat{\bm{g}}\bm{\eta}\hat{\bm{g}}^{-1},\bm{L}^{\bar{\eta}}=\hat{\bm{g}}\bar{\bm{\eta}}\hat{\bm{g}}^{-1};\bm{Q})$
and the equations

$\displaystyle\pi_{1}\bm{L}^{\eta}(e^{\wedge\pi_{1}\hat{\bm{g}}(e^{\wedge\Phi_{NS\textrm{-}NS}})})\
=$ $\displaystyle\ 0\,,$ (5.4a)
$\displaystyle\pi_{1}\bm{L}^{\bar{\eta}}(e^{\wedge\pi_{1}\hat{\bm{g}}(e^{\wedge\Phi_{NS\textrm{-}NS}})})\
=$ $\displaystyle\ 0\,,$ (5.4b) $\displaystyle
Q\pi_{1}\hat{\bm{g}}(e^{\wedge\Phi_{NS\textrm{-}NS}})\ =$ $\displaystyle\
0\,,$ (5.4c)

respectively, which characterize the WZW-like formulation. The constraint
equations,

$\displaystyle\pi_{1}\bm{L}^{\eta}(e^{\wedge G_{\eta\bar{\eta}}(V)})\ =$
$\displaystyle\ 0\,,$ (5.5a)
$\displaystyle\pi_{1}\bm{L}^{\bar{\eta}}(e^{\wedge G_{\eta\bar{\eta}}(V)})\ =$
$\displaystyle\ 0\,,$ (5.5b)

are identically satisfied by the pure-gauge (functional) string field111111
The method how we find the explicit form of $G_{\eta\bar{\eta}}(V)$ and
$B_{d}(V)$ is given in [13].

$G_{\eta\bar{\eta}}(V)\ =\
\eta\bar{\eta}V+\frac{1}{2}\left(L_{2}^{\eta}(\eta\bar{\eta}V,\bar{\eta}V)+\eta
L_{2}^{\bar{\eta}}(\eta\bar{\eta}V,V)\right)+\cdots\,,$ (5.6)

with the string field $V$ of the WZW-like formulation having the ghost number
$0$ and the picture number $(0,0)$ . Therefore, if we identify

$G_{\eta\bar{\eta}}(V)\ =\
\pi_{1}\hat{\bm{g}}(e^{\wedge\Phi_{NS\textrm{-}NS}})\,,$ (5.7)

it gives a map between the string fields in two formulations. By this
identification, the last equation (5.2c) becomes the equation of motion of the
WZW-like formulation:

$QG_{\eta\bar{\eta}}(V)\ =\ 0\,.$ (5.8)

In order to write down the WZW-like action, it is also necessary to introduce
another important functional field called the associated string field,

$B_{d}(V(t))\ =\
-dV(t)+\frac{1}{2}\left(L^{\eta}_{2}(V(t),\bar{\eta}dV(t))+L^{\bar{\eta}}(\eta
V(t),dV(t))\right)+\cdots\,,\quad(d=\partial_{t},\delta,Q).$ (5.9)

Here, $V(t)$ is an extension of $V$ by a parameter $t\in[0,1]$ satisfying
$V(0)=0$ and $V(1)=V$ . The identification (5.7) induces the map

$B_{d}(V(t))\ =\
\pi_{1}\hat{\bm{g}}\bm{D_{\xi\bar{\xi}}}(e^{\wedge\Phi_{NS\textrm{-}NS}})\,,$
(5.10)

where $\bm{D_{\xi\bar{\xi}}}$ is the coderivation derived from
$\pi_{1}\bm{D_{\xi\bar{\xi}}}=\xi\bar{\xi}\pi_{1}D$ with $D=d$ for
$d=\partial_{t},\delta$ and $D=\pi_{1}\bm{L}^{NS\textrm{-}NS}$ for $d=Q$ .
These maps (5.7) and (5.10) are consistent with the identities characterizing
the associated field,

$\displaystyle\ dG_{\eta\bar{\eta}}(V(t))\ =\
D_{\bar{\eta}}(t)D_{\eta}(t)B_{d}(V(t))\,,$ (5.11a) $\displaystyle\
D_{\bar{\eta}}(t)D_{\eta}(t)\Big{(}\partial_{t}B_{\delta}(V(t))-\delta
B_{\partial_{t}}(V(t))\Big{)}\ =\ 0\,,$ (5.11b)

where we introduced the nilpotent linear operators $D_{\eta}(t)$ and
$D_{\bar{\eta}}(t)$ as

$D_{\eta}(t)\varphi\ =\ \pi_{1}\bm{L}^{\eta}(e^{\wedge
G_{\eta\bar{\eta}}(V(t))}\wedge\varphi)\,,\qquad D_{\bar{\eta}}(t)\varphi\ =\
\pi_{1}\bm{L}^{\bar{\eta}}(e^{\wedge
G_{\eta\bar{\eta}}(V(t))}\wedge\varphi)\,,$ (5.12)

for a general string field $\varphi\in\mathcal{H}_{l}^{NS\textrm{-}NS}$ .
Then, we can map the action to the WZW-like action

$I_{\textrm{WZW}}^{NS\textrm{-}NS}\ =\
\int^{1}_{0}dt\,\omega_{l}\Big{(}B_{\partial_{t}}(V(t)),QG_{\eta\bar{\eta}}(V(t))\Big{)}\,.$
(5.13)

