aainstitutetext: NHETC and Department of Physics and Astronomy, Rutgers
University, 126 Frelinghuysen Rd., Piscataway NJ 08855, USA

# Algebraic Structures In Closed Superstring Field Theory, Homotopy Transfer
And Effective Actions

Ranveer Kumar Singh<EMAIL_ADDRESS>
###### Abstract

A consistent action for heterotic and type II superstring field theory was
recently proposed by Sen. We give an algebraic formulation of this action in
terms of certain twisted $L_{\infty}$-algebra. We further show that Sen’s
Wilsonian effective superstring field action can be obtained using homotopy
transfer and the effective theory also possesses the algebraic structure of a
twisted $L_{\infty}$-algebra.

##  1 Introduction

String field theory is an attempt to formulate a non-perturbative description
of string theory. Consistent construction of string field theories exists for
all string theories: bosonic open and closed strings Siegel:1984ogw ;
Kaku1974FieldTO ; Witten:1985cc ; Zwiebach:1992ie ; Zwiebach:1997fe ,
heterotic strings and type II strings Sen:2015uaa ; Sen:2015hha ; Sen:2014dqa
and open-closed superstrings FarooghMoosavian:2019yke , see reviews
Erbin:2021smf ; Erler:2019loq ; deLacroix:2017lif ; Sen-Zweibach  for a
comprehensive introduction111See also Maccaferri:2023vns  for a quick exposure
to the recent results in string field theory. Most of these string field
theories are succinctly described in terms of algebraic structures called
homotopy algebras first introduced in mathematics by J. Stasheff
58e41a3e-2db4-3e2f-8edb-8effbf4ad0ce . The two basic structures describing
open and closed string field theories are $A_{\infty}$ and
$L_{\infty}$-algebras Zwiebach:1992ie ; Zwiebach:1997fe ; Gaberdiel:1997ia ;
Kajiura:2003ax ; Kajiura:2004xu ; Markl:1997bj ; Kajiura:2006mt ;
Kajiura:2001ng . The basic structure for these algebras is the existence of
$n$-multilinear maps, called $n$-ary products (or string products) and denoted
by $\\{m_{n}\\}_{n\geq 1}$ for $A_{\infty}$ algebras and
$\\{\ell_{n}\\}_{n\geq 1}$ for $L_{\infty}$-algebras, which satisfy a strong
homotopy Jacobi identity which generalises associativity. String vertices in
closed and open string theories satisfy similar identity, called the main
identity in string field theory literature. These string vertices can then be
used to define string products which then give rise to homotopy algebras.

Formulation of string field theory opens up possibilities to apply the methods
of quantum field theory to resolve various problems in perturbative string
theory. In particular, string field theory has been used to regularise IR
divergences in string theory and compute D-instanton amplitudes Sen:2014pia ;
Sen:2015hia ; Sen:2016qap ; Sen:2019qqg ; Sen:2020eck ; Sen:2021jbr ;
Sen:2021qdk ; Sen:2021tpp ; Alexandrov:2021shf . Another important application
is to write down Wilsonian effective superstring actions by integrating out
heavy modes to get an effective action for light fields. By integrating out
massive modes in open (super)string field theory, one can extract the
(super)Yang-Mills action Berkovits:2003ny ; Asada:2017ykq . It turns out that
effective actions can be understood in terms of homotopy transfer of homotopy
algebras. For string field theories based on $A_{\infty}$ and
$L_{\infty}$-algebras, a beautiful demonstration of this procedure was given
in Erbin:2020eyc  (see also Arvanitakis:2020rrk ; Arvanitakis:2021ecw ).A
broad brush sketch of this story goes as follows: Let $\mathcal{H}$ be the
(super)string Hilbert space.

  1. 1.

One begins by formulating the $A_{\infty}$ ($L_{\infty}$)-algebra underlying
the theory in terms of tensor colagebra $T\mathcal{H}=\bigoplus_{n\geq
0}\mathcal{H}^{\otimes n}$ (symmetrized tensor coalgebra
$S\mathcal{H}=\bigoplus_{n\geq 0}\mathcal{H}^{\wedge n}$), see Appendix A for
definitions. The string products $m_{n}:\mathcal{H}^{\otimes n}\to\mathcal{H}$
($\ell_{n}:\mathcal{H}^{\wedge n}\to\mathcal{H}$) on $\mathcal{H}$ then
extends 222See (LABEL:eq:coderbmci) for the construction of this extension to
$S\mathcal{H}$. to coderivations $\bm{m}_{n}:T\mathcal{H}\to T\mathcal{H}$
($\bm{\ell}_{n}:S\mathcal{H}\to S\mathcal{H}$). The strong homotopy Jacobi
identity of the $n$-ary product is then equivalent to the total $n$-ary
product $\bm{m}=\sum_{n\geq 1}\bm{m}_{n}$ ($\bm{\ell}=\sum_{n\geq
1}\bm{\ell}_{n}$) being nilpotent: $\bm{m}^{2}=0$ ($\bm{\ell}^{2}=0$).

  2. 2.

The free theory is described by a (symmetrized) tensor coalgebra with only the
1-ary product, which is nilpotent. In string field theory, this nilpotent
operator is the BRST charge $m_{1}:=Q_{B}$ ($\ell_{1}:=Q_{B}$).

  3. 3.

Suppose $P$ is the projection operator such that the Wilsonian effective
action is obtained by integrating out states not in the image of $P$. Then the
Wilsonian effective free theory is trivially obtained by mapping the BRST
charge to its compression 333The compression of a linear operator $T$ on a
Hilbert space $H$ to a subspace $K$ is the operator $P_{K}T|_{K}:K\rightarrow
K,$ where $P_{K}:H\rightarrow K$ is the orthogonal projection onto $K$.
$PQ_{B}|_{P\mathcal{H}}$. The free theory and the free effective theory can be
described by a strong deformation retract (SDR)
$(T\mathcal{H},\bm{Q_{B}})\to(TP\mathcal{H},\mathbf{P}\bm{Q_{B}}|_{TP\mathcal{H}})$
in case of $A_{\infty}$-algebra and
$(S\mathcal{H},\bm{Q_{B}})\to(SP\mathcal{H},\mathbf{P}\bm{Q_{B}}|_{SP\mathcal{H}})$
in case of $L_{\infty}$-algebra. An important axiom for an SDR is the
nilpotency of the operators:

$\begin{split}&\bm{Q_{B}}^{2}=0,\quad(\mathbf{P}\bm{Q_{B}}|_{TP\mathcal{H}})^{2}=0\leavevmode\nobreak\
,\\\
&\bm{Q_{B}}^{2}=0,\quad(\mathbf{P}\bm{Q_{B}}|_{SP\mathcal{H}})^{2}=0\leavevmode\nobreak\
.\end{split}$ (1)

  4. 4.

The interacting theory is obtained from the free theory by introducing a
perturbation $\delta\bm{m}=\sum_{n\geq 2}\bm{m}_{n}$ for $A_{\infty}$-algebra
and $\delta\bm{\ell}=\sum_{n\geq 2}\bm{\ell}_{n}$ for $L_{\infty}$-algebra.
The main step is the application of the homological perturbation lemma which
gives a new SDR if the perturbation satisfies certain conditions, see
Erbin:2020eyc  for the precise statement. Thus we obtain a new SDR
$(T\mathcal{H},\bm{Q_{B}}+\delta\bm{m})\to(TP\mathcal{H},\mathbf{P}\bm{Q_{B}}|_{TP\mathcal{H}}+\delta\widetilde{\bm{m}})$
in case of $A_{\infty}$-algebra and
$(S\mathcal{H},\bm{Q_{B}}+\delta\bm{\ell})\to(SP\mathcal{H},\mathbf{P}\bm{Q_{B}}|_{SP\mathcal{H}}+\delta\widetilde{\bm{\ell}})$
in case of $L_{\infty}$-algebra. The $n$-ary products
$\\{\widetilde{m}_{n}\\}_{n\geq 1},\\{\widetilde{\ell}_{n}\\}_{n\geq 1}$ for
the effective theory is given by the expansion

$\begin{split}&\mathbf{P}\bm{Q_{B}}|_{TP\mathcal{H}}+\delta\widetilde{\bm{m}}=\sum_{n\geq
1}\widetilde{\bm{m}}_{n}\leavevmode\nobreak\ ,\\\
&\mathbf{P}\bm{Q_{B}}|_{SP\mathcal{H}}+\delta\widetilde{\bm{\ell}}=\sum_{n\geq
1}\widetilde{\bm{\ell}}_{n}.\end{split}$ (2)

They satisfy the strong homotopy Jacobi identity by the axioms of an SDR. The
effective action can then be written using the effective string products. This
procedure of constructing new homotopy algebra from old is called homotopy
transfer.

The aim of this paper is to demonstrate the same for heterotic and type II
superstring field theory. The procedure of integrating out heavy modes to
obtain Wilsonian effective action for light modes in superstring field theory
was given in Sen:2016qap . Algebraic construction of various superstring field
theories based on homotopy algebras appears in Kunitomo:2021wiz ;
Kunitomo:2022qqp ; Kunitomo:2019glq ; Erler:2017onq ; Konopka:2016grr ;
Erler:2016ybs ; Kunitomo:2015usa . It turns out that the formulations in these
references do not capture Sen’s formulation of the type II and heterotic
superstring action and hence is not very useful to demonstrate the
applicability of homotopy transfer to obtain Sen’s Wilsonian effective
actions.

In this paper, we define the notion of a $g$-twisted $L_{\infty}$-algebra,
where $g:V^{\star}\to V$ is a linear map and $V^{\star}$ is the dual of the
underlying vector space $V$. We then show that the description of heterotic
and type II superstring theory given in Sen:2015uaa ; deLacroix:2017lif ;
FarooghMoosavian:2019yke  is naturally captured by a $\mathcal{G}$-twisted
$L_{\infty}$-algebra where $\mathcal{G}$ is given in terms of the zero-mode of
the picture changing operator. We recast the discussion of Wilsonian effective
action in terms of an SDR and then apply a slightly modified version of the
homological perturbation lemma to show that Sen’s Wilsonian effective action
in Sen:2016qap  can be obtained using homotopy transfer.

The paper is organised as follows: in Section 2, more specifically in Section
2.1 and Section 2.2 we review the basics of closed superstring field theory.
In Section 2.3 and Section 2.4, we define the notion of $g$-twisted
$L_{\infty}$-algebra and show that the closed superstring field theory can be
described in terms of these homotopy algebras. In Section 3 we review
Wilsonian effective theory and perform some preliminary calculations. Finally
in Section 4 we show that the effective string products can be obtained using
homotopy transfer. In Appendix A we recall some basic notions about
(symmetrised) tensor coalgebras and recast $L_{\infty}$-algebras in the
language of symmetrised tensor coalgebras. In Appendix B we review the
definition of an SDR and state the version of homological perturbation lemma
used in this paper to obtain effective string products.

##  2 Closed superstring field theory as an $L_{\infty}$-algebra

###  2.1 The Hilbert space of superstring theory

Let us recall the basic ingredients of closed superstring theory. We follow
deLacroix:2017lif ; Sen:2015uaa  for this exposition.

Let $\mathcal{H}_{T}$ be the Hilbert space of GSO even states in the small
Hilbert space Friedan:1985ey ; Friedan:1985ge  of the matter-ghost CFT with
arbitrary ghost and picture number satisfying

$b_{0}^{-}|s\rangle=0,\quad
L_{0}^{-}|s\rangle=0,\quad|s\rangle\in\mathcal{H}_{T}\leavevmode\nobreak\ ,$
(3)

where we have defined

$b_{0}^{\pm}=b_{0}\pm\bar{b}_{0},\quad L_{0}^{\pm}=L_{0}\pm\bar{L}_{0},\quad
c_{0}^{\pm}=\frac{1}{2}\left(c_{0}\pm\bar{c}_{0}\right).$ (4)

$\mathcal{H}_{T}$ decomposes into NS and R sectors for the heterotic string
and into four sectors for type II:

$\begin{array}[]{ll}\mathcal{H}_{T}=\mathcal{H}_{\mathrm{NS}}\oplus\mathcal{H}_{\mathrm{R}},&\quad\text{
Heterotic },\\\
\mathcal{H}_{T}=\mathcal{H}_{\mathrm{NSNS}}\oplus\mathcal{H}_{\mathrm{NSR}}\oplus\mathcal{H}_{\mathrm{RNS}}\oplus\mathcal{H}_{\mathrm{RR}},&\quad\text{
Type II }.\end{array}$ (5)

For heterotic string, let $\mathcal{H}_{p}$ be the subspace of
$\mathcal{H}_{T}$ with picture number $p$. For NS sector, $p\in\mathbb{Z}$ and
for R sector $p\in\mathbb{Z}+\frac{1}{2}$. For type II, let
$\mathcal{H}_{p,q}$ denote the subspace of $\mathcal{H}_{T}$ with left moving
picture number $p$ and right-moving picture number $q$. Define the spaces

$\begin{split}&\widehat{\mathcal{H}}_{T}=\begin{cases}\mathcal{H}_{-1}\oplus\mathcal{H}_{-\frac{1}{2}},&\quad\text{Heterotic},\\\
\mathcal{H}_{-1,-1}\oplus\mathcal{H}_{-\frac{1}{2},-1}\oplus\mathcal{H}_{-1,-\frac{1}{2}}\oplus\mathcal{H}_{-\frac{1}{2},-\frac{1}{2}},&\quad\text{Type
II},\end{cases}\\\
&\widetilde{\mathcal{H}}_{T}=\begin{cases}\mathcal{H}_{-1}\oplus\mathcal{H}_{-\frac{3}{2}},&\quad\text{Heterotic},\\\
\mathcal{H}_{-1,-1}\oplus\mathcal{H}_{-\frac{3}{2},-1}\oplus\mathcal{H}_{-1,-\frac{3}{2}}\oplus\mathcal{H}_{-\frac{3}{2},-\frac{3}{2}},&\quad\text{Type
II}.\end{cases}\end{split}$ (6)

We will denote the Hilbert space of the theory by

$\mathcal{H}_{T}:=\widetilde{\mathcal{H}}_{T}\oplus\widehat{\mathcal{H}}_{T}.$
(7)

Let
$\left\\{\left|\varphi_{r}\right\rangle\right\\}\subset\widehat{\mathcal{H}}_{T},\left\\{\left|\varphi_{r}^{c}\right\rangle\right\\}\subset\widetilde{\mathcal{H}}_{T}$
be a basis of $\widehat{\mathcal{H}}_{T}$ and $\widetilde{\mathcal{H}}_{T}$
satisfying

$\left\langle\varphi_{r}^{c}\left|c_{0}^{-}\right|\varphi_{s}\right\rangle=\delta_{rs},\quad\left\langle\varphi_{s}\left|c_{0}^{-}\right|\varphi_{r}^{c}\right\rangle=\delta_{rs}\leavevmode\nobreak\
.$ (8)

Here $\langle\cdot|\cdot\rangle$ is the BPZ inner product and the $c_{0}^{-}$
insertion is needed for a nonzero inner product because of the anticommutator
$\\{b_{0}^{-},c_{0}^{-}\\}=1$ and the condition (3). The first relation in (8)
implies the second relation in (8) and also a completeness relation

$\sum_{r}\left|\varphi_{r}\rangle\langle\varphi_{r}^{c}\right|c_{0}^{-}=\mathds{1},\quad\sum_{r}\left|\varphi_{r}^{c}\rangle\langle\varphi_{r}\right|c_{0}^{-}=\mathds{1}.$
(9)

The basis states $\varphi_{r},\varphi_{r}^{c}$ carry nontrivial grassmann
parities which we shall denote by $(-1)^{\gamma_{r}}$ and
$(-1)^{\gamma_{r}^{c}}$ respectively. In the NS sector of the heterotic string
and NSNS and RR sector of type II, the grassmann parity of $\varphi_{r}$ and
$\varphi_{r}^{c}$ is odd and even respectively if the ghost number of
$\varphi_{r}$ and $\varphi_{r}^{c}$ is odd and even respectively. In the R
sector of the heterotic string and NSR and RNS sector of type II, the
grassmann parity of $\varphi_{r}$ and $\varphi_{r}^{c}$ is odd and even
respectively if the ghost number of $\varphi_{r}$ and $\varphi_{r}^{c}$ is
even and odd respectively. Moreover one has

