# Open string field theory with stubs

Martin Schnabl and Georg Stettinger

###### Abstract

There are various reasons why adding stubs to the vertices of open string
field theory (OSFT) is interesting: Not only the stubs can tame certain
singularities and make the theory more well-behaved, but also the new theory
shares a lot of similarities with closed string field theory, which helps to
improve our understanding of its structure and possible solutions. In this
paper we explore two natural ways of implementing stubs into the framework of
OSFT, resulting in an $A_{\infty}$-algebra giving rise to infinitely many
vertices. We find two distinct consistent actions, both generated by a field
redefinition, interestingly sharing the same equations of motion. In the last
section we illustrate their relationship and physical meaning by applying our
construction to nearly marginal solutions.

_CEICO, Institute of Physics of the Czech Academy of Sciences,_

_Na Slovance 2, 182 00 Prague 8, Czech Republic_

###### Contents

  1. 1 Introduction and motivation
  2. 2 Deforming the Witten action using stubs
    1. 2.1 Elements of homological perturbation theory
    2. 2.2 Transferring algebraic structure: SDR-case
    3. 2.3 Transferring algebraic structure: Non-SDR-case
    4. 2.4 Higher products using tree diagrams
    5. 2.5 Proof of cyclicity to all orders
    6. 2.6 Geometric picture
  3. 3 Analytic solutions and action(s)
    1. 3.1 Projection cohomomorphism from the HPL
    2. 3.2 The action
    3. 3.3 Elements from closed string field theory
    4. 3.4 Constructing the field redefinition
    5. 3.5 Proof of the ansatz for the infinitesimal field redefinition
    6. 3.6 Finite field redefinition
  4. 4 Physical interpretation
    1. 4.1 Nearly marginal solutions
    2. 4.2 The $A_{\infty}$-action
  5. 5 Conclusion and outlook

##  1 Introduction and motivation

It is a well-known fact that the algebraic and geometric structures of open
and closed string field theory are fundamentally different. The OSFT action
(Witten action) consists of a standard kinetic term and a single cubic
interaction and reads [1]

$S\left(\Psi\right)=\frac{1}{2}\omega\left(\Psi,Q\Psi\right)+\frac{1}{3}\omega\left(\Psi,\Psi\ast\Psi\right).$
(1)

The algebraic ingredients are a nilpotent differential given by the BRST-
operator, the star product and a symplectic form $\omega$, which all together
form a cyclic differential graded algebra. In the following, we will use the
coalgebra notation of [2] with
$m_{2}\left(\Psi_{1},\Psi_{2}\right):=\left(-\right)^{d\left(\Psi_{1}\right)}\Psi_{1}\ast\Psi_{2}$
and the shifted degree given by $d\left(\Psi\right)=gh\left(\Psi\right)+1$.
Now the defining algebraic properties are

$\displaystyle Q^{2}$ $\displaystyle=0,$ (2) $\displaystyle
Qm_{2}\left(\Psi_{1},\Psi_{2}\right)$
$\displaystyle=-m_{2}\left(Q\Psi_{1},\Psi_{2}\right)-\left(-\right)^{d\left(\Psi_{1}\right)}m_{2}\left(\Psi_{1},Q\Psi_{2}\right),$
(3) $\displaystyle
m_{2}\left(\Psi_{1},m_{2}\left(\Psi_{2},\Psi_{3}\right)\right)$
$\displaystyle=-\left(-\right)^{d\left(\Psi_{1}\right)}m_{2}\left(m_{2}\left(\Psi_{1},\Psi_{2}\right),\Psi_{3}\right),$
(4)
$\displaystyle\omega\left(\Psi_{1},m_{2}\left(\Psi_{2},\Psi_{3}\right)\right)$
$\displaystyle=-\left(-\right)^{d\left(\Psi_{1}\right)}\omega\left(m_{2}\left(\Psi_{1},\Psi_{2}\right),\Psi_{3}\right),$
(5) $\displaystyle\omega\left(\Psi_{1},Q\Psi_{2}\right)$
$\displaystyle=-\left(-\right)^{d\left(\Psi_{1}\right)}\omega\left(Q\Psi_{1},\Psi_{2}\right).$
(6)

In contrast, the classical closed string field theory (CSFT) action [3]
contains infinitely many vertices and reads

$S\left(\Psi\right)=\frac{1}{2}\omega^{\prime}\left(\Psi,Q\Psi\right)+\sum_{n=3}^{\infty}\frac{\kappa^{n-2}}{n!}\left\\{\Psi^{n}\right\\},$
(7)

where the vertices

$\left\\{\Psi^{n}\right\\}=\underset{\mathcal{V}{}_{0,n}}{\int}\Omega_{\Psi^{n}}^{\left(0\right)0,n}$
(8)

are given by integrating certain differential forms over vertex regions
$\mathcal{V}{}_{0,n}$ in the moduli space of $n$-punctured Riemann surfaces.
At quantum level there are also vertices of higher genus to consider, i. e.
intrinsic loops already present within the elementary vertices. However, they
will not be of interest here for now.

The vertices also can be decomposed into a symplectic form $\omega^{\prime}$
111The symplectic form $\omega^{\prime}$ differs from the $\omega$ contained
in the Witten action by an additional insertion of $c_{0}^{-}.$ and higher
products $l_{n}$:

$\left\\{\Psi^{n+1}\right\\}=\omega^{\prime}\left(\Psi,l_{n}\left(\Psi,\Psi,...,\Psi\right)\right).$
(9)

In contrast to the open string star product, the $l_{n}$ are all totally
symmetric. The objects $Q=:l_{1},$ all higher $l_{n}$ and $\omega$ give rise
to a cyclic $L_{\infty}$-algebra:

$\displaystyle
l_{n}l_{1}\left(\Psi_{1},...,\Psi_{n}\right)+l_{n-1}l_{2}\left(\Psi_{1},...,\Psi_{n}\right)+...+l_{1}l_{n}\left(\Psi_{1},...,\Psi_{n}\right)=0,\,\,\,\,\,\,\,\,\,\forall
n$ (10)
$\displaystyle\omega^{\prime}\left(\Psi_{1},l_{n}\left(\Psi_{2},...,\Psi_{n+1}\right)\right)=-\left(-\right)^{d\left(\Psi_{1}\right)}\omega^{\prime}\left(l_{n}\left(\Psi_{1},...,\Psi_{n}\right),\Psi_{n+1}\right).\,\,\,\,\,\,\,\,\,\forall
n$ (11)

The multiplication of the multilinear products in the first line is defined as

$l_{k}l_{l}\left(\Psi_{1},...,\Psi_{k+l-1}\right)=\sum_{\sigma}\frac{\left(-1\right)^{\epsilon\left(\sigma\right)}}{l!\left(k-1\right)!}l_{k}\left(l_{l}\left(\Psi_{\sigma\left(1\right)},...,\Psi_{\sigma\left(l\right)}\right),\Psi_{\sigma\left(l+1\right)},...,\Psi_{\sigma\left(k+l-1\right)}\right),$
(12)

where the sum runs over all permutations of $k+l-1$ elements and the factor
$\left(-1\right)^{\epsilon\left(\sigma\right)}$ is just the sign picked up
when permuting the $\Psi$s.

The different algebraic structures have a geometric origin: For a unitary
theory it is necessary that the Feynman string diagrams computed from the
action cover the full moduli space of $n$-punctured Riemann surfaces exactly
once. For the open string where one needs to consider surfaces with boundaries
it has been shown by Zwiebach [4] that this is indeed the case for the action
(1) with only one cubic interaction. In contrast, for the closed string, i. e.
surfaces without boundaries, such a simple cubic representation of the action
seems not to be possible, see [5].

The simplicity of the OSFT action allowed for the discovery of analytic
classical solutions, most importantly the open string tachyon vacuum [6]. In
classical CSFT, no such analytic solutions have been found yet [7, 8]. To make
some effort into this direction, we want to propose the following strategy: If
CSFT cannot be simplified easily, maybe we can make OSFT “more complicated” in
the sense that its algebraic and geometric structure resembles that of CSFT?
There are two motivations for that: First, it could help to make the very
abstract language of CSFT more intuitive and tractable. Second, if we can
translate the known analytic solutions to the new, deformed OSFT, it could
give an idea how CSFT solutions might look like. In total there are three
steps to do:

  1. 1.

Find a consistent deformation of the Witten OSFT such that its mathematical
structure resembles CSFT

  2. 2.

Find analytic solutions of this deformed theory

  3. 3.

Make an educated guess for analytic solutions of CSFT

The first two tasks will be worked out in this paper while the third one is
left for the future.

The deformation we will consider has many additional interesting aspects:
First, OSFT exhibits some issues with singularities or ill-defined quantities:
For example, there exist “identity-based” solutions [9] (for a recent
discussion see [10]) which solve the equations of motion but do not have a
well defined action. There are hints that these solutions could be better
behaved in the deformed theory. Moreover, the deformed products
$M_{2},M_{3},\dots$ defined in the next section could allow for a natural
definition of a Banach type $A_{\infty}$-algebra over the space of string
fields.

The paper is organized as follows: In the second section the deformation using
stubs [11] is discussed in detail, resulting in an explicit definition of the
higher products of the $A_{\infty}$-algebra. The necessary mathematical
ingredients, namely homological perturbation theory and homotopy transfer, are
introduced as well. As a side result, we give a consistent method of applying
homotopy transfer to general homotopy equivalences which are not deformation
retracts. Section three deals with analytical solutions and the action of the
stubbed theory. Surprisingly, we find two consistent actions with the same
equations of motion, both generated by a field redefinition. The two field
redefinitions, one coming from homological perturbation theory and one
motivated by closed string field theory, are derived explicitly. The last
section is devoted to the physical interpretation of our results. We apply the
whole construction to a suitable class of solutions and compute the action up
to first non-trivial order in the field expansion.

##  2 Deforming the Witten action using stubs

We have seen that the algebraic structure of CSFT is that of an
$L_{\infty}$-algebra, so naively we should try to find an $L_{\infty}$-based
deformation of OSFT. However, this would imply a commutative product of open
strings, which would hence be fundamentally different from the Witten product.
We still want the deformation be equivalent to Witten theory, so the best we
can do is to aim for an $A_{\infty}$-algebra with products $M_{n}$ obeying

$M_{n}M_{1}\left(\Psi_{1},\ldots,\Psi_{n}\right)+M_{n-1}M_{2}\left(\Psi_{1},\ldots,\Psi_{n}\right)+\ldots+M_{1}M_{n}\left(\Psi_{1},\ldots,\Psi_{n}\right)=0\,\,\,\,\,\,\,\,\,\forall
n$ (13)

and the multiplication given by

$\displaystyle
M_{k}M_{l}\left(\Psi_{1},\ldots,\Psi_{k+l-1}\right)=M_{k}\left(M_{l}\left(\Psi_{1},\ldots,\Psi_{l}\right),\Psi_{l+1},\ldots,\Psi_{k+l-1}\right)+$
$\displaystyle+\left(-\right)^{d\left(\Psi_{1}\right)}M_{k}\left(\Psi_{1},M_{l}\left(\Psi_{2},\ldots,\Psi_{l+1}\right),\Psi_{l+2},\ldots,\Psi_{k+l-1}\right)+$
$\displaystyle\cdots+$
$\displaystyle+\left(-\right)^{d\left(\Psi_{1}\right)+\ldots+d\left(\Psi_{k-1}\right)}M_{k}\left(\Psi_{1},\ldots,\Psi_{k-1},M_{l}\left(\Psi_{k},\ldots,\Psi_{k+l-1}\right)\right).$
(14)

From now on we will use the tensor coalgebra notation of [2] where the
$A_{\infty}$-relations take the simple form

$\sum_{k=1}^{n}\mathbf{M_{k}M_{n+1-k}}=0.$ (15)

Geometrically, to mimic the situation of CSFT, we would like to have a three-
vertex that does _not_ give rise to a full moduli space cover, such that
higher vertices are necessary. One modification that achieves both of these
requirements is to attach _stubs_ on the three vertex, which means, small
pieces of propagating strings on each input. In this way, there will appear
some regions of moduli space that are not covered by Feynman diagrams and
hence have to be taken over by elementary vertices. The higher products which
will give rise to those vertices will indeed form an $A_{\infty}$-algebra.