Using the identities (5.11), we can calculate an arbitrary variation of the
action as

$\delta I^{NS\textrm{-}NS}_{\textrm{WZW}}\ =\
\omega_{l}\Big{(}B_{\delta}(V),QG_{\eta\bar{\eta}}(V)\Big{)}\,,$ (5.14)

which derives the equation of motion (5.8) and the gauge invariance under the
transformation

$B_{\delta_{g}}(V)\ =\ Q\Lambda+D_{\eta}\Omega+D_{\bar{\eta}}\bar{\Omega}\,.$
(5.15)

The gauge transformation generated by $\Lambda$ is mapped from that generated
by $\Lambda_{NS\textrm{-}NS}$ in the formulation with the $L_{\infty}$
structure with the identification

$\Lambda\ =\
\pi_{1}\hat{\bm{g}}\Big{(}e^{\wedge\Phi_{NS\textrm{-}NS}}\wedge\xi\bar{\xi}\Lambda_{NS\textrm{-}NS}\Big{)}\,.$
(5.16)

The extra gauge invariances come from the fact that the map (5.7) is not one-
to-one due to the invariance of $G_{\eta\bar{\eta}}(V)$ under the variation
$B_{\delta}(V)=D_{\eta}\Omega+D_{\bar{\eta}}\bar{\Omega}$ following from the
identity (5.11a).

###  5.2 Complete WZW-like action and gauge invariance

It is straightforward to extend the results on the NS-NS sector to all four
sectors. We first note that the similarity transformation by $\hat{\bm{F}}$ is
trivial except for the NS-NS sector, and thus, the string product $\bm{L}$ we
constructed reduces to

$\bm{L}|^{(0,0)}\ =\ \bm{Q}+\sum_{p=0}^{\infty}\bm{B}^{(p,p)}_{p+1}|^{(0,0)}\
\equiv\ \bm{Q}+\bm{B}|^{(0,0)}\,,$ (5.17)

in the NS-NS sector. Then, the differential equations (3.20) implies that the
generating functional
$\bm{L}|^{(0,0)}(s,\bar{s},t)=\bm{Q}+\bm{B}|^{(0,0)}(s,\bar{s},t)$ of string
product in the NS-NS sector satisfies

$\displaystyle\ \partial_{t}\bm{L}|^{(0,0)}(s,\bar{s},t)\ =\
[\,\bm{L}|^{(0,0)}(s,\bar{s},t)\,,(\bm{\lambda}+\bar{\bm{\lambda}})|^{(0,0)}(s,\bar{s},t)\,]\,,$
(5.18a) $\displaystyle\ \partial_{s}\bm{L}|^{(0,0)}(s,\bar{s},t)\ =\
[\,\bm{\eta}\,,\bm{\lambda}|^{(0,0)}(s,\bar{s},t)\,]\,,\qquad[\,\bm{\eta}\,,\bar{\bm{\lambda}}|^{(0,0)}(s,\bar{s},t)\,]\
=\ 0\,,$ (5.18b) $\displaystyle\
\partial_{\bar{s}}\bm{L}|^{(0,0)}(s,\bar{s},t)\ =\
[\,\bar{\bm{\eta}}\,,\bar{\bm{\lambda}}|^{(0,0)}(s,\bar{s},t)\,]\,,\qquad[\,\bar{\bm{\eta}}\,,\bm{\lambda}|^{(0,0)}(s,\bar{s},t)\,]\
=\ 0\,,$ (5.18c)

where

$\displaystyle\bm{B}|^{(0,0)}(s,\bar{s},t)\ =$ $\displaystyle\
\sum_{p,\bar{p},m,\bar{m}=0}^{\infty}t^{p+\bar{p}}s^{m}\bar{s}^{\bar{m}}\bm{B}^{(p,\bar{p})}_{p+m+1,\bar{p}+\bar{m}+1}|^{(0,0)}\,,$
(5.19a) $\displaystyle\bm{\lambda}|^{(0,0)}(s,\bar{s},t)\ =$ $\displaystyle\
\sum_{p,\bar{p},m,\bar{m}=0}^{\infty}t^{p+\bar{p}}s^{m}\bar{s}^{\bar{m}}\bm{\lambda}^{(p+1,\bar{p})}_{p+m+2,\bar{p}+\bar{m}+1}|^{(0,0)}\,,$
(5.19b) $\displaystyle\bar{\bm{\lambda}}|^{(0,0)}(s,\bar{s},t)\ =$
$\displaystyle\
\sum_{p,\bar{p},m,\bar{m}=0}^{\infty}t^{p+\bar{p}}s^{m}\bar{s}^{\bar{m}}\bar{\bm{\lambda}}^{(p,\bar{p}+1)}_{p+m+1,\bar{p}+\bar{m}+2}|^{(0,0)}\,.$
(5.19c)

Since the equations (5.18) agrees with the equations (5.1), we can identify
the new string product in the NS-NS sector,
$\bm{L}|^{(0,0)}=\bm{L}|^{(0,0)}(0,0,1)$ , to that constructed in [10]. The
cohomomorphism (5.3) is then given by

$\hat{\bm{g}}\ =\
\vec{\mathcal{P}}\exp\left(\int_{0}^{1}dt(\bm{\lambda}+\bar{\bm{\lambda}})|^{(0,0)}(0,0,t)\right)\,.$
(5.20)