$(-1)^{\gamma_{r}+\gamma_{r}^{c}}=-1\leavevmode\nobreak\ .$ (10)

Introduce the zero modes of the PCOs:

$\mathcal{X}_{0}\equiv\oint\frac{dz}{z}\mathcal{X}(z),\quad\bar{\mathcal{X}}_{0}=\oint\frac{d\bar{z}}{\bar{z}}\bar{\mathcal{X}}(\bar{z}),$
(11)

where the contour integral is normalised such that

$\oint_{C}\frac{dz}{z}=1,\quad\oint_{\bar{C}}\frac{d\bar{z}}{\bar{z}}=1\leavevmode\nobreak\
,$ (12)

and $C,\bar{C}$ is a contour around $z=0,\bar{z}=0$ respectively, see
deLacroix:2017lif  for the definition of the operator $\mathcal{X}$. Define
the operator $\mathcal{G}$ as

$\mathcal{G}=\begin{cases}\mathds{1}\oplus\mathcal{X}_{0},&\mathcal{H}_{T}=\mathcal{H}_{\mathrm{NS}}\oplus\mathcal{H}_{\mathrm{R}}\leavevmode\nobreak\
,\\\
\mathds{1}\oplus\mathcal{X}_{0}\oplus\bar{\mathcal{X}}_{0}\oplus\mathcal{X}_{0}\bar{\mathcal{X}}_{0},&\mathcal{H}_{T}=\mathcal{H}_{\mathrm{NSNS}}\oplus\mathcal{H}_{\mathrm{RNS}}\oplus\mathcal{H}_{\mathrm{NSR}}\oplus\mathcal{H}_{\mathrm{RR}}.\end{cases}$
(13)

on the heterotic and type II Hilbert space. One has that

$\left[\mathcal{G},L_{0}^{\pm}\right]=0,\quad\left[\mathcal{G},b_{0}^{\pm}\right]=0,\quad\left[\mathcal{G},Q_{B}\right]=0\text{.
}$ (14)

It also satisfies

$\langle\widetilde{\Psi}|c_{0}^{-}\mathcal{G}|\widetilde{\Psi}^{\prime}\rangle=\langle\mathcal{G}\widetilde{\Psi}|c_{0}^{-}|\widetilde{\Psi}^{\prime}\rangle\leavevmode\nobreak\
,\quad|\widetilde{\Psi}\rangle,|\widetilde{\Psi}^{\prime}\rangle\in\widetilde{\mathcal{H}}_{T}.$
(15)

###  2.2 Superstring field theory action

A string field is a grassmann even element of
$|\Psi\rangle\in\widehat{\mathcal{H}}_{T}$. We also need another non-dynamical
grassmann even string field
$|\widetilde{\Psi}\rangle\in\widetilde{\mathcal{H}}_{T}$ to write down the
action. The 1PI effective action is written as

$S_{1PI}=\frac{1}{g_{s}^{2}}\left[-\frac{1}{2}\left\langle\widetilde{\Psi}\left|c_{0}^{-}Q_{B}\mathcal{G}\right|\widetilde{\Psi}\right\rangle+\left\langle\widetilde{\Psi}\left|c_{0}^{-}Q_{B}\right|\Psi\right\rangle+\sum_{n=1}^{\infty}\frac{1}{n!}\left\\{\Psi^{n}\right\\}\right]\leavevmode\nobreak\
,$ (16)

where $\left\\{\Psi^{n}\right\\}$ is the $n$-string vertex summed over all 1PI
Feynman diagrams, see deLacroix:2017lif ; Sen:2015uaa  for precise definition
in terms of integral over moduli space of Riemann surfaces. This action has
infinite dimensional gauge invariance given by

$|\delta\Psi\rangle=Q_{B}|\Lambda\rangle+\sum_{n=0}^{\infty}\frac{1}{n!}\mathcal{G}\left[\Psi^{n}\Lambda\right],\quad|\delta\widetilde{\Psi}\rangle=Q_{B}|\widetilde{\Lambda}\rangle+\sum_{n=0}^{\infty}\frac{1}{n!}\left[\Psi^{n}\Lambda\right],$
(17)

where
$\left[\Psi^{n}\Lambda\right]:=[\Psi\Psi\dots\Psi\Lambda]\in\widetilde{\mathcal{H}}_{T}$
is defined by the relation

$\left\langle\Psi\left|c_{0}^{-}\right|[\Psi_{1}\Psi_{2}\dots\Psi_{n}]\right\rangle=\\{\Psi\Psi_{1}\Psi_{2}\dots\Psi_{n}\\},\quad\forall\leavevmode\nobreak\
\leavevmode\nobreak\ \Psi\in\widehat{\mathcal{H}}_{T}.$ (18)

We now list some of the properties of the $n$-string vertex
$\\{\Psi_{1}\Psi_{2}\dots\Psi_{n}\\}$ and the state
$[\Psi_{1}\Psi_{2}\dots\Psi_{n}]$.

  1. 1.

Ghost number anomaly implies that the a correlation function of vertex
operators on a genus $g$ Riemann surface is nonzero only if the total ghost
number of vertex operators in the correlator is $6-6g$. This implies that the
$n$-string vertex $\\{\Psi_{1}\Psi_{2}\dots\Psi_{n}\\}$ vanishes unless
(deLacroix:2017lif, , Eq. (3.7))

$\sum_{i=1}^{n}(\text{gh}(\Psi_{i})-2)=0\leavevmode\nobreak\ ,$ (19)

where $\text{gh}(\Psi_{i})$ denotes the ghost number of $\Psi_{i}$. This means
that on a genus $g$ surface

$\text{gh}\left([\Psi_{1}\Psi_{2}\dots\Psi_{n}]\right)=3-6g+\sum_{i=1}^{n}(\text{gh}(\Psi_{i})-2).$
(20)

  2. 2.

The grassmann parity of $[\Psi_{1}\Psi_{2}\dots\Psi_{n}]$ is opposite to the
sum of the grassmann parities of $\Psi_{i}$’s.

  3. 3.

The following relations hold Sen:2015uaa :

$\begin{gathered}\left\\{\Psi_{1}\Psi_{2}\cdots\Psi_{i-1}\Psi_{i+1}\Psi_{i}\Psi_{i+2}\cdots\Psi_{n}\right\\}=(-1)^{\gamma_{i}\gamma_{i+1}}\left\\{\Psi_{1}\Psi_{2}\cdots\Psi_{n}\right\\}\leavevmode\nobreak\
,\\\
{\left[\Psi_{1}\cdots\Psi_{i-1}\Psi_{i+1}\Psi_{i}\Psi_{i+2}\cdots\Psi_{n}\right]=(-1)^{\gamma_{i}\gamma_{i+1}}\left[\Psi_{1}\cdots\Psi_{n}\right]\leavevmode\nobreak\
,}\end{gathered}$ (21)

where $\gamma_{i}$ is the grassmannality of $\left|\Psi_{i}\right\rangle$.

  4. 4.

The following identity, called the main identity, is satisfied Sen:2015uaa :

$\displaystyle\sum_{i=1}^{n}(-1)^{\gamma_{1}+\dots+\gamma_{i-1}}\left\\{\Psi_{1}\cdots\Psi_{i-1}\left(Q_{B}\Psi_{i}\right)\Psi_{i+1}\cdots\Psi_{n}\right\\}$
(22) $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{\begin{subarray}{c}\ell,k\geq 2\\\
\ell+k=n\end{subarray}}\sum_{\begin{subarray}{c}\left\\{i_{a};a=1,\dots,\ell\right\\},\left\\{j_{b};b=1,\dots,k\right\\}\\\
\\{ia\\}\cup\left\\{j_{b}\right\\}=\\{1,\dots,n\\}\end{subarray}}\sigma\left(\left\\{i_{a}\right\\},\left\\{j_{b}\right\\}\right)\left\\{\mathcal{G}\left[\Psi_{j_{1}}\cdots\Psi_{j_{k}}\right]\Psi_{i_{1}}\cdots\Psi_{i_{\ell}}\right\\}\leavevmode\nobreak\
,$

and

$\displaystyle
Q_{B}\left[\Psi_{1}\cdots\Psi_{n}\right]+\sum_{i=1}^{n}(-1)^{\gamma_{1}+\dots+\gamma_{i-1}}\left[\Psi_{1}\cdots\Psi_{i-1}\left(Q_{B}\Psi_{i}\right)\Psi_{i+1}\cdots\Psi_{n}\right]$
(23) $\displaystyle=$ $\displaystyle-\sum_{\begin{subarray}{c}\ell,k\geq 2\\\
\ell+k=n\end{subarray}}\sum_{\begin{subarray}{c}\left\\{i_{a};a=1,\dots,\ell\right\\},\left\\{j_{b};b=1,\dots,k\right\\}\\\
\left\\{i_{a}\right\\}\cup\left\\{j_{b}\right\\}=\\{1,\dots,n\\}\end{subarray}}\sigma\left(\left\\{i_{a}\right\\},\left\\{j_{b}\right\\}\right)\left[\mathcal{G}\left[\Psi_{j_{1}}\cdots\Psi_{j_{k}}\right]\Psi_{i_{1}}\cdots\Psi_{i_{\ell}}\right]\leavevmode\nobreak\
,$

where $\sigma\left(\left\\{i_{a}\right\\},\left\\{j_{b}\right\\}\right)$ is
the sign that we get while rearranging

$\Psi_{1},\cdots\Psi_{n}\to\Psi_{j_{1}},\cdots\Psi_{j_{k}}\Psi_{i_{1}},\cdots\Psi_{i_{\ell}},\leavevmode\nobreak\
.$ (24)

Note in the inner sum above, the sum is only over partitions without
permutations over the individual subpartitions. For example for $n=4$, the sum
is over

$\begin{split}\\{\\{1,2\\},\\{3,4\\}\\},\\{\\{1,3\\},\\{2,4\\}\\},\\{\\{1,2\\},\\{3,4\\}\\},\\{\\{1\\},\\{2,3,4\\}\\},\\{\\{2\\},\\\
\\{1,3,4\\}\\},\\{\\{3\\},\\{1,2,4\\}\\},\\{\\{4\\},\\{1,2,3\\}\\},\end{split}$
(25)

see the notion of unshuffles in (30) for a precise definition.

  5. 5.

The following relation holds Sen:2015uaa :

$\begin{split}\left\\{\Psi_{1}\cdots\Psi_{k}\mathcal{G}\left[\Psi^{\prime}_{1}\cdots\Psi^{\prime}_{\ell}\right]\right\\}&=(-1)^{\gamma+\gamma^{\prime}+\gamma\gamma^{\prime}}\left\\{\Psi^{\prime}_{1}\cdots\Psi^{\prime}_{\ell}\mathcal{G}\left[\Psi_{1}\cdots\Psi_{k}\right]\right\\}\leavevmode\nobreak\
,\end{split}$ (26)

where $\gamma$ and $\gamma^{\prime}$ are the total grassmannalities of
$\Psi_{1},\cdots\Psi_{k}$ and $\Psi^{\prime}_{1},\cdots\Psi^{\prime}_{\ell}$
respectively.

The equation of motion is given by

$\displaystyle
Q_{B}(|\Psi\rangle-\mathcal{G}|\widetilde{\Psi}\rangle)=0\leavevmode\nobreak\
,$ (27) $\displaystyle
Q_{B}|\widetilde{\Psi}\rangle+\sum_{n=1}^{\infty}\frac{1}{(n-1)!}\left[\Psi^{n-1}\right]=0\leavevmode\nobreak\
.$

Acting on the second equation by $\mathcal{G}$ from the left and adding it to
the first equation we get

$Q_{B}|\Psi\rangle+\sum_{n=1}^{\infty}\frac{1}{(n-1)!}\mathcal{G}\left[\Psi^{n-1}\right]=0\leavevmode\nobreak\
.$ (28)

For classical action, the sum starts from $n=3$ since the one and two point
function on sphere vanishes for all SCFTs. So the classical action, gauge
transformation and equations of motion takes the form

$\begin{split}\text{Action:}\quad&S_{\text{cl}}=\frac{1}{g_{s}^{2}}\left[-\frac{1}{2}\left\langle\widetilde{\Psi}\left|c_{0}^{-}Q_{B}\mathcal{G}\right|\widetilde{\Psi}\right\rangle+\left\langle\widetilde{\Psi}\left|c_{0}^{-}Q_{B}\right|\Psi\right\rangle+\sum_{n=3}^{\infty}\frac{1}{n!}\left\\{\Psi^{n}\right\\}_{0}\right]\leavevmode\nobreak\
,\\\ \text{Gauge
transformation:}\quad&|\delta\Psi\rangle=Q_{B}|\Lambda\rangle+\sum_{n=2}^{\infty}\frac{1}{n!}\mathcal{G}\left[\Psi^{n}\Lambda\right]_{0},\quad|\delta\widetilde{\Psi}\rangle=Q_{B}|\widetilde{\Lambda}\rangle+\sum_{n=2}^{\infty}\frac{1}{n!}\left[\Psi^{n}\Lambda\right]_{0},\\\
\text{Eq. of
motion:}\quad&Q_{B}(|\Psi\rangle-\mathcal{G}|\widetilde{\Psi}\rangle)=0,\quad
Q_{B}|\widetilde{\Psi}\rangle+\sum_{n=3}^{\infty}\frac{1}{(n-1)!}\left[\Psi^{n-1}\right]_{0}=0,\\\
\implies\quad&Q_{B}|\Psi\rangle+\sum_{n=3}^{\infty}\frac{1}{(n-1)!}\mathcal{G}\left[\Psi^{n-1}\right]_{0}=0,\end{split}$
(29)

where $\left\\{\Psi^{n}\right\\}_{0}$ denotes the genus zero $n$-string vertex
and $[\Psi^{n}]_{0}$ is defined as in (18) with the genus zero string vertex.
It also satisfies the identities in (23) and (22). We will omit the subscript
0 from now on.

###  2.3 $g$-twisted $L_{\infty}$-algebras

In this section, we slightly generalise the mathematical definition of
$L_{\infty}$-algebras suitable for application to closed superstring field
theory. We refer the reader to 10.1007/BFb0101184 ; Zwiebach:1992ie  for
material on $L_{\infty}$-algebras.