The Hamiltonian of our CFT which generates time evolution is $L_{0}$, so the
operator which inserts a strip of length $\lambda$ into a string diagram is
$e^{-\lambda L_{0}}$. The natural modification of the star product which
attaches stubs symmetrically on all three inputs is then given by [12]:

$m_{2}\left(\cdot,\cdot\right)\rightarrow
M_{2}\left(\cdot,\cdot\right)=e^{-\lambda L_{0}}m_{2}\left(e^{-\lambda
L_{0}}\cdot,e^{-\lambda L_{0}}\cdot\right).$ (16)

$M_{2}$ is cyclic with respect to the simplectic form $\omega$ and
$Q$-invariant:

$\omega\left(\Psi_{1},M_{2}\left(\Psi_{2},\Psi_{3}\right)\right)=-\left(-\right)^{d\left(\Psi_{1}\right)}\omega\left(M_{2}\left(\Psi_{1},\Psi_{2}\right),\Psi_{3}\right),$
(17)
$QM_{2}\left(\Psi_{1},\Psi_{2}\right)=-M_{2}\left(Q\Psi_{1},\Psi_{2}\right)-\left(-\right)^{d\left(\Psi_{1}\right)}M_{2}\left(\Psi_{1},Q\Psi_{2}\right).$
(18)

Those relations follow from $\left[Q,L_{0}\right]=0$ and the fact that $L_{0}$
is BPZ-even. It is easy to see that $M_{2}$ is not associative as expected for
an $A_{\infty}$-algebra:

$\displaystyle e^{-\lambda L_{0}}m_{2}\left(e^{-\lambda
L_{0}}\Psi_{1},e^{-2\lambda L_{0}}m_{2}\left(e^{-\lambda
L_{0}}\Psi_{2},e^{-\lambda L_{0}}\Psi_{3}\right)\right)$
$\displaystyle+\,e^{-\lambda L_{0}}m_{2}\left(e^{-2\lambda
L_{0}}m_{2}\left(e^{-\lambda L_{0}}\Psi_{1},e^{-\lambda
L_{0}}\Psi_{2}\right),e^{-\lambda L_{0}}\Psi_{3}\right)$ $\displaystyle\neq
0.$ (19)

The task is now to explicitly determine all the higher products and prove that
they satisfy all relevant properties. This will be done in two ways, first in
a formal algebraic way using homological perturbation theory and second in a
more heuristic way using tree diagrams.

###  2.1 Elements of homological perturbation theory

The starting point of homological perturbation theory (HPT) is a _chain
homotopy equivalence_. Let $V$, $W$ be two chain complexes, i. e. graded
vector spaces with a nilpotent differential of degree one denoted by
$d_{V}\left(d_{W}\right)$. Furthermore, we are given two chain maps
$p:\,V\rightarrow W$ and $i:\,W\rightarrow V$ of degree zero as well as a
homotopy map $h:V\rightarrow V$ of degree minus one obeying the relations

$\displaystyle pd_{V}$ $\displaystyle=d_{W}p$ (20) $\displaystyle id_{W}$
$\displaystyle=d_{V}i$ (21) $\displaystyle ip-1$
$\displaystyle=hd_{V}+d_{V}h.$ (22)

This ensures that $i$ and $p$ as well as the combination $ip$ leave cohomology
classes invariant. The most common case in literature is that of a _special
deformation retract_(SDR) where $W$ is isomorphic to a subspace of $V$, see
e.g. [13]. The maps $i$ and $p$ are then given by the canonical inclusion and
projection, respectively. It follows trivially that $pi=1$ and additionally
the following _annihilation conditions_ are demanded _:_

$hh=0,\,\,\,\,\,\,hi=0,\,\,\,\,\,\,ph=0.$ (23)

It can be shown that any deformation retract with $pi=1$ can be made an SDR by
a redefinition of the maps [14].

Lets assume that the differential $d_{V}$ is perturbed by some $\delta$ of
degree one such that $\left(d_{V}+\delta\right)^{2}=0$ again. The _homological
perturbation lemma_(HPL) now states that one can construct a new SDR using the
perturbed differential with the other maps given by

$\displaystyle D_{V}$ $\displaystyle=d_{V}+\delta$ (24) $\displaystyle D_{W}$
$\displaystyle=d_{W}+p\delta\left(1-h\delta\right)^{-1}i$ (25) $\displaystyle
I$ $\displaystyle=\left(1-h\delta\right)^{-1}i$ (26) $\displaystyle P$
$\displaystyle=p\left(1-\delta h\right)^{-1}$ (27) $\displaystyle H$
$\displaystyle=\left(1-h\delta\right)^{-1}h.$ (28)

The expression $\left(1-h\delta\right)^{-1}$ requires some explanation:
Usually it should be understood as a geometric series

$\left(1-h\delta\right)^{-1}=1+h\delta+h\delta h\delta+h\delta h\delta
h\delta+...\,\,\,\,\,\,\,\,\,.$ (29)

We demand for the validity of the HPL that this series either converges or
terminates, which will be the case in our examples. The proof of the lemma
including all relevant relations is straight forward but tedious and can be
found in [14, 2].

###  2.2 Transferring algebraic structure: SDR-case

Our main purpose for using the HPL will be transferring some algebraic
structure from one side of the homotopy equivalence to the other. Lets
consider an SDR with an associative multiplication defined on $V$. It is a
natural question to ask if there exists some “induced product” defined on $W$:
For example, for $a,$ $b$ $\in$ $W$ the product $ia\cdot ib$ in $V$ need not
be in $W$ anymore. The only thing that could be done is to project it to $W$,
i. e. define $a\cdot b\mid_{W}=p\left(ia\cdot ib\right)$, but this product
will in general not be associative anymore. What happens is that an
associative algebra on one side of the equivalence becomes an
$A_{\infty}$-algebra after transferring to the other side. A formal way to
construct all the higher products is given by the so-called _tensor trick:_

One defines a new SDR over the tensor coalgebra222See the appendix for a short
summary on tensor coalgebras.  of $V$ ($W$) with the tensorial versions of the
maps given by

$\displaystyle\mathbf{d_{V\left(W\right)}}=\underset{n=1}{\overset{\infty}{\sum}}\underset{k=0}{\overset{n-1}{\sum}}1^{\otimes
k}\otimes d_{V\left(W\right)}\otimes 1^{\otimes n-k-1}$ (30)
$\displaystyle\mathbf{i}=\underset{n=1}{\overset{\infty}{\sum}}i^{\otimes n}$
(31) $\displaystyle\mathbf{p}=\underset{n=1}{\overset{\infty}{\sum}}p^{\otimes
n}$ (32)
$\displaystyle\mathbf{h}=\underset{n=1}{\overset{\infty}{\sum}}\underset{k=0}{\overset{n-1}{\sum}}1^{\otimes
k}\otimes h\otimes\left(ip\right)^{\otimes n-k-1}.$ (33)

$\mathbf{d_{V\left(W\right)}}$ is the standard tensor coderivation associated
to $d_{V\left(W\right)}$, see (138), $\mathbf{i}$ and $\mathbf{p}$ are
cohomomorphisms while $\mathbf{h}$ is defined somewhat asymmetrically which
will turn out to be crucial for the formalism to work. One can check directly
that those definitions indeed give rise to an SDR again. The product on $V$,
denoted by $m_{2}$, can now be treated as a perturbation $\delta$ of the
coderivative $\mathbf{d_{V}}$,

$\mathbf{D_{V}}=\mathbf{d_{V}}+\mathbf{m_{2}},$ (34)

where $\mathbf{m_{2}}$ is the coderivation associated to $m_{2}$.
$\mathbf{D_{V}}^{2}=0$ will be fulfilled as a consequence of $m_{2}$ being
associative and $d_{V}$ obeying the Leibniz rule. According to the HPL we will
be given a new, perturbed complex with the maps

$\displaystyle\mathbf{\mathbf{D_{W}}=d_{W}+pm_{2}\left(1-hm_{2}\right)^{-1}i}$
(35)
$\displaystyle\mathbf{\boldsymbol{I}}=\mathbf{\left(1-hm_{2}\right)^{-1}i}$
(36)
$\displaystyle\mathbf{\boldsymbol{P}}=\mathbf{\boldsymbol{p}\left(1-m_{2}h\right)^{-1}}$
(37)
$\displaystyle\mathbf{\boldsymbol{H}}=\mathbf{\boldsymbol{h}\left(1-hm_{2}\right)^{-1}},$
(38)

where $\mathbf{D_{W}}$ squares to zero and therefore defines an
$A_{\infty}$-algebra on $W$. The higher products can be obtained by expanding
$\mathbf{D_{W}}$, i. e. projecting onto $n$ inputs and one output:

$M_{n}=\pi_{1}\mathbf{D_{W}}\pi_{n}.$ (39)

For example we get

$M_{2}\left(\cdot,\cdot\right)=pm_{2}\left(i\cdot,i\cdot\right)$ (40)

as already anticipated above.

###  2.3 Transferring algebraic structure: Non-SDR-case

We want to apply those concepts now to the problem of attaching stubs in OSFT.
The space $V$ should be given by the space of string fields
$\mathcal{H}_{BCFT}$ with its grading
$d\left(\Psi\right)=gh\left(\Psi\right)+1$ and the differential $Q.$ By
comparing (16) with (40) it seems natural to define

$i=p=e^{-\lambda L_{0}}.$ (41)

However, we see immediately that this definition does not give rise to an SDR:
$i$ and $p$ are not an inclusion and projection anymore and $pi\neq 1.$ Still,
in principle the HPL holds for arbitrary homotopy equivalences, so there is a
chance to succeed anyway.

Our choice is so far completely symmetric, so lets define
$W=V=\mathcal{H}_{BCFT},$ $d_{W}=d_{V}=Q$ and $h_{W}=h_{V}=h$ where we have to
solve

$e^{-2\lambda L_{0}}-1=hQ+Qh$ (42)

The simplest (although not unique) solution for $h$ is

$h=\frac{e^{-2\lambda L_{0}}-1}{L_{0}}b_{0}.$ (43)

It is important to stress that $h$ is well-behaved also for $L_{0}=0$ and does
not have any pole.

One could now try to proceed with the tensor trick as above and eventually
compute the map $\mathbf{D_{W}}$, but this runs into problems: Although
$\mathbf{D_{W}}^{2}=0$ still, as guaranteed by the HPL, $\mathbf{D_{W}}$ is
not a coderivation anymore! This means, it does not obey the co-Leibniz rule
(136). As explained in the Appendix of [15], the condition $pi=1$ as well as
the annihilation relations are necessary (and sufficient) for the tensor trick
to work. So there has to be some modification to account for that and it will
actually turn out to be surprisingly simple: We can just take the expression
we get for $\mathbf{D_{W}}$ and while expanding to calculate the higher
products, _pretend_ that all SDR-relations (23) are satisfied. More precisely,
we define an operator $P_{SDR}$ acting on the space of maps from
$T\mathcal{H}\rightarrow T\mathcal{H}$ which projects on maps in which all
SDR-relations hold. This means, every time that $pi$ occurs, it will be
replaced by $1$ and every time $hh,$ $hi$ or $ph$ occurs, the term will be
discarded:

$P_{SDR}\left(....hh....\right)=0,\,\,\,\,\,\,\,P_{SDR}\left(....hi....\right)=0,\,\,\,\,\,\,\,\,P_{SDR}\left(....ph....\right)=0,\,\,\,\,\,\,\,\,P_{SDR}\left(....pi....\right)=P_{SDR}\left(....1....\right)$

Now our new higher products will be given as

$M_{n}=P_{SDR}\pi_{1}\mathbf{D_{W}}\pi_{n}.$ (44)

Their associated coderivations can be added together to form a total map
called $\mathbf{M}$,

$\mathbf{M}=\underset{n=1}{\overset{\infty}{\sum}}\underset{m=n}{\overset{\infty}{\sum}}\underset{k=0}{\overset{m-n}{\sum}}1^{\otimes
k}\otimes M_{n}\otimes 1^{\otimes m-k-n}.$ (45)

It is a coderivation by construction and moreover it squares to zero because
in the proof of the HPL, where it is shown that $D_{W}^{2}=0$ (Eq. 1.2.16 of
[2]), the SDR-relations are never used. Since we know that
$\mathbf{D_{W}}^{2}=0$ independently of the SDR-relations, also
$\mathbf{M}^{2}=0$ and $\mathbf{M}$ defines the desired $A_{\infty}$-algebra.