With this identification, the $L_{\infty}$ triplet
$(\bm{\eta},\bar{\bm{\eta}};\bm{L})$ can transform to the dual triplet
$(\bm{L}^{\eta},\bm{L}^{\bar{\eta}};\tilde{\bm{L}})$ , where

$\displaystyle\pi_{1}\tilde{\bm{L}}\ =$ $\displaystyle\
\pi_{1}\hat{\bm{g}}\bm{L}\hat{\bm{g}}^{-1}\ =\
\pi_{1}\bm{Q}+\mathcal{G}\pi_{1}\tilde{\bm{b}}\,,$ (5.21)
$\displaystyle\pi_{1}\tilde{\bm{b}}\ =$ $\displaystyle\
\pi_{1}\hat{\bm{g}}(\bm{b}-\bm{B}|^{(0,0)})\hat{\bm{g}}^{-1}\,.$ (5.22)

Since $\hat{\bm{g}}$ is identity except in the NS-NS sector, we find that

$\pi_{1}\hat{\bm{g}}(e^{\wedge\Phi})\ =\
\pi_{1}\hat{\bm{g}}(e^{\wedge\Phi_{NS\textrm{-}NS}})+\Phi_{R\textrm{-}NS}+\Phi_{NS\textrm{-}R}+\Phi_{R\textrm{-}R}\,,$
(5.23)

and can identify $\Phi_{R\textrm{-}NS}$ , $\Phi_{NS\textrm{-}R}$ , and
$\Phi_{R\textrm{-}R}$ with the corresponding string fields $\Psi$ ,
$\bar{\Psi}$ , and $\Sigma$ , in the WZW-like formulation, respectively:

$\Phi_{R\textrm{-}NS}\ =\ \Psi\,,\quad\Phi_{R\textrm{-}NS}\ =\
\bar{\Psi}\,,\quad\Phi_{R\textrm{-}R}\ =\ \Sigma\,.$ (5.24)

The identification (5.7) is now extended as

$\pi_{1}\hat{\bm{g}}(e^{\wedge\Phi})=\ \
G_{\eta\bar{\eta}}(V)+\Psi+\bar{\Psi}+\Sigma\ \equiv\ G(\mathcal{V})\,,$
(5.25)

where we denoted the string fields in the WZW-formulation as $\mathcal{V}$
collectively. The dual constraint equations (5.4a) and (5.4b) map to

$\displaystyle 0\ =$ $\displaystyle\ \pi_{1}\bm{L}^{\eta}(e^{\wedge
G(\mathcal{V})})\ =\ \pi_{1}\bm{L}^{\eta}(e^{\wedge
G_{\eta\bar{\eta}}(V)})+\eta\Psi+\eta\bar{\Psi}+\eta\Sigma\,,$ (5.26)
$\displaystyle 0\ =$ $\displaystyle\ \pi_{1}\bm{L}^{\bar{\eta}}(e^{\wedge
G(\mathcal{V})})=\ \pi_{1}\bm{L}^{\bar{\eta}}(e^{\wedge
G_{\eta\bar{\eta}}(V)})+\bar{\eta}\Psi+\bar{\eta}\bar{\Psi}+\bar{\eta}\Sigma\,,$
(5.27)

which decompose to the Maurer-Cartan equations (5.5) and the constraints that
$\Psi$ , $\bar{\Psi}$ , and $\Sigma$ are the fields in the small Hilbert
space. The map for the associated string field (5.10) can also be extended to
that for all four sectors as

$\pi_{1}\hat{\bm{g}}\bm{D_{\xi{\bar{\xi}}}}(e^{\wedge\Phi})\ =\
B_{d}(V(t))+\xi\bar{\xi}d\Psi(t)+\xi\bar{\xi}d\bar{\Psi}(t)+\xi\bar{\xi}d\Sigma(t)\
\equiv\mathcal{B}_{d}(\mathcal{V}(t))\,.$ (5.28)

We can find that the identities (5.11) are then extended to

$\displaystyle\ dG(\mathcal{V}(t))\ =\
D_{\bar{\eta}}(t)D_{\eta}(t)\mathcal{B}_{d}(\mathcal{V}(t))\,,$ (5.29)
$\displaystyle\
D_{\bar{\eta}}(t)D_{\eta}(t)\Big{(}\partial_{t}\mathcal{B}_{\delta}(\mathcal{V}(t))-\delta\mathcal{B}_{\delta}(\mathcal{V}(t))\Big{)}\
=\ 0\,.$ (5.30)

Here, for general string field $\varphi\in\mathcal{H}$ ,

$\displaystyle D_{\eta}(t)\varphi\ =$ $\displaystyle\
\pi_{1}^{(0,0)}\bm{L}^{\eta}(e^{\wedge
G_{\eta\bar{\eta}}(V(t))}\wedge\varphi_{NS\textrm{-}NS})+\eta\varphi_{R\textrm{-}NS}+\eta\varphi_{R\textrm{-}NS}+\eta\varphi_{R\textrm{-}R}\,,$
(5.31) $\displaystyle D_{\bar{\eta}}(t)\varphi\ =$ $\displaystyle\
\pi_{1}^{(0,0)}\bm{L}^{\bar{\eta}}(e^{\wedge
G_{\eta\bar{\eta}}(V(t))}\wedge\varphi_{NS\textrm{-}NS})+\bar{\eta}\varphi_{R\textrm{-}NS}+\bar{\eta}\varphi_{NS\textrm{-}R}+\bar{\eta}\varphi_{R\textrm{-}R}\,,$
(5.32)

with

$\varphi_{NS\textrm{-}NS}\ =\
\pi^{(0,0)}\varphi\,,\quad\varphi_{R\textrm{-}NS}\ =\
\pi^{(1,0)}\varphi\,,\quad\varphi_{NS\textrm{-}R}\ =\
\pi^{(0,1)}\varphi\,,\quad\varphi_{R\textrm{-}R}\ =\ \pi^{(1,1)}\varphi\,.$
(5.33)