For $p,q\in\mathbb{N}$, a $(p,q)$-unshuffle is a permutation

$\left(\mu_{1},\cdots,\mu_{p},\nu_{1},\cdots,\nu_{q}\right)\leavevmode\nobreak\
,$ (30)

of $(1,2,\dots,p+q)$ such that

$\mu_{1}<\mu_{2}<\cdots<\mu_{p},\quad\nu_{1}<\nu_{2}<\cdots<v_{q}.$ (31)

We denote the set of all $(p,q)$-unshuffles by UnShuff$(p,q)$.
Let $V=\oplus_{i\in\mathbb{Z}}V_{i}$ be a $\mathbb{Z}$-graded vector space.
For a homogenous vector $v$, we denote the degree by $|v|$. Let

$\chi\left(\sigma,v_{1},\cdots,v_{n}\right)\in\\{-1,+1\\}\leavevmode\nobreak\
,$ (32)

denote the products of $(-1)^{\left|v_{i}\right|\left|v_{i+1}\right|}$ for
each interchange of neighbours

$\left(\dots,v_{i},v_{i+1},\dots\right)\longrightarrow\left(\dots,v_{i+1},v_{i},\dots\right)\leavevmode\nobreak\
,$ (33)

involved in a decomposition of the permutation $\sigma$ into transpositions.
Note that this is well defined since the number of transposition in a
decomposition of a permutation is either even or odd.
Let $V^{\wedge n}$ denote the symmetrized tensor product obtained by linear
span of vectors of the form

$v_{1}\wedge\dots\wedge v_{n}:=\sum_{\sigma\in
S_{n}}\chi(\sigma,v_{1},\dots,v_{n})(v_{\sigma(1)}\otimes\dots\otimes
v_{\sigma(n)}).$ (34)

Clearly for any $\sigma\in S_{n}$

$v_{\sigma(1)}\wedge\dots\wedge
v_{\sigma(n)}=\chi(\sigma,v_{1},\dots,v_{n})(v_{1}\wedge\dots\wedge v_{n}).$
(35)

The symmetrized tensor product $V^{\wedge n}$ naturally acquires a grading
from that of $V$ by the condition

$V_{i_{1}}\wedge\dots\wedge V_{i_{n}}\subset(V^{\wedge
n})_{i_{1}+\dots+i_{n}}.$ (36)

A multilinear map $f:V^{\times n}\longrightarrow V$ is called graded-symmetric
if

$f(v_{\sigma(1)},\dots,v_{\sigma(n)})=\chi(\sigma,v_{1},\dots,v_{n})(v_{1},\dots,v_{n}),\quad\sigma\in
S_{n}\leavevmode\nobreak\ .$ (37)

By (35), any multilinear map $f:V^{\wedge n}\longrightarrow V$ is
automatically graded-symmetric. Let $c_{j}:V^{\wedge j}\longrightarrow V$ and
$d_{i}:V^{\wedge i}\longrightarrow V$ be multilinear symmetric maps. We define
the product $c_{j}d_{i}:V^{\wedge(i+j-1)}\longrightarrow V$ as follows: for
$(i+j-1=n)$-tuples $\left(v_{1},\cdots,v_{n}\right)$ of homogeneous vectors
$v_{i}\in V_{\left|v_{i}\right|}$

$\begin{split}c_{j}d_{i}(v_{1},\cdots,v_{n})=\sum_{\sigma\in\operatorname{UnShuff}(i,j-1)}&\chi\left(\sigma,v_{1},\cdots,v_{n}\right)\\\
&\times
c_{j}\left(d_{i}\left(v_{\sigma(1)},\cdots,v_{\sigma(i)}\right)),v_{\sigma(i+1)},\cdots,v_{\sigma(n)}\right).\end{split}$
(38)

This product can equivalently be written as

$c_{j}d_{i}=c_{j}\circ\left(d_{i}\wedge\mathds{1}_{V^{\wedge j-1}}\right),$
(39)

where for multilinear maps $\alpha:V^{\wedge k}\longrightarrow V^{\wedge
l},\leavevmode\nobreak\ \beta:V^{\wedge m}\longrightarrow V^{\wedge n}$, we
can define their wedge product

$\alpha\wedge\beta:V^{\wedge k+m}\longrightarrow V^{\wedge
l+n}\leavevmode\nobreak\ ,$ (40)

by writing

$\displaystyle\alpha\wedge\beta\left(v_{1},\ldots,v_{k+m}\right)=\sum_{\sigma\in\operatorname{UnShuff}(k,m)}$
$\displaystyle\chi\left(\sigma,v_{1},\cdots,v_{k+m}\right)\times$ (41)
$\displaystyle\alpha\left(v_{\sigma(1)},\cdots,v_{\sigma(k)}\right)\wedge\beta\left(v_{\sigma(k+1)},\cdots,v_{\sigma(k+m)}\right),$

and

$\mathds{1}_{V^{\wedge n}}=\frac{1}{n!}\left(\mathds{1}_{V}\right)^{\wedge
n}=\left(\mathds{1}_{V}\right)^{\otimes n}\leavevmode\nobreak\ ,$ (42)

is the identity operator on $V^{\wedge n}$. We also define the commutator of
two multilinear maps by

$[c_{j},d_{i}]:=c_{j}d_{i}-(-1)^{|c_{j}||d_{i}|}d_{i}c_{j}\leavevmode\nobreak\
,$ (43)

where $|c_{j}|$ is the degree of 444Recall that a map $f:V\to W$ between
$\mathbb{Z}$-graded vector spaces is of degree $n$ if $f(V_{i})\subset
W_{i+n}$ for all $i\in\mathbb{Z}$, where $V_{i}$ is the degree $i$ homogenous
subspace. the map $c_{j}$.

Let $V^{\star}$ be the dual of $V$ with some choice of $\mathbb{Z}$-grading.
We now define twisted $L_{\infty}$-algebras which are slight generalisation of
classical $L_{\infty}$-algebras. Let $V$ be a $\mathbb{Z}$-graded vector space
with dual $V^{\star}$ and let $g:V^{\star}\longrightarrow V$ be a linear map.
A $g$-twisted $L_{\infty}$-algebra is a tuple
$(V,\\{\ell_{n}\\}_{n=1}^{\infty},g)$ where $\ell_{1}:V\longrightarrow
V,\leavevmode\nobreak\ \ell_{1}:V^{\star}\longrightarrow V^{\star}$ is a
degree $-1$ nilpotent linear map: $\ell_{1}^{2}=0$, and
$\\{\ell_{n}\\}_{n=2}^{\infty}$ is a family of multilinear maps called the
$n$-ary bracket:

$\ell_{n}(\cdots):=[-,-,\dots,-]_{n}:V^{\wedge n}\longrightarrow
V^{\star}\leavevmode\nobreak\ ,$ (44)

of degree $n-2$ such that the following conditions hold:

  1. 1.

Graded-symmetry: each $\ell_{n}$ is graded symmetric: for homogeneous vectors
$v_{i}$ and a permutation $\sigma$

$\ell_{n}\left(v_{\sigma(1)},v_{\sigma(2)},\cdots,v_{\sigma(n)}\right)=\chi\left(\sigma,v_{1},\cdots,v_{n}\right)\cdot\ell_{n}\left(v_{1},v_{2},\cdots
v_{n}\right).$ (45)

  2. 2.

Strong homotopy Jacobi identity: for all $k\in\mathbb{N}$,

$\sum_{\begin{subarray}{c}i,j\in\mathbb{N}\\\
i+j=k+1\end{subarray}}\ell_{j}^{g}\ell^{g}_{i}=0,$ (46)

where $\ell_{i}^{g}:V^{\wedge i}\longrightarrow V$ is given by

$\ell^{g}_{1}:=\ell_{1},\leavevmode\nobreak\ \ell^{g}_{i}:=g\circ\ell_{i},$
(47)

and the product is defined in (38).

The first few strong homotopy Jacobi identities are

$\begin{split}k=1:\leavevmode\nobreak\ \leavevmode\nobreak\
&\ell_{1}\left(\ell_{1}\left(v_{1}\right)\right)=0,\\\
k=2:\leavevmode\nobreak\ \leavevmode\nobreak\
&\ell_{1}\left(\ell_{2}^{g}\left(v_{1},v_{2}\right)\right)+\ell_{2}^{g}\left(\ell_{1}\left(v_{1}\right),v_{2}\right)+(-1)^{|v_{1}||v_{2}|}\ell_{2}^{g}\left(\ell_{1}\left(v_{2}\right),v_{1}\right)=0,\\\
k=3:\leavevmode\nobreak\ \leavevmode\nobreak\
&\ell_{1}\left(\ell_{3}^{g}\left(v_{1},v_{2},v_{3}\right)\right)+\ell_{2}^{g}\left(\ell_{2}^{g}\left(v_{1},v_{2}\right),v_{3}\right)\\\
&+(-1)^{|v_{1}|\left(|v_{2}|+|v_{3}|\right)}\ell_{2}^{g}\left(\ell_{2}^{g}\left(v_{2},v_{3}\right),v_{1}\right)\\\
&+(-1)^{|v_{3}|\left(|v_{1}|+|v_{2}|\right)}\ell_{2}^{g}\left(\ell_{2}^{g}\left(v_{3},v_{1}\right),v_{2}\right)\\\
&+\ell_{3}^{g}\left(\ell_{1}\left(v_{1}\right),v_{2},v_{3}\right)+(-1)^{|v_{1}|}\ell_{3}^{g}\left(v_{1},\ell_{1}\left(v_{2}\right),v_{3}\right)\\\
&+(-1)^{|v_{1}|+|v_{2}|}\ell_{3}^{g}\left(v_{1},v_{2},\ell_{1}\left(v_{3}\right)\right)=0.\end{split}$
(48)

It is easy to see that if $(V,\\{\ell_{n}\\},g)$ is a $g$-twisted
$L_{\infty}$-algebra then $(V,\\{\ell^{g}_{n}\\})$ is a usual
$L_{\infty}$-algebra. One can generalise the notion of $g$-twisted
$L_{\infty}$-algebra to the case when $V$ is a supervector space, i.e. a
$\mathbb{Z}_{2}$-graded vector space. Let us denote the
$\mathbb{Z}_{2}$-degree of a vector $v$ by
$d_{\mathbb{Z}_{2}}(v)\in\\{\bar{0},\bar{1}\\}=\mathbb{Z}_{2}$. A $g$-twisted
$L_{\infty}$-superalgebra is a triple $(V,\\{\ell_{n}\\},g)$, where $V$ is a
supervector space and $\ell_{n},g$ are as above, satisfying the axioms of
$g$-twisted $L_{\infty}$-algebra with the sign
$\chi(\sigma,v_{1},\dots,v_{n})\in\\{+1,-1\\}$ obtained by replacing the
degree $|v|$ of a vector $v$ by555Note that in the sum in (49),
$d_{\mathbb{Z}_{2}}(v)$ is considered as the integer $0,1$ rather than an
equivalence class.

$d(v):=|v|+(d_{\mathbb{Z}_{2}}(v)+1).$ (49)

The degree of $\ell_{n}$ is $(n-2)$ with respect to the $\mathbb{Z}$-grading
$|\cdot|$ on $V$ and is odd degree with respect to $d(\cdot)$. Suppose each
$\ell_{n}^{g}$ has odd degree $d(\ell_{n}^{g})$ as a map from $V\to V$, then
the strong homotopy Jacobi identity can also be written as

$\sum_{\begin{subarray}{c}i,j\in\mathbb{N}\\\
i+j=k+1\end{subarray}}\ell_{j}^{g}\ell^{g}_{i}=\frac{1}{2}\sum_{\begin{subarray}{c}i,j\in\mathbb{N}\\\
i+j=k+1\end{subarray}}[\ell_{j}^{g},\ell^{g}_{i}]=0.$ (50)

From now on, we assume that $V$ is a supervector space. To reduce to the usual
case, we can take the odd subspace of $V$ to be trivial. There is a natural
pairing

$\begin{split}&V^{\star}\otimes V\longrightarrow\mathbb{C}\\\
&(\phi,v)\mapsto\phi(v).\end{split}$ (51)

Suppose now that $V$ is a reflexive vector space666Recall that a vector space
is called reflexive if the natural map $V\to V^{\star\star}$ given by
$x\mapsto[\widehat{x}:f\mapsto\widehat{x}(f):=f(x)]$ is a linear isomorphism.
Thus finite dimensional vector spaces are reflexive.. Define a bilinear
pairing $\omega:V^{\star}\otimes V\longrightarrow\mathbb{C}$ by

$\omega(v_{1},v_{2}):=(-1)^{d(v_{1})}v_{1}(v_{2})\leavevmode\nobreak\ ,$ (52)

on homogeneous elements $v_{1}\in V^{\star}$ and any $v_{2}\in V$ and then
extend it linearly to all of $V^{\star}$. The bilinear pairing
$\omega:V^{\star}\otimes V\longrightarrow\mathbb{C}$ defined above is said to
be graded anti-symmetric if

$\omega\left(v_{1},v_{2}\right)=-(-1)^{d(v_{1})d(v_{2})}\omega\left(v_{2},v_{1}\right),$
(53)

where the right hand side is the bilinear pairing on $V^{\star\star}\otimes
V^{\star}$ with the natural identification of $V^{\star\star}$ with $V$, see
footnote 6. The $n$-ary brackets $\ell_{n},\leavevmode\nobreak\ n\geq 2$ are
called cyclic with respect to $\omega$ if they satisfy

$\omega\left(\ell_{n}\left(v_{1},\dots,v_{n}\right),v_{n+1}\right)=-(-1)^{d(v_{1})}\omega\left(v_{1},\ell_{n}\left(v_{2},\ldots,v_{n+1}\right)\right).$
(54)

We say that $(V,\\{\ell_{n}\\}_{n=1}^{\infty},g)$ is a cyclic $g$-twisted
$L_{\infty}$-(super)algebra with respect to $\omega$ if every
$\ell_{n},\leavevmode\nobreak\ n\geq 2$ is cyclic with respect to $\omega$.

###  2.4 Closed superstring field theory and $L_{\infty}$-superalgebra

We now show that the superstring theory Hilbert space
$\widehat{\mathcal{H}}_{T}$ along with the $n$-string vertices is a cyclic
$L_{\infty}$-superalgebra with respect to (modified) BPZ inner product. Let us
first define the grading on the full Hilbert space $\mathcal{H}_{T}$ by
declaring

$|\Psi|=1-\text{gh}(\Psi)\leavevmode\nobreak\ ,$ (55)

where $\text{gh}(\Psi)$ is the ghost number of $\Psi$. Next the supervector
space structure of $\widehat{\mathcal{H}}_{T}$ is defined by putting a
$\mathbb{Z}_{2}$ grading on the full Hilbert space $\mathcal{H}_{T}$ as
follows:

$d_{\mathbb{Z}_{2}}(\Psi)=\begin{cases}0,&\Psi\in\mathcal{H}_{\mathrm{NSNS}},\mathcal{H}_{\mathrm{RR}}\quad\text{for
Type II},\\\ 0,&\Psi\in\mathcal{H}_{\mathrm{NS}}\quad\text{for Heterotic},\\\
1,&\Psi\in\mathcal{H}_{\mathrm{NSR}},\mathcal{H}_{\mathrm{RNS}}\quad\text{for
Type II},\\\ 1,&\Psi\in\mathcal{H}_{\mathrm{R}}\quad\text{for
Heterotic}.\end{cases}$ (56)

With this definition, it is clear that the grassmannality of states in
$\widehat{\mathcal{H}}_{T}$ and $\widetilde{\mathcal{H}}_{T}$ and the grading
is related by

$d(\Psi):=|\Psi|+d_{\mathbb{Z}_{2}}(\Psi)+1=d_{\mathbb{Z}_{2}}(\Psi)-\text{gh}(\Psi)+2\equiv\gamma\bmod
2\leavevmode\nobreak\ ,$ (57)

where $\gamma$ is the grassmanality of $\Psi$. Next, the relations (8)
identifies $\widetilde{\mathcal{H}}_{T}$ with a subspace of the dual
space777Note that for finite dimensional vector spaces, this identification is
an isomorphism. But since our spaces are necessarily infinite dimensional this
defines only a subspace. $\widehat{\mathcal{H}}_{T}^{\star}$ via the map

$\widetilde{\mathcal{H}}_{T}\ni\widetilde{\Psi}\mapsto\left\langle\widetilde{\Psi}|c_{0}^{-}|\cdot\right\rangle,$
(58)

where $\left\langle\widetilde{\Psi}|c_{0}^{-}|\cdot\right\rangle$ is a linear
functional on $\widehat{\mathcal{H}}_{T}$ acting via the BPZ inner product:

$\widehat{\mathcal{H}}_{T}\ni\Psi\mapsto\left\langle\widetilde{\Psi}|c_{0}^{-}|\Psi\right\rangle\in\mathbb{C}.$
(59)

Moreover since Hilbert spaces are reflexive i.e.
$\widehat{\mathcal{H}}_{T}\cong\widehat{\mathcal{H}}_{T}^{\star\star}$ under
the map (see footnote 6)

$\begin{split}&\Psi\mapsto\langle\cdot|c_{0}^{-}|\Psi\rangle\in\widehat{\mathcal{H}}_{T}^{\star\star}\leavevmode\nobreak\
,\end{split}$ (60)

which acts as

$\widetilde{\Psi}\mapsto\left\langle\widetilde{\Psi}|c_{0}^{-}|\Psi\right\rangle\leavevmode\nobreak\
,$ (61)

the second relation in (8) can be understood as the restriction of the pairing
$\widehat{\mathcal{H}}_{T}^{\star\star}\otimes\widehat{\mathcal{H}}_{T}^{\star}\to\mathbb{C}$.

