SISSA 04/2024/FISI

Adding stubs to quantum string field theories

C. Maccaferri(a)111Email: maccafer at unito.it, R. Poletti(a)222Email:
polettiricc at gmail.com, A. Ruffino(a)333Email: ruffinoalb at gmail.com and
B. Valsesia(b)444Email: bvalsesi at sissa.it
(a)Dipartimento di Fisica, Università di Torino and
INFN Sezione di Torino
Via Pietro Giuria 1, I-10125 Torino, Italy
(b)International School of Advanced Studies (SISSA),
via Bonomea 265, 34136 Trieste and
INFN Sezione di Trieste, Italy

Abstract

Generalizing recent work by Schnabl-Stettinger and Erbin-Fırat, we outline a
universal algebraic procedure for ‘adding stubs’ to string field theories
obeying the BV quantum master equation. We apply our results to classical and
quantum closed string field theory as well as to open-closed string field
theory. We also clarify several aspects of the integration-out process in the
co-algebraic formulation of string field theory at the quantum level.

###### Contents

  1. 1 Introduction and summary
  2. 2 Systematic approach for adding stubs
    1. 2.1 Classical stubs
    2. 2.2 Quantum stubs
    3. 2.3 Example: adding quantum stubs to a cubic theory
    4. 2.4 Closed string field theory with stubs
    5. 2.5 Open-closed string field theory with stubs
  3. 3 Equivalence of amplitudes
    1. 3.1 Example: open string one-loop tadpole in OC-SFT
  4. 4 Conclusions
  5. A Path integral and co-algebraic homotopy transfer
  6. B  $SL(2,\mathbb{C})$ vertices for the open string one-loop tadpole
    1. B.1 Open string non-separating plumbing fixture
    2. B.2 Closed string separating plumbing fixture
    3. B.3 Fundamental vertex and decoupling of BRST exact states

##  1 Introduction and summary

String Field Theory (SFT) has witnessed a notable boost in recent times, with
advances in concrete physical applications [1, 2, 3, 4, 5, 6, 7, 8] as well as
fundamental progress both at the geometric [10, 11, 9, 12, 13, 14, 15] and at
the algebraic level [18, 19, 17, 16]. See [20, 21, 22, 23, 24] for recent
reviews on the subject.

One noticeable recent progress has been a better understanding of the so-
called stub deformation [25, 26, 27]. This deformation is associated to one of
the core aspects of SFT: the decomposition of the moduli space of Riemann
surfaces associated to string amplitudes in ‘vertex regions’, which are
encoded into the fundamental interaction vertices of the SFT action and
‘Feynman’ regions which are the portions of moduli space which are covered
through the Feynman diagrams, obtained by gluing lower-order vertices with
propagators. Essentially by construction, the Feynman regions always contain
the part of moduli space where the Riemann surface degenerates and this
provides a systematic way of dealing with divergences by suitably interpreting
the SFT propagator [28, 29, 30], which is always associated to the
degenerating part of the Riemann surface. The distinction between vertex and
Feynman regions is arbitrary (up to the obvious requirement that degenerations
should be included in the Feynman regions) and it is part of the off-shell
data of SFT. Once a choice of consistent vertices is given, there is still the
possibility of adjusting the size of the local coordinate patches by rescaling
them with canonical ‘stub’ operators $e^{-\lambda L_{0}}$ (for open strings)
or $e^{-\lambda(L_{0}+\bar{L}_{0})}$ (for closed strings). For $\lambda>0$
this has the effect of increasing the part of moduli space which is covered by
the fundamental vertices and reducing the part covered by the propagators in
the Feynman diagrams. In particular for very large $\lambda$, almost all the
moduli space is covered by the fundamental vertices except the regions where
the surface degenerates (which are always covered by the Schwinger parameters
of the propagators). This possibility of varying $\lambda$ is the stub
deformation. This deformation can be also understood as a smooth way of
integrating out off-shell degrees freedom, in that in the $\lambda\to\infty$
limit the stub operator $e^{-\lambda L_{0}}$ approaches the projector $P_{0}$
into the kernel of $L_{0}$ 555This is certainly true as long as we act on
$h\geq 0$ states. When tachyons are present, the stub deformation amplifies
them, instead of acting as a suppression factor. It would be interesting to
define a different stub operator which can also suppress tachyons. An example
would be $1/(1+\lambda L_{0})$ which however doesn’t work well when there are
tachyonic nearly-marginal states which accumulate at $L_{0}\to 0^{-}$.

$\displaystyle\lim_{\lambda\to+\infty}e^{-\lambda L_{0}}\sim P_{0}.$ (1.1)

By construction this deformation amounts in a change of the SFT data and thus
it falls into the general schemes of field redefinitions. A qualitative
‘physical’ effect of this deformation is to push (for large $\lambda$) the
effect of the higher level states inside the fundamental vertices, leaving the
Feynman diagrams to be dominated by the contribution of the massless fields.
However, for any finite $\lambda$, the off-shell degrees of freedom are
strictly conserved and therefore one should not expect a qualitative change in
the theory, in particular in the vacuum structure [33]. In this respect the
stub deformation appears to be conceptually different from the standard
integration-out, where some off-shell (or high energy) degrees of freedom
completely disappear, although the two operations clearly share similarities.
This similarity with the integrating-out process has been a key-idea in the
recent works [25, 26] which gave two related algebraic descriptions of the
deformation in the context of (classical) Witten OSFT using the by-now-famous
homotopy transfer machinery [18, 19, 17, 31, 32].