It is easy to show that an arbitrary variation of the action

$I_{\textrm{WZW}}\ =\
\int^{1}_{0}dt\,\omega_{l}\Big{(}\mathcal{B}_{\partial_{t}}(\mathcal{V}(t))\,,\mathcal{G}^{-1}\pi_{1}\tilde{\bm{L}}(e^{\wedge
G(\mathcal{V}(t))})\Big{)}$ (5.34)

gives

$\delta I_{\textrm{WZW}}\ =\
\omega_{l}\Big{(}\mathcal{B}_{\delta}(\mathcal{V}),\mathcal{G}^{-1}\pi_{1}\tilde{\bm{L}}(e^{\wedge
G(\mathcal{V})})\Big{)}\,.$ (5.35)

It derives the equations of motion

$\tilde{\bm{L}}(e^{\wedge G(\mathcal{V})})\ =\ 0$ (5.36)

and yields the invariance of the action under the gauge transformation

$\mathcal{B}_{\delta_{\Lambda}}(\mathcal{V})\ =\
\pi_{1}^{(0,0)}\tilde{\bm{L}}(e^{\wedge
G}\wedge\hat{\Lambda})+\xi\bar{\xi}(\pi_{1}^{(1,0)}+\pi_{1}^{(0,1)}+\pi_{1}^{(1,1)})\tilde{\bm{L}}(e^{\wedge
G}\wedge\bar{\eta}\eta\hat{\Lambda})\,,$ (5.37)

where

$\pi_{1}^{(0,0)}\hat{\Lambda}=\Lambda\,,\quad\pi_{1}^{(1,0)}\hat{\Lambda}=\xi\bar{\xi}\lambda\,,\quad\pi_{1}^{(0,0)}\hat{\Lambda}=\xi\bar{\xi}\bar{\lambda}\,,\quad\pi_{1}^{(0,0)}\hat{\Lambda}=\xi\bar{\xi}\rho\,,$
(5.38)

with
$\eta\lambda=\bar{\eta}\lambda=\eta\bar{\lambda}=\bar{\eta}\bar{\lambda}=\eta\rho=\bar{\eta}\rho=0$
. The identification with the parameters of the formulation with $L_{\infty}$
structure is given by

$\tilde{\Lambda}\ =\
\pi_{1}^{(0,0)}\hat{\bm{g}}(e^{\wedge\Phi_{NS\textrm{-}NS}}\wedge\xi\bar{\xi}\Lambda_{NS\textrm{-}NS})\,,\quad\lambda\
=\ \Lambda_{R\textrm{-}NS}\,,\quad\bar{\lambda}\ =\
\Lambda_{NS\textrm{-}R}\,,\quad\rho\ =\ \Lambda_{R\textrm{-}R}\,.$ (5.39)

##  6 Conclusion and discussion

In this paper, we have revisited the type II superstring field theory with
$L_{\infty}$ structure and proposed an alternative method to construct string
products, which is symmetric with respect to the left- and right-moving
sectors. The symmetric method makes transparent not only the construction of
string products but also the proof that the tree-level S-matrix agrees with
that calculated using the first-quantization method. Another advantage of the
symmetric construction is that it enables us to write down a WZW-like action
through a map between the string fields of the two formulations, which was not
possible with the previous (asymmetric) construction method. The complete WZW-
like action of type II superstring field theory, which was the only missing
piece, has now been constructed. We have completed the superstring field
theory for all three complementary formulations, which allows us to use a
convenient formulation depending on what we are studying.

There is another interesting superstring field theory to consider: the
(oriented) open-closed superstring field theory. This is the one that should
be derived as type II superstring field theory on a non-trivial D-brane
background but is worth constructing independently as it helps to calculate
non-perturbative effects. In fact, the one based on the formulation with extra
free field has already been constructed [25] and used for studying some non-
perturbative effect [26, 27, 28, 29, 30]. It is interesting to construct it
based on the open-closed homotopy algebra (OCHA) [31] and study the relation
to the one in the WZW-like formulation.

Acknowledgments

This work is supported in part by JSPS Grant-in-Aid for Scientific Research
(C) Grant Number JP18K03645.