As an example, lets explicitly calculate $M_{3}:$

$\displaystyle P_{SDR}\pi_{1}\mathbf{D_{W}}\pi_{3}$
$\displaystyle=P_{SDR}\pi_{1}\mathbf{pm_{2}h\mathbf{m_{2}}i}\pi_{3}=P_{SDR}\left(pm_{2}\left(1\otimes
h+h\otimes ip\right)\left(1\otimes m_{2}+m_{2}\otimes 1\right)i^{\otimes
3}\right)$
$\displaystyle=P_{SDR}\left(pm_{2}\left(i\cdot,hm_{2}\left(i\cdot,i\cdot\right)\right)-pm_{2}\left(m_{2}\left(i\cdot,i\cdot\right),hi\cdot\right)+pm_{2}\left(hi\cdot,ipm_{2}\left(i\cdot,i\cdot\right)\right)+pm_{2}\left(hm_{2}\left(i\cdot,i\cdot\right),ipi\cdot\right)\right)$
$\displaystyle=pm_{2}\left(i\cdot,hm_{2}\left(i\cdot,i\cdot\right)\right)+pm_{2}\left(hm_{2}\left(i\cdot,i\cdot\right),i\cdot\right).$
(46)

The second and third term in the second line contained an $hi$ and were
deleted whereas in the last term $ipi$ was replaced by $i$.

This whole procedure including the proof of the statement may seem quite
handwavy, however, there exists a more formal and precise way of arriving at
the same result using operad theory [16]. A combinatorial proof using tree
diagrams will be given in the next section.

###  2.4 Higher products using tree diagrams

The proposal is that $M_{n}$ is equal to the sum of all distinct, rooted, full
binary trees with $n$ leaves such that every leaf represents one input and the
root is the output. With every leaf there is one factor of $i$ associated,
with every node the product $m_{2}$, with every internal line $h$ and with the
root $p$. In [17] it is argued that this is true for SDRs, since we construct
the products in the same way as for an SDR, we conclude that the proposal also
holds for our non-SDR case.

In the tree language, the $A_{\infty}$-relations can be proven directly: The
commutator of $\mathbf{Q}$ with an $n$-leaved tree gives a sum of $n-2$ terms,
in which one of the $n-2$ internal lines $h$ is replaced by $1-ip$. The
$1$-terms actually cancel away because of the associativity of $m_{2}$: The
propagator associated with unity connects two nodes $m_{2}$ which leads to an
expression of the form $m_{2}\left(m_{2}\left(A,B\right),C\right)$ where
$A,B,C$ are three subtrees. In the sum there always exists a second tree with
another propagator turned into unity giving rise to the expression
$m_{2}\left(A,m_{2}\left(B,C\right)\right)$. These two trees cancel away such
that only the -$ip$-factors remain in total. Now the other terms occuring in
the relation

$-\left[\mathbf{Q,M_{n}}\right]=\mathbf{M_{2}M_{n-1}}+\mathbf{M_{3}M_{n-2}}+...+\mathbf{M_{n-1}M_{2}}$
(47)

can be interpreted as follows: If we project on one output,
$\pi_{1}\mathbf{M_{k}M_{n+1-k}}$ gives a sum of trees where one of the terms
in $M_{n+1-k}$ is connected with its root to one of the $k$ leaves of one of
the trees in $M_{k}$ and this is done in all possible combinations. The result
is a sum of trees with in total $n$ leaves, where one of the internal lines
does not contain $h$ but $ip$; the $i$ from the leaf of the left tree and the
$p$ from the root of the right tree. But that is exactly the same sum of terms
we have on the l.h.s., indeed it is easy to see that each tree occuring on the
r.h.s. must also occur on the l.h.s. and vice versa.

An interesting crosscheck of the $\pi_{1}$-projection of (47) can be done by
comparing the total number of trees on both sides: The number of full binary
trees with $n+1$ leaves is given by the Catalan number

$C_{n}=\frac{1}{n+1}\begin{pmatrix}2n\\\ n\end{pmatrix}.$ (48)

This means that on the l.h.s. there are $C_{n-1}$ trees in $M_{n}$ and $n-2$
internal lines that can be changed, hence a total of $C_{n-1}\left(n-2\right)$
trees. On the r.h.s. we have $2\cdot C_{1}C_{n-2}+3\cdot
C_{2}C_{n-3}+...+\left(n-1\right)C_{n-2}C_{1}$ trees in total, leading to the
equation

$\overset{n-1}{\underset{k=2}{\sum}}\left(n+1-k\right)C_{n-k}C_{k-1}=\left(n-2\right)C_{n-1}.$
(49)

The Catalan numbers fulfill the following useful recursive relations:

$C_{n+1}=\frac{2\left(2n+1\right)}{n+2}C_{n},\,\,\,\,\,\,\,\,\,\,C_{n+1}=\overset{n}{\underset{k=0}{\sum}}C_{n-k}C_{k}.$
(50)

Using them one can proceed by induction: Assuming equation (49) is valid for
some $n$, then

$\displaystyle\overset{n}{\underset{k=2}{\sum}}\left(n+2-k\right)C_{n+1-k}C_{k-1}$
$\displaystyle=$
$\displaystyle\,\overset{n}{\underset{k=2}{\sum}}\left(n+2-k\right)\frac{2\left(2n-2k+1\right)}{n-k+2}C_{n-k}C_{k-1}$
$\displaystyle=$
$\displaystyle\,4\overset{n-1}{\underset{k=2}{\sum}}\left(n+1-k\right)C_{n-k}C_{k-1}-2\overset{n-1}{\underset{k=2}{\sum}}C_{n-k}C_{k-1}+2C_{0}C_{n-1}$
$\displaystyle=$
$\displaystyle\left(4n-8\right)C_{n-1}-2\overset{n-2}{\underset{k=1}{\sum}}C_{n-1-k}C_{k}+2C_{n-1}$
$\displaystyle=$
$\displaystyle\left(4n-6\right)C_{n-1}-2\left(C_{n}-C_{0}C_{n-1}-C_{n-1}C_{0}\right)$
$\displaystyle=$
$\displaystyle\left(4n-2\right)\frac{n+1}{2\left(2n-1\right)}C_{n}-2C_{n}$
$\displaystyle=$ $\displaystyle\left(n-1\right)C_{n}$ (51)

as it should be to complete the induction. This shows that the number of terms
in the equation (47) is the same on both sides.

###  2.5 Proof of cyclicity to all orders

Using the tree language it is possible to prove that all higher products
$M_{n}$ are cyclic with respect to the BPZ-product. We have to show

$\omega\left(\Psi_{1},M_{n}\left(\Psi_{2},...,\Psi_{n+1}\right)\right)=-\left(-\right)^{d\left(\Psi_{1}\right)}\omega\left(M_{n}\left(\Psi_{1},...,\Psi_{n}\right),\Psi_{n+1}\right),$
(52)

hence we start with a sum of trees on the l.h.s. and use the BPZ-properties of
$m_{2}$ and $h$ as well as $p=i^{\dagger}=i$ to rewrite it as the sum of trees
on the r.h.s.. The explicit steps are:

  1. 1.

Take the $p$ from the root of the tree and write it to the left side of the
product where it can be interpreted as $i$, acting on $\Psi_{1}$.

  2. 2.

Take the $m_{2}$ from the root of the tree and use cyclicity of $m_{2}$ to
apply it on the first two arguments inside of $\omega$. This gives a sign
factor of $-\left(-\right)^{d\left(\Psi_{1}\right)}$. In general one will be
left with two subtrees then with $n+1$ leaves in total.

  3. 3.

Take the $h$ from the right subtree and use that it is BPZ-even to apply it on
the left subtree. This gives an additional sign factor of
$\left(-\right)^{d\left(left\,\,subtree\right)}$.

  4. 4.

Take the $m_{2}$ from the root of the right subtree and apply it on the first
two arguments inside of $\omega$. It gives a sign factor of
$-\left(-\right)^{d\left(left\,\,subtree\right)+1}$ (the $+1$ comes from the
$h$ that was shifted in step 3) which cancels the sign factor of step 3.
Again, one is left with two subtrees.

  5. 5.

Repeat steps 3 and 4 until the right subtree only consist of $i$ acting on one
input. The total sign factor remains
$-\left(-\right)^{d\left(\Psi_{1}\right)}$.

  6. 6.

Remove the $i$ acting on $\Psi_{n+1}$ and let it act as a $p$ on the left
subtree.

Now the left subtree fulfills all requirements to be an element of $M_{n}$ and
since we also have the right sign factor, we have obtained a term contained in
the r.h.s. of (52). The manipulations are all uniquely invertible so we can
conclude that the map between the trees is one-to-one and all terms we need
are constructed exactly once. As a result, Eq. (52) holds and all higher
products are cyclic. Moreover, as already suggested by the name, Eq. (52)
together with the antisymmetry of $\omega$ implies invariance of the vertices
under cyclic permutations.

###  2.6 Geometric picture

We have now shown that the higher products fulfill all the algebraic
requirements but we do not know anything yet about the geometric picture, if
they indeed give rise to a full single cover of the moduli space. To answer
this question, the tree description of the products turns out to be very
useful. Lets consider an arbitrary string tree diagram using the stubbed three
vertex: As long as the external states are on-shell, the stubs make no
difference because the external legs consist of a semiinfinite strip anyway.
On the internal lines instead, the stubs make a difference because all
internal strips with a length smaller than $2\lambda$ do not appear. We can
conclude that the additional elemantary vertices we need should consist of all
tree diagrams with all internal strips having a length smaller than
$2$$\lambda$. The Siegel-gauge string propagator in the Schwinger
parametrization is given by

$\int_{0}^{\infty}dt\,e^{-tL_{0}}b_{0}=\frac{b_{0}}{L_{0}}.$ (53)

The integral over $t$ can be thought of an integral over strips of propagating
strings of all different lengths. Following this logic, the propagators in our
new vertices should be given as

$\int_{0}^{2\lambda}dt\,e^{-tL_{0}}b_{0}=-\frac{e^{-2\lambda
L_{0}}-1}{L_{0}}b_{0}=-h$ (54)

and hence be equal to minus the homotopy!333It is interesting to notice at
this point that the simple but not unique choice of $h$ corresponds to
choosing Siegel gauge for the propagator. The minus sign is just a convention
and can be absorbed in the definition of $h$. We have constructed the higher
products by drawing all binary tree diagrams with Witten vertices, $h$ as
propagators and $e^{-\lambda L_{0}}$ on the leaves and the root. Those are in
one-to-one correspondence with all the Feynman tree diagrams that should make
up the new elementary vertices. This shows that our higher products $M_{n}$
indeed define a set of vertices which gives rise to a full cover of the moduli
space.

##  3 Analytic solutions and action(s)

###  3.1 Projection cohomomorphism from the HPL

Using the definition

$\mathbf{m}=\mathbf{Q}+\mathbf{m_{2}}$ (55)

the equations of motion of the original Witten theory can be written in
coalgebra language as

$\mathbf{m}\frac{1}{1-\Psi}=0,$ (56)

i. e. solutions are Maurer-Cartan elements of the $A_{2}$-algebra defining the
theory. In the same spirit, to find solutions of the deformed theory, we have
to solve the equation

$\mathbf{M}\frac{1}{1-\Psi}=0.$ (57)

In fact, the homological perturbation lemma already gave us an operator which
maps solutions of the Witten theory to solutions of the stubbed theory. This
can be seen as follows: The perturbed projection $\mathbf{P}$ is a chain map
and hence obeys

$\mathbf{Pm=MP.}$ (58)

Now lets assume $\Psi^{\ast}$ is a solution of the Witten theory, then

$\left(\Psi^{*}\right)^{\prime}=\pi_{1}\mathbf{P}\frac{1}{1-\Psi^{*}}$ (59)

obeys

$\mathbf{M}\frac{1}{1-\left(\Psi^{*}\right)^{\prime}}=\mathbf{M}\frac{1}{1-\pi_{1}\mathbf{P}\frac{1}{1-\Psi^{*}}}=\mathbf{MP}\frac{1}{1-\Psi^{*}}=\mathbf{Pm}\frac{1}{1-\Psi^{*}}=0,$
(60)

where Eq. (146) was used. It remains to determine the cohomorphism
$\mathbf{P}$ for the case where the homotopy equivalence is not an SDR. Again,
as for the higher products above, the simplest way is to take the expression
from the HPL

$\mathbf{P}_{HPL}=\mathbf{p}\left(\mathbf{1-m_{2}h}\right)^{-1}$ (61)

and apply the operator $P_{SDR}$ on its components to get

$P_{n}=P_{SDR}\pi_{1}\mathbf{P}_{HPL}\pi_{n}.$ (62)

The resulting maps can then be packaged into a cohomorphism $\mathbf{P}$
again. More explicitly, we get for the first few orders