Now define the multilinear maps $\ell_{1}:=Q_{B}$ and (see (18))

$\begin{split}&\ell_{n}:\widehat{\mathcal{H}}_{T}^{\wedge
n}\longrightarrow\widetilde{\mathcal{H}}_{T}\leavevmode\nobreak\ ,\\\
&\Psi_{1}\wedge\dots\wedge\Psi_{n}\mapsto[\Psi_{1}\Psi_{2}\dots\Psi_{n}],\quad
n>1.\end{split}$ (62)

Since $Q_{B}$ has ghost number $-1$, we have

$|Q_{B}\Psi|=1-\text{gh}(Q_{B}\Psi)=1-(\text{gh}(\Psi)+1)=(1-\text{gh}(\Psi))-1=|\Psi|-1.$
(63)

Thus $\ell_{1}$ is a degree $-1$ map. Next, from (20) we see that

$\begin{split}|\ell_{n}(\Psi_{1},\dots,\Psi_{n})|&=1-\text{gh}(\ell_{n}(\Psi_{1},\dots,\Psi_{n}))\\\
&=1-3-\sum_{i=1}^{n}(\text{gh}(\Psi_{i})-2)\\\
&=(n-2)+\sum_{i=1}^{n}(1-\text{gh}(\Psi_{i}))\\\
&=(n-2)+\sum_{i=1}^{n}|\Psi_{i}|,\end{split}$ (64)

which means that $\ell_{n}$ is a degree $n-2$ map as required. We also define
the $\mathbb{Z}_{2}$-parity of the string product to be888This is consistent
with the fact that the product of two NSR or RNS states in Type II is an NSNS
or RR state. While the product of an NSNS or RR state with an NSR or RNS state
is NSR or RNS. Similarly the product of two R states is an NS state in
Heterotic while the product of an NS state with an R state is an R state.

$d_{\mathbb{Z}_{2}}(\ell_{n}(\Psi_{1},\dots,\Psi_{n})):=\sum_{i=1}^{n}d_{\mathbb{Z}_{2}}(\Psi_{i})\in\mathbb{Z}_{2}\leavevmode\nobreak\
,$ (65)

so that

$d(\ell_{n}(\Psi_{1},\dots,\Psi_{n}))=|\ell_{n}(\Psi_{1},\dots,\Psi_{n})|+\sum_{i=1}^{n}d_{\mathbb{Z}_{2}}(\Psi_{i})+1.$
(66)

This implies that

$d(\ell_{n}(\Psi_{1},\dots,\Psi_{n}))=\sum_{i=1}^{n}d(\Psi_{i})-1\leavevmode\nobreak\
,$ (67)

so that the degree of $\ell_{n}$ with respect to $d(\cdot)$ is $-1$ and
$\ell_{n}$ is an odd degree operator.

From (57) and (64), we see that the grassmanality of
$\ell_{n}(\Psi_{1},\dots,\Psi_{n})$ is opposite of the sum of the
grassmanalities of $\Psi_{i}$:

$\gamma_{\ell_{n}}\equiv 1+\sum_{i=1}^{n}\gamma_{i}\bmod 2$ (68)

where $\gamma_{\ell_{n}}$ is the grassmanality of
$\ell_{n}(\Psi_{1},\dots,\Psi_{n})$. From (21), we see that $\ell_{n}$ is
graded-symmetric multilinear map. Indeed, decomposing any permutation $\sigma$
into transpositions and then using (21) gives the required relation. Finally,
an application of the PCO $\mathcal{G}$ on the main identity (23) gives

$\displaystyle
Q_{B}\mathcal{G}\left[\Psi_{1}\cdots\Psi_{n}\right]+\sum_{i=1}^{n}(-1)^{\gamma_{1}+\dots+\gamma_{i-1}}\mathcal{G}\left[\Psi_{1}\cdots\Psi_{i-1}\left(Q_{B}\Psi_{i}\right)\Psi_{i+1}\cdots\Psi_{n}\right]$
(69) $\displaystyle+\sum_{\begin{subarray}{c}\ell,k\geq 2\\\
\ell+k=n\end{subarray}}\sum_{\begin{subarray}{c}\left\\{i_{a};a=1,\dots,\ell\right\\},\left\\{j_{b};b=1,\dots,k\right\\}\\\
\left\\{i_{a}\right\\}\cup\left\\{j_{b}\right\\}=\\{1,\dots,n\\}\end{subarray}}\sigma\left(\left\\{i_{a}\right\\},\left\\{j_{b}\right\\}\right)\mathcal{G}\left[\mathcal{G}\left[\Psi_{j_{1}}\cdots\Psi_{j_{k}}\right]\Psi_{i_{1}}\cdots\Psi_{i_{\ell}}\right]=0,$

where we used the fact that $[Q_{B},\mathcal{G}]=0$. This is exactly the same
as the strong homotopy Jacobi identity once we realise that the sum over
partition of $\\{1,\dots,n\\}$ in (23) is the same as the sum over unshuffles.
The first two terms in the above identity correspond to the terms
$(i,j)=(1,n),(n,1)$. The bilinear pairing between
$\widetilde{\mathcal{H}}_{T}$ and $\widehat{\mathcal{H}}_{T}$ is given by

$\omega\left(\widetilde{\Psi},\Psi\right)=(-1)^{d(\widetilde{\Psi})}\left\langle\widetilde{\Psi}\big{|}c_{0}^{-}\big{|}\Psi\right\rangle.$
(70)

We also have

$\omega\left(\Psi,\widetilde{\Psi}\right)=(-1)^{d(\Psi)}\left\langle\Psi\big{|}c_{0}^{-}\big{|}\widetilde{\Psi}\right\rangle.$
(71)

The BPZ inner product is symmetric up to a sign dependent on grassmanalities
of the states:

$\langle\widetilde{\Psi}|c_{0}^{-}|\Psi^{\prime}\rangle=-(-1)^{\widetilde{\gamma}+\gamma^{\prime}+\widetilde{\gamma}\gamma^{\prime}}\langle\Psi^{\prime}|c_{0}^{-}|\widetilde{\Psi}\rangle\leavevmode\nobreak\
,$ (72)

where $\widetilde{\gamma}$ and $\gamma^{\prime}$ are the grassmannality of
$\widetilde{\Psi}\in\widetilde{\mathcal{H}}_{T}$ and
$\Psi^{\prime}\in\widehat{\mathcal{H}}_{T}$ respectively. The factor comes
from commuting the vertex operators of the states past $c_{0}^{-}$ and past
each other and the additional negative sign comes from the
SL(2,$\mathbb{C}$)-action $I:z\to-1/z$ on the integral for the zero mode of
$c(z),\overline{c}(\bar{z})$. We have from (57)

$\omega\left(\widetilde{\Psi},\Psi\right)=-(-1)^{2d(\widetilde{\Psi})+d(\Psi)}(-1)^{d(\widetilde{\Psi})d(\Psi)}\omega\left(\Psi,\widetilde{\Psi}\right).$
(73)

This implies that $\omega$ is graded anti-symmetric:

$\omega\left(\widetilde{\Psi},\Psi\right)=-(-1)^{d(\widetilde{\Psi})d(\Psi)}\omega\left(\Psi,\widetilde{\Psi}\right).$
(74)

From (15) we see that

$\omega\left(\widetilde{\Psi},\mathcal{G}\widetilde{\Psi}^{\prime}\right)=\omega\left(\mathcal{G}\widetilde{\Psi},\widetilde{\Psi}^{\prime}\right)\leavevmode\nobreak\
.$ (75)

We now show that with respect to $\omega$, $\ell_{n}$ is cyclic:

$\omega\left(\ell_{n}\left(\Psi_{1},\Psi_{2},\dots,\Psi_{n}\right),\Psi_{n+1}\right)=-(-1)^{d(\Psi_{1})}\omega\left(\Psi_{1},\ell_{n}\left(\Psi_{2},\dots,\Psi_{n+1}\right)\right)\leavevmode\nobreak\
,$ (76)

for $\Psi_{i}\in\widehat{\mathcal{H}}_{T}$. This is proved as follows:

$\displaystyle\omega\left(\Psi_{1},\ell_{n}\left(\Psi_{2},\dots,\Psi_{n+1}\right)\right)$
$\displaystyle=(-1)^{d(\Psi_{1})}\left\langle\Psi_{1}|c_{0}^{-}|\ell_{n}\left(\Psi_{2}\ldots\Psi_{n+1}\right)\right\rangle$
(77)
$\displaystyle=(-1)^{d(\Psi_{1})}\left\\{\Psi_{1}\Psi_{2}\ldots\Psi_{n+1}\right\\}$
$\displaystyle=(-1)^{d(\Psi_{1})}(-1)^{\gamma_{1}\gamma_{n+1}+\ldots+\gamma_{n}\gamma_{n+1}}\left\\{\Psi_{n+1}\Psi_{1}\Psi_{2}\Psi_{2}\ldots\Psi_{n}\right\\}$
$\displaystyle=(-1)^{d(\Psi_{1})}(-1)^{(\gamma_{1}+\dots+\gamma_{n})\gamma_{n+1}}(-1)^{d(\Psi_{n+1})}$
$\displaystyle\times\omega\left(\Psi_{n+1},\ell_{n}\left(\Psi_{1},\dots,\Psi_{n}\right)\right),$

where we used (21). Here $(-1)^{\gamma_{i}}$ is the grassmannality of
$\Psi_{i}$. Now using (57) and (67) it is easy to see that

$\gamma_{1}+\dots+\gamma_{n}\equiv(d(\ell_{n}(\Psi_{1},\dots,\Psi_{n}))-1)\bmod
2$ (78)

This gives using (57)

$\begin{split}\omega\left(\Psi_{1},\ell_{n}\left(\Psi_{2},\dots,\Psi_{n+1}\right)\right)&=(-1)^{d(\Psi_{1})}(-1)^{(d(\ell_{n}(\Psi_{1},\dots,\Psi_{n})-1)d(\Psi_{n+1})}(-1)^{d(\Psi_{n+1})}\\\
&\times\omega\left(\Psi_{n+1},\ell_{n}\left(\Psi_{1},\dots,\Psi_{n}\right)\right)\\\
&=-(-1)^{d(\Psi_{1})}\omega\left(\ell_{n}\left(\Psi_{1},\dots,\Psi_{n}\right),\Psi_{n+1}\right)\end{split}$
(79)

where we used (74).

The superstring action can then be written using $\omega$ as

$S_{\text{cl}}=\frac{1}{g_{s}^{2}}\left[-\frac{1}{2}\omega(\widetilde{\Psi},Q_{B}\mathcal{G}\widetilde{\Psi})+\omega(\widetilde{\Psi},Q_{B}\Psi)+\sum_{n=1}^{\infty}\frac{1}{(n+1)!}\omega\left(\Psi,\ell_{n}(\Psi^{n})\right)\right],$
(80)

where
$\widetilde{\Psi}\in\widetilde{\mathcal{H}}_{T},\Psi\in\widehat{\mathcal{H}}_{T}$
are grassmann even string fields and

$\ell_{n}(\Psi^{n}):=\ell_{n}(\underbrace{\Psi,\Psi,\dots,\Psi}_{n\leavevmode\nobreak\
\text{times}}).$ (81)

Note that the first term in the sum is zero since the BPZ inner product
vanishes for two states in $\widehat{\mathcal{H}}_{T}$. Choosing a smooth
interpolation $\Psi(t)$ for $0\leq t\leq 1$ with $\Psi(0)=0$ and
$\Psi(1)=\Psi$, we can write the action as

$S_{\text{cl}}=\frac{1}{g_{s}^{2}}\left[-\frac{1}{2}\omega(\widetilde{\Psi},Q_{B}\mathcal{G}\widetilde{\Psi})+\omega(\widetilde{\Psi},Q_{B}\Psi)+\sum_{n=1}^{\infty}\frac{1}{n!}\int_{0}^{1}dt\leavevmode\nobreak\
\omega(\dot{\Psi}(t),\ell_{n}(\Psi(t)^{n}))\right]\leavevmode\nobreak\ .$ (82)

The EOM takes the form

$\displaystyle\operatorname{EOM}(\widetilde{\Psi})=Q_{B}(\Psi-\mathcal{G}\widetilde{\Psi}),$
(83)
$\displaystyle\operatorname{EOM}(\Psi)=Q_{B}\widetilde{\Psi}+\sum_{n=2}^{\infty}\frac{1}{n!}\ell_{n}\left(\Psi^{n}\right)=Q_{B}\widetilde{\Psi}+\mathcal{J}^{\prime}(\Psi),$

where we have introduced the notation

$\mathcal{J}^{\prime}(\Psi)=\sum_{n=2}^{\infty}\frac{1}{n!}\ell_{n}\left(\Psi^{n}\right).$
(84)

Using these two equations, we obtain

$\operatorname{EOM}(\Psi)=Q_{B}\Psi+\mathcal{J}(\Psi)\leavevmode\nobreak\ ,$
(85)

where

$\mathcal{J}(\Psi):=\mathcal{G}\mathcal{J}^{\prime}(\Psi)=\sum_{n=2}^{\infty}\frac{1}{n!}\ell_{n}^{\mathcal{G}}\left(\Psi^{n}\right),$
(86)

where $\ell_{n}^{\mathcal{G}}=\mathcal{G}\circ\ell_{n}$. The linearized gauge
transformation is given by

$\displaystyle\delta_{\Lambda}\Psi=Q_{B}\Lambda+\sum_{n=2}^{\infty}\frac{1}{n!}\ell_{n+1}^{\mathcal{G}}\left(\Lambda,\Psi^{n}\right)\leavevmode\nobreak\
,$ (87)
$\displaystyle\delta_{\Lambda}\widetilde{\Psi}=Q_{B}\widetilde{\Lambda}+\sum_{n=2}^{\infty}\frac{1}{n!}\ell_{n+1}\left(\Lambda,\Psi^{n}\right)\leavevmode\nobreak\
,$

where $|\Lambda\rangle\in\widehat{\mathcal{H}}_{T}$ and
$|\widetilde{\Lambda}\rangle\in\widetilde{\mathcal{H}}_{T}$ both grassmann
odd.

##  3 Wilsonian effective action in superstring field theory

In this section, we review the procedure of integrating out heavy fields to
obtain the Wilsonian effective action as described in Sen:2016qap . We then
show that the effective action can be written equivalently in terms of certain
effective string products.

###  3.1 The string field theory description

Let us suppose we have a projection operator $P$ on a subset of string fields
satisfying the conditions

$\left[P,b_{0}^{\pm}\right]=0,\quad\left[P,c_{0}^{\pm}\right]=0,\quad\left[P,L_{0}^{\pm}\right]=0,\quad[P,\mathcal{G}]=0,\quad\left[P,Q_{B}\right]=0\text{.
}$ (88)

We also assume that the projection operator is grassmann even and self-
conjugate with respect to the BPZ inner product:

$\left\langle\widetilde{\Psi}\left|c_{0}^{-}\right|P\Psi\right\rangle=\left\langle
P\widetilde{\Psi}\left|c_{0}^{-}\right|\Psi\right\rangle,\quad\widetilde{\Psi}\in\widetilde{\mathcal{H}}_{T},\leavevmode\nobreak\
\leavevmode\nobreak\ \Psi\in\widehat{\mathcal{H}}_{T}.$ (89)

The usual way to get the Wilsonian effective action is by integrating out the
states in
$\bar{P}\left(\widetilde{\mathcal{H}}_{T}\oplus\widehat{\mathcal{H}}_{T}\right)$,
where $\bar{P}=1-P$. To write down the effective action, we construct new
string products $\\{\ldots\\}_{e}$ by using the propagator

$\left\langle\varphi_{s}^{c}\left|c_{0}^{-}b_{0}^{+}\left(L_{0}^{+}\right)^{-1}\mathcal{G}\bar{P}\right|\varphi_{r}^{c}\right\rangle\text{.
}$ (90)

The effective action is then given by

$S_{e}=\frac{1}{g_{s}^{2}}\left[-\frac{1}{2}\left\langle\widetilde{\psi}\left|c_{0}^{-}Q_{B}\mathcal{G}\right|\widetilde{\psi}\right\rangle+\left\langle\widetilde{\psi}\left|c_{0}^{-}Q_{B}\right|\psi\right\rangle+\sum_{n=3}^{\infty}\frac{1}{n!}\left\\{\psi^{n}\right\\}_{e}\right]\leavevmode\nobreak\
,$ (91)

where $\psi\in P\widehat{\mathcal{H}}_{T},\quad\widetilde{\psi}\in
P\widetilde{\mathcal{H}}_{T}$.