##  Appendix A Projected commutators

The definition of the projected commutator introduced in (3.14) is given by

$\displaystyle[\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}]^{11}\ =$
$\displaystyle\
\sum_{n,r,\bar{r}}\sum_{m,s,\bar{s}}[\,\bm{\mathfrak{D}}_{n}|^{(2r,2\bar{r})}\,,\bm{\mathfrak{D}}^{\prime}_{m}|^{(2s,2\bar{s})}]\mid^{(2(r+s),2(\bar{r}+\bar{s}))}\,,$
(A.1a) $\displaystyle[\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}]^{21}\ =$
$\displaystyle\
\sum_{n,r,\bar{r}}\sum_{m,s,\bar{s}}[\,\bm{\mathfrak{D}}_{n}|^{(2r,2\bar{r})}\,,\bm{\mathfrak{D}}^{\prime}_{m}|^{(2s,2\bar{s})}]\mid^{(2(r+s-1),2(\bar{r}+\bar{s}))}\,,$
(A.1b) $\displaystyle[\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}]^{12}\ =$
$\displaystyle\
\sum_{n,r,\bar{r}}\sum_{m,s,\bar{s}}[\,\bm{\mathfrak{D}}_{n}|^{(2r,2\bar{r})}\,,\bm{\mathfrak{D}}^{\prime}_{m}|^{(2s,2\bar{s})}]\mid^{(2(r+s),2(\bar{r}+\bar{s}-1))}\,,$
(A.1c) $\displaystyle[\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}]^{22}\ =$
$\displaystyle\
\sum_{n,r,\bar{r}}\sum_{m,s,\bar{s}}[\,\bm{\mathfrak{D}}_{n}|^{(2r,2\bar{r})}\,,\bm{\mathfrak{D}}_{m}|^{(2s,2\bar{s})}]\mid^{(2(r+s-1),2(\bar{r}+\bar{s}-1))}\,,$
(A.1d)

for coderivations
$\bm{\mathfrak{D}}=\sum_{n,r,\bar{r}}\bm{\mathfrak{D}}_{n}|^{(2r,2\bar{r})}$
and
$\bm{\mathfrak{D}}^{\prime}=\sum_{m,s,\bar{s}}\bm{\mathfrak{D}}^{\prime}_{m}|^{(2s,2\bar{s})}$
. An alternative expressions obtained by projecting the intermediate state
[24],

$\displaystyle[\,\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}]^{11}\ =$
$\displaystyle\
\sum_{n}\left(\bm{\mathfrak{D}}_{n}\big{(}\pi_{1}^{(0,0)}\bm{\mathfrak{D}}^{\prime}\wedge\mathbb{I}_{n-1}\big{)}-(-)^{|\mathfrak{D}||\mathfrak{D}^{\prime}|}\bm{\mathfrak{D}}^{\prime}_{n}\big{(}\pi_{1}^{(0,0)}\bm{\mathfrak{D}}\wedge\mathbb{I}_{n-1}\big{)}\right)\,,$
(A.2a) $\displaystyle[\,\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}]^{21}\
=$ $\displaystyle\
\sum_{n}\left(\bm{\mathfrak{D}}_{n}\big{(}\pi_{1}^{(1,0)}\bm{\mathfrak{D}}^{\prime}\wedge\mathbb{I}_{n-1}\big{)}-(-)^{|\mathfrak{D}||\mathfrak{D}^{\prime}|}\bm{\mathfrak{D}}^{\prime}_{n}\big{(}\pi_{1}^{(1,0)}\bm{\mathfrak{D}}\wedge\mathbb{I}_{n-1}\big{)}\right)\,,$
(A.2b) $\displaystyle[\,\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}]^{12}\
=$ $\displaystyle\
\sum_{n}\left(\bm{\mathfrak{D}}_{n}\big{(}\pi_{1}^{(0,1)}\bm{\mathfrak{D}}^{\prime}\wedge\mathbb{I}_{n-1}\big{)}-(-)^{|\mathfrak{D}||\mathfrak{D}^{\prime}|}\bm{\mathfrak{D}}^{\prime}_{n}\big{(}\pi_{1}^{(0,1)}\bm{\mathfrak{D}}\wedge\mathbb{I}_{n-1}\big{)}\right)\,,$
(A.2c) $\displaystyle[\,\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}]^{22}\
=$ $\displaystyle\
\sum_{n}\left(\bm{\mathfrak{D}}_{n}\big{(}\pi_{1}^{(1,1)}\bm{\mathfrak{D}}^{\prime}\wedge\mathbb{I}_{n-1}\big{)}-(-)^{|\mathfrak{D}||\mathfrak{D}^{\prime}|}\bm{\mathfrak{D}}^{\prime}_{n}\big{(}\pi_{1}^{(1,1)}\bm{\mathfrak{D}}\wedge\mathbb{I}_{n-1}\big{)}\right)\,,$
(A.2d)

are also useful. Here, $|\mathfrak{D}|$ ($|\mathfrak{D}^{\prime}|$) is equal
to 0 or 1 when the degree of coderivation $\bm{\mathfrak{D}}$
($\bm{\mathfrak{D}}^{\prime}$) is even or odd, respectively. The square
bracket with subscript $\mathfrak{X}=X$ or $\bar{X}$ is defined by inserting
$\mathfrak{X}$ at the intermediate state121212 These brackets can only be
applicable for the case considered since $X$ and $\bar{X}$ are the PCO’s
acting on the states with picture number $-3/2$ .:

$[\,\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}]^{22}_{\mathfrak{X}}\ =\
\sum_{n}\left(\bm{\mathfrak{D}}_{n}\big{(}\mathfrak{X}\pi_{1}^{(1,1)}\bm{\mathfrak{D}}^{\prime}\wedge\mathbb{I}_{n-1}\big{)}-(-)^{|\mathfrak{D}||\mathfrak{D}^{\prime}|}\bm{\mathfrak{D}}^{\prime}_{n}\big{(}\mathfrak{X}\pi_{1}^{(1,1)}\bm{\mathfrak{D}}\wedge\mathbb{I}_{n-1}\big{)}\right)\,.\\\
$ (A.3)

Similarly, the square bracket with subscript $\mathcal{O}=\Xi$ or $\bar{\Xi}$
is defined by