$\displaystyle P_{1}$ $\displaystyle=p$ $\displaystyle P_{2}$
$\displaystyle=pm_{2}\left(\cdot,h\cdot\right)+pm_{2}\left(h\cdot,ip\cdot\right)$
$\displaystyle P_{3}$
$\displaystyle=pm_{2}\left(\cdot,hm_{2}\left(\cdot,h\cdot\right)\right)+pm_{2}\left(\cdot,hm_{2}\left(h\cdot,ip\cdot\right)\right)+pm_{2}\left(h\cdot,hm_{2}\left(ip\cdot,ip\cdot\right)\right)+pm_{2}\left(hm_{2}\left(\cdot,h\cdot\right),ip\cdot\right)$
$\displaystyle\,\,\,\,\,\,+pm_{2}\left(hm_{2}\left(h\cdot,ip\cdot\right),ip\cdot\right)+pm_{2}\left(h\cdot,ipm_{2}\left(\cdot,h\cdot\right)\right)+pm_{2}\left(h\cdot,ipm_{2}\left(h\cdot,ip\cdot\right)\right)$
(63) $\displaystyle...\,\,\,\,\,\,\,\,\,\,\,\,\,\,.$

One can now check the equation $\pi_{1}\mathbf{Pm}=\pi_{1}\mathbf{MP}$ order
by order:

$\displaystyle\pi_{1}\mathbf{Pm}\pi_{1}$
$\displaystyle=pQ=Qp=\pi_{1}\mathbf{MP}\pi_{1}$
$\displaystyle\pi_{1}\mathbf{Pm}\pi_{2}$
$\displaystyle=P_{2}\left(Q\cdot,\cdot\right)+P_{2}\left(\cdot,Q\cdot\right)+pm_{2}$
$\displaystyle=-pm_{2}\left(Q\cdot,h\cdot\right)+pm_{2}\left(hQ\cdot,ip\cdot\right)+pm_{2}\left(\cdot,hQ\cdot\right)+pm_{2}\left(h\cdot,ipQ\cdot\right)+pm_{2}$
$\displaystyle=-pm_{2}\left(Q\cdot,h\cdot\right)-pm_{2}\left(Qh\cdot,ip\cdot\right)+pm_{2}\left(\left(ip-1\right)\cdot,ip\cdot\right)$
$\displaystyle\,\,\,\,\,-pm_{2}\left(\cdot,Qh\cdot\right)+pm_{2}\left(\cdot,\left(ip-1\right)\cdot\right)+pm_{2}\left(h\cdot,Qip\cdot\right)+pm_{2}$
$\displaystyle=Qpm_{2}\left(\cdot,h\cdot\right)+Qpm_{2}\left(h\cdot,ip\cdot\right)+pm_{2}\left(ip\cdot,ip\cdot\right)$
$\displaystyle=QP_{2}+M_{2}\left(p\cdot,p\cdot\right)=\pi_{1}\mathbf{MP}\pi_{2}.$
(64)

For order three the calculation is already very tedious but it also turns out
to work. The important point is that in the manipulations that are necessary,
the SDR-relations were never used. This implies, since we know that (63) works
for SDRs, it also works in our case and (63) is a valid definition. Now we
have a cohomorphism by construction that obeys the chain map relation (58)
such that we can construct analytic solutions of the deformed theory.444It is
actually a non-trivial counting problem to determine the number of terms of
$P_{n}$. It grows faster than the Catalan numbers because at order three we
have seven terms and at order four already 33.

###  3.2 The action

The on-shell action is one of the most important observables in OSFT, for
example for the tachyon vacuum its value is equal to minus the energy of the
D-brane which has decayed. Since the stubbed theory should be physically
equivalent to the original Witten theory, we expect that the values for the
on-shell action we get in the two theories coincide. The Witten action can be
written in coalgebra notation [2] as

$S\left(\Psi\right)=\int_{0}^{1}dt\,\omega\left(\pi_{1}\boldsymbol{\partial_{t}}\frac{1}{1-\Psi\left(t\right)},\pi_{1}\mathbf{m}\frac{1}{1-\Psi\left(t\right)}\right),$
(65)

where $\Psi\left(t\right)$ is any smooth interpolation between
$\Psi\left(0\right)=0$ and $\Psi\left(1\right)=\Psi$ and
$\boldsymbol{\partial_{t}}$ the coderivation associated to $\partial_{t}$.
Similarly, the stubbed action reads

$S^{\prime}\left(\Psi\right)=\int_{0}^{1}dt\,\omega\left(\pi_{1}\boldsymbol{\partial_{t}}\frac{1}{1-\Psi\left(t\right)},\pi_{1}\mathbf{M}\frac{1}{1-\Psi\left(t\right)}\right).$
(66)

We would expect now a relation

$S^{\prime}\left(\left(\Psi^{*}\right)^{\prime}\right)\stackrel{{\scriptstyle?}}{{=}}S\left(\Psi^{*}\right)$
(67)

with $\Psi^{*}$ a MC-element of $\mathbf{m}.$ However, this relation turns out
to be non-obvious: Instead

$\displaystyle S\left(\Psi\right)$
$\displaystyle=\int_{0}^{1}dt\,\omega\left(\pi_{1}\boldsymbol{\partial_{t}}\frac{1}{1-\Psi\left(t\right)},\pi_{1}\mathbf{m}\frac{1}{1-\Psi\left(t\right)}\right)=\int_{0}^{1}dt\,\omega\left(\pi_{1}\boldsymbol{\partial_{t}}\mathbf{P^{-1}P}\frac{1}{1-\Psi\left(t\right)},\pi_{1}\mathbf{P^{-1}MP}\frac{1}{1-\Psi\left(t\right)}\right)$
$\displaystyle=\int_{0}^{1}dt\,\omega\left(\pi_{1}\boldsymbol{\partial_{t}}\mathbf{P^{-1}}\frac{1}{1-\pi_{1}\mathbf{P}\frac{1}{1-\Psi\left(t\right)}},\pi_{1}\mathbf{P^{-1}M}\frac{1}{1-\pi_{1}\mathbf{P}\frac{1}{1-\Psi\left(t\right)}}\right)$
$\displaystyle=\int_{0}^{1}dt\,\omega\left(\pi_{1}\mathbf{P^{-1}}\boldsymbol{\partial_{t}}\frac{1}{1-\left(\Psi\right)^{\prime}\left(t\right)},\pi_{1}\mathbf{P^{-1}M}\frac{1}{1-\left(\Psi\right)^{\prime}\left(t\right)}\right)=:\tilde{S}\left(\Psi^{\prime}\right),$
(68)

where the last line differs from $S^{\prime}\left(\Psi^{\prime}\right)$ by the
insertions of $\mathbf{P^{-1}}$ on both inputs of $\omega$.

The invertibility of $\mathbf{P}$ is actually a delicate question: In general,
a cohomomorphism is invertible iff its linear component, i. e. $P_{1}=p$ is
invertible. Now $p=e^{-\lambda L_{0}}$ inserts a strip of length $\lambda$, so
one would expect the inverse $e^{\lambda L_{0}}$ to remove a strip of length
$\lambda$ from the world sheet, which is not always possible. On the other
hand, $e^{\lambda L_{0}}$ makes sense on a string field expanded in
eigenstates of $L_{0}$ as long as the eigenvalues are finite. From now on, we
shall assume that this is the case and $e^{\lambda L_{0}}$ is well-defined on
all string fields in question.

Explicitly, the inversion of $\mathbf{P}$ works as follows:
$\mathbf{PP^{-1}}=\mathbf{P^{-1}P}$ should be equal to the identity
cohomomorphism, which is identity in its linear component and zero in all
higher components. One can now solve the components
$P_{n}^{-1}=\pi_{1}\mathbf{P^{-1}}\pi_{n}$ order by order:

$\displaystyle P_{1}^{-1}$ $\displaystyle=p^{-1}$ $\displaystyle P_{2}^{-1}$
$\displaystyle=-m_{2}\left(p^{-1}\cdot,hp^{-1}\cdot\right)-m_{2}\left(hp^{-1}\cdot,i\cdot\right)$
$\displaystyle P_{3}^{-1}$
$\displaystyle=-m_{2}\left(hp^{-1}\cdot,hm_{2}\left(i\cdot,i\cdot\right)\right)+m_{2}\left(m_{2}\left(p^{-1}\cdot,hp^{-1}\cdot\right),hp^{-1}\cdot\right)+m_{2}\left(m_{2}\left(hp^{-1}\cdot,i\cdot\right),hp^{-1}\cdot\right)$
$\displaystyle...\,\,\,\,\,\,\,\,\,\,\,\,\,.$ (69)

One can now write out in more detail:

$\displaystyle\tilde{S}\left(\Psi\right)=\,$
$\displaystyle\frac{1}{2}\omega\left(p^{-1}\Psi,p^{-1}Q\Psi\right)$
$\displaystyle+\,$
$\displaystyle\frac{1}{3}\omega\left(p^{-1}\Psi,p^{-1}M_{2}\left(\Psi,\Psi\right)\right)+\frac{1}{3}\omega\left(p^{-1}\Psi,P_{2}^{-1}\left(\left(Q\Psi,\Psi\right)+\left(\Psi,Q\Psi\right)\right)\right)+\frac{1}{3}\omega\left(P_{2}^{-1}\left(\Psi,\Psi\right),p^{-1}Q\Psi\right)$
$\displaystyle+\,$
$\displaystyle\frac{1}{4}\omega\left(p^{-1}\Psi,p^{-1}M_{3}\left(\Psi,\Psi,\Psi\right)\right)+\frac{1}{4}\omega\left(p^{-1}\Psi,P_{2}^{-1}\left(\left(M_{2}\left(\Psi,\Psi\right),\Psi\right)+\left(\Psi,M_{2}\left(\Psi,\Psi\right)\right)\right)\right)$
$\displaystyle+\,$
$\displaystyle\frac{1}{4}\omega\left(p^{-1}\Psi,P_{3}^{-1}\left(\left(Q\Psi,\Psi,\Psi\right)+\left(\Psi,Q\Psi,\Psi\right)+\left(\Psi,\Psi,Q\Psi\right)\right)\right)+\frac{1}{4}\omega\left(P_{2}^{-1}\left(\Psi,\Psi\right),p^{-1}M_{2}\left(\Psi,\Psi\right)\right)$
$\displaystyle+\,$
$\displaystyle\frac{1}{4}\omega\left(P_{2}^{-1}\left(\Psi,\Psi\right),P_{2}^{-1}\left(\left(Q\Psi,\Psi\right)+\left(\Psi,Q\Psi\right)\right)\right)+\frac{1}{4}\omega\left(P_{3}^{-1}\left(\Psi,\Psi,\Psi\right),p^{-1}Q\Psi\right)+\mathcal{O}\left(\Psi^{\otimes
5}\right).$ (70)

It seems that the cohomomorphism $\mathbf{P}$ does not define the field
redefinition we were looking for, instead it relates the Witten theory to a
theory defined by $\tilde{S}\left(\Psi\right)$. The equations of motion
derived from $\tilde{S}$ are the same as for $S^{\prime}$, namely
$\mathbf{M}\frac{1}{1-\Psi}=0$, but it is not at all obvious that the two
actions agree even on-shell.

The reason is that the HPL does not know anything about the symplectic form
$\omega$: To get an invariant action with
$S^{\prime}\left(\Psi^{\prime}\right)\stackrel{{\scriptstyle}}{{=}}S\left(\Psi\right)$,
not only $\mathbf{m}$ has to transform accordingly, but also $\omega$ would
have to go to

$\omega^{\prime}\left(\Psi_{1},\Psi_{2}\right)=\omega\left(\pi_{1}\mathbf{P^{-1}}\frac{1}{1-\Psi_{1}},\pi_{1}\mathbf{P^{-1}}\frac{1}{1-\Psi_{2}}\right),$
(71)

otherwise $\mathbf{P}$ would fail to be cyclic. As it can be seen from (68),
$\tilde{S}\left(\Psi\right)$ is just a fancy rewriting of the Witten action
and therefore defines an equivalent theory. But since we would like to keep
the original $\omega$, the field redefinition induced by $\mathbf{P}$ from the
HPL and giving rise to $\tilde{S}\left(\Psi\right)$ is not exactly what we
want. However, since it shares the same equations of motion as
$S^{\prime}\left(\Psi\right)$, it might provide a new family of gauge-
invariant observables for solutions of $S^{\prime}\left(\Psi\right)$,
parametrized by $\lambda.$ In the last section we will check this explicitly
on a special class of solutions. For now the next task is to derive the
originally desired field redefinition which relates the actions
$S\left(\Psi\right)$ and $S^{\prime}\left(\Psi\right)$. 555It would also be an
interesting direction to examine if there exists some kind of “dual” HPL that
directly yields this correct field redefinition.