We will show that this effective action can be obtained using homotopy
transfer of the $\mathcal{G}$-twisted cyclic $L_{\infty}$-superalgebra
$\left(\widehat{\mathcal{H}}_{T},\\{\ell_{n}\\},\mathcal{G}\right)$.

###  3.2 Effective equation of motions

Let us split the state $\widetilde{\Psi},\Psi$ as

$\widetilde{\Psi}=\widetilde{\psi}+\widetilde{R},\quad\Psi=\psi+R\leavevmode\nobreak\
,$ (92)

where
$\widetilde{\psi}=P\widetilde{\Psi},\psi=P\Psi,\widetilde{R}=\bar{P}\Psi,R=\bar{P}\Psi$.
Next note that (89) implies that

$\omega\left(P\Psi,\Psi^{\prime}\right)=\omega\left(\Psi,P\Psi^{\prime}\right)\leavevmode\nobreak\
,$ (93)

since $P$ is grassmann even and commutes with $\mathcal{G}$. Next define the
degree-odd 999It is degree-odd since $b_{0}^{+}$ is grassmann odd. operator

$h=\frac{b_{0}^{+}}{L_{0}^{+}}\bar{P}.$ (94)

From the property (88) we have

$[h,\mathcal{G}]=0\leavevmode\nobreak\
,\quad\\{h,c_{0}^{-}\\}=0\leavevmode\nobreak\ ,$ (95)

thus $h:\widetilde{\mathcal{H}}_{T}\longrightarrow\widetilde{\mathcal{H}}_{T}$
and also
$h:\widehat{\mathcal{H}}_{T}\longrightarrow\widehat{\mathcal{H}}_{T}$. Then we
have the Hodge-Kodaira decomposition

$\left\\{h,Q_{B}\right\\}=\bar{P}\text{. }$ (96)

Indeed using

$\\{AB,C\\}=A[B,C]+\\{A,C\\}B\leavevmode\nobreak\ ,$ (97)

we get

$\displaystyle\left\\{\frac{b_{0}^{+}}{L_{0}^{+}}\bar{P},Q_{B}\right\\}$
$\displaystyle=\frac{\left\\{b_{0}^{+},Q_{B}\right\\}}{L_{0}^{+}}\bar{P}$ (98)
$\displaystyle=\frac{L_{0}^{+}}{L_{0}^{+}}\bar{P}=\bar{P}\leavevmode\nobreak\
.$

Also

$\displaystyle h^{2}$
$\displaystyle=\left(b_{0}^{+}\left(L_{0}^{+}\right)^{-1}\bar{P}\right)\left(b_{0}^{+}\left(L_{0}^{+}\right)^{-1}\bar{P}\right)$
(99)
$\displaystyle=b_{0}^{+}\left(L_{0}^{+}\right)^{-1}b_{0}^{+}\left(L_{0}^{+}\right)^{-1}\bar{P}^{2}$
$\displaystyle=\left(b_{0}^{+}\right)^{2}\left(L_{0}^{+}\right)^{-2}\bar{P}^{2}$
$\displaystyle=0.$

We also have

$\displaystyle Ph$
$\displaystyle=b_{0}^{+}\left(L_{0}^{+}\right)^{-1}P\bar{P}$ (100)
$\displaystyle=0=hP.$

Thus we have the annihilation conditions

$h^{2}=Ph=hP=0\leavevmode\nobreak\ .$ (101)

Substituting the decomposition (92) in the action (80) and varying separately
with respect to $\widetilde{\psi},\widetilde{R},\psi,R$, we obtain

$\displaystyle\operatorname{EOM}_{\widetilde{\psi}}(\widetilde{\psi},\widetilde{R},\psi)=P\operatorname{EOM}(\widetilde{\psi}+\widetilde{R})=Q_{B}(\psi-\mathcal{G}\widetilde{\psi})\leavevmode\nobreak\
,$ (102)
$\displaystyle\operatorname{EOM}_{\widetilde{R}}(\widetilde{\psi},\widetilde{R},R)=\bar{P}\operatorname{EOM}(\widetilde{\psi}+\widetilde{R})=Q_{B}(R-\mathcal{G}\widetilde{R})\leavevmode\nobreak\
,$
$\displaystyle\operatorname{EOM}_{\psi}(\psi,R,\widetilde{\psi})=P\operatorname{EOM}(\psi+R)=Q_{B}\widetilde{\psi}+P\mathcal{J}^{\prime}(\psi+R)\leavevmode\nobreak\
,$
$\displaystyle\operatorname{EOM}_{R}(\psi,R,\widetilde{R})=\bar{P}\operatorname{EOM}(\psi+R)=Q_{B}\widetilde{R}+\bar{P}\mathcal{J}^{\prime}(\psi+R).$

Again combining these EOMs we obtain the following two equations:

$\displaystyle\operatorname{EOM}_{\psi}(\psi,R)=P\operatorname{EOM}(\psi+R)=Q_{B}\psi+P\mathcal{J}(\psi+R)\leavevmode\nobreak\
,$ (103)
$\displaystyle\operatorname{EOM}_{R}(\psi,R)=\bar{P}\operatorname{POM}(\psi+R)=Q_{B}R+\bar{P}\mathcal{J}(\psi+R).$

###  3.3 Fixing the Siegel gauge

The Siegel gauge condition is

$b_{0}^{+}|\widetilde{\Psi}\rangle=0,\quad b_{0}^{+}|\Psi\rangle=0\implies
b_{0}^{+}\left(|\Psi\rangle-\frac{1}{2}\mathcal{G}|\widetilde{\Psi}\rangle\right)=0\text{.
}$ (104)

To integrate out $R$ and $\widetilde{R}$ from the dynamics, we need to use the
EOM and solve for $R,\widetilde{R}$ and plug it back in the action. But before
that we need to fix gauge for $\widetilde{R},R$. Define the Siegel gauge
projector $P_{s}=b_{0}^{+}c_{0}^{+}$. Then

$\bar{P}_{s}\equiv 1-P_{s}=c_{0}^{+}b_{0}^{+},$ (105)

since

$\left\\{b_{0}^{+},c_{0}^{+}\right\\}=\left\\{b_{0}^{-},c_{0}^{-}\right\\}=1.$
(106)

Then decompose
$\widetilde{R}=\widetilde{R}_{1}+\widetilde{R}_{2},R=R_{1}+R_{2}$ where

$\displaystyle\widetilde{R}_{1}=P_{s}\widetilde{R},\quad\widetilde{R}_{2}=\bar{P}_{s}\widetilde{R}\leavevmode\nobreak\
,$ (107) $\displaystyle R_{1}=P_{s}R,\quad
R_{2}=\bar{P}_{s}R\leavevmode\nobreak\ .$

The Siegel gauge condition on $\widetilde{R},R$ is then

$\widetilde{R}_{2}=R_{2}=0.$ (108)

The EOM then decomposes into a set of equations in
$\widetilde{R}_{1},\widetilde{R}_{2}$ and $R_{1},R_{2}$. Next we have

$[P,P_{s}]=\left[P,b_{0}^{+}c_{0}^{+}\right]=0\leavevmode\nobreak\ .$ (109)

Thus using (109) and the EOMR and EOM${}_{\widetilde{R}}$ we obtain

$\displaystyle\operatorname{EOM}_{\widetilde{R}_{1}}\left(\widetilde{\psi},\widetilde{R}_{1},R_{1}\right)=\bar{P}_{s}Q_{B}\left(R_{1}-\mathcal{G}\widetilde{R}_{1}\right)\leavevmode\nobreak\
,$ (110)
$\displaystyle\operatorname{EOM}_{R_{1}}\left(\psi,R_{1},\widetilde{R}_{1}\right)=\bar{P}_{s}Q_{B}\widetilde{R}_{1}+\bar{P}_{s}\bar{P}\mathcal{J}^{\prime}\left(\psi+R_{1}\right)\leavevmode\nobreak\
,$
$\displaystyle\operatorname{EOM}_{\widetilde{R}_{2}}\left(\widetilde{\psi},\widetilde{R}_{1},R_{1}\right)=P_{s}Q_{B}\left(R_{1}-\mathcal{G}\widetilde{R}_{1}\right)\leavevmode\nobreak\
,$
$\displaystyle\operatorname{EOM}_{R_{2}}\left(\psi,R_{1},\widetilde{R}_{1}\right)=P_{s}Q_{B}\widetilde{R}_{1}+P_{s}\bar{P}\mathcal{J}^{\prime}\left(\psi+R_{1}\right).$

Again combining these equations we obtain

$\displaystyle\operatorname{EOM}_{R_{1}}\left(\psi,R_{1}\right)=\bar{P}_{s}Q_{B}R_{1}+\bar{P}_{s}\bar{P}\mathcal{J}\left(\psi+R_{1}\right)$
(111)
$\displaystyle\operatorname{EOM}_{R_{2}}\left(\psi,R_{1}\right)=P_{s}Q_{B}R_{1}+P_{s}\bar{P}\mathcal{J}\left(\psi+R_{1}\right).$

We now want to solve for $\widetilde{R}_{1},R_{1}$ using the first equation
above. Given a solution $R_{1}$ to the first equation of (111),
$\widetilde{R}_{1}$ is determined by the second equation of (110) up to a
$Q_{B}$-closed state. We have

$\bar{P}_{s}Q_{B}R_{1}+\bar{P}_{s}\bar{P}\mathcal{J}\left(\psi+R_{1}\right)=0.$
(112)

Since

$\left\\{b_{0}^{\pm},Q_{B}\right\\}=L_{0}^{\pm}\text{ and
}\left(b_{0}^{+}\right)^{2}=0\leavevmode\nobreak\ ,$ (113)

$\big{(}$follows from
$\left\\{b_{n},Q_{B}\right\\}=L_{n},\left\\{\bar{b}_{n},Q_{B}\right\\}=\bar{L}_{n}$
and $\left\\{b_{n},\bar{b}_{m}\right\\}=0\big{)},$ we have

$\bar{P}_{s}Q_{B}R_{1}=c_{0}^{+}L_{0}^{+}R_{1}\leavevmode\nobreak\ ,$ (114)

where we used the fact
$b_{0}^{+}R_{1}=b_{0}^{+}P_{s}R=\left(b_{0}^{+}\right)^{2}c_{0}^{+}R=0$. Thus
$R_{1}$ must satisfy

$\displaystyle
c_{0}^{+}L_{0}^{+}R_{1}+c_{0}^{+}b_{0}^{+}\bar{P}J\left(\psi+R_{1}\right)=0$
(115) $\displaystyle\implies$ $\displaystyle
c_{0}^{+}L_{0}^{+}\left(R_{1}+\left(L_{0}^{+}\right)^{-1}b_{0}^{+}\bar{P}\mathcal{J}\left(\psi+R_{1}\right)\right)=0$
$\displaystyle\implies$ $\displaystyle
R_{1}(\psi)=-\left.h\mathcal{J}(\Psi)\right|_{\Psi=\psi+R_{1}}.$

One can obtain recursive solution to this as follows: let
$G=-h\circ\mathcal{J}$. Then assuming $R_{1}(0)=0$, we get

$R_{1}(\psi)=G(\psi+G(\psi+G(\psi+\ldots)))\text{. }$ (116)

More explicitly from $\Psi=\psi+R_{1}(\psi)$ we have

$\displaystyle\Psi(\psi)=$
$\displaystyle\psi-\frac{1}{2!}h\ell_{2}^{\mathcal{G}}(\psi,\psi)-\frac{1}{3!}h\ell_{3}^{\mathcal{G}}(\psi,\psi,\psi)+\frac{2}{(2!)^{2}}h\ell_{2}^{\mathcal{G}}\left(h\ell_{2}^{\mathcal{G}}(\psi,\psi),\psi\right)+$
(117)
$\displaystyle-\frac{1}{4!}h\ell_{4}^{\mathcal{G}}(\psi,\psi,\psi,\psi)+\frac{2}{2!3!}h\ell_{2}^{\mathcal{G}}\left(h\ell_{3}^{\mathcal{G}}(\psi,\psi,\psi),\psi\right)+\frac{3}{2!3!}h\ell_{3}^{\mathcal{G}}\left(h\ell_{2}^{\mathcal{G}}(\psi,\psi),\psi,\psi\right)+$
$\displaystyle-\frac{1}{(2!)^{3}}h\ell_{2}^{\mathcal{G}}\left(h\ell_{2}^{\mathcal{G}}(\psi,\psi),h\ell_{2}^{\mathcal{G}}(\psi,\psi)\right)-\frac{2^{2}}{(2!)^{3}}h\ell_{2}^{\mathcal{G}}\left(h\ell_{2}^{\mathcal{G}}\left(h\ell_{2}^{\mathcal{G}}(\psi,\psi),\psi\right),\psi\right)+O\left(\psi^{5}\right).$

One can show that this solution satisfies the second equation of (111) as in
Erbin:2020eyc . It is easy to see that all the four equations in (110) are
satisfied since (111) is equivalent to (110). Substituting the recursive
solution (117) in the first equation of (103) we obtain the eom for $\psi$:

$\operatorname{eom}(\psi)=\sum_{n=1}^{\infty}\frac{1}{n!}\widetilde{\ell}^{\mathcal{G}}_{n}\left(\psi^{n}\right)$
(118)

where $\widetilde{\ell}^{\mathcal{G}}_{1}=\widetilde{\ell}_{1}$ and
$\widetilde{\ell}^{\mathcal{G}}_{n}=\mathcal{G}\circ\widetilde{\ell}_{n}$ for
$n>1$ where

$\displaystyle\widetilde{\ell}_{1}\left(\Psi_{1}\right)=PQ_{B}\Psi_{1},$ (119)
$\displaystyle\widetilde{\ell}_{2}\left(\Psi_{1},\Psi_{2}\right)=P\ell_{2}\left(\Psi_{1},\Psi_{2}\right),$
$\displaystyle\widetilde{\ell}_{3}\left(\Psi_{1},\Psi_{2},\Psi_{3}\right)=P\ell_{3}\left(\Psi_{1},\Psi_{2},\Psi_{3}\right)-P\ell_{2}\left(\Psi_{1},h\ell_{2}^{\mathcal{G}}\left(\Psi_{2},\Psi_{3}\right)\right)$
$\displaystyle\hskip
75.39963pt-(-1)^{d(\Psi_{1})\left(d(\Psi_{2})+d(\Psi_{3})\right)}P\ell_{2}\left(\Psi_{2},h\ell_{2}^{\mathcal{G}}\left(\Psi_{3},\Psi_{1}\right)\right)$
$\displaystyle\hskip
75.39963pt-(-1)^{d(\Psi_{3})\left(d(\Psi_{1})+d(\Psi_{2})\right)}P\ell_{2}\left(\Psi_{3},h\ell_{2}^{\mathcal{G}}\left(\Psi_{1},\Psi_{2}\right)\right)\leavevmode\nobreak\
,$

and so on. In the next section, we will prove that
$\left(P\widehat{\mathcal{H}}_{T},\\{\widetilde{\ell}_{n}\\}_{n=1}^{\infty},\mathcal{G}\right)$
is a $\mathcal{G}$-twisted $L_{\infty}$-superalgebra.