Our work starts from here, in particular from the Erbin-Fırat (EF) formulation
where the stub deformation is obtained by integrating-out some auxiliary
fields in an appropriately extended theory. Our first observation consists in
a simple reformulation of what Erbin and Fırat did. This simple reformulation
is summarized as follows. Given a theory with dynamical field $\rho$, to which
we want to add stubs, we add a free auxiliary sector with no propagating
degrees of freedom, which we denote $\chi$

$\displaystyle\rho$ $\displaystyle\to(\rho,\chi)$ (1.2) $\displaystyle
S[\rho]=\frac{1}{2}\omega(\rho,Q\rho)+S_{int}[\rho]\,$
$\displaystyle\to\,S^{\prime}[\rho,\chi]=\frac{1}{2}\omega(\rho,Q\rho)+S_{int}[\rho]+\frac{1}{2}\omega(\chi,Q\chi).$
(1.3)

The field $\chi$ is completely decoupled and, in addition, is taken to live in
a subspace where the cohomology of $Q$ is absent. In particular (if we are
considering a stub operator of the form $e^{-\lambda L_{0}}$, having in mind
open strings for definitness) the field $\chi$ is taken to live outside the
kernel of $L_{0}$

$\displaystyle\chi\in(1-P_{0}){\cal H}.$ (1.4)

It is clear that integrating out $\chi$ doesn’t change the original theory, if
not for an unimportant constant (the one-loop determinant of the $\chi$
quadratic term, which can be subtracted away in the definition of the extended
action, if one wishes). To perform the stub deformation we simply do a linear
field redefinition which mixes the free auxiliary field $\chi$ with the
physical interacting field $\rho$

$\displaystyle\left(\begin{matrix}\rho\\\
\chi\end{matrix}\right)=\left(\begin{matrix}c_{\lambda}&-\sqrt{1-c_{\lambda}^{2}}\\\
\sqrt{1-c_{\lambda}^{2}}&c_{\lambda}\end{matrix}\right)\left(\begin{matrix}\psi\\\
\Sigma\end{matrix}\right),$ (1.5)

where $c_{\lambda}$ is the stub operator, for example $c_{\lambda}=e^{-\lambda
L_{0}}$. Being a linear and orthogonal field redefinition this does not change
the path integral measure and it also leaves the total kinetic term for the
new fields $\psi$ and $\Sigma$ invariant

$\displaystyle
S^{\prime}[\rho,\chi]=S[\psi,\Sigma]=\frac{1}{2}\omega(\psi,Q\psi)+\frac{1}{2}\omega(\Sigma,Q\Sigma)+S_{int}\left[c_{\lambda}\psi-\sqrt{1-c_{\lambda}^{2}}\Sigma\right].$
(1.6)

This action is very closely related to the extended action considered by EF,
as we are going to discuss in the next section. It is not difficult to see
that the field $\Sigma$ still lives outside of the $Q$ cohomology, in
particular

$\displaystyle\Sigma\in(1-P_{0}){\cal H},$ (1.7)

therefore it can be completely integrated-out. The stubbed (quantum) theory is
the resulting effective action

$\displaystyle e^{-S_{\rm stub}^{(\lambda)}[\psi]}=\int_{L}{\cal
D}\Sigma\,e^{-S[\psi,\Sigma]},$ (1.8)

where $L$ is a lagrangian submanifold corresponding to an appropriate gauge
fixing. In this paper we will make this simple idea more precise and we will
give explicit constructions in the context of classical and quantum closed
bosonic SFT as well as open-closed SFT.

The paper is organized as follows. In section 2 we describe in detail our
construction at the classical and quantum level and we apply it to a cubic
theory, assuming the theory makes sense at the (perturbative) quantum level.
Then we apply our construction to proper solutions to the BV quantum master
equation such as closed SFT and open-closed SFT. In section 3 we give a
general proof that tree and loop amplitudes are invariant under the stub
deformation. As a non-trivial example we discuss in detail how the stub
deformation changes the decomposition of moduli space in the one-loop open
string tadpole in open-closed SFT. In the concluding section 4 we put forward
some possible explorations that our present work opens up. Two appendices
contain important technical material. In appendix A we prove that the quantum
homotopy transfer at the co-algebraic level indeed computes the BV path
integral. This is done by carefully translating the work of Doubek, Jurčo and
Pulmann [34] from the BV to the co-algebra language. The second appendix B
contains an explicit construction of the $SL(2,\mathbb{C})$ vertices which are
needed for the one-loop open string tadpole amplitude in open-closed SFT.

##  2 Systematic approach for adding stubs

In this section we provide a systematic algebraic procedure for introducing
stubs into a generic quantum theory within the BV formalism. Then we will
apply it to classical and quantum closed SFT and finally to open-closed SFT.

###  2.1 Classical stubs

We begin by considering a generic (euclidean for definiteness) classical SFT
described by the following action

$S[\rho]=\frac{1}{2}\omega(\rho,Q\rho)+S_{{\rm int}}[\rho],$ (2.1)

where $\rho$ is a degree zero string field belonging to the Hilbert space
${\cal H}$, $Q$ is nilpotent and $S_{{\rm int}}[\rho]$ is the interacting part
of the starting un-stubbed theory. Moreover, $\omega(\cdot,\cdot)$ is a
symplectic form, a graded anti-symmetric inner product

$\omega(\rho_{1},\rho_{2})=-(-)^{d(\rho_{1})d(\rho_{2})}\omega(\rho_{2},\rho_{1}),$
(2.2)

in which $d(\cdot)$ is the degree. The symplectic form is non-vanishing if
$d(\rho_{1})+d(\rho_{2})=1$. To introduce stubs we consider an operator
$c_{\lambda}:\mathcal{H}\to\mathcal{H}$, which can be written as

$c_{\lambda}=e^{-\lambda L},$ (2.3)

where $\lambda$ is the stub parameter and $L:\mathcal{H}\to\mathcal{H}$ is a
Grassmann even operator satisfying $[L,Q]=0$666In this paper we consider for
definiteness $c_{\lambda}=e^{-\lambda L}$ but it is clear that one can
consider more general stubs of the kind $c(\lambda,L)$, see [25] for explicit
examples. . In this completely general context, we also require $L$ to be BPZ
even, in the sense that 777In [35] this condition has been relaxed, it would
be interesting to see how our formalism can be accordingly extended to this
case.