$[\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}]^{22}_{\mathcal{O}}\ =\
\sum_{n}\left(\bm{\mathfrak{D}}_{n}(\mathcal{O}\pi_{1}^{(1,1)}\bm{\mathfrak{D}}^{\prime}\wedge\mathbb{I}^{\wedge(n-1)})+(-1)^{(|\mathfrak{D}|+1)(|\mathcal{D}^{\prime}|+1)}\bm{\mathfrak{D}}^{\prime}_{n}(\mathcal{O}\pi_{1}^{(1,1)}\bm{\mathfrak{D}}\wedge\mathbb{I}^{\wedge(n-1)})\right)\,.$
(A.4)

The projected commutators (A.1) satisfy the Jacobi identities

$[\,\bm{\mathfrak{D}}\,,[\,\bm{\mathfrak{D}}^{\prime}\,,\bm{\mathfrak{D}}^{\prime\prime}\,]^{ab}]^{cd}+[\,\bm{\mathfrak{D}}\,,[\,\bm{\mathfrak{D}}^{\prime}\,,\bm{\mathfrak{D}}^{\prime\prime}\,]^{cd}]^{ab}+(\textrm{cyclic}\
\textrm{perm.})\ =\ 0\,,$ (A.5)

which reduce to the conventional ones if two projected commutators are the
same type. Similar identities also hold even if they include projected
commutator(s) with subscript $\mathfrak{X}=X$ or $\bar{X}$ :

$\displaystyle\
[\,\bm{\mathfrak{D}}\,,[\,\bm{\mathfrak{D}}^{\prime}\,,\bm{\mathfrak{D}}^{\prime\prime}\,]^{22}_{\mathfrak{X}}]^{ab}+[\,\bm{\mathfrak{D}}\,,[\,\bm{\mathfrak{D}}^{\prime}\,,\bm{\mathfrak{D}}^{\prime\prime}\,]^{ab}]^{22}_{\mathfrak{X}}+(\textrm{cyclic}\
\textrm{perm.})\ =\ 0\,,$ (A.6) $\displaystyle\
[\,\bm{\mathfrak{D}}\,,[\,\bm{\mathfrak{D}}^{\prime}\,,\bm{\mathfrak{D}}^{\prime\prime}\,]^{22}_{\mathfrak{X}}]^{22}_{\mathfrak{X}^{\prime}}+[\,\bm{\mathfrak{D}}\,,[\,\bm{\mathfrak{D}}^{\prime}\,,\bm{\mathfrak{D}}^{\prime\prime}\,]^{22}_{\mathfrak{X}^{\prime}}]^{22}_{\mathfrak{X}}+(\textrm{cyclic}\
\textrm{perm.})\ =\ 0\,.$ (A.7)

The additional sign factor is necessary for the Jacobi identities including
the one with subscript $\mathcal{O}=\Xi$ or $\bar{\Xi}$ since $\Xi$ and
$\bar{\Xi}$ are the Grassmann odd operators. For example,

$\displaystyle\
[\,\bm{\mathfrak{D}}\,,[\,\bm{\mathfrak{D}}^{\prime}\,,\bm{\mathfrak{D}}^{\prime\prime}\,]^{22}_{\mathcal{O}}]^{ab}+(-1)^{|\mathfrak{D}^{\prime}|}[\,\bm{\mathfrak{D}}\,,[\,\bm{\mathfrak{D}}^{\prime}\,,\bm{\mathfrak{D}}^{\prime\prime}\,]^{ab}]^{22}_{\mathcal{O}}+(\textrm{cyclic}\
\textrm{perm.})\ =\ 0\,,$ (A.8) $\displaystyle\
[\,\bm{\mathfrak{D}}\,,[\,\bm{\mathfrak{D}}^{\prime}\,,\bm{\mathfrak{D}}^{\prime\prime}\,]^{22}_{\mathcal{O}}]^{22}_{\mathfrak{X}}+(-1)^{|\mathfrak{D}^{\prime}|}[\,\bm{\mathfrak{D}}\,,[\,\bm{\mathfrak{D}}^{\prime}\,,\bm{\mathfrak{D}}^{\prime\prime}\,]^{22}_{\mathfrak{X}}]^{22}_{\mathcal{O}}+(\textrm{cyclic}\
\textrm{perm.})\ =\ 0\,.$ (A.9)

We should note that $\mathcal{O}$ is not commutative to $Q$ and either $\eta$
or $\bar{\eta}$ . Thus, for example,

$\displaystyle[\,\bm{Q}\,,[\,\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}^{\prime}}\,]^{22}_{\Xi}\,]\
=$ $\displaystyle\
[\,[\,\bm{Q}\,,\bm{\mathfrak{D}}\,]\,,\bm{\mathfrak{D}}^{\prime}\,]^{22}_{\Xi}+(-1)^{|\mathfrak{D}|+1}[\,\bm{\mathfrak{D}}\,,[\,\bm{Q}\,,\bm{\mathfrak{D}}^{\prime}\,]\,]^{22}_{\Xi}+(-1)^{|\mathfrak{D}|}[\,\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}\,]^{22}_{X}\,,$
(A.10)
$\displaystyle[\,\bm{\eta}\,,[\,\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}^{\prime}}\,]^{22}_{\Xi}\,]\
=$ $\displaystyle\
[\,[\,\bm{\eta}\,,\bm{\mathfrak{D}}\,]\,,\bm{\mathfrak{D}}^{\prime}\,]^{22}_{\Xi}+(-1)^{|\mathfrak{D}|+1}[\,\bm{\mathfrak{D}}\,,[\,\bm{\eta}\,,\bm{\mathfrak{D}}^{\prime}\,]\,]^{22}_{\Xi}+(-1)^{|\mathfrak{D}|}[\,\bm{\mathfrak{D}}\,,\bm{\mathfrak{D}}^{\prime}\,]^{22}\,.$
(A.11)