Notice that $h$ is BPZ self-conjugate:

$\omega\left(\Psi_{1},h\Psi_{2}\right)=(-1)^{d(\Psi_{1})}\omega\left(h\Psi_{1},\Psi_{2}\right),$
(120)

the factor of $(-1)^{d(\Psi_{1})}$ appears because $h$ is grassmann odd. Let
us prove that $\widetilde{\ell}_{n}$ are cyclic with respect to
$\tilde{\omega}:=\omega|_{P\widehat{\mathcal{H}}_{T}}$. We have for
$\psi_{i}\in P\widehat{\mathcal{H}}$

$\displaystyle\tilde{\omega}\left(\psi_{1},\widetilde{\ell}_{2}\left(\psi_{2},\psi_{3}\right)\right)$
$\displaystyle=\tilde{\omega}\left(\psi_{1},P\ell_{2}\left(\psi_{2},\psi_{3}\right)\right)$
(121)
$\displaystyle=\omega\left(P\psi_{1},\ell_{2}\left(\psi_{2},\psi_{3}\right)\right)$
$\displaystyle=-(-1)^{d\left(\psi_{1}\right)}\omega\left(\ell_{2}\left(\psi_{1},\psi_{2}\right),\psi_{3}\right)$
$\displaystyle=-(-1)^{d\left(\psi_{1}\right)}\omega\left(\ell_{2}\left(\psi_{1},\psi_{2}\right),P\psi_{3}\right)$
$\displaystyle=-(-1)^{d\left(\psi_{1}\right)}\omega\left(P\ell_{2}\left(\psi_{1},\psi_{2}\right),\psi_{3}\right)$
$\displaystyle=-(-1)^{d\left(\psi_{1}\right)}\tilde{\omega}\left(\widetilde{\ell}_{2}\left(\psi_{1},\psi_{2}\right),\psi_{3}\right),$

where we used the cyclicity of $\ell_{n}$ (76), $P^{2}=P$ and (93). The same
argument implies

$\displaystyle\tilde{\omega}\left(\psi_{1},P\ell\left(\psi_{2},\ldots,\psi_{n+1}\right)\right)$
$\displaystyle=-(-1)^{d\left(\psi_{1}\right)}\tilde{\omega}\left(P\ell_{n}\left(\psi_{1},\ldots,\psi_{n}\right),\psi_{n+1}\right)$
(122)
$\displaystyle=-(-1)^{d\left(\psi_{1}\right)}\tilde{\omega}\left(\ell_{n}\left(\psi_{1},\ldots,\psi_{n}\right),P\psi_{n+1}\right)$
$\displaystyle=-(-1)^{d\left(\psi_{1}\right)}\tilde{\omega}\left(\ell_{n}\left(\psi_{1},\ldots,\psi_{n}\right),\psi_{n+1}\right).$

At next order, using (122) we have

$\begin{split}\tilde{\omega}\left(\psi_{1},\widetilde{\ell}_{3}\left(\psi_{2},\psi_{3},\psi_{4}\right)\right)&=-(-1)^{d\left(\psi_{1}\right)}\tilde{\omega}\left(P\ell_{3}\left(\psi_{1},\psi_{2},\psi_{3}\right),\psi_{4}\right)\\\
&+(-1)^{d\left(\psi_{1}\right)}\tilde{\omega}\left(\ell_{2}\left(\psi_{1},\psi_{2}\right),h\mathcal{G}\ell_{2}\left(\psi_{3},\psi_{4}\right)\right)\\\
&+(-1)^{d\left(\psi_{1}\right)}(-1)^{d\left(\psi_{2}\right)d\left(\psi_{3}\right)}\tilde{\omega}\left(\ell_{2}\left(\psi_{1},\psi_{3}\right),h\mathcal{G}\ell_{2}\left(\psi_{2},\psi_{4}\right)\right)\\\
&+(-1)^{d\left(\psi_{1}\right)}\tilde{\omega}\left(\ell_{2}\left(\psi_{1},h\ell_{2}^{\mathcal{G}}\left(\psi_{2},\psi_{3}\right)\right),\psi_{4}\right)\\\
&=-(-1)^{d\left(\psi_{1}\right)}\big{[}\tilde{\omega}\left(P\ell_{3}\left(\psi_{1},\psi_{2},\psi_{3}\right),\psi_{4}\right)-\tilde{\omega}\left(\ell_{2}\left(h\ell_{2}^{\mathcal{G}}\left(\psi_{1},\psi_{2}\right),\psi_{3}\right),\psi_{4}\right)\\\
&\hskip
56.9055pt-(-1)^{d\left(\psi_{2}\right)d\left(\psi_{3}\right)}\tilde{\omega}\left(\ell_{2}\left(h\ell_{2}^{\mathcal{G}}\left(\psi_{1},\psi_{3}\right),\psi_{2}\right),\psi_{4}\right)\\\
&\hskip
56.9055pt-\tilde{\omega}\left(\ell_{2}\left(\psi_{1},h\ell_{2}^{\mathcal{G}}\left(\psi_{2},\psi_{3}\right)\right),\psi_{4}\right)\big{]}\\\
&=-(-1)^{d\left(\psi_{1}\right)}\tilde{\omega}\left(\widetilde{\ell}_{3}\left(\psi_{1},\psi_{2},\psi_{3}\right),\psi_{4}\right)\leavevmode\nobreak\
,\end{split}$ (123)

where we used the fact that $h,\ell_{2}$ are both degree odd with respect
$d(\cdot)$ and graded-antisymmetry (45) of $\ell_{n}$ in the first step and
the cyclicity (76) of $\ell_{n}$, its graded antisymmetry, $[h,\mathcal{G}]=0$
and (120) in the second and third step.

As before, we can use (110) and the solution $\psi$ to determine
$\widetilde{\psi}$ up to a $Q_{B}$-exact field. The effective action which
captures the dynamics of $\psi$ determined by the eom (118) of $\psi$ can be
written as

$\begin{split}S^{\text{eff}}\left(\psi,\widetilde{\psi}\right)&=\frac{1}{g_{s}^{2}}\left[-\frac{1}{2}\tilde{\omega}\left(\widetilde{\psi},Q_{B}\mathcal{G}\widetilde{\psi}\right)+\tilde{\omega}\left(\widetilde{\psi},Q_{B}\psi\right)+\sum_{n=1}^{\infty}\frac{1}{(n+1)!}\tilde{\omega}\left(\psi,\widetilde{\ell}_{n}(\psi^{n})\right)\right]\\\
&=\frac{1}{g_{s}^{2}}\left[-\frac{1}{2}\tilde{\omega}(\widetilde{\psi},Q_{B}\mathcal{G}\widetilde{\Psi})+\tilde{\omega}(\widetilde{\Psi},Q_{B}\psi)+\sum_{n=1}^{\infty}\frac{1}{n!}\int_{0}^{1}dt\leavevmode\nobreak\
\tilde{\omega}(\dot{\psi}(t),\ell_{n}(\psi(t)^{n}))\right]\leavevmode\nobreak\
,\end{split}$ (124)

where $\psi(t)$ is a smooth interpolation for $0\leq t\leq 1$ with $\psi(0)=0$
and $\psi(1)=\psi$. Again $\widetilde{\psi}\in P\widetilde{\mathcal{H}}_{T}$
is a non-dynamical field. It can easily be checked that this effective action
has an infinite dimensional gauge symmetry given by

$\displaystyle\delta_{\lambda}\psi=Q_{B}\lambda+\sum_{n=2}^{\infty}\frac{1}{n!}\widetilde{\ell}_{n+1}^{\mathcal{G}}\left(\lambda,\psi^{n}\right)\leavevmode\nobreak\
,$ (125)
$\displaystyle\delta_{\lambda}\widetilde{\psi}=Q_{B}\widetilde{\lambda}+\sum_{n=2}^{\infty}\frac{1}{n!}\widetilde{\ell}_{n+1}\left(\lambda,\psi^{n}\right)\leavevmode\nobreak\
,$

where $|\lambda\rangle\in P\widehat{\mathcal{H}}_{T}$ and
$|\widetilde{\lambda}\rangle\in P\widetilde{\mathcal{H}}_{T}$ both grassmann
odd.

We can capture the system in terms of a strong deformation retract (SDR), see
Appendix B for definitions. We have the projection operator

$\Pi:\widehat{\mathcal{H}}_{T}\longrightarrow P\widehat{\mathcal{H}}_{T},$
(126)

and the natural inclusion map

$\mathrm{I}:P\widehat{\mathcal{H}}_{T}\longrightarrow\widehat{\mathcal{H}}_{T}.$
(127)

The Hodge-Kodaira decomposition (96) and the annihilation conditions (101)
imply that we have a strong deformation retract (SDR)

$\rotatebox[origin={c}]{270.0}{${{{{\circlearrowright}}}}$}\leavevmode\nobreak\
(-h)\left(\widehat{\mathcal{H}}_{T},Q_{B}\right)\underset{\mathrm{I}}{\stackrel{{\scriptstyle\Pi}}{{\mathrel{\begin{subarray}{c}\xrightarrow{\rule{56.9055pt}{0.0pt}}\\\\[-3.87495pt]
\xleftarrow{\rule{56.9055pt}{0.0pt}}\end{subarray}}}}}\left(P\widehat{\mathcal{H}}_{T},\Pi
Q_{B}\mathrm{I}\right),$ (128)

where the propagator $h$ is (minus) the homotopy operator, see Appendix B for
definitions. We also have maps

$\Pi:\widetilde{\mathcal{H}}_{T}\longrightarrow
P\widetilde{\mathcal{H}}_{T},\quad\mathrm{I}:P\widetilde{\mathcal{H}}_{T}\longrightarrow\widetilde{\mathcal{H}}_{T},\quad
h:\widetilde{\mathcal{H}}_{T}\longrightarrow\widetilde{\mathcal{H}}_{T}.$
(129)

This means that we have another SDR

$\begin{split}\rotatebox[origin={c}]{270.0}{${{{{\circlearrowright}}}}$}\leavevmode\nobreak\
(-h)\left(\widetilde{\mathcal{H}}_{T},Q_{B}\right)\underset{\mathrm{I}}{\stackrel{{\scriptstyle\Pi}}{{\mathrel{\begin{subarray}{c}\xrightarrow{\rule{56.9055pt}{0.0pt}}\\\\[-3.87495pt]
\xleftarrow{\rule{56.9055pt}{0.0pt}}\end{subarray}}}}}\left(P\widetilde{\mathcal{H}}_{T},\Pi
Q_{B}\mathrm{I}\right)\leavevmode\nobreak\ .\end{split}$ (130)

##  4 Effective action from homotopy transfer

In this section, we will prove that
$\left(P\widehat{\mathcal{H}}_{T},\\{\widetilde{\ell}_{n}\\}_{n=1}^{\infty},\mathcal{G}\right)$
is a $\mathcal{G}$-twisted $L_{\infty}$-superalgebra. We will prove this using
the homological perturbation lemma and symmetrized tensor coalgebra language.
See Appendix A and Appendix B for discussion on these topics.

###  4.1 Closed superstring field theory as symmetrized tensor coalgebra

We begin by describing the closed superstring field theory in terms of
symmetrized tensor coalgebra, see Appendix A for review of definitions. Let
$S\widetilde{\mathcal{H}}_{T},S\widehat{\mathcal{H}}_{T}$ be the symmetric
algebra given by

$\begin{split}&S\widetilde{\mathcal{H}}_{T}=\widetilde{\mathcal{H}}_{T}^{\wedge
0}\oplus\widetilde{\mathcal{H}}_{T}^{\wedge
1}\oplus\widetilde{\mathcal{H}}_{T}^{\wedge 2}\oplus\ldots\\\
&S\widehat{\mathcal{H}}_{T}=\widehat{\mathcal{H}}_{T}^{\wedge
0}\oplus\widehat{\mathcal{H}}_{T}^{\wedge
1}\oplus\widehat{\mathcal{H}}_{T}^{\wedge 2}\oplus\ldots\leavevmode\nobreak\
,\end{split}$ (131)

where $\widetilde{\mathcal{H}}_{T}^{\wedge
0}=\mathbb{C}\textbf{1}_{S\widetilde{\mathcal{H}}_{T}},\leavevmode\nobreak\
\widehat{\mathcal{H}}_{T}^{\wedge
0}=\mathbb{C}\textbf{1}_{S\widehat{\mathcal{H}}_{T}}$ and
$\textbf{1}_{S\widetilde{\mathcal{H}}_{T}},\textbf{1}_{S\widehat{\mathcal{H}}_{T}}$
is the identity of the coalgebra
$S\widetilde{\mathcal{H}}_{T},S\widehat{\mathcal{H}}_{T}$ respectively. As
described in Appendix A, these are coalgebras with the counit and coproduct
given by (174) and (176).

Let us now express the SDR (128) in terms of tensor coalgebra. Let
$\bm{Q_{B}}$ denote the coderivation on $S\widehat{\mathcal{H}}_{T}$ and
$S\widetilde{\mathcal{H}}_{T}$ obtained from (LABEL:eq:coderbmci) using the
BRST charge $Q_{B}$:

$\bm{Q_{B}}|_{S\widehat{\mathcal{H}}}=\sum_{n\geq
1}\left(Q_{B}\wedge\mathds{1}_{\widehat{\mathcal{H}}^{\wedge
n-1}}\right)\pi_{n},\quad\bm{Q_{B}}|_{S\widetilde{\mathcal{H}}}=\sum_{n\geq
1}\left(Q_{B}\wedge\mathds{1}_{\widetilde{\mathcal{H}}^{\wedge
n-1}}\right)\pi_{n}\leavevmode\nobreak\ ,$ (132)

where $\pi_{n}$ is the projection onto the $n^{th}$ homogenous subspace. We
will simply denote it by $\bm{Q_{B}}$, the space it acts on will be clear from
context. The projection $P$ and the maps in (126), (127) gives rise to linear
maps on $S\widehat{\mathcal{H}}_{T}$ and $S\widetilde{\mathcal{H}}_{T}$.
Define

$\mathbf{P}:S\widehat{\mathcal{H}}_{T}\longrightarrow
S\widehat{\mathcal{H}}_{T},\quad\mathbf{I}:SP\widehat{\mathcal{H}}_{T}\longrightarrow
S\widehat{\mathcal{H}}_{T},\quad\bm{\Pi}:S\widehat{\mathcal{H}}_{T}\longrightarrow
SP\widehat{\mathcal{H}}_{T},$ (133)

where $SP\widehat{\mathcal{H}}_{T}$ is the symmetrized tensor coalgebra on the
image $P\widehat{\mathcal{H}}_{T}$ of $P$ in $\widehat{\mathcal{H}}_{T}$, by

$\mathbf{P}=\sum_{k=0}^{\infty}\mathbf{P}_{k},\quad\mathbf{I}=\sum_{k=0}^{\infty}\mathbf{I}_{k},\quad\bm{\Pi}=\sum_{k=0}^{\infty}\bm{\Pi}_{k}\leavevmode\nobreak\
,$ (134)

where

$\displaystyle\mathbf{P}_{k}:=\mathbf{P}\pi_{k}:=\frac{1}{k!}P^{\wedge
k}\pi_{k}\leavevmode\nobreak\ ,$ (135)
$\displaystyle\mathbf{I}_{k}:=\mathbf{I}\pi_{k}:=\frac{1}{k!}\mathrm{I}^{\wedge
k}\pi_{k}\leavevmode\nobreak\ ,$
$\displaystyle\bm{\Pi}_{k}:=\bm{\Pi}\pi_{k}:=\frac{1}{k!}\Pi^{\wedge
k}\pi_{k}.$

Similarly we get linear maps $\mathbf{P},\bm{\Pi},\textbf{I}$ on
$S\widetilde{\mathcal{H}}_{T}$ corresponding to $P,\Pi,\mathrm{I}$
respectively. It is easy to check that

$\mathbf{P}=\mathbf{I}\bm{\Pi},\quad\bm{\Pi}\mathbf{I}=\mathds{1},$ (136)

where $\mathds{1}$ denotes the identity operator on
$SP\widehat{\mathcal{H}}_{T}$ or $SP\widetilde{\mathcal{H}}_{T}$. We also want
to define the propagator $\bm{h}:S\widehat{\mathcal{H}}_{T}\longrightarrow
S\widehat{\mathcal{H}}_{T}$ and
$\bm{h}:S\widetilde{\mathcal{H}}_{T}\longrightarrow
S\widetilde{\mathcal{H}}_{T}$ which satisfies the Hodge-Kodaira decomposition
as operators on the symmetrized tensor coalgebra:

$\bm{Q_{B}h}+\bm{hQ_{B}}=\mathds{1}_{S\widehat{\mathcal{H}}_{T}}-\mathbf{P}\leavevmode\nobreak\
.$ (137)

The following choice does the trick Erbin:2020eyc

$\bm{h}:=\sum_{k=0}^{\infty}\bm{h}_{k},$ (138)

where

$\bm{h}_{k}:=\bm{h}\pi_{k}:=\frac{1}{k!}\sum_{j=0}^{k-1}\left[h\wedge
P^{\wedge
j}\wedge\left(\mathds{1}_{\widehat{\mathcal{H}}_{T}}\right)^{\wedge(k-1-j)}\right]\pi_{k}.$
(139)