$\omega(\rho_{1},L\rho_{2})=\omega(L\rho_{1},\rho_{2}).$ (2.4)

Furthermore, having in mind the case when $L=L_{0}$ or $L_{0}^{+}$, we require
the existence of a Grassmann odd operator $b$ such that $[Q,b]=L$. Moreover we
also assume that $b=0$ is a ‘good’ gauge fixing, meaning that the cohomology
of $Q$ can be systematically searched for inside ${\rm ker}(b)$ and therefore
is contained in ${\rm ker}(L)$.
Let us now introduce a decoupled auxiliary field $\chi$, without cohomology.
This condition can be achieved by requiring that
$\chi\in(1-P)\mathcal{H}\coloneqq\bar{P}\mathcal{H}$, where $P$ projects onto
${\rm ker}(L)$. With this trivial addiction the action 2.1 becomes

$S^{\prime}[\rho,\chi]=\frac{1}{2}\omega(\rho,Q\rho)+\frac{1}{2}\omega(\chi,Q\chi)+S_{{\rm
int}}[\rho].$ (2.5)

Notice that this is not adding any new propagating degrees of freedom because
$\chi\in\bar{P}\mathcal{H}$, where the cohomology is absent. In particular,
classically integrating out $\chi$ (i.e. solving its equation of motion) is
the same as setting it to zero.

Now, let us do a trivial field redefinition, a rotation induced by the
orthogonal matrix

$\begin{pmatrix}\rho\\\ \chi\end{pmatrix}=O_{\lambda}\begin{pmatrix}\psi\\\
\Sigma\end{pmatrix}=\begin{pmatrix}c_{\lambda}&-\sqrt{1-c_{\lambda}^{2}}\\\
\sqrt{1-c_{\lambda}^{2}}&c_{\lambda}\end{pmatrix}\begin{pmatrix}\psi\\\
\Sigma\end{pmatrix},$ (2.6)

which leaves the kinetic terms unchanged but mixes the interactions

$S[\psi,\Sigma]\coloneqq
S^{\prime}[\rho,\chi]=\frac{1}{2}\omega(\psi,Q\psi)+\frac{1}{2}\omega(\Sigma,Q\Sigma)+S_{{\rm
int}}\left[c_{\lambda}\psi-\sqrt{1-c_{\lambda}^{2}}\Sigma\right].$ (2.7)

Notice that since $\chi\in\bar{P}\mathcal{H}$, then
$\Sigma\in\bar{P}\mathcal{H}$ as well, given that
$\sqrt{1-c_{\lambda}^{2}}\,\psi\sim(2\lambda
L)^{1/2}\,\psi\in\bar{P}\mathcal{H}$. Therefore $\Sigma$ can be integrated out
completely without loosing physical degrees of freedom. This time however this
is not the same as setting it equal to zero due to the non-trivial couplings
between $\Sigma$ and $\psi$. However, since $\Sigma$ doesn’t contain physical
degrees of freedom, classically integrating it out (i.e. solving the $\Sigma$
equation of motion as a function of $\psi$) should give a theory with the same
physical content as the initial one. This new theory is by definition the
(classical) stubbed theory

$\tilde{S}[\psi]=S[\psi,\Sigma(\psi)]=\frac{1}{2}\omega(\psi,Q\psi)+\tilde{S}_{\rm
int}[\psi].$ (2.8)

The procedure we have outlined is just a convenient re-writing of what has
been done by Erbin and Fırat in [25] in the particular context of Witten
theory. Indeed in this case (2.7) reduces to the extended theory described in
[25] by renaming $\Sigma_{\rm
EF}=\sqrt{1-c_{\lambda}^{2}}\Sigma\in\bar{P}{\cal H}$, namely

$\displaystyle S_{\rm EF}[\psi,\Sigma_{\rm EF}]\coloneqq S[\psi,\Sigma]$ (2.9)
$\displaystyle=\frac{1}{2}\omega\left(\psi,Q\psi\right)+\frac{1}{2}\omega\left(\Sigma_{\rm
EF},\frac{Q}{1-c_{\lambda}^{2}}\Sigma_{\rm
EF}\right)-\frac{1}{3}\omega\left((\Sigma_{\rm
EF}-c_{\lambda}\psi),m_{2}\left((\Sigma_{\rm EF}-c_{\lambda}\psi),(\Sigma_{\rm
EF}-c_{\lambda}\psi)\right)\right).$

Furthermore in [25] the authors consider a reduction of the theory (2.9) onto
a “line” defined by the linear relation

$\Sigma_{\rm
EF}=\left(c_{\lambda}-c_{\lambda}^{-1}\right)\psi\longleftrightarrow\Sigma=-\frac{\sqrt{1-c_{\lambda}^{2}}}{c_{\lambda}}\psi,$
(2.10)

which gives Witten’s theory without stubs. In our language this means to undo
the field redefinition 2.6 and then integrate out the free field $\chi$ (which
is the same as setting it to zero) thus getting back the original theory
without stubs. Thanks to our reformulation, we will be able to extend these
results to the quantum theory in a particularly transparent way.

###  2.2 Quantum stubs

Let us now repeat the above procedure in the quantum case. This time the
integration out of the auxiliary fields really means to perform the path
integral over $\chi$ or the ‘stub-rotated’ $\Sigma$. Again we assume to start
from a theory without stubs described by the action $S[\rho]$, which we
suppose satisfies the quantum BV master equation. Then we add the non-
interacting auxiliary field $\chi$. As a simple warm-up let us review how to
perform the BV path integral over the free auxiliary field $\chi$. Remember
that $\chi\in\bar{P}{\cal H}$, which can be decomposed as

$\displaystyle\bar{P}{\cal H}=hQ{\cal H}\oplus Qh{\cal H},$ (2.11)

where we have defined the homotopy operator / propagator

$\displaystyle h\coloneqq\frac{b}{L}\,\bar{P}.$ (2.12)

The BV path integral has to be done over a lagrangian submanifold (a maximal
subspace where the symplectic form $\omega$ vanishes) of the total space
$\bar{P}{\cal H}$. A gauge fixing is a choice of lagrangian submanifold. In
the case at hand we use the obvious gauge fixing $h\chi=0$ and therefore the
lagrangian submanifold is the (component-dual of the) vector space $hQ{\cal
H}$, on which the symplectic form correctly vanishes. Let $\\{c_{i}\\}$ be a
basis of $hQ{\cal H}$ and $\\{b_{j}\\}$ a basis of $Qh{\cal H}$. Notice that
only the $c_{i}$’s give a non-trivial contribution to the kinetic term, so
that expressing $\chi=\gamma^{i}c_{i}+\beta^{j}b_{j}$, we remain with

$\displaystyle\omega(\chi,Q\chi)=\gamma^{i}Q_{ij}\gamma^{j},$ (2.13)

where $Q_{ij}=\omega(c_{i},Qc_{j})$. Doing the BV path integral thus means to
perform the (super) gaussian integration over the BV fields $\gamma^{i}$
according to their grassmanality

$\int_{h\chi=0}\mathcal{D}\chi\,e^{-S[\rho]-\frac{1}{2}\omega(\chi,Q\chi)}=e^{-{\cal
A}-S[\rho]},$ (2.14)

where ${\cal A}=\frac{1}{2}\log({\rm sdet}(Q_{ij}))$. Therefore, if we start
from the extended action

$S^{\prime}[\rho,\chi]=\frac{1}{2}\omega(\rho,Q\rho)+\frac{1}{2}\omega(\chi,Q\chi)+S_{{\rm
int}}[\rho]-\mathcal{A},$ (2.15)

the BV path integral over $\chi$ will precisely reproduce the original action
$S(\rho)$. This sets the difference in vacuum energies between the initial and
the extended theory, which are now fully equivalent also at the quantum level.