##  Appendix B Expansion of $\bm{B}_{3}^{(\bullet,\bullet)}(s,\bar{s})$,
$\bm{\lambda}_{3}^{(\bullet,\bullet)}(s,\bar{s})$ and
$\bar{\bm{\lambda}}_{3}^{(\bullet,\bullet)}(s,\bar{s})$

Figure 4: The flow of how each component of 3-string products are determined
1. Figure 5: The flow of how each component of 3-string products are
determined 2.

In section 3, we show that how the generating functionals of 3-string and
gauge products with specific picture number are determined. In this appendix,
we illustrate the flow to be determined the 3-string (gauge) products with
specific cyclic Ramond numbers, which are given as coefficients further
expanded by $s$ and $\bar{s}$ :

$\displaystyle\bm{B}_{3}^{(0,0)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{3}^{(0,0)}\mid^{(4,4)}+s\bm{B}_{3}^{(0,0)}\mid^{(2,4)}+\bar{s}\bm{B}_{3}^{(0,0)}\mid^{(4,2)}$
$\displaystyle\
+s^{2}\bm{B}_{3}^{(0,0)}\mid^{(0,4)}+s\bar{s}\bm{B}_{3}^{(0,0)}\mid^{(2,2)}+\bar{s}^{2}\bm{B}_{3}^{(0,0)}\mid^{(4,0)}$
$\displaystyle\
+s^{2}\bar{s}\bm{B}_{3}^{(0,0)}\mid^{(0,2)}+s\bar{s}^{2}\bm{B}_{3}^{(0,0)}\mid^{(2,0)}+s^{2}\bar{s}^{2}\bm{B}_{3}^{(0,0)}\mid^{(0,0)}\,,$
(B.1) $\displaystyle\bm{B}_{3}^{(1,0)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{3}^{(1,0)}\mid^{(2,4)}+s\bm{B}_{3}^{(1,0)}\mid^{(0,4)}+\bar{s}\bm{B}_{3}^{(1,0)}\mid^{(2,2)}$
$\displaystyle\
+s\bar{s}\bm{B}_{3}^{(1,0)}\mid^{(0,2)}+\bar{s}^{2}\bm{B}_{3}^{(1,0)}\mid^{(2,0)}+s\bar{s}^{2}\bm{B}_{3}^{(1,0)}\mid^{(0,0)}\,,$
(B.2) $\displaystyle\bm{B}_{3}^{(0,1)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{3}^{(0,1)}\mid^{(4,2)}+s\bm{B}_{3}^{(0,1)}\mid^{(2,2)}+\bar{s}\bm{B}_{3}^{(0,1)}\mid^{(4,0)}$
$\displaystyle\
+s^{2}\bm{B}_{3}^{(0,1)}\mid^{(0,2)}+s\bar{s}\bm{B}_{3}^{(0,1)}\mid^{(2,0)}+s^{2}\bar{s}\bm{B}_{3}^{(0,1)}\mid^{(0,0)}\,,$
(B.3) $\displaystyle\bm{B}_{3}^{(2,0)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{3}^{(2,0)}\mid^{(0,4)}+\bar{s}\bm{B}_{3}^{(2,0)}\mid^{(0,2)}+\bar{s}^{2}\bm{B}_{3}^{(2,0)}\mid^{(0,0)}\,,$
(B.4) $\displaystyle\bm{B}_{3}^{(1,1)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{3}^{(1,1)}\mid^{(2,2)}+s\bm{B}_{3}^{(1,1)}\mid^{(0,2)}+\bar{s}\bm{B}_{3}^{(1,1)}\mid^{(2,0)}+s\bar{s}\bm{B}_{3}^{(1,1)}\mid^{(0,0)}\,,$
(B.5) $\displaystyle\bm{B}_{3}^{(0,2)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{3}^{(0,2)}\mid^{(4,0)}+s\bm{B}_{3}^{(0,2)}\mid^{(2,0)}+s^{2}\bm{B}_{3}^{(0,2)}\mid^{(0,0)}\,,$
(B.6) $\displaystyle\bm{B}_{3}^{(2,1)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{3}^{(2,1)}\mid^{(0,2)}+\bar{s}\bm{B}_{3}^{(2,1)}\mid^{(0,0)}\,,$ (B.7)
$\displaystyle\bm{B}_{3}^{(1,2)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{3}^{(1,2)}\mid^{(2,0)}+s\bm{B}_{3}^{(1,2)}\mid^{(0,0)}\,,$ (B.8)
$\displaystyle\bm{B}_{3}^{(2,2)}(s,\bar{s})\ =$ $\displaystyle\
\bm{B}_{3}^{(2,2)}\mid^{(0,0)}\,,$ (B.9)