###  3.3 Elements from closed string field theory

In [18], Zwiebach and Hata have shown how to relate slightly different,
consistent sets of vertices in CSFT via an infinitesimal field redefinition.
Our strategy is now to apply their method to our problem in OSFT and integrate
the result to the finite case. This will not only provide us the field
redefinition we are looking for, but also give some insight into the rather
abstract formalism of CSFT. First, it is useful to collect some basic
information about the structure of CSFT.

As already explained in the introduction, the vertices are given by
integrating _basic differential forms_
$\Omega_{\Psi_{1}\Psi_{2}...\Psi_{n}}^{g,n}$ defined by

$\Omega_{\Psi_{1}\Psi_{2}...\Psi_{n}}^{g,n}\left(\hat{V}_{1},\hat{V}_{2},...,\hat{V}_{6g-6+2n}\right)=\left(2\pi
i\right)^{-\left(3g-3+n\right)}\langle\Sigma|\mathbf{b\left(v_{1}\right)}...\mathbf{b\left(v_{6g-6+2n}\right)}|\Psi_{1}\rangle...|\Psi_{n}\rangle.$
(72)

They are living in the tangent space of the fibre bundle
$\mathcal{\hat{P}}_{g,n}$ over the moduli space $\mathcal{M}_{g,n}$, with the
fiber being the space of local coordinates around the punctures modulo phase
rotations. The dimension of this bundle is infinite, but the degree of
$\Omega_{\Psi_{1}\Psi_{2}...\Psi_{n}}^{g,n}$ is just the real dimension of the
base space $\mathcal{M}_{g,n}$. It takes as arguments tangent vectors
$\hat{V}_{i}\in T\mathcal{\hat{P}}_{g,n}$, which represent deformations of the
world sheet Riemann surface $\Sigma$, either by changing the moduli or the
local coordinates. The $\mathbf{v_{i}}$ are Schiffer vectors on $\Sigma$,
supported around the punctures, which generate those deformations. This means
that the local coordinate around the $n$th puncture transforms as

$z^{\left(n\right)}\rightarrow z^{\left(n\right)}+\epsilon
v^{\left(n\right)}\left(z^{\left(n\right)}\right)$ (73)

for some small $\epsilon$. The $b$-ghost insertions are then defined as

$\mathbf{b\left(v\right)}=\sum_{i=1}^{n}\left(\oint\frac{dz_{i}}{2\pi
i}b\left(z_{i}\right)v^{\left(i\right)}\left(z_{i}\right)+\oint\frac{d\bar{z}_{i}}{2\pi
i}\bar{b}\left(\bar{z}_{i}\right)\bar{v}^{\left(i\right)}\left(\bar{z}_{i}\right)\right).$
(74)

We will be only interested in the classical action without the loop vertices,
so the genus $g$ shall be zero from now on. The basic forms can now be
integrated over sections of $\mathcal{\hat{P}}_{0,n}$ defining the vertices
$\mathcal{V}{}_{0,n}$.

The quantization procedure makes use of the Batalin-Vilkovisky formalism;
although we are not interested in quantum effects, the BV-antibracket is used
in constructing the symmetry generator. It is defined as

$\left\\{A,B\right\\}=\frac{\partial_{r}A}{\partial\Psi^{i}}\frac{\partial_{l}B}{\partial\Psi_{i}^{*}}-\frac{\partial_{r}A}{\partial\Psi_{i}^{*}}\frac{\partial_{l}B}{\partial\Psi^{i}},$
(75)

where the $\Psi_{i}^{*}$ are antifields of opposite parity associated to each
basis element of $\mathcal{H}$. The BV-master action takes the same form as
(7) with the only difference that the $\Psi$ are not restricted in ghost
number and run over fields as well as antifields.

###  3.4 Constructing the field redefinition

We are looking for a non-linear field redefinition of the form

$\Psi^{\prime}=F\left(\Psi\right)=\stackrel{{\scriptstyle[}}{{n}}=1]{\infty}{\sum}F_{n}\left(\Psi^{\otimes
n}\right)=\pi_{1}\mathbf{F}\frac{1}{1-\Psi}$ (76)

that relates the Witten action to the stubbed $A_{\infty}$-action in the form
(66). To be consistent with the results of Zwiebach and Hata [18] we demand

$S\left(\Psi^{\prime}\right)=S^{\prime}\left(\Psi\right).$ (77)

Since the kinetic term is identical, we immediately find

$F_{1}\left(\Psi\right)=\Psi.$ (78)

In [18] it is shown that under an infinitesimal field redefinition of the form

$\Psi\rightarrow\Psi+t\left\\{\Psi,e\right\\}+\mathcal{O}\left(t^{2}\right),$
(79)

the classical action transforms as

$S\left(\Psi\right)\rightarrow
S\left(\Psi\right)+t\left\\{S,e\right\\}+\mathcal{O}\left(t^{2}\right),$ (80)

where $\left\\{\right\\}$ denotes the BV-antibracket. In the paper it is now
argued that for any small change of vertices, the change of the action indeed
takes this form and the generator $e$ is constructed explicitly:

$e\left(u_{0}\right)=-\sum_{n}\kappa^{n-2}\frac{1}{n!}\underset{\mathcal{V}_{0,n}\left(u_{0}\right)}{\int}\Omega_{\mathbf{b\left(u\right)}\Psi^{\otimes
n}}^{\left(0\right)0,n}.$ (81)

Here we assume that there exists some family of consistent vertex sets
$\mathcal{V}_{0,n}\left(u\right)$ parametrized by some real number $u$ and
everything is evaluated at the point $u_{0}$. The vector $\mathbf{u}$ is some
Schiffer vector which generates a deformation of the
$\mathcal{V}_{0,n}\left(u_{0}\right)$ in the direction of $u$, i. e. it
generates diffeomorphisms which push $\mathcal{V}_{0,n}\left(u_{0}\right)$
into $\mathcal{V}_{0,n}\left(u_{0}+\delta u\right)$. For the case of varying
the stub length, this Schiffer vector takes a particular simple form: First,
lets notice that the stub length $\lambda$ for closed strings is defined as
the geodesic distance from the location $\mid z\mid=1$ of the local coordinate
to the begin of the semiinfinite cylinder associated with the puncture. This
implies that $\lambda$ can be changed by just rescaling the coordinate:
Sending $z$ to $z^{\prime}=z+\epsilon z$, the location of $\mid
z^{\prime}\mid=1$ corresponds to $\mid z\mid=1-\epsilon$, such that $\lambda$
is increased by $\epsilon.$ By comparing with Eq. (73) we read off

$u^{\left(i\right)}=z^{\left(i\right)}.$ (82)

The $b$-ghost insertion is then given by

$\mathbf{b\left(u\right)}=\sum_{i=1}^{n}\left(\oint\frac{dz_{i}}{2\pi
i}z_{i}b\left(z_{i}\right)+\oint\frac{d\bar{z}_{i}}{2\pi
i}\bar{z}_{i}\bar{b}\left(\bar{z}_{i}\right)\right)=\sum_{i=1}^{n}b_{0}^{\left(i\right)}+\overline{b}_{0}^{\left(i\right)}=\sum_{i=1}^{n}b_{0}^{+\left(i\right)}.$
(83)

We want to use the above expression for $e$ in the context of changing the
stub length for open strings, hence a few modifications and simplifications
are necessary: First, the combinatorial factor $n!$ originates from total
symmetrization of the vertices and is not necessary for open strings. Second,
the insertions of $b_{0}^{+}$ should get replaced simply by $b_{0}$ since
there is no antiholomorphic sector. Moreover, the string coupling $\kappa$
will be set to one. Now the generator simplifies to

$e\left(\lambda\right)=-\sum_{n}\underset{\mathcal{V}\left(\lambda\right)_{0,n}}{\int}\Omega_{\mathbf{b_{0}}\Psi^{\otimes
n}}^{0,n}.$ (84)

If we make the ansatz

$\Psi^{\prime}=\Psi+\delta\lambda\stackrel{{\scriptstyle[}}{{n}}=2]{\infty}{\sum}f_{n}\left(\Psi^{\otimes
n}\right)$ (85)

as the infinitesimal version of (76), then the $f_{n}$ are determined as

$f_{n}\left(\Psi^{\otimes n}\right)=\left\\{\Psi,e^{\left(n\right)}\right\\},$
(86)

where $\delta\lambda$ plays the role of $t$ in (79).

To find $f_{2}\left(\Psi,\Psi\right)$ we need to consider
$e\left(\lambda\right)$ for $n=3$: The vertex
$\mathcal{V}\left(\lambda\right)_{0,3}$ is zero dimensional, so there is no
integral and the surface state $\langle\Sigma|$ is just the Witten vertex with
stubs of length $\lambda$,

$\langle
V_{3}\left(\lambda\right)|=\omega\left(\cdot,M_{2}\left(\cdot,\cdot\right)\right)=\omega\left(e^{-\lambda
L_{0}}\cdot,e^{-\lambda L_{0}}\cdot\ast e^{-\lambda L_{0}}\cdot\right).$ (87)

Inserting into (84) yields

$e^{\left(3\right)}\left(\lambda,\Psi\right)=-\left(\omega\left(b_{0}\Psi,M_{2}\left(\Psi,\Psi\right)\right)+\omega\left(\Psi,M_{2}\left(b_{0}\Psi,\Psi\right)\right)+\omega\left(\Psi,M_{2}\left(\Psi,b_{0}\Psi\right)\right)\right).$
(88)

The BV-bracket with $\Psi$ can be straightforwardly evaluated; after carefully
checking the signs the result is

$f_{2}\left(\Psi,\Psi\right)=-b_{0}M_{2}\left(\Psi,\Psi\right)+M_{2}\left(b_{0}\Psi,\Psi\right)+M_{2}\left(\Psi,b_{0}\Psi\right).$
(89)

We expect now the relation

$S^{\prime}\left(\Psi+\delta\lambda
f_{2}\left(\Psi,\Psi\right),\lambda\right)=S^{\prime}\left(\Psi,\lambda+\delta\lambda\right)$
(90)

to hold up to order 3 in $\Psi$; by directly inserting we can compute
explicitly

$\displaystyle S^{\prime}\left(\Psi+\delta\lambda
f_{2}\left(\Psi,\Psi\right),\lambda\right)$
$\displaystyle=\frac{1}{2}\omega\left(\Psi,Q\Psi\right)+\frac{1}{3}\omega\left(\Psi,M_{2}\left(\Psi,\Psi\right)\right)+\delta\lambda\,\omega\left(\Psi,Qf_{2}\left(\Psi,\Psi\right)\right)+\mathcal{O}\left(\Psi^{\otimes
4}\right).$ (91)

The last and most interesting term yields

$\displaystyle\left(-\omega\left(\Psi,Qb_{0}M_{2}\left(\Psi,\Psi\right)\right)+\omega\left(\Psi,QM_{2}\left(b_{0}\Psi,\Psi\right)\right)+\omega\left(\Psi,QM_{2}\left(\Psi,b_{0}\Psi\right)\right)\right)\delta\lambda$
$\displaystyle=$
$\displaystyle\left(-\omega\left(\Psi,L_{0}M_{2}\left(\Psi,\Psi\right)\right)+\omega\left(\Psi,b_{0}QM_{2}\left(\Psi,\Psi\right)\right)-\omega\left(Q\Psi,M_{2}\left(b_{0}\Psi,\Psi\right)\right)-\omega\left(Q\Psi,M_{2}\left(\Psi,b_{0}\Psi\right)\right)\right)\delta\lambda$
$\displaystyle=$
$\displaystyle-\omega\left(\Psi,L_{0}M_{2}\left(\Psi,\Psi\right)\right)\delta\lambda,$
(92)

where the last three terms in the second line cancel after applying the
Leibniz rule and cyclicity. On the other hand,

$\displaystyle S^{\prime}\left(\Psi,\lambda+\delta\lambda\right)=$
$\displaystyle\,S^{\prime}\left(\Psi,\lambda\right)+\delta\lambda\frac{d}{d\lambda}S^{\prime}\left(\Psi,\lambda\right)=\frac{1}{2}\omega\left(\Psi,Q\Psi\right)+\frac{1}{3}\omega\left(\Psi,M_{2}\left(\Psi,\Psi\right)\right)$
$\displaystyle-\frac{1}{3}\delta\lambda\,\omega\left(L_{0}\Psi,M_{2}\left(\Psi,\Psi\right)\right)-\frac{1}{3}\delta\lambda\,\omega\left(\Psi,M_{2}\left(L_{0}\Psi,\Psi\right)\right)-\frac{1}{3}\delta\lambda\,\omega\left(\Psi,M_{2}\left(\Psi,L_{0}\Psi\right)\right)+\mathcal{O}\left(\Psi^{\otimes
4}\right)$ $\displaystyle=$
$\displaystyle\,\frac{1}{2}\omega\left(\Psi,Q\Psi\right)+\frac{1}{3}\,\omega\left(\Psi,M_{2}\left(\Psi,\Psi\right)\right)-\delta\lambda\,\omega\left(L_{0}\Psi,M_{2}\left(\Psi,\Psi\right)\right)+\mathcal{O}\left(\Psi^{\otimes
4}\right),$ (93)

which is the same expression up to order $\Psi^{\otimes 3}$ (Again, cyclicity
was used in the last line.).