We proceed now with the second step, namely redefining the fields as described
in (2.6), obtaining

$S[\psi,\Sigma]\coloneqq
S^{\prime}[\rho(\psi,\Sigma),\chi(\psi,\Sigma)]=\frac{1}{2}\omega(\psi,Q\psi)+\frac{1}{2}\omega(\Sigma,Q\Sigma)+S_{{\rm
int}}\left[c_{\lambda}\psi-\sqrt{1-c_{\lambda}^{2}}\Sigma\right]-\mathcal{A}.$
(2.16)

The stubbed theory is given by integrating out $\Sigma$

$\tilde{S}[\psi]=-\log\int_{h\Sigma=0}\mathcal{D}{\Sigma}\,e^{-S^{\prime}[\psi,\Sigma]}=\Lambda+\frac{1}{2}\omega(\psi,Q\psi)+\tilde{S}_{\rm
int}[\psi],$ (2.17)

where $\Lambda$ is a constant associated with the vacuum bubbles. Notice that
the 1-loop bubble that one gets by integrating out $\Sigma$ is just the same
as the one obtained by integrating out $\chi$ (which has been subtracted away
in the definition of the extended action), so the vacuum energy $\Lambda$ only
includes (interacting) vacuum diagrams from two loops onwards.
From now on, we will ignore these extra constants (but see the comments in the
conclusions) and focus on the dynamical part of the stubbed action, which will
obviously depend on the peculiar interactions of the original theory.

###  2.3 Example: adding quantum stubs to a cubic theory

Let us begin with a simple example, a cubic theory

$S[\rho]=\frac{1}{2}\omega(\rho,Q\rho)+\frac{1}{3}\omega\left(\rho,m_{2}(\rho,\rho)\right),$
(2.18)

which we have written starting from a cyclic differential graded associative
algebra $(Q,m_{2})$, which is the simplest non-trivial example of a cyclic
$A_{\infty}$ algebra (see for example [18] for quick review). The action
satisfies the classical BV master equation thanks to the $A_{\infty}$
relations

$\displaystyle Q^{2}=$ $\displaystyle 0$ (2.19) $\displaystyle Qm_{2}+m_{2}Q=$
$\displaystyle 0$ (2.20) $\displaystyle m_{2}m_{2}=$ $\displaystyle 0.$ (2.21)

But, in general, it does not satisfy the quantum BV master equation.
Decomposing the original string field $\rho=r^{a}\,v_{a}$ along a basis
$\\{v_{a}\\}$, the BV bracket $(\cdot,\cdot)$ and the BV laplacian $\Delta$
are defined as follows

$\displaystyle\left(\cdot,\cdot\right)=\dfrac{\overleftarrow{\partial}}{\partial
r^{a}}(\omega)^{ab}\dfrac{\overrightarrow{\partial}}{\partial r^{b}},$ (2.22)
$\displaystyle\Delta=\frac{(-)^{r^{a}}}{2}(\omega)^{ab}\dfrac{\overrightarrow{\partial}}{\partial
r^{a}}\dfrac{\overrightarrow{\partial}}{\partial r^{b}}.$ (2.23)

where

$(\omega)^{ab}\coloneqq\omega(v^{a},v^{b}),$ (2.24)

with $\\{v^{a}\\}$ being the $\omega$-canonical dual basis

$\displaystyle\omega(v^{a},v_{b})=-\omega(v_{b},v^{a})=\delta^{a}_{b}.$ (2.25)

Focusing on the BV laplacian, in the case of the present cubic theory, we get

$\Delta{S}(\rho)=\omega\left(\rho,m_{2}(U)\right),$ (2.26)

where we have introduced the Poisson bi-vector

$\displaystyle U=\frac{(-1)^{v^{a}}}{2}v_{a}\wedge v^{a}\in{\cal H}^{\wedge
2},$ (2.27)

where $\wedge$ is the (graded) symmetrized tensor product. Since, thanks to
the associativity of $m_{2}$, we have $(S,S)=0$, the quantum BV master
equation reduces to

$\frac{1}{2}(S,S)+\Delta S=\omega\left(\rho,m_{2}(U)\right)$ (2.28)

and is thus violated unless $m_{2}(U)=0$. As it is well-known (see for example
[36]) this quantity is non-vanisihing and ill-defined in Witten theory.
Formally it receives contributions from the emission of physical closed string
states, which appear as poles in the one-loop two point open string amplitude
[37]. But, for the sake of simplicity (and leaving open the possibility that
there could be very peculiar D-branes in very peculiar closed string
backgrounds where Witten theory makes sense quantum-mechanically), we assume
here that we have an associative two-product $m_{2}$ such that $m_{2}(U)=0$.
In later examples we will work with proper solutions to the quantum master
equation, but the structures we will encounter will be essentially the same as
in this simpler example, after we assume $m_{2}(U)=0$. That said, let us then
introduce the decoupled auxiliary field $\chi$

$S^{\prime}[\rho,\chi]=\frac{1}{2}\omega(\rho,Q\rho)+\frac{1}{2}\omega(\chi,Q\chi)+\frac{1}{3}\omega\left(\rho,m_{2}(\rho,\rho)\right).$
(2.29)