With this choice, the annihilation conditions are satisfied:

$\bm{h}\mathbf{I}=\mathbf{P}\bm{h}=\bm{h}^{2}=0.$ (140)

Thus we have an SDR of the free theory

$\rotatebox[origin={c}]{270.0}{${{{{\circlearrowright}}}}$}\leavevmode\nobreak\
(-\bm{h})\left(S\widehat{\mathcal{H}}_{T},\bm{Q_{B}}\right)\underset{\mathbf{I}}{\stackrel{{\scriptstyle\bm{\Pi}}}{{\mathrel{\begin{subarray}{c}\xrightarrow{\rule{56.9055pt}{0.0pt}}\\\\[-3.87495pt]
\xleftarrow{\rule{56.9055pt}{0.0pt}}\end{subarray}}}}}\left(SP\widehat{\mathcal{H}}_{T},\bm{\Pi
Q_{B}}\mathbf{I}\right).$ (141)

As described in Appendix A (Theorem A), the string products
$\ell_{n}^{\mathcal{G}}:\widehat{\mathcal{H}}_{T}^{\wedge
n}\longrightarrow\widehat{\mathcal{H}}_{T}$ for $n\geq 1$ with
$\ell_{1}^{\mathcal{G}}=Q_{B}$ gives us a coderivation

$\bm{\ell^{\mathcal{G}}}=\mu\circ(\ell^{\mathcal{G}}\boxtimes\mathds{1}_{S\widehat{\mathcal{H}}_{T}})\circ\Delta_{S\widehat{\mathcal{H}}_{T}}:S\widehat{\mathcal{H}}_{T}\longrightarrow
S\widehat{\mathcal{H}}_{T},\quad\ell^{\mathcal{G}}=\sum_{n=1}^{\infty}\ell^{\mathcal{G}}_{n}\leavevmode\nobreak\
,$ (142)

where $\mu:S\widehat{\mathcal{H}}_{T}\boxtimes
S\widehat{\mathcal{H}}_{T}\longrightarrow S\widehat{\mathcal{H}}_{T}$ is the
multiplication map and
$\Delta_{S\widehat{\mathcal{H}}_{T}}:S\widehat{\mathcal{H}}_{T}\longrightarrow
S\widehat{\mathcal{H}}_{T}\boxtimes S\widehat{\mathcal{H}}_{T}$ is the
coproduct. Then the strong homotopy Jacobi identity is equivalent to the
vanishing of the commutator

$(\bm{\ell^{\mathcal{G}}})^{2}=\frac{1}{2}[\bm{\ell^{\mathcal{G}}},\bm{\ell^{\mathcal{G}}}]=0,$
(143)

For an even-degree vector $A$, introduce the notation

$e^{\wedge A}:=\sum_{n=0}^{\infty}\frac{A^{\wedge
n}}{n!}=\textbf{1}_{S\widehat{\mathcal{H}}_{T}}+A+\frac{A\wedge
A}{2!}+\frac{A\wedge A\wedge A}{3!}+\dots$ (144)

To write the action (82) in the coalgebra langauge, we introduce the bra
action of $\omega$ as:

$\begin{split}&\langle\omega|:\widehat{\mathcal{H}}_{T}\otimes\widetilde{\mathcal{H}}_{T}\longrightarrow\mathbb{C},\\\
&\langle\omega|\widetilde{\Psi}\otimes\Psi=\omega(\widetilde{\Psi},\Psi)\leavevmode\nobreak\
.\end{split}$ (145)

Similarly
$\langle\omega|:\widetilde{\mathcal{H}}_{T}\otimes\widehat{\mathcal{H}}_{T}\longrightarrow\mathbb{C}$
is also defined. The action (82) can then be written as

$S_{\text{cl}}=\frac{1}{g_{s}^{2}}\left[-\frac{1}{2}\langle\omega|\widetilde{\Psi}\otimes
Q_{B}\mathcal{G}\widetilde{\Psi}+\langle\omega|\widetilde{\Psi}\otimes
Q_{B}\Psi+\int_{0}^{1}dt\leavevmode\nobreak\
\langle\omega|\pi_{1}\bm{\partial_{t}}e^{\wedge\Psi(t)}\otimes\mu\circ(\pi_{1}\boxtimes\pi_{0})\circ\delta\bm{\ell}e^{\wedge\Psi(t)}\right]\leavevmode\nobreak\
,$ (146)

where $\bm{\partial_{t}}$ is the coderivation corresponding to the linear map
$\partial_{t}$. Then the equation of motion obtained from this action takes
the form

$\displaystyle\operatorname{EOM}(\widetilde{\Psi})=Q_{B}(\Psi-\mathcal{G}\widetilde{\Psi})\leavevmode\nobreak\
,$ (147)
$\displaystyle\operatorname{EOM}(\Psi)=Q_{B}\widetilde{\Psi}+\mu\circ(\pi_{1}\boxtimes\pi_{0})\circ\delta\bm{\ell}e^{\wedge\Psi(t)}\leavevmode\nobreak\
,$

which can be combined to give

$\pi_{1}\bm{\ell^{\mathcal{G}}}e^{\wedge\Psi}=0.$ (148)

This equation is identical to the EOM (85).

###  4.2 Perturbed SDR and effective string products

We will now use the homological perturbation lemma to deduce that the
effective string products $\widetilde{\ell}_{n}$ in (119) satisfy the strong
homotopy Jacobi identity. Introduce the perturbation

$\begin{split}\delta\bm{\ell}:&S\widehat{\mathcal{H}}_{T}\longrightarrow
S\widetilde{\mathcal{H}}_{T}\boxtimes S\widehat{\mathcal{H}}_{T}\\\
&\delta\bm{\ell}=\sum_{k>1}(\ell_{k}\boxtimes\mathds{1}_{S\widehat{\mathcal{H}}_{T}})\circ\Delta_{S\widehat{\mathcal{H}}_{T}}\leavevmode\nobreak\
,\end{split}$ (149)

to the free SDR (141). The PCO $\mathcal{G}$ gives rise to a linear map

$\bm{\mathcal{G}}:S\widetilde{\mathcal{H}}_{T}\longrightarrow
S\widehat{\mathcal{H}}_{T}\leavevmode\nobreak\ ,$ (150)

as follows:

$\bm{\mathcal{G}}=\sum_{k=0}^{\infty}\bm{\mathcal{G}}_{k},\quad\bm{\mathcal{G}}_{k}:=\bm{\mathcal{G}}\pi_{k}:=\frac{1}{k!}\mathcal{G}^{\wedge
k}\pi_{k}\leavevmode\nobreak\ .$ (151)

Then from (143), the map

$\mu\circ(\bm{\mathcal{G}}\boxtimes\mathds{1}_{S\widehat{\mathcal{H}}_{T}}):S\widetilde{\mathcal{H}}_{T}\boxtimes
S\widehat{\mathcal{H}}_{T}\longrightarrow
S\widehat{\mathcal{H}}_{T}\leavevmode\nobreak\ ,$ (152)

satisfies

$(\bm{Q_{B}}+\delta\bm{\ell^{\mathcal{G}}})^{2}=(\bm{\ell^{\mathcal{G}}})^{2}=0\leavevmode\nobreak\
,$ (153)

where

$\delta\bm{\ell^{\mathcal{G}}}:=\mu\circ(\bm{\mathcal{G}}\boxtimes\mathds{1}_{S\widehat{\mathcal{H}}_{T}})\circ\delta\bm{\ell}\leavevmode\nobreak\
.$ (154)

Moreover, since $\mathcal{G}$ commutes with $P$, we also have the restriction
map

$\bm{\mathcal{G}}:SP\widetilde{\mathcal{H}}_{T}\longrightarrow
SP\widehat{\mathcal{H}}_{T}\leavevmode\nobreak\ ,$ (155)

and it satisfies

$\mu\circ(\bm{\mathcal{G}}\boxtimes\mathds{1}_{SP\widehat{\mathcal{H}}_{T}})\circ(\bm{\Pi}\boxtimes\bm{\Pi})=\bm{\Pi}\circ\mu\circ(\bm{\mathcal{G}}\boxtimes\mathds{1}_{S\widehat{\mathcal{H}}_{T}})\leavevmode\nobreak\
.$ (156)

Thus by homological perturbation Theorem B, we obtain the perturbed SDR:

$\rotatebox[origin={c}]{270.0}{${{{{\circlearrowright}}}}$}\leavevmode\nobreak\
(-\widetilde{\bm{h}})\left(S\widehat{\mathcal{H}}_{T},\bm{\ell^{\mathcal{G}}}\right)\underset{\widetilde{\mathbf{I}}}{\stackrel{{\scriptstyle\widetilde{\bm{\Pi}}}}{{\mathrel{\begin{subarray}{c}\xrightarrow{\rule{56.9055pt}{0.0pt}}\\\\[-3.87495pt]
\xleftarrow{\rule{56.9055pt}{0.0pt}}\end{subarray}}}}}\left(SP\widehat{\mathcal{H}}_{T},\bm{\Pi
Q_{B}}\mathbf{I}+\delta\widetilde{\bm{\ell^{\mathcal{G}}}}\right),$ (157)

where

$\displaystyle\delta\widetilde{\bm{\ell}}=(\bm{\Pi}\boxtimes\bm{\Pi})\delta\bm{\ell}\frac{1}{\mathds{1}_{S\widehat{\mathcal{H}}_{T}}+\bm{h}\delta\bm{\ell^{\mathcal{G}}}}\mathbf{I},$
(158)
$\displaystyle\widetilde{\bm{h}}=\bm{h}-\bm{h}\delta\bm{\ell^{\mathcal{G}}}\frac{1}{\mathds{1}_{S\widehat{\mathcal{H}}_{T}}+\bm{h}\delta\bm{\ell^{\mathcal{G}}}}\bm{h},$
$\displaystyle\widetilde{\mathbf{I}}=\mathbf{I}-\bm{h}\delta\bm{\ell^{\mathcal{G}}}\frac{1}{\mathds{1}_{S\widehat{\mathcal{H}}_{T}}+\bm{h}\delta\bm{\ell^{\mathcal{G}}}}\mathbf{I},$
$\displaystyle\widetilde{\bm{\Pi}}=\bm{\Pi}-\bm{\Pi}\delta\bm{\ell^{\mathcal{G}}}\frac{1}{\mathds{1}_{S\widehat{\mathcal{H}}_{T}}+\bm{h}\delta\bm{\ell^{\mathcal{G}}}}\bm{h}.$

and

$\delta\widetilde{\bm{\ell^{\mathcal{G}}}}=\mu\circ(\bm{\mathcal{G}}\boxtimes\mathds{1}_{SP\widehat{\mathcal{H}}_{T}})\circ\delta\widetilde{\bm{\ell}}.$
(159)

Using the definition

$\frac{1}{1+A}=\sum_{n=0}^{\infty}(-A)^{n},$ (160)

we get

$\displaystyle\widetilde{\bm{h}}$
$\displaystyle=\frac{1}{\mathds{1}_{S\widehat{\mathcal{H}}_{T}}+\bm{h}\delta\bm{\ell^{\mathcal{G}}}}\bm{h},$
(161) $\displaystyle\widetilde{\mathbf{I}}$
$\displaystyle=\frac{1}{\mathds{1}_{S\widehat{\mathcal{H}}_{T}}+\bm{h}\delta\bm{\ell^{\mathcal{G}}}}\mathbf{I},$
$\displaystyle\widetilde{\bm{\Pi}}$
$\displaystyle=\bm{\Pi}\frac{1}{\mathds{1}_{S\widehat{\mathcal{H}}_{T}}+\delta\bm{\ell^{\mathcal{G}}}\bm{h}}.$

Since (157) is an SDR, we have

$\left(\widetilde{\bm{\ell^{\mathcal{G}}}}\right)^{2}=\frac{1}{2}\left[\widetilde{\bm{\ell^{\mathcal{G}}}},\widetilde{\bm{\ell^{\mathcal{G}}}}\right]=0,$
(162)

where $\widetilde{\bm{\ell^{\mathcal{G}}}}=\bm{\Pi
Q_{B}}\mathbf{I}+\delta\widetilde{\bm{\ell^{\mathcal{G}}}}$. The effective
products are given by

$\widetilde{\ell}_{1}=PQ_{B},\quad\widetilde{\ell}_{n}:=\mu\circ(\pi_{1}\boxtimes\pi_{0})\circ\delta\widetilde{\bm{\ell}}\circ\pi_{n},\quad
n>1\leavevmode\nobreak\ ,$ (163)

where the image $\widehat{\mathcal{H}}^{\wedge 0}\cong\mathbb{C}$ of $\pi_{0}$
is considered as the space $\widetilde{\mathcal{H}}^{\wedge
0}\cong\mathbb{C}$. (162) then implies that the effective products
$\widetilde{\ell}_{n}$ satisfies the strong homotopy Jacobi identity. It is
also straightforward to check that $\widetilde{\ell}_{n}$ defined above agrees
with (119). As in (Erbin:2020eyc, , Appendix B.2.3), we have

$\begin{split}\Psi(\psi)&=\pi_{1}\widetilde{\mathbf{I}}e^{\wedge\psi}=\pi_{1}\frac{1}{\mathds{1}_{S\widehat{\mathcal{H}}_{T}}+\bm{h}\delta\bm{\ell^{\mathcal{G}}}}\mathbf{I}e^{\wedge\psi}\\\
&=\psi-\frac{1}{2!}h\ell_{2}^{\mathcal{G}}(\psi,\psi)-\frac{1}{3!}h\ell_{3}^{\mathcal{G}}(\psi,\psi,\psi)+\frac{2}{(2!)^{2}}h\ell_{2}^{\mathcal{G}}\left(h\ell_{2}^{\mathcal{G}}(\psi,\psi),\psi\right)+\\\
&-\frac{1}{4!}h\ell_{4}^{\mathcal{G}}(\psi,\psi,\psi,\psi)+\frac{2}{2!3!}h\ell_{2}^{\mathcal{G}}\left(h\ell_{3}^{\mathcal{G}}(\psi,\psi,\psi),\psi\right)+\frac{3}{2!3!}h\ell_{3}^{\mathcal{G}}\left(h\ell_{2}^{\mathcal{G}}(\psi,\psi),\psi,\psi\right)+\\\
&-\frac{1}{(2!)^{3}}h\ell_{2}^{\mathcal{G}}\left(h\ell_{2}^{\mathcal{G}}(\psi,\psi),h\ell_{2}^{\mathcal{G}}(\psi,\psi)\right)-\frac{2^{2}}{(2!)^{3}}h\ell_{2}^{\mathcal{G}}\left(h\ell_{2}^{\mathcal{G}}\left(h\ell_{2}^{\mathcal{G}}(\psi,\psi),\psi\right),\psi\right)+O\left(\psi^{5}\right).\end{split}$
(164)

Thus we have shown that
$\left(P\widehat{\mathcal{H}}_{T},\\{\widetilde{\ell}_{n}\\}_{n=1}^{\infty},\mathcal{G}\right)$
is a $\mathcal{G}$-twisted $L_{\infty}$-superalgebra.

The maps $\widetilde{\mathbf{I}},\widetilde{\bm{\Pi}}$ can be shown to be
cohomomorphisms provided that we can prove that

$\Delta_{S\widehat{\mathcal{H}}}(\bm{h}\delta\bm{\ell^{\mathcal{G}}})^{k}\mathbf{I}=\sum_{n=0}^{k}\left[(\bm{h}\delta\bm{\ell^{\mathcal{G}}})^{n}\mathbf{I}\boxtimes(\bm{h}\delta\bm{\ell^{\mathcal{G}}})^{k-n}\mathbf{I}\right]\Delta_{SP\widehat{\mathcal{H}}}.$
(165)

An order-by-order proof of (165) appeared in (Erbin:2020eyc, , Appendix B).
Since $\bm{\mathcal{G}}$ commutes with $\mathbf{\Pi}$, using (165), one can
prove that $\delta\widetilde{\bm{\ell^{\mathcal{G}}}}$ and hence
$\widetilde{\bm{\ell^{\mathcal{G}}}}$ is a coderivation. The effective action
can be written using the effective string product as (cf. (146)):

$S^{\text{eff}}=\frac{1}{g_{s}^{2}}\left[-\frac{1}{2}\langle\tilde{\omega}|\widetilde{\psi}\otimes
Q_{B}\mathcal{G}\widetilde{\psi}+\langle\tilde{\omega}|\widetilde{\psi}\otimes
Q_{B}\psi+\int_{0}^{1}dt\leavevmode\nobreak\
\langle\tilde{\omega}|\pi_{1}\bm{\partial_{t}}e^{\wedge\psi(t)}\otimes\mu\circ(\pi_{1}\boxtimes\pi_{0})\circ\delta\widetilde{\bm{\ell}}e^{\wedge\psi(t)}\right]\leavevmode\nobreak\
.$ (166)

Unpacking this reproduces the action (124).