The explicit form of $f_{2}\left(\Psi,\Psi\right)$ suggests the following
general structure: We can guess the ansatz

$f_{n}\left(\Psi^{\otimes n}\right)=-b_{0}M_{n}\left(\Psi^{\otimes
n}\right)+M_{n}\left(\mathbf{b_{0}}\left(\Psi^{\otimes n}\right)\right),$ (94)

where $\mathbf{b_{0}}$ again denotes the coderivation associated to $b_{0}$.
At first sight, Eq.(94) looks a bit strange now from the coalgebra perspective
because it is a commutator of two odd objects, so one would more naturally
expect an anticommutator. However, the first term in (94) stems from the
application of $b_{0}$ on the first argument of the symplectic form $\omega$,
hence the sign contains implicit information about $\omega$. From the
discussion about $\mathbf{P}$ from the HPL we could anticipate that $\omega$
has to enter the calculation at some point. The more natural looking
expression $\pi_{1}\left[\mathbf{b_{0}},\mathbf{M_{n}}\right]$ would have been
independent of $\omega$.

One can prove now that the ansatz (94) is indeed correct by directly inserting
into the action.

###  3.5 Proof of the ansatz for the infinitesimal field redefinition

If we focus solely on terms of order $n+1$ in $\Psi$ we get

$S^{\prime\left(n+1\right)}\left(\Psi^{\prime}\right)=\omega\left(\Psi,Qf_{n}\left(\Psi^{\otimes
n}\right)\right)\delta\lambda\,+\,\sum_{k=2}^{n-1}\omega\left(\Psi,M_{k}\left(f_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes
k-1}\right)\right)\delta\lambda\,+\,\frac{1}{n+1}\omega\left(\Psi,M_{n}\left(\Psi^{\otimes
n}\right)\right).$ (95)

The last term is the contribution from the original $S\left(\Psi\right)$, so
the first two terms denoted by $\delta S^{\left(n+1\right)}$ should yield the
infinitesimal variation

$\delta
S^{\prime\left(n+1\right)}\overset{?}{=}\delta\lambda\frac{d}{d\lambda}\frac{1}{n+1}\omega\left(\Psi,M_{n}\left(\Psi^{\otimes
n}\right)\right).$ (96)

Inserting the ansatz and performing some straight forward manipulations gives

$\displaystyle\frac{\delta S^{\prime\left(n+1\right)}}{\delta\lambda}=$
$\displaystyle\,\omega\left(\Psi,Q\left(-b_{0}M_{n}+M_{n}\mathbf{b_{0}}\right)\left(\Psi^{\otimes
n}\right)\right)+\sum_{k=2}^{n-1}\omega\left(\Psi,M_{k}\left(\left(-b_{0}M_{n+1-k}+M_{n+1-k}\mathbf{b_{0}}\right)\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-1}\right)\right)$ $\displaystyle=$
$\displaystyle\,-\omega\left(\Psi,L_{0}M_{n}\left(\Psi^{\otimes
n}\right)\right)+\omega\left(\Psi,b_{0}\left[\mathbf{Q,M_{n}}\right]\left(\Psi^{\otimes
n}\right)\right)-\omega\left(\Psi,b_{0}M_{n}\mathbf{Q}\left(\Psi^{\otimes
n}\right)\right)+\omega\left(\Psi,QM_{n}\mathbf{b_{0}}\left(\Psi^{\otimes
n}\right)\right)$
$\displaystyle\,+\sum_{k=2}^{n-1}\left(-\omega\left(\Psi,M_{k}\left(b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes
k-1}\right)\right)+\omega\left(\Psi,M_{k}\left(M_{n+1-k}\mathbf{b_{0}}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-1}\right)\right)\right).$ (97)

The last two terms of the second line actually cancel each other:

$\displaystyle-\omega\left(\Psi,b_{0}M_{n}\mathbf{Q}\left(\Psi^{\otimes
n}\right)\right)+\omega\left(\Psi,QM_{n}\mathbf{b_{0}}\left(\Psi^{\otimes
n}\right)\right)$ $\displaystyle=$
$\displaystyle-\omega\left(b_{0}\Psi,M_{n}\mathbf{Q}\left(\Psi^{\otimes
n}\right)\right)-\omega\left(Q\Psi,M_{n}\mathbf{b_{0}}\left(\Psi^{\otimes
n}\right)\right)$ $\displaystyle=$
$\displaystyle-\omega\left(b_{0}\Psi,M_{n}\left(Q\Psi,\Psi,...,\Psi\right)\right)-\omega\left(b_{0}\Psi,M_{n}\left(\Psi,Q\Psi,...,\Psi\right)\right)-...-\omega\left(b_{0}\Psi,M_{n}\left(\Psi,\Psi,...,Q\Psi\right)\right)$
$\displaystyle-\omega\left(Q\Psi,M_{n}\left(b_{0}\Psi,\Psi,...,\Psi\right)\right)-\omega\left(Q\Psi,M_{n}\left(\Psi,b_{0}\Psi,...,\Psi\right)\right)-...-\omega\left(Q\Psi,M_{n}\left(\Psi,\Psi,...,b_{0}\Psi\right)\right).$
(98)

Because of cyclicity of the $\left(n+1\right)$-vertex the last two lines
contain the same terms, just differing by a sign which comes from commuting
$Q$ with $b_{0}$. Therefore they add to zero. The second term in the second
line of (97) can be further manipulated using the $A_{\infty}$-relations:

$\displaystyle\frac{\delta S^{\prime\left(n+1\right)}}{\delta\lambda}=$
$\displaystyle-\omega\left(\Psi,L_{0}M_{n}\left(\Psi^{\otimes
n}\right)\right)-\sum_{k=2}^{n-1}\omega\left(\Psi,M_{k}\left(b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-1}\right)\right)$
$\displaystyle+\sum_{k=2}^{n-1}\left(\omega\left(\Psi,M_{k}\left(M_{n+1-k}\left(\mathbf{b_{0}}\left(\Psi^{\otimes
n+1-k}\right)\right),\Psi^{\otimes
k-1}\right)\right)-\omega\left(\Psi,b_{0}\mathbf{M_{k}M_{n+1-k}}\left(\Psi^{\otimes
n}\right)\right)\right).$ (99)

Now again, the terms in the last line cancel after using cyclicity:

$\displaystyle\omega\left(\Psi,b_{0}\mathbf{M_{k}M_{n+1-k}}\left(\Psi^{\otimes
n}\right)\right)=\omega\left(b_{0}\Psi,M_{k}\mathbf{M_{n+1-k}}\left(\Psi^{\otimes
n}\right)\right)$ $\displaystyle=\,$
$\displaystyle\omega\left(M_{k}\left(b_{0}\Psi,M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes
k-2}\right),\Psi\right)+\omega\left(M_{k}\left(b_{0}\Psi,\Psi,M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-3}\right),\Psi\right)$
$\displaystyle+...+\omega\left(M_{k}\left(b_{0}\Psi,\Psi^{\otimes
k-2},M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right)\right),\Psi\right)+\omega\left(M_{k}\left(b_{0}\Psi,\Psi^{\otimes
k-1}\right),M_{n+1-k}\left(\Psi^{\otimes n+1-k}\right)\right).$ (100)

All terms except for the last one can be further manipulated using the
antisymmetry of $\omega$ and cyclicity of $M_{k}$. For example,

$\displaystyle\omega\left(M_{k}\left(b_{0}\Psi,M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes
k-2}\right),\Psi\right)=-\omega\left(\Psi,M_{k}\left(b_{0}\Psi,M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-2}\right)\right)$ $\displaystyle=\,$
$\displaystyle\omega\left(M_{k}\left(\Psi,b_{0}\Psi,M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes
k-3}\right),\Psi\right)=...=\omega\left(M_{k}\left(\Psi^{\otimes
k-1},b_{0}\Psi\right),M_{n+1-k}\left(\Psi^{\otimes n+1-k}\right)\right),$
(101)

so in all of the terms the $M_{n+1-k}$ can be moved to the outermost right.
After all, the terms can be summed up as

$\displaystyle\omega\left(M_{k}\left(\Psi^{\otimes
k-1},b_{0}\Psi\right),M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right)\right)+\omega\left(M_{k}\left(\Psi^{\otimes
k-2},b_{0}\Psi,\Psi\right),M_{n+1-k}\left(\Psi^{\otimes n+1-k}\right)\right)$
$\displaystyle+...+\omega\left(M_{k}\left(b_{0}\Psi,\Psi^{\otimes
k-1}\right),M_{n+1-k}\left(\Psi^{\otimes n+1-k}\right)\right)$
$\displaystyle=\,$
$\displaystyle\omega\left(M_{k}\left(\mathbf{b_{0}}\left(\Psi^{\otimes
k}\right)\right),M_{n+1-k}\left(\Psi^{\otimes n+1-k}\right)\right)$
$\displaystyle=\,$
$\displaystyle\omega\left(\Psi,M_{n+1-k}\left(M_{k}\left(\mathbf{b_{0}}\left(\Psi^{\otimes
k}\right)\right),\Psi^{\otimes n-k}\right)\right),$ (102)

which is after the summation over $k$ identical to the first term in the
second line of (99), just with opposite sign. So we arrive at the expression

$\frac{\delta
S^{\prime\left(n+1\right)}}{\delta\lambda}=-\omega\left(\Psi,L_{0}M_{n}\left(\Psi^{\otimes
n}\right)\right)-\sum_{k=2}^{n-1}\omega\left(\Psi,M_{k}\left(b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-1}\right)\right),$ (103)

which should now be compared to the result of formula (96).

The derivative with respect to $\lambda$ can act on the stubs as well as on
the homotopy $h.$ The action on $e^{-\lambda L_{0}}$ inserts a factor of
$-L_{0}$ on every input string field of the $\left(n+1\right)$-vertex. Since
the vertices are cyclically symmetric, we get $n+1$ identical terms, which
cancels the prefactor $\frac{1}{n+1}.$ The result is

$\frac{d}{d\lambda}\frac{1}{n+1}\omega\left(\Psi,M_{n}\left(\Psi^{\otimes
n}\right)\right)\supset-\omega\left(L_{0}\Psi,M_{n}\left(\Psi^{\otimes
n}\right)\right),$ (104)

which is equal to the first term of (103). To compute the action on $h$, the
tree representation turns out to be useful again: First of all,

$\frac{d}{d\lambda}h=-2b_{0}e^{-2\lambda L_{0}},$ (105)

hence we get a sum of all possible tree diagrams with one propagator replaced
by $-2b_{0}e^{-2\lambda L_{0}}$. We can cut through the diagram along this
replaced propagator and think of the factor $e^{-2\lambda L_{0}}$ as arising
from two $e^{-\lambda L_{0}}$-stubs from the leaf and the root of the two
subtrees. Both subtrees are now part of a higher product $M_{k}$ for some $k$,
$2\leq k\leq n-1$. So the whole expression can be written as a combination of
two higher products $M_{k}$, $M_{n+1-k}$ with a factor $-2b_{0}$ inserted:

$\displaystyle\frac{d}{d\lambda}\frac{1}{n+1}\omega\left(\Psi,M_{n}\left(\Psi^{\otimes
n}\right)\right)\supset-\frac{1}{n+1}\sum_{k=2}^{n-1}$
$\displaystyle\omega\left(\Psi,M_{k}\left(2b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-1}\right)\right)+$
$\displaystyle\omega\left(\Psi,M_{k}\left(\Psi,2b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-2}\right)\right)+...+$
$\displaystyle\omega\left(\Psi,M_{k}\left(\Psi^{\otimes
k-1},2b_{0}M_{n+1-k}\left(\Psi^{\otimes n+1-k}\right)\right)\right).$ (106)