This action can be conveniently packaged as

$S^{\prime}[R]=\frac{1}{2}\Omega\left(R,m^{\prime}_{1}(R)\right)+\frac{1}{3}\Omega\left(R,m^{\prime}_{2}(R,R)\right),$
(2.30)

by organizing the string field and the auxiliary field in a vector

$R\coloneqq\begin{pmatrix}\rho\\\
\chi\end{pmatrix}\in\mathcal{H}^{\prime}\coloneqq\mathcal{H}\oplus\bar{P}\mathcal{H},$
(2.31)

and redefining the symplectic form and the string products as follows

$\begin{split}&m^{\prime}_{1}(R)\coloneqq\begin{pmatrix}Q\rho\\\
Q\chi\end{pmatrix},\\\
&m^{\prime}_{2}(R_{1},R_{2})\coloneqq\begin{pmatrix}m_{2}(\rho_{1},\rho_{2})\\\
0\end{pmatrix},\\\ &\Omega(R_{1},R_{2})\coloneqq{\rm
Tr}\begin{pmatrix}\omega(\rho_{1},\rho_{2})&0\\\
0&\omega\left(\chi_{1},\chi_{2}\right)\end{pmatrix}.\end{split}$ (2.32)

Let’s also define the Poisson bi-vector $U^{\prime}$ associated to the
symplectic form $\Omega$ on $\cal{H}^{\prime}$, which satisfies

$\displaystyle(\Omega\otimes\bm{1}_{{\cal H}^{\prime}})(1_{{\cal
H}^{\prime}}\otimes U^{\prime})$ $\displaystyle=\bm{1}_{{\cal H}^{\prime}},$
(2.33)

and is explicitly written as

$U^{\prime}=\frac{(-1)^{v^{\prime a}}}{2}v^{\prime}_{a}\wedge v^{\prime
a}\in{\cal H}^{\prime\wedge 2},$ (2.34)

where the $\\{v^{\prime}_{a}\\}$ and $\\{v^{\prime a}\\}$ are two canonically
normalized basis of $\mathcal{H^{\prime}}$

$\displaystyle\Omega(v^{\prime
a},v^{\prime}_{b})=-\Omega(v^{\prime}_{b},v^{\prime a})=\delta^{a}_{b},$
(2.35)

from which we can write the completeness relation

$1_{{\cal H}^{\prime}}=v^{\prime}_{a}\Omega(v^{\prime a},\cdot)=-v^{\prime
a}\Omega(v^{\prime}_{a},\cdot).$ (2.36)

Now, let us perform the field redefinition (2.6), $R=O_{\lambda}\Psi$,
obtaining

$S[\Psi]\coloneqq
S^{\prime}[O_{\lambda}\Psi]=\frac{1}{2}\Omega\left(\Psi,m^{\prime}_{1}(\Psi)\right)+\frac{1}{3}\Omega\left(\Psi,O^{-1}_{\lambda}m^{\prime}_{2}(O_{\lambda}\Psi,O_{\lambda}\Psi)\right),$
(2.37)

where we used the fact that we can move $O_{\lambda}$ from the left to the
right of $\Omega$ by taking its inverse (since $L$ is bpz even)

$\Omega\left(O_{\lambda}\Psi_{1},\Psi_{2}\right)=\Omega\left(\Psi_{1},O^{-1}_{\lambda}\Psi_{2}\right)$
(2.38)

and the fact that this field redefinition leaves the kinetic term unchanged
$m^{\prime}_{1}=O^{-1}_{\lambda}m^{\prime}_{1}O_{\lambda}$. Therefore, by
defining

$\begin{split}&M_{1}\coloneqq m^{\prime}_{1}\\\
&M_{2}(\Psi_{1},\Psi_{2})\coloneqq
O^{-1}_{\lambda}m^{\prime}_{2}(O_{\lambda}\Psi_{1},O_{\lambda}\Psi_{2})\\\
&\qquad\qquad\quad=\begin{pmatrix}c_{\lambda}m_{2}\left(c_{\lambda}\psi-\sqrt{1-c_{\lambda}^{2}}\Sigma,c_{\lambda}\psi-\sqrt{1-c_{\lambda}^{2}}\Sigma\right)\\\
-\sqrt{1-c_{\lambda}^{2}}m_{2}\left(c_{\lambda}\psi-\sqrt{1-c_{\lambda}^{2}}\Sigma,c_{\lambda}\psi-\sqrt{1-c_{\lambda}^{2}}\Sigma\right)\end{pmatrix},\end{split}$
(2.39)

the action (2.37) becomes

$S[\Psi]=\frac{1}{2}\Omega\left(\Psi,M_{1}(\Psi)\right)+\frac{1}{3}\Omega\left(\Psi,M_{2}(\Psi,\Psi)\right).$
(2.40)

Let us now uplift this formalism to the level of tensor co-algebras, by
extending the structures to the tensor product space $\mathcal{TH^{\prime}}$,
as explained in detail in [18]. In particular, the string products on
$\mathcal{H^{\prime}}$ are promoted to coderivations and the map $O_{\lambda}$
to a cohomorphism. This means that if we introduce the co-product
$\Delta_{\mathcal{TH^{\prime}}}:\mathcal{TH^{\prime}}\to\mathcal{TH^{\prime}}\otimes^{\prime}\mathcal{TH^{\prime}}$
we have the following relations

$\displaystyle\Delta_{\mathcal{TH^{\prime}}}\bm{m}^{\prime}=\left(\bm{m}^{\prime}\otimes^{\prime}1_{{\cal
TH}^{\prime}}+1_{{\cal
TH}^{\prime}}\otimes^{\prime}\bm{m}^{\prime}\right)\Delta_{\mathcal{TH^{\prime}}},$
(2.41)
$\displaystyle\Delta_{\mathcal{TH^{\prime}}}\bm{O}_{\lambda}=(\bm{O}_{\lambda}\otimes^{\prime}\bm{O}_{\lambda})\Delta_{\mathcal{TH^{\prime}}},$
(2.42)

where we defined $\bm{m}^{\prime}=\bm{m}^{\prime}_{1}+\bm{m}^{\prime}_{2}$ and

$\displaystyle\bm{m}^{\prime}_{1}\pi_{k}=\sum_{i=0}^{k-1}1^{\otimes i}\otimes
M_{1}\otimes 1^{\otimes(k-i-1)}$ (2.43)
$\displaystyle\bm{m}^{\prime}_{2}\pi_{k}=\sum_{i=0}^{k-2}1^{\otimes i}\otimes
M_{2}\otimes 1^{\otimes(k-i-2)},$ (2.44)

where $\pi_{k}$ projects onto the $k$-th tensor power of
$\mathcal{TH^{\prime}}$. Similarly, the cohomomorphism $\bm{O}_{\lambda}$ is
defined as