and

$\displaystyle\bm{\lambda}_{3}^{(1,0)}(s,\bar{s})\ =$ $\displaystyle\
\bm{\lambda}_{3}^{(1,0)}\mid^{(2,4)}+s\bm{\lambda}_{3}^{(1,0)}\mid^{(0,4)}+\bar{s}\bm{\lambda}_{3}^{(1,0)}\mid^{(2,2)}$
$\displaystyle\
+s\bar{s}\bm{\lambda}_{3}^{(1,0)}\mid^{(0,2)}+\bar{s}^{2}\bm{\lambda}_{3}^{(1,0)}\mid^{(2,0)}+s\bar{s}^{2}\bm{\lambda}_{3}^{(1,0)}\mid^{(0,0)}\,,$
(B.10) $\displaystyle\bm{\lambda}^{(2,0)}(s,\bar{s})\ =$ $\displaystyle\
\bm{\lambda}_{3}^{(2,0)}\mid^{(0,4)}+\bar{s}\bm{\lambda}_{3}^{(2,0)}\mid^{(0,2)}+\bar{s}^{2}\bm{\lambda}_{3}^{(2,0)}\mid^{(0,0)}\,,$
(B.11) $\displaystyle\bm{\lambda}_{3}^{(1,1)}(s,\bar{s})\ =$ $\displaystyle\
\bm{\lambda}_{3}^{(1,1)}\mid^{(2,2)}+s\bm{\lambda}_{3}^{(1,1)}\mid^{(0,2)}+\bar{s}\bm{\lambda}_{3}^{(1,1)}\mid^{(2,0)}+s\bar{s}\bm{\lambda}_{3}^{(1,1)}\mid^{(0,0)}\,,$
(B.12) $\displaystyle\bm{\lambda}_{3}^{(2,1)}(s,\bar{s})\ =$ $\displaystyle\
\bm{\lambda}_{3}^{(2,1)}\mid^{(0,2)}+\bar{s}\bm{\lambda}_{3}^{(2,1)}\mid^{(0,0)}\,,$
(B.13) $\displaystyle\bm{\lambda}_{3}^{(1,2)}(s,\bar{s})\ =$ $\displaystyle\
\bm{\lambda}_{3}^{(1,2)}\mid^{(2,0)}+s\bm{\lambda}_{3}^{(1,2)}\mid^{(0,0)}\,,$
(B.14) $\displaystyle\bm{\lambda}_{3}^{(2,2)}(s,\bar{s})\ =$ $\displaystyle\
\bm{\lambda}_{3}^{(2,2)}\mid^{(0,0)}\,,$ (B.15)
$\displaystyle\bar{\bm{\lambda}}_{3}^{(0,1)}(s,\bar{s})\ =$ $\displaystyle\
\bar{\bm{\lambda}}_{3}^{(0,1)}\mid^{(4,2)}+s\bar{\bm{\lambda}}_{3}^{(0,1)}\mid^{(2,2)}+\bar{s}\bar{\bm{\lambda}}_{3}^{(0,1)}\mid^{(4,0)}$
$\displaystyle\
+s^{2}\bar{\bm{\lambda}}_{3}^{(0,1)}\mid^{(0,2)}+s\bar{s}\bar{\bm{\lambda}}_{3}^{(0,1)}\mid^{(2,0)}+s\bar{s}^{2}\bar{\bm{\lambda}}_{3}^{(0,1)}\mid^{(0,0)}\,,$
(B.16) $\displaystyle\bar{\bm{\lambda}}_{3}^{(1,1)}(s,\bar{s})\ =$
$\displaystyle\
\bar{\bm{\lambda}}_{3}^{(1,1)}\mid^{(2,2)}+s\bar{\bm{\lambda}}_{3}^{(1,1)}\mid^{(0,2)}+\bar{s}\bar{\bm{\lambda}}_{3}^{(1,1)}\mid^{(2,0)}+s\bar{s}\bar{\bm{\lambda}}_{3}^{(1,1)}\mid^{(0,0)}\,,$
(B.17) $\displaystyle\bar{\bm{\lambda}}^{(0,2)}(s,\bar{s})\ =$ $\displaystyle\
\bar{\bm{\lambda}}_{3}^{(0,2)}\mid^{(4,0)}+s\bar{\bm{\lambda}}_{3}^{(0,2)}\mid^{(2,0)}+s^{2}\bar{\bm{\lambda}}_{3}^{(0,2)}\mid^{(0,0)}\,,$
(B.18) $\displaystyle\bar{\bm{\lambda}}_{3}^{(2,1)}(s,\bar{s})\ =$
$\displaystyle\
\bar{\bm{\lambda}}_{3}^{(2,1)}\mid^{(0,2)}+\bar{s}\bar{\bm{\lambda}}_{3}^{(2,1)}\mid^{(0,0)}\,,$
(B.19) $\displaystyle\bar{\bm{\lambda}}_{3}^{(1,2)}(s,\bar{s})\ =$
$\displaystyle\
\bar{\bm{\lambda}}_{3}^{(1,2)}\mid^{(2,0)}+s\bar{\bm{\lambda}}_{3}^{(1,2)}\mid^{(0,0)}\,,$
(B.20) $\displaystyle\bar{\bm{\lambda}}_{3}^{(2,2)}(s,\bar{s})\ =$
$\displaystyle\ \bar{\bm{\lambda}}_{3}^{(2,2)}\mid^{(0,0)}\,.$ (B.21)

The flow of each product is shown in Figs. 4 and 5. The 3-products at the end
of flows are used to make the $L_{\infty}$ structure for writing down the
action, which are those with no picture number deficit.

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