Because of cyclicity of the $k+1$-vertex, the different lines contain the same
terms so we have

$\frac{d}{d\lambda}\frac{1}{n+1}\omega\left(\Psi,M_{n}\left(\Psi^{\otimes
n}\right)\right)\supset-\frac{1}{n+1}\sum_{k=2}^{n-1}2k\cdot\omega\left(\Psi,M_{k}\left(b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-1}\right)\right).$ (107)

The last bracket can be further manipulated:

$\displaystyle\omega\left(\Psi,M_{k}\left(b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-1}\right)\right)$
$\displaystyle=-\omega\left(M_{k}\left(b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-1}\right),\Psi\right)$
$\displaystyle=\omega\left(b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),M_{k}\left(\Psi^{\otimes k}\right)\right)$
$\displaystyle=-\omega\left(M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),b_{0}M_{k}\left(\Psi^{\otimes k}\right)\right)$
$\displaystyle=\omega\left(\Psi,M_{n+1-k}\left(\Psi^{\otimes
n-k},b_{0}M_{k}\left(\Psi^{\otimes k}\right)\right)\right)$
$\displaystyle=\omega\left(\Psi,M_{n+1-k}\left(b_{0}M_{k}\left(\Psi^{\otimes
k}\right),\Psi^{\otimes n-k}\right)\right).$ (108)

In the last step cyclicity of the $\left(n+1-k\right)$-vertex was used again.
We see that in the sum of (107) the $k$th term and the $\left(n+1-k\right)$th
term are identical so the sum can be rewritten as

$\displaystyle-\frac{1}{n+1}\sum_{k=2}^{n-1}\left(n+1\right)\cdot\omega\left(\Psi,M_{k}\left(b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-1}\right)\right)$ $\displaystyle=$
$\displaystyle\sum_{k=2}^{n-1}\omega\left(\Psi,M_{k}\left(b_{0}M_{n+1-k}\left(\Psi^{\otimes
n+1-k}\right),\Psi^{\otimes k-1}\right)\right),$ (109)

which is precisely the second term in (103).

This completes the proof that the ansatz

$f_{n}\left(\Psi^{\otimes n}\right)=-b_{0}M_{n}\left(\Psi^{\otimes
n}\right)+M_{n}\left(\mathbf{b_{0}}\left(\Psi^{\otimes n}\right)\right)$ (110)

indeed yields the correct infinitesimal field redefinition.

###  3.6 Finite field redefinition

So far we have only been concerned with infinitesimal variations of $\lambda$,
now we want to generalize the results to finite changes. We know

$\Psi_{\lambda+\delta\lambda}=\Psi_{\lambda}+\delta\lambda
f_{2}^{\lambda}\left(\Psi_{\lambda},\Psi_{\lambda}\right)+\delta\lambda
f_{3}^{\lambda}\left(\Psi_{\lambda},\Psi_{\lambda},\Psi_{\lambda}\right)+...=\Psi_{\lambda}+\delta\lambda\frac{d}{d\lambda}\Psi_{\lambda},$
(111)

where we have written the superscript $\lambda$ to indicate that the $f_{n}$
also depend on $\lambda$ explicitly. This equation can be integrated to

$\Psi_{\lambda}=\Psi_{0}+\int_{0}^{\lambda}dtf_{2}^{t}\left(\Psi_{t},\Psi_{t}\right)+\int_{0}^{\lambda}dtf_{3}^{t}\left(\Psi_{t},\Psi_{t},\Psi_{t}\right)+...\,\,\,\,\,\,\,\,\,.$
(112)

Inserting $\Psi_{\lambda}$ back we get a perturbative expansion in the
original solution $\Psi_{0}:$

$\displaystyle\Psi_{\lambda}=$
$\displaystyle\Psi_{0}+\int_{0}^{\lambda}dtf_{2}^{t}\left(\Psi_{0},\Psi_{0}\right)+\int_{0}^{\lambda}dtf_{3}^{t}\left(\Psi_{0},\Psi_{0},\Psi_{0}\right)+$
$\displaystyle\int_{0}^{\lambda}dtf_{2}^{t}\left(\int_{0}^{t}dsf_{2}^{s}\left(\Psi_{0},\Psi_{0}\right),\Psi_{0}\right)+\int_{0}^{\lambda}dtf_{2}^{t}\left(\Psi_{0},\int_{0}^{t}dsf_{2}^{s}\left(\Psi_{0},\Psi_{0}\right)\right)+\mathcal{O}\left(\Psi_{0}^{\otimes
4}\right).$ (113)

This formula provides an algorithm to find the associated
$A_{\infty}$-solution to each known solution of the Witten OSFT.

##  4 Physical interpretation

To summarize, we found two distinct field redefinitions

$\tilde{\Psi}=\stackrel{{\scriptstyle[}}{{n}}=1]{\infty}{\sum}P_{n}\left(\Psi^{\otimes
n}\right),\,\,\,\,\,\,\,\,\,\,\Psi^{\prime}=\stackrel{{\scriptstyle[}}{{n}}=1]{\infty}{\sum}F_{n}\left(\Psi^{\otimes
n}\right),$ (114)

which generate two different actions $\tilde{S}$ and $S^{\prime}$ via

$\tilde{S}\left(\tilde{\Psi}\right)=S\left(\Psi\right),\,\,\,\,\,\,\,\,\,\,S^{\prime}\left(\Psi\right)=S\left(\Psi^{\prime}\right).$
(115)

However, both actions share the same equations of motion, namely the Maurer-
Cartan equation of the stubbed $A_{\infty}$-algebra

$\pi_{1}\mathbf{M}\frac{1}{1-\Psi}=0.$ (116)

It remains to examine what the physical meaning of those two actions is, most
importantly, if they yield the same on-shell value for a given solution
$\Psi^{\ast}$.

In [19, 20], a special class of solutions containing a nearly marginal vertex
operator is introduced which serves as a useful playground to analyze this
question.

###  4.1 Nearly marginal solutions

Consider a matter conformal primary field $V$ with weight $h$ smaller but very
close to one. The string field

$\Psi_{1}=\mu\cdot cV\left(0\right)|0\rangle,$ (117)

with some real coupling constant $\mu$ obeys the Siegel gauge condition

$b_{0}\Psi_{1}=0$ (118)

and will serve as a starting point to find the full solution
$\Psi=\sum\Psi_{n}$ as a perturbative series in the expansion parameter
$y=1-h$. One can solve for the string coupling $\mu$ using the Witten
equations of motion,

$Q\Psi+\Psi\ast\Psi=0,$ (119)

to obtain [20]

$\mu=\frac{y}{C_{VVV}}+\mathcal{O}\left(y^{3}\right),$ (120)

where $C_{VVV}$ denotes the three-point function constant of $V$. We can
deduce that $\Psi_{1}=\mathcal{O}\left(y\right)$ and from the perturbative
algorithm for the full solution one can also show that in general
$\Psi_{n}=\mathcal{O}\left(y^{n}\right)$.

The on-shell action in Witten theory can be written compactly as

$S\left(\Psi\right)=-\frac{1}{6}\left\langle\,\Psi,Q\Psi\,\right\rangle.$
(121)

From

$Q\Psi_{1}=\mu y\cdot c\partial cV\left(0\right)|0\rangle$ (122)

we see that the action will be of leading order $y^{3}$ and given by

$S\left(\Psi\right)=-\frac{1}{6}\frac{y^{3}}{C_{VVV}^{2}}\left\langle\,cV,c\partial
cV\,\right\rangle=\frac{1}{6}\frac{y^{3}}{C_{VVV}^{2}}+\mathcal{O}\left(y^{4}\right),$
(123)

if we assume that $V$ is conveniently normalized,
$\left\langle\,V\left(z_{1}\right),V\left(z_{2}\right)\,\right\rangle=z_{12}^{-2h}$.

###  4.2 The $A_{\infty}$-action

The first important observation is that the cohomomorphism $\mathbf{P}$
simplifies significantly for string fields in Siegel gauge: Since $h$ is
proportional to $b_{0}$, it annihilates $\Psi$ and (63) collapses to

$\tilde{\Psi}=\stackrel{{\scriptstyle[}}{{n}}=1]{\infty}{\sum}P_{n}\left(\Psi^{\otimes
n}\right)=p\Psi.$ (124)

We already know that $\tilde{S}\left(\tilde{\Psi}\right)$ yields the original
value $S\left(\Psi\right)$, so now we want to check the expression
$S^{\prime}\left(\tilde{\Psi}\right)$: $\Psi_{1}$ is an $L_{0}$-eigenstate so
we straightforwardly get

$p\Psi_{1}=e^{-\lambda L_{0}}\Psi_{1}=e^{\lambda y}\Psi_{1}.$ (125)

For cubic order in $y$ we just have to insert this into the kinetic term (121)
and get

$S^{\prime}\left(\tilde{\Psi}\right)\mid_{y^{3}}=-\frac{1}{6}\left\langle\,p\Psi_{1},Qp\Psi_{1}\,\right\rangle=\frac{1}{6}\frac{y^{3}}{C_{VVV}^{2}}e^{2\lambda
y}\mid_{y^{3}}=\frac{1}{6}\frac{y^{3}}{C_{VVV}^{2}},$ (126)

which agrees with the result above up to terms of
$\mathcal{O}\left(y^{4}\right)$.

For a more non-trivial check we can collect the terms of quartic order in $y$:
In the action we have to consider the first three terms

$S^{\prime}\left(\tilde{\Psi}\right)\mid_{y^{4}}=\left(-\frac{1}{2}\left\langle\,\tilde{\Psi},Q\tilde{\Psi}\,\right\rangle-\frac{1}{3}\left\langle\,\tilde{\Psi},M_{2}\left(\tilde{\Psi},\tilde{\Psi}\right)\,\right\rangle-\frac{1}{4}\left\langle\,\tilde{\Psi},M_{3}\left(\tilde{\Psi},\tilde{\Psi},\tilde{\Psi}\right)\,\right\rangle\right)\mid_{y^{4}}.$
(127)

The equations of motion however tell us that

$\left\langle\,\tilde{\Psi},Q\tilde{\Psi}\,\right\rangle+\left\langle\,\tilde{\Psi},M_{2}\left(\tilde{\Psi},\tilde{\Psi}\right)\,\right\rangle+\left\langle\,\tilde{\Psi},M_{3}\left(\tilde{\Psi},\tilde{\Psi},\tilde{\Psi}\right)\,\right\rangle=\mathcal{O}\left(y^{5}\right),$
(128)

so the expression simplifies to

$S^{\prime}\left(\tilde{\Psi}\right)\mid_{y^{4}}=\left(-\frac{1}{4}\left\langle\,\tilde{\Psi},Q\tilde{\Psi}\,\right\rangle-\frac{1}{12}\left\langle\,\tilde{\Psi},M_{2}\left(\tilde{\Psi},\tilde{\Psi}\right)\,\right\rangle\right)\mid_{y^{4}}.$
(129)

Plugging in $\tilde{\Psi}=e^{\lambda y}\Psi_{1}$ and isolating $y^{4}$-terms
yields

$\displaystyle S^{\prime}\left(\tilde{\Psi}\right)\mid_{y^{4}}=$
$\displaystyle\left(-\frac{1}{4}\left\langle\,e^{\lambda
y}\Psi_{1},Qe^{\lambda
y}\Psi_{1}\,\right\rangle-\frac{1}{12}\left\langle\,e^{\lambda
y}\Psi_{1},e^{-\lambda L_{0}}\left(e^{-\lambda L_{0}}e^{\lambda y}\Psi_{1}\ast
e^{-\lambda L_{0}}e^{\lambda
y}\Psi_{1}\right)\,\right\rangle\right)\mid_{y^{4}}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{4}e^{2\lambda
y}\left\langle\,\Psi_{1},Q\Psi_{1}\,\right\rangle-\frac{1}{12}e^{6\lambda
y}\left\langle\,\Psi_{1},\left(\Psi_{1}\ast\Psi_{1}\right)\,\right\rangle\right)\mid_{y^{4}}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\frac{\lambda
y^{4}}{C_{VVV}^{2}}\left\langle\,cV,c\partial
cV\,\right\rangle-\frac{1}{2}\frac{\lambda
y^{4}}{C_{VVV}^{3}}\left\langle\,cV,\left(cV\ast cV\right)\,\right\rangle.$
(130)

The correlation functions can be calculated by using standard CFT methods, see
e.g. [21, 20]; the result is

$\left\langle\,cV,c\partial
cV\,\right\rangle=-1,\,\,\,\,\,\,\,\,\,\,\,\,\left\langle\,cV,\left(cV\ast
cV\right)\,\right\rangle=C_{VVV}\left(\frac{3\sqrt{3}}{4}\right)^{3y}=C_{VVV}\left(1+3y\cdot\text{ln}\left(\frac{3\sqrt{3}}{4}\right)\right)+\mathcal{O}\left(y^{2}\right).$
(131)

We see by inserting into (130) that
$S^{\prime}\left(\tilde{\Psi}\right)\mid_{y^{4}}$ indeed vanishes and the
value of the on-shell action is the same as the original $S\left(\Psi\right)$
to order $y^{4}$. In principle, terms containing $\Psi_{2}\propto y^{2}$ also
contribute at this order. However, since any appearance of $\lambda$
automatically comes with a factor $y$, there are no terms of order $y^{4}$
containing $\lambda$ as well as $\Psi_{2}$. All $\Psi_{2}$-contributions are
just the ones already present in $S\left(\Psi\right)$ and were studied in
detail in [20].