$\displaystyle\bm{O}_{\lambda}\pi_{k}\coloneqq\left(O_{\lambda}\right)^{\otimes
k}.$ (2.45)

The action (2.30) can be rewritten in Wess-Zumino-Witten (WZW) form

$S^{\prime}[R]=\int_{0}^{1}dt\,\Omega\left(\pi_{1}\bm{\partial}_{t}\frac{1}{1-\otimes
R(t)}\,,\,\pi_{1}\bm{m}^{\prime}\frac{1}{1-\otimes R(t)}\right),$ (2.46)

where

$\frac{1}{1-\otimes x}\coloneqq\sum_{k=0}^{\infty}x^{\otimes k}$

is the group element and where $R(t)$ is an interpolation such that $R(0)=0$
and $R(1)=R$. Following [16], the BV master equation can be expressed as

$\displaystyle\frac{1}{2}{\big{(}}S^{\prime},S^{\prime}{\big{)}}+\Delta
S^{\prime}=\int_{0}^{1}dt\,\Omega\left(\pi_{1}\bm{\partial}_{t}\frac{1}{1-\otimes
R(t)}\,,\,\pi_{1}(\bm{m}^{\prime
2}+\bm{m}^{\prime}\bm{U}^{\prime})\frac{1}{1-\otimes R(t)}\right).$ (2.47)

Here $\bm{U}^{\prime}$ is the uplift of $U^{\prime}$ to
$\mathcal{TH^{\prime}}$ (see for example [17, 16] )

$\displaystyle\bm{U}^{\prime}=\frac{1}{2}(-1)^{v^{\prime
a}}\bm{v}^{\prime}_{a}\bm{v}^{\prime a},$ (2.48)

where $\bm{v}$ is the zero-product coderivation associated to the vector $v$,
which inserts $v$ in all possible ways inside a tensor product, [18]. Upon use
of the classical $A_{\infty}$ relations $(\bm{m}^{\prime})^{2}=0$, the quantum
consistency reduces to $\bm{m}^{\prime}\bm{U}^{\prime}=0$, whose only non-
trivial implication is $m_{2}(U)=0$, which is the quantum consistency of the
initial un-stubbed theory we are assuming since the beginning.
In this formalism, the field redefinition (2.6) can be written as a map
between group like elements, via the cohomomorphism $\bm{O}_{\lambda}$

$\frac{1}{1-\otimes\Psi}=\bm{O}_{\lambda}\frac{1}{1-\otimes R},$ (2.49)

and thus, using (2.38), the above action becomes

$S^{\prime}[O_{\lambda}\Psi]=S[\Psi]=\int_{0}^{1}dt\,\Omega\left(\pi_{1}\bm{\partial}_{t}\frac{1}{1-\otimes\Psi(t)}\,,\,\pi_{1}\bm{M}\frac{1}{1-\otimes\Psi(t)}\right),$
(2.50)

where we consistently defined

$\bm{M}\coloneqq\bm{O}^{-1}_{\lambda}\bm{m}^{\prime}\bm{O}_{\lambda}=\bm{m}^{\prime}_{1}+\bm{O}^{-1}_{\lambda}\bm{m}^{\prime}_{2}\bm{O}_{\lambda}$
(2.51)

that is a coderivation

$\Delta_{\mathcal{TH^{\prime}}}\bm{M}=\left(\bm{M}\otimes^{\prime}1_{{\cal
TH}^{\prime}}+1_{{\cal
TH}^{\prime}}\otimes^{\prime}\bm{M}\right)\Delta_{\mathcal{TH^{\prime}}},$
(2.52)

as a consequence of (2.41) and (2.42).
Clearly, the coderivation $\bm{M}$ satisfies the $A_{\infty}$ relations
$\bm{M}^{2}=0$, whereas verifying that $\bm{M}\bm{U}^{\prime}=0$ requires just
few further steps. The first essential property is that in complete
generality, given an even cohomorphism constructed from a linear operator $F$
on ${\cal H}$, $\bm{F}=\oplus_{k}F^{\otimes k}\pi_{k}$ , we easily find

$\bm{F}\bm{U}=\bm{U}_{F}\bm{F},$ (2.53)

where $\bm{U}_{F}$ corresponds to the co-algebra uplifting of the bi-vector

$U_{F}\coloneqq\bm{F}U=\frac{(-)^{v^{a}}}{2}Fv_{a}\wedge
Fv^{a}\coloneqq\frac{(-)^{f^{a}}}{2}f_{a}\wedge f^{a},$ (2.54)

namely

$\bm{U}_{F}=\frac{(-)^{f^{a}}}{2}\bm{f}_{a}\bm{f}^{a}.$ (2.55)

In our context we are interested in $U^{\prime}_{O_{\lambda}}$ which can be
manipulated using the completeness relation (2.36) and (2.38)

$\begin{split}U^{\prime}_{O_{\lambda}}&=\frac{(-)^{v^{\prime
a}}}{2}O_{\lambda}v^{\prime}_{a}\wedge O_{\lambda}v^{\prime a}\\\
&=\frac{(-)^{v^{\prime a}}}{2}v^{\prime}_{b}\Omega(v^{\prime
b},O_{\lambda}v^{\prime}_{a})\wedge O_{\lambda}v^{\prime a}\\\
&=\frac{(-)^{v^{\prime}_{a}}}{2}v^{\prime}_{b}\wedge O_{\lambda}v^{\prime
a}\Omega(v^{\prime}_{a},O^{-1}_{\lambda}v^{\prime b})\\\
&=\frac{(-)^{v^{\prime b}}}{2}v^{\prime}_{b}\wedge
O_{\lambda}O^{-1}_{\lambda}v^{\prime b}\\\ &=U^{\prime}.\end{split}$ (2.56)