Acknowledgements. RKS would like to thank Ashoke Sen for suggesting this
problem and numerous discussions on superstring field theory. RKS would also
like to thank Carlo Maccaferri for comments which improved the presentation of
the paper. RKS thanks ICTS, Bangalore for hospitality where this work began.
The work of RKS was supported by the US Department of Energy under grant DE-
SC0010008.

##  Appendix A (Symmetrized) Tensor coalgebra

We begin by describing the general construction of the symmetrized tensor
coalgebra, the reader is referred to bourbaki1998algebra  for more details.

Throughout this section $\mathbb{F}$ is field of characteristic $\neq 2$. A
coalgebra over a field $\mathbb{F}$ is an $\mathbb{F}$-vector space $C$ along
with $\mathbb{F}$-linear maps

$\Delta:C\longrightarrow C\otimes
C,\quad\varepsilon:C\longrightarrow\mathbb{F}\leavevmode\nobreak\ ,$ (167)

called coproduct and counit respectively, such that

$\begin{split}&(\mathds{1}_{C}\otimes\Delta)\circ\Delta=(\Delta\otimes\mathds{1}_{C})\circ\Delta\leavevmode\nobreak\
,\\\
&(\mathds{1}_{C}\otimes\varepsilon)\circ\Delta=\mathds{1}_{C}=(\varepsilon\otimes\mathds{1}_{C})\circ\Delta.\end{split}$
(168)

The first and second relations in (168) are called coassociativity and counit
property respectively. Let $(C,\Delta,\varepsilon)$ and
$(C^{\prime},\Delta^{\prime},\varepsilon^{\prime})$ be two coalgebras. Then a
cohomomorphism between $(C,\Delta,\varepsilon)$ and
$(C^{\prime},\Delta^{\prime},\varepsilon^{\prime})$ is an $\mathbb{F}$-linear
map $f:C\longrightarrow C^{\prime}$ such that

$\Delta^{\prime}\circ f=(f\otimes f)\circ\Delta,\quad\varepsilon^{\prime}\circ
f=\varepsilon,$ (169)

that is, the following diagram commutes:

$\tikzcd C&{C^{\prime}}C{C^{\prime}}\\\ {C\otimes C}{C^{\prime}\otimes
C^{\prime}}\mathbb{F}\arrow[from=1-1,to=1-3]\arrow["\Delta"^{\prime},from=1-1,to=2-1]\arrow[from=2-1,to=2-3]\arrow["{\Delta^{\prime}}",from=1-3,to=2-3]\arrow["f",from=1-5,to=1-7]\arrow["\varepsilon"^{\prime},from=1-5,to=2-6]\arrow["{\varepsilon^{\prime}}",from=1-7,to=2-6]$

The identity map $\mathds{1}_{C}$ is clearly a cohomomorphism. An
$\mathbb{F}$-linear endomorphism $d:C\longrightarrow C$ is called a
coderivation if it satisfies the co-Leibnitz rule

$\Delta\circ d=(d\otimes\mathds{1}_{C}+\mathds{1}_{C}\otimes d)\circ\Delta.$
(170)

The space of all coderivations forms a vector space which we denote by
$\text{Coder}(V)$. An element $c\in C$ is called a group-like element if

$\Delta(c)=c\otimes c,\quad\varepsilon(c)=1.$ (171)

The canonical example of coalgebra associated to a vector space is the tensor
coalgebra. Let $V$ be an $\mathbb{F}$-vector space and let $TV$ denote the
tensor algebra

$TV=\bigoplus_{n=0}^{\infty}V^{\otimes n},$ (172)

where $V^{\otimes 0}=\mathbb{F}\mathbf{1}_{TV}\cong\mathbb{F}$ and
$\mathbf{1}_{TV}$ should be considered as the identity of the tensor algebra.
The coproduct is given by

$\begin{split}\Delta_{TV}:TV&\longrightarrow TV\boxtimes TV\\\ v_{1}\otimes
v_{2}\otimes\dots\otimes v_{n}\mapsto\Delta_{TV}(v_{1}\otimes
v_{2}\otimes\dots\otimes v_{n})&:=\sum_{k=0}^{n}(v_{1}\otimes\dots\otimes
v_{k})\boxtimes(v_{k+1}\otimes\dots\otimes v_{n})\end{split}$ (173)

where we have used $\boxtimes$ to denote an “external” tensor product required
to define the coproduct and differentiate it from the “internal” tensor
product already used in defining the $TV$. The counit is given by

$\begin{split}&\varepsilon:TV\longrightarrow\mathbb{F},\\\ a\mapsto
a\textbf{1}_{TV},\leavevmode\nobreak\ \leavevmode\nobreak\ v\mapsto
0,\quad&a\textbf{1}_{TV}\in T^{0}V=\mathbb{F}\textbf{1}_{TV},\quad
v\in\bigoplus_{n=1}^{\infty}T^{n}V.\end{split}$ (174)

This notion of tensor coalgebra is useful to describe the open string field
theory Gaberdiel:1997ia ; Kajiura:2003ax ; Kajiura:2004xu ; Erbin:2020eyc .

We now come to the main example relevant for our purposes. We will assume that
$V$ is $\mathbb{Z}$-graded. Let $SV$ denote the symmetrized tensor algebra
defined by (see Subsection 2.3 for notations)

$SV=\bigoplus_{n=0}^{\infty}V^{\wedge n}.$ (175)

Here $V^{\wedge 0}=\mathbb{F}\mathbf{1}_{SV}\cong\mathbb{F}$. Here as before
$\mathbf{1}_{SV}$ should be considered as the identity of the symmetric tensor
algebra. The coproduct is given by

$\begin{split}\Delta_{SV}:&SV\longrightarrow SV\boxtimes
SV\leavevmode\nobreak\ ,\\\ \Delta_{SV}\left(v_{1}\wedge\ldots\wedge
v_{n}\right)&=\sum_{\begin{subarray}{c}i,j\in\mathbb{N}_{0}\\\
i+j=n\end{subarray}}\sum_{\sigma\in
S_{n}}\frac{\chi(\sigma,v_{1},\dots,v_{n})}{i!j!}\\\ &\hskip
28.45274pt\times\left(v_{\sigma(1)}\wedge\ldots\wedge
v_{\sigma\left(i\right)}\right)\boxtimes\left(v_{\sigma(i+1)}\wedge\ldots\wedge
v_{\sigma\left(i+j\right)}\right)\\\
&=\sum_{\begin{subarray}{c}i,j\in\mathbb{N}_{0}\\\
i+j=n\end{subarray}}\sum_{\sigma\in\operatorname{UnShuff}(i,j)}\chi\left(\sigma,v_{1},\cdots,v_{n}\right)\\\
&\hskip 28.45274pt\times\left(v_{\sigma(1)}\wedge\cdots\wedge
v_{\sigma(i)}\right)\boxtimes\left(v_{\sigma(i+1)}\wedge\cdots\wedge
v_{\sigma(n)}\right)\end{split}$ (176)

where $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$ and $\boxtimes$ again denotes the
external tensor product. The counit is same as for the tensor coalgebra (174).
It turns out that the vector space Coder($SV$) of coderivations of $SV$ is
isomorphic to the vector space of linear maps Hom$(SV,V)$. The precise map is
given by the following theorem. The following map

$\begin{split}\mathrm{Hom}(SV,V)&\longrightarrow\mathrm{Coder}(SV),\\\
f&\longmapsto\mu\circ(f\boxtimes\mathds{1}_{SV})\circ\Delta_{SV}\leavevmode\nobreak\
,\end{split}$ (177)

where $\mu:SV\boxtimes SV\longrightarrow SV$ is the multiplication map, is an
isomorphism of vector spaces with the inverse given by

$\begin{split}\mathrm{Coder}(SV)&\longrightarrow\mathrm{Hom}(SV,V)\\\
d&\longmapsto\pi_{1}\circ d\leavevmode\nobreak\ ,\end{split}$ (178)

where

$\begin{split}&\pi_{n}:SV\longrightarrow V^{\wedge n}.\end{split}$ (179)

is the projection of $SV$ to the $n^{\text{th}}$-homogenous component
$V^{\wedge n}$. Linear maps in $\mathrm{Hom}(SV,V)$ are called symmetric
multilinear maps.

The sum of the $n$-ary brackets $\ell_{n}^{g}:V^{\wedge n}\longrightarrow V$
for101010Recall that $\ell_{1}^{g}=\ell_{1}$. $n\geq 1$

$\ell^{g}=\sum_{n=1}^{\infty}\ell_{n}^{g}\leavevmode\nobreak\ ,$ (180)

gives us a linear map $\ell^{g}:SV\longrightarrow V$. Then by Theorem A, we
get a coderivation $\bm{\ell^{g}}:SV\longrightarrow SV$. Then the strong
homotopy Jacobi identity is equivalent to the equation

$(\bm{\ell^{g}})^{2}=\frac{1}{2}[\bm{\ell^{g}},\bm{\ell^{g}}]=0\leavevmode\nobreak\
,$ (181)

where $[\cdot,\cdot]$ is the graded commutator defined by (43) (see Remark
2.3).

##  Appendix B Homological perturbation lemma

In this section, we review the definitions of (strong) deformation retracts
and the homological perturbation lemma. We follow crainic2004perturbation ,
(Erbin:2020eyc, , Appendix A) for this description. Let $V,W$ be two
$\mathbb{Z}$-graded vector spaces. A homotopy equivalence (HE) is a
pair111111The general notion of homotopy equivalence is defined for arbitrary
pair of complexes. We will consider the pair $(V,d_{V})$ as a (co)chain
complex with a single nontrivial (co)chain group. $(V,d_{V})$ and $(W,d_{W})$
with degree-odd linear maps $d_{V}:V\longrightarrow V$ and
$d_{W}:W\longrightarrow W$ along with two linear maps $\pi:V\longrightarrow W$
and $\iota:W\longrightarrow V$ satisfying the following conditions

  1. 1.

$d_{V}^{2}=d_{W}^{2}=0$ and $\pi,\iota$ are quasi-isomorphisms.121212A quasi-
isomorphism is a chain map between (co)chain complexes which descends to an
isomorphism of (co)homology. Here $\pi,\iota$ are isomorphisms of the
(co)homology on $V,W$ induced by $d_{V},d_{W}$ respectively.

  2. 2.

There exists a degree-odd linear map $\eta:V\longrightarrow V$ called the
homotopy map satisfying the Hodge-Kodaira decomposition

$\iota\pi-\mathds{1}_{V}=\eta d_{V}+d_{V}\eta.$ (182)

A deformation retract (DR) is an HE with the requirement
$\pi\iota=\mathds{1}_{W}$. A strong deformation retract (SDR) is a DR such
that the homotopy map $\eta$ satisfies the annihilation conditions

$\eta^{2}=\eta\iota=\pi\eta=0.$ (183)

We will represent an HE, DR or SDR by the compact diagram

$\rotatebox[origin={c}]{270.0}{${{{{\circlearrowright}}}}$}\leavevmode\nobreak\
\eta\left(V,d_{V}\right)\underset{\iota}{\stackrel{{\scriptstyle\pi}}{{\mathrel{\begin{subarray}{c}\xrightarrow{\rule{56.9055pt}{0.0pt}}\\\\[-3.87495pt]
\xleftarrow{\rule{56.9055pt}{0.0pt}}\end{subarray}}}}}\left(W,d_{W}\right)$
(184)

and specify whether it is an HE, DR or an SDR. The condition that $\pi,\iota$
be quasi-isomorphisms automatically require them to be (co)chain maps. In
particular, they must satisfy

$\displaystyle d_{W}\pi$ $\displaystyle=\pi d_{V},$ (185) $\displaystyle\iota
d_{W}$ $\displaystyle=d_{V}\iota.$

The homological perturbation lemma gives us a recipe to construct a new SDR
from a given SDR. We are interested in the following version of the
homological perturbation lemma. Let

$\rotatebox[origin={c}]{270.0}{${{{{\circlearrowright}}}}$}\leavevmode\nobreak\
\eta\left(V,d_{V}\right)\underset{\iota}{\stackrel{{\scriptstyle\pi}}{{\mathrel{\begin{subarray}{c}\xrightarrow{\rule{56.9055pt}{0.0pt}}\\\\[-3.87495pt]
\xleftarrow{\rule{56.9055pt}{0.0pt}}\end{subarray}}}}}\left(W,d_{W}\right)$
(186)

be an HE (resp. DR, resp. SDR), $V^{\prime},W^{\prime}$ vector spaces and

$\begin{split}&\delta_{V}:V\longrightarrow V^{\prime}\otimes V,\quad
g:V^{\prime}\otimes V\longrightarrow V\\\
&\pi^{\prime}:V^{\prime}\longrightarrow W^{\prime},\quad
g^{\prime}:W^{\prime}\otimes W\longrightarrow W\leavevmode\nobreak\
,\end{split}$ (187)

be linear maps satisfying the following conditions:

  1. 1.

$\tilde{d}_{V}^{2}=(d_{V}+g\delta_{V})^{2}=0$.

  2. 2.

The operator $(\mathds{1}_{V}-\eta g\delta_{V})$ is invertible.

  3. 3.

$g^{\prime}(\pi^{\prime}\otimes\pi)=\pi g$.

Then the perturbed data

$\rotatebox[origin={c}]{270.0}{${{{{\circlearrowright}}}}$}\leavevmode\nobreak\
\tilde{\eta}\left(V,\tilde{d}_{V}\right)\underset{\tilde{\iota}}{\stackrel{{\scriptstyle\tilde{\pi}}}{{\mathrel{\begin{subarray}{c}\xrightarrow{\rule{56.9055pt}{0.0pt}}\\\\[-3.87495pt]
\xleftarrow{\rule{56.9055pt}{0.0pt}}\end{subarray}}}}}\left(W,\tilde{d}_{W}\right)\leavevmode\nobreak\
,$ (188)

is again an HE (resp. DR, resp. SDR) with the following definitions:

$\displaystyle\delta_{W}$
$\displaystyle=(\pi^{\prime}\otimes\pi)\delta_{V}\frac{1}{\mathds{1}_{V}-\eta
g\delta_{V}}\iota,$ (189) $\displaystyle\tilde{\iota}$
$\displaystyle=\iota+\eta g\delta_{V}\frac{1}{\mathds{1}_{V}-\eta
g\delta_{V}}\iota,$ $\displaystyle\tilde{\pi}$ $\displaystyle=\pi+\pi
g\delta_{V}\frac{1}{\mathds{1}_{V}-\eta g\delta_{V}}\eta,$
$\displaystyle\tilde{\eta}$ $\displaystyle=\eta+\eta
g\delta_{V}\frac{1}{\mathds{1}_{V}-\eta g\delta_{V}}\eta,$

and $\tilde{d}_{W}=d_{W}+\tilde{g}\delta_{W}$, where

$(\mathds{1}_{V}-\eta g\delta_{V})^{-1}=\frac{1}{\mathds{1}_{V}-\eta
g\delta_{V}}=\sum_{n=0}^{\infty}(\eta g\delta_{V})^{n}.$ (190)

The map $\delta_{V}$ is called a _perturbation_ of the HE (resp. DR, resp.
SDR) (186). The proof is identical to (Erbin:2020eyc, , Appendix A.3), and
crainic2004perturbation  with perturbation $g\delta_{V}$.

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