We see that to the first two leading orders, the actions
$S^{\prime}\left(\tilde{\Psi}\right)$ and
$\tilde{S}\left(\tilde{\Psi}\right)=S\left(\Psi\right)$ give the same on-shell
result. Since
$S^{\prime}\left(\tilde{\Psi}\right)=S\left(\tilde{\Psi}^{\prime}\right)$,
where

$\tilde{\Psi}^{\prime}=\stackrel{{\scriptstyle[}}{{n}}=1]{\infty}{\sum}F_{n}\left(\tilde{\Psi}^{\otimes
n}\right)=\pi_{1}\mathbf{FP}\frac{1}{1-\Psi},$ (132)

this suggests that the combination $\mathbf{FP}$ gives rise to a gauge
transformation rather than a physically distinct solution. However, a full
proof of this statement will be left for future publications.

##  5 Conclusion and outlook

We succeeded in providing an explicit consistent description of OSFT with
stubs and interestingly found two possible actions with the same equations of
motion. The field redefinitions used to convert solutions of Witten OSFT to
the new theory are given in explicit form. We hope that the analysis of
solutions to the stubbed equations of motion can teach us more general
properties of solutions to Maurer-Cartan equations for $A_{\infty}$\- or
$L_{\infty}$-algebras, in particular about the solutions of closed string
field theory. One way to proceed would be to transform the whole construction
to the sliver frame, where many analytic solutions of OSFT are formulated.

Another possible future direction is to examine wether the stubbed theory is
“more well-behaved” in the sense that some typical singularities and
ambiguities, for example connected to identity-like solutions, are
ameliorated.

#### Acknowledgements

We thank Ted Erler, Jakub Vošmera, Branislav Jurčo, Igor Khavkine and Martin
Markl for useful discussions. Our work has been funded by the Grant Agency of
Czech Republic under the grant EXPRO 20-25775X.

### Appendix

#### Tensor coalgebras

The tensor coalgebra $TV$ associated to a (graded) vector space $V$ is defined
as the Fock space

$V^{\otimes 0}+V^{\otimes 1}+V^{\otimes 2}+...$ (133)

together with the comultiplication $\Delta:\,\,TV\rightarrow
TV\otimes^{\prime}TV$ given by

$\Delta\left(v_{1}\otimes...\otimes
v_{n}\right)=\sum_{k=0}^{n}\left(v_{1}\otimes...\otimes
v_{k}\right)\otimes^{\prime}\left(v_{k+1}\otimes...\otimes v_{n}\right)$ (134)

on homogeneous elements and extended by linearity. Here the $v_{i}$ are
elements of $V$ and $\otimes^{\prime}$ denotes the tensor product arising from
a comultiplication, in contrast to the usual $\otimes$. We define the
projection operator $\pi_{n}:\,\,TV\rightarrow TV$ to project any element on
its $n$th tensor power component,

$\pi_{n}TV=V^{\otimes n}.$ (135)

A linear map $\mathbf{d}:\,\,TV\rightarrow TV$ is called a coderivation if it
satisfies the co-Leibniz rule:

$\Delta\mathbf{d}=\left(\mathbf{d}\otimes^{\prime}\mathbf{1}+\mathbf{1}\otimes^{\prime}\mathbf{d}\right)\Delta.$
(136)

Linear combinations of coderivations are again coderivations as well as their
graded commutator

$\left[\mathbf{d}_{1},\mathbf{d}_{2}\right]=\mathbf{d}_{1}\mathbf{d}_{2}-\left(-1\right)^{deg\left(\mathbf{d}_{1}\right)deg\left(\mathbf{d}_{2}\right)}\mathbf{d}_{2}\mathbf{d}_{1}.$
(137)

The product $\mathbf{d}_{1}\mathbf{d}_{2}$ is in general not a coderivation.
For any $m$-linear map $d_{m}:\,\,V^{\otimes m}\rightarrow V$ one can
construct an associated coderivation by the formula

$\mathbf{d}=\underset{n=m}{\overset{\infty}{\sum}}\underset{k=0}{\overset{n-m}{\sum}}1^{\otimes
k}\otimes d_{m}\otimes 1^{\otimes n-k-m}.$ (138)

The co-Leibniz rule guarantees that any coderivation is a sum of terms of this
form for different $m$. The individual $m$-products can be recovered as

$d_{m}=\pi_{1}\mathbf{d}\pi_{m}.$ (139)

If an odd coderivation $\mathbf{d}$ obeys

$\mathbf{d}^{2}=0$ (140)

then its components $d_{m}$ form an $A_{\infty}$-algebra.

A linear map $\mathbf{f}$ is called a cohomomorphism if it fulfills

$\Delta\mathbf{f}=\left(\mathbf{f}\otimes^{\prime}\mathbf{f}\right)\Delta.$
(141)

Linear combinations and products of cohomomorphisms are again cohomomorphisms.
Given a family of $m$-products $f_{m}$ one can construct a unique
cohomomorphism via

$\mathbf{f}=\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\sum_{m_{1}+...+m_{j}=k}f_{m_{1}}\otimes...\otimes
f_{m_{j}}.$ (142)

Again, the individual products can be recovered from $\mathbf{f}$ as

$f_{m}=\pi_{1}\mathbf{f}\pi_{m}.$ (143)

Of special importance are elements of $TV$ of the form

$1+v+v\otimes v+v\otimes v\otimes v+...=:\frac{1}{1-v}$ (144)

for some $v\in V.$ They fulfill the following useful properties:

$\pi_{1}\mathbf{f}\frac{1}{1-v}=\sum_{m=1}^{\infty}f_{m}\left(v^{\otimes
m}\right),$ (145)
$\mathbf{f}\frac{1}{1-v}=\frac{1}{1-\pi_{1}\mathbf{f}\frac{1}{1-v}}$ (146)

for any cohomomorphism $\mathbf{f}.$

A bilinear map $\langle\omega|$: $TV\times TV\rightarrow\mathbb{C}$ is called
a symplectic form if it satisfies

$\langle\omega|v_{1}\otimes
v_{2}=:\omega\left(v_{1},v_{2}\right)=-\left(-1\right)^{deg\left(v_{1}\right)deg\left(v_{2}\right)}\omega\left(v_{2},v_{1}\right).$
(147)

A multilinear product $m_{k}$ is called cyclic with respect to $\omega$ if it
fulfills

$\omega\left(v_{1},m_{k}\left(v_{2},...,v_{k+1}\right)\right)=-\left(-1\right)^{deg\left(v_{1}\right)deg\left(m_{k}\right)}\omega\left(m_{k}\left(v_{1},...,v_{k}\right),v_{k+1}\right).$
(148)

A coderivation $\mathbf{d}$ is cyclic if all of its components
$d_{m}=\pi_{1}\mathbf{d}\pi_{m}$ are cyclic or equivalently

$\langle\omega|\pi_{2}\mathbf{d}=0.$ (149)

Given two symplectic forms $\langle\omega|$, $\langle\omega^{\prime}|$, a
cohomomorphism $\mathbf{f}$ is cyclic if

$\langle\omega^{\prime}|\pi_{2}\mathbf{f}=\langle\omega|\pi_{2}.$ (150)

## References

  * [1] E. Witten, “Noncommutative Geometry and String Field Theory,” Nucl. Phys. B 268 (1986), 253-294 doi:10.1016/0550-3213(86)90155-0
  * [2] J. Vošmera, “Selected topics in string field theory and physics of D-branes,”
  * [3] B. Zwiebach, “Closed string field theory: Quantum action and the B-V master equation,” Nucl. Phys. B 390 (1993), 33-152 doi:10.1016/0550-3213(93)90388-6 [arXiv:hep-th/9206084 [hep-th]].
  * [4] B. Zwiebach, “A Proof that Witten’s open string theory gives a single cover of moduli space,” Commun. Math. Phys. 142 (1991), 193-216 doi:10.1007/BF02099176
  * [5] H. Sonoda and B. Zwiebach, “COVARIANT CLOSED STRING THEORY CANNOT BE CUBIC,” Nucl. Phys. B 336 (1990), 185-221 doi:10.1016/0550-3213(90)90108-P
  * [6] M. Schnabl, “Analytic solution for tachyon condensation in open string field theory,” Adv. Theor. Math. Phys. 10 (2006) no.4, 433-501 doi:10.4310/ATMP.2006.v10.n4.a1 [arXiv:hep-th/0511286 [hep-th]].
  * [7] T. Erler, “Four Lectures on Closed String Field Theory,” Phys. Rept. 851 (2020), 1-36 doi:10.1016/j.physrep.2020.01.003 [arXiv:1905.06785 [hep-th]].
  * [8] T. Erler, “The closed string field theory action vanishes,” JHEP 10 (2022), 055 doi:10.1007/JHEP10(2022)055 [arXiv:2204.12863 [hep-th]].
  * [9] T. Takahashi and S. Tanimoto, “Marginal and scalar solutions in cubic open string field theory,” JHEP 03 (2002), 033 doi:10.1088/1126-6708/2002/03/033 [arXiv:hep-th/0202133 [hep-th]].
  * [10] T. Erler, “Four lectures on analytic solutions in open string field theory,” Phys. Rept. 980 (2022), 1-95 doi:10.1016/j.physrep.2022.06.004 [arXiv:1912.00521 [hep-th]].
  * [11] B. Zwiebach, “Oriented open - closed string theory revisited,” Annals Phys. 267 (1998), 193-248 doi:10.1006/aphy.1998.5803 [arXiv:hep-th/9705241 [hep-th]].
  * [12] T. Takezaki, “Open superstring field theory including the Ramond sector based on the supermoduli space,” [arXiv:1901.02176 [hep-th]].
  * [13] H. Erbin, C. Maccaferri, M. Schnabl and J. Vošmera, “Classical algebraic structures in string theory effective actions,” JHEP 11 (2020), 123 doi:10.1007/JHEP11(2020)123 [arXiv:2006.16270 [hep-th]].
  * [14] M. Crainic; “On the perturbation lemma, and deformations,” [arXiv:math/0403266]
  * [15] S. Konopka, “The S-Matrix of superstring field theory,” JHEP 11 (2015), 187 doi:10.1007/JHEP11(2015)187 [arXiv:1507.08250 [hep-th]].
  * [16] J. L. Loday, B. Vallette; “Algebraic operads,” Springer Berlin, Heidelberg, 2012 doi:10.1007/978-3-642-30362-3
  * [17] S. Li and K. Zeng, “Homotopy Algebras in Higher Spin Theory,” Adv. Theor. Math. Phys. 24 (2020) no.3, 757-819 doi:10.4310/ATMP.2020.v24.n3.a5 [arXiv:1807.06037 [hep-th]].
  * [18] H. Hata and B. Zwiebach, “Developing the covariant Batalin-Vilkovisky approach to string theory,” Annals Phys. 229 (1994), 177-216 doi:10.1006/aphy.1994.1006 [arXiv:hep-th/9301097 [hep-th]].
  * [19] S. Mukherji and A. Sen, “Some all order classical solutions in nonpolynomial closed string field theory,” Nucl. Phys. B 363 (1991), 639-664 doi:10.1016/0550-3213(91)80037-M
  * [20] J. Scheinpflug and M. Schnabl, “Conformal perturbation theory from open string field theory,” [arXiv:2301.05216 [hep-th]].
  * [21] Y. Okawa, “Analytic methods in open string field theory,” Prog. Theor. Phys. 128 (2012), 1001-1060 doi:10.1143/PTP.128.1001