Therefore, from these two results we can state that

$\bm{O}_{\lambda}\bm{U}^{\prime}=\bm{U}^{\prime}\bm{O}_{\lambda},$ (2.57)

and similarly

$\bm{O}^{-1}_{\lambda}\bm{U}^{\prime}=\bm{U}^{\prime}\bm{O}^{-1}_{\lambda}.$
(2.58)

Thus, we can easily check that (2.50) is consistent at quantum level

$\bm{M}\bm{U}^{\prime}=\bm{O}_{\lambda}^{-1}\bm{m}^{\prime}\bm{O}_{\lambda}\bm{U}^{\prime}=\bm{O}_{\lambda}^{-1}\bm{m}^{\prime}\bm{U}^{\prime}\bm{O}_{\lambda}=0,$
(2.59)

because $\bm{m^{\prime}}\bm{U^{\prime}}=0$, as a consequence of the original
assumption $m_{2}(U)=0$.
Now we want to integrate-out the auxiliary field $\Sigma$ as in (2.17). As
discussed in detail in appendix A (see in particular the final equation
(A.70)), this will generate constant terms (which we ignore here) and
effective string products for the remaining string fields $\psi$, which can be
conveniently obtained by the (quantum) homotopy transfer, starting from the
free strong deformation retract

$\circlearrowright(-h)\left(\mathcal{H}^{\prime},M_{1}\right)\xtofrom[\text{$\iota$}]{\text{$\pi$}}\left(\mathcal{H},Q\right)$
(2.60)

where the canonical projector $\pi$ and the canonical inclusion $\iota$ are
defined as follows

$\begin{split}&\pi(\Psi)=\psi\in\mathcal{H},\\\
&\iota(\psi)=\begin{pmatrix}\psi\\\
0\end{pmatrix}\in\mathcal{H}^{\prime},\end{split}$ (2.61)

and clearly satisfy $(\iota\pi)^{2}=\iota\pi$ and $\pi\iota=\bm{1}$. The
needed propagator $h$ is defined as

$h(\Psi)=\begin{pmatrix}0\\\ \frac{b}{L}\Sigma\end{pmatrix},$ (2.62)

which obeys the annihilation relations and the Hodge-Kodaira decomposition

$\begin{split}&\pi h=h\iota=h^{2}=0,\\\
&M_{1}h+hM_{1}=1_{\mathcal{H}^{\prime}}-\iota\pi.\end{split}$ (2.63)

Notice that in (2.62) the propagator is well defined even without the
$\bar{P}$ because $\Sigma\in\bar{P}\cal{H}$.
Now, we can promote all these objects at the co-algebra level. In doing this
$\pi$ and $\iota$ are promoted to cohomomorphisms and the propagator is
uplifted as [18]

$\bm{h}\pi_{n}=\sum_{i=0}^{n}1_{\mathcal{H^{\prime}}}^{\otimes i}\otimes
h\otimes(\iota\pi)^{\otimes n-i-1}\pi_{n}.$ (2.64)

This establishes the existence of the strong deformation retract at the level
of the tensor co-algebra

$\circlearrowright(-\bm{h})\left(\mathcal{TH}^{\prime},\bm{M}_{1}\right)\xtofrom[\text{$\bm{\iota}$}]{\text{$\bm{\pi}$}}\left(\mathcal{TH},\bm{Q}\right).$
(2.65)

The result of integrating out $\Sigma$ from the interacting theory is obtained
by deforming the above strong deformation retract through
$\bm{M}_{2}+\bm{U}^{\prime}$ and the quantum homological perturbation lemma
gives us the explicit expression for the effective products, that is

$\tilde{\bm{m}}=\bm{\pi}(\bm{M}_{1}+\bm{M}_{2})\dfrac{1}{\bm{1}_{\cal
TH}+\bm{h}\left(\bm{M}_{2}+\bm{U}^{\prime}\right)}\bm{\iota}=\bm{Q}+\bm{\pi}\bm{M}_{2}\dfrac{1}{\bm{1}_{\cal
TH}+\bm{h}\left(\bm{M}_{2}+\bm{U}^{\prime}\right)}\bm{\iota}.$ (2.66)

As explained in the appendix A, these products define the dynamical part of
the effective action, which can be written as

$\tilde{S}[\psi]=-\log\int_{h\Sigma=0}\mathcal{D}{\Sigma}\,e^{-S[\psi,\Sigma]}=\sum_{n=0}^{\infty}\frac{1}{n+1}\omega\left(\psi,\tilde{m}_{n}\left(\psi^{\otimes
n}\right)\right)+\Lambda,$ (2.67)

where

$\tilde{m}_{n}=\pi_{1}\tilde{\bm{m}}\pi_{n}$ (2.68)

and $\Lambda$ contains the (connected) vacuum bubbles generated by the path
integral.

Let us now study in detail the effective products by making explicit the
dependence on the starting coderivations $\bm{m^{\prime}}$, which refers to
the theory without stubs (with the addition of the trivial free and auxiliary
field $\chi$)

$\begin{split}\tilde{\bm{m}}&=\bm{Q}+\bm{\pi}\bm{M}_{2}\dfrac{1}{\bm{1}_{\cal
TH^{\prime}}+\bm{h}\left(\bm{M}_{2}+\bm{U}^{\prime}\right)}\bm{\iota}\\\
&=\bm{Q}+\bm{\pi}\bm{O}^{-1}_{\lambda}\bm{m}^{\prime}_{2}\bm{O}_{\lambda}\dfrac{1}{\bm{1}_{\cal
TH^{\prime}}+\bm{h}\left(\bm{O}^{-1}_{\lambda}\bm{m}^{\prime}_{2}\bm{O}_{\lambda}+\bm{U}^{\prime}\right)}\bm{\iota}\\\
&=\bm{Q}+\bm{\pi}\bm{O}^{-1}_{\lambda}\bm{m}^{\prime}_{2}\dfrac{1}{\bm{1}_{\cal
TH^{\prime}}+\bm{O}_{\lambda}\bm{h}\bm{O}^{-1}_{\lambda}\left(\bm{m}^{\prime}_{2}+\bm{O}_{\lambda}\bm{U}^{\prime}\bm{O}^{-1}_{\lambda}\right)}\bm{O}_{\lambda}\bm{\iota}\\\
&=\bm{Q}+\bm{\pi}\bm{O}^{-1}_{\lambda}\bm{m}^{\prime}_{2}\dfrac{1}{\bm{1}_{\cal
TH^{\prime}}+\bm{O}_{\lambda}\bm{h}\bm{O}^{-1}_{\lambda}\left(\bm{m}^{\prime}_{2}+\bm{U}^{\prime}\right)}\bm{O}_{\lambda}\bm{\iota},\end{split}$
(2.69)

where in the last step we used (2.57). Therefore, we can consistently define

$\displaystyle\bm{\pi}_{\lambda}\coloneqq\bm{\pi}\bm{O}^{-1}_{\lambda},$
(2.70) $\displaystyle\bm{\iota}_{\lambda}\coloneqq\bm{O}_{\lambda}\bm{\iota},$
(2.71)
$\displaystyle\bm{h}_{\lambda}\coloneqq\bm{O}_{\lambda}\bm{h}\bm{O}^{-1}_{\lambda},$
(2.72)

obtaining

$\tilde{\bm{m}}=\bm{Q}+\bm{\pi}_{\lambda}\bm{m}^{\prime}_{2}\dfrac{1}{\bm{1}_{\cal
TH^{\prime}}+\bm{h}_{\lambda}\left(\bm{m}^{\prime}_{2}+\bm{U}^{\prime}\right)}\bm{\iota}_{\lambda}.$
(2.73)

Notice that $\bm{\pi}_{\lambda}$ and $\bm{\iota}_{\lambda}$ are
cohomomorphisms because are given by the composition of two cohomomorphisms
and they trivially satisfy the retract property, namely
$\bm{\pi}_{\lambda}\bm{\iota}_{\lambda}=1_{\mathcal{TH}}$ and

$\bm{P}_{\lambda}\coloneqq\bm{\iota}_{\lambda}\bm{\pi}_{\lambda},$ (2.74)

is a projector. In particular, $\bm{P}_{\lambda}$ is the cohomomorphism
associated with the following projector acting on $\mathcal{H}^{\prime}$

$P_{\lambda}=\begin{pmatrix}c^{2}_{\lambda}&c_{\lambda}\sqrt{1-c_{\lambda}^{2}}\\\
c_{\lambda}\sqrt{1-c_{\lambda}^{2}}&1-c^{2}_{\lambda}\end{pmatrix}.$ (2.75)

Let us now focus on the propagator $\bm{h}_{\lambda}$, which clearly satisfies
the annihilation relations
$\bm{h}_{\lambda}^{2}=\bm{\pi}_{\lambda}\bm{h}_{\lambda}=\bm{h}_{\lambda}\bm{\iota}_{\lambda}=0$,
and it acts on a bunch of $n$ states as follows

$\bm{h}_{\lambda}\pi_{n}=\sum_{i=0}^{n}1_{\mathcal{TH^{\prime}}}^{\otimes
i}\otimes h_{\lambda}\otimes P_{\lambda}^{\otimes n-i-1}\pi_{n},$ (2.76)

that is the usual structure of a propagator lifted to the co-algebras
formalism. Moreover, the explicit expression of the propagator is

$h_{\lambda}=\begin{pmatrix}\frac{b}{L}(1-c^{2}_{\lambda})&-\frac{b}{L}c_{\lambda}\sqrt{1-c_{\lambda}^{2}}\\\
-\frac{b}{L}c_{\lambda}\sqrt{1-c_{\lambda}^{2}}&\frac{b}{L}c^{2}_{\lambda}\end{pmatrix}.$
(2.77)

Therefore, adding stubs through the procedure described above can be
equivalently implemented by applying this peculiar homotopy transfer referred
to the action (2.30), namely considering the following strong deformation
retract

$\circlearrowright(-\bm{h}_{\lambda})\left(\mathcal{TH}^{\prime},\bm{m_{1}}^{\prime}\right)\xtofrom[\text{$\bm{\iota}_{\lambda}$}]{\text{$\bm{\pi}_{\lambda}$}}\left(\mathcal{TH},\bm{Q}\right)$
(2.78)

and deforming it with $\bm{m_{1}}^{\prime}\to\bm{m}^{\prime}+\bm{U}^{\prime}$.
This describes the transfer between the two nilpotent structures

$\displaystyle\tilde{\bm{m}}+\bm{U}=\,$
$\displaystyle\bm{\pi}_{\lambda}(\bm{m}^{\prime}+\bm{U}^{\prime})\dfrac{1}{\bm{1}_{\cal
TH^{\prime}}+\bm{h}_{\lambda}\left(\bm{m}^{\prime}_{2}+\bm{U}^{\prime}\right)}\bm{\iota}_{\lambda}$
(2.79) $\displaystyle=\,$
$\displaystyle\bm{\pi}(\bm{M}+\bm{U}^{\prime})\dfrac{1}{\bm{1}_{\cal
TH^{\prime}}+\bm{h}\left(\bm{M}_{2}+\bm{U}^{\prime}\right)}\bm{\iota}.$ (2.80)

Notice that, as explained in appendix A for a generic integration-out, the
transfer of $\bm{U}^{\prime}$ just gives $\bm{U}$

$\displaystyle\bm{U}=\bm{\pi}_{\lambda}(\bm{U}^{\prime})\dfrac{1}{\bm{1}_{\cal
TH^{\prime}}+\bm{h}_{\lambda}\left(\bm{m}^{\prime}_{2}+\bm{U}^{\prime}\right)}\bm{\iota}_{\lambda},$
(2.81)

without contaminations from the interactions.

It is important to realize that, although the quantum cubic action is supposed
to be the same as the classical one, the resulting quantum action with stubs
is different from its classical counterpart [25, 26]. This is because it has
to satisfy the quantum BV master equation (under the assumption that
$m_{2}(U)=0$) and not the classical. For instance, we can notice a tadpole
term that is not present in the classical treatment but which is necessary for
the final theory to satisfy the BV quantum master equation. In particular if
we consider the action weighted by the string coupling constant $\kappa$
according to the topological expansion we get

$\begin{split}\tilde{m}_{0}&=-\kappa\bm{\pi}_{\lambda}\bm{m}^{\prime}_{2}\bm{h}_{\lambda}\bm{U}^{\prime}\bm{1}_{\cal
TH^{\prime}}+O(\kappa^{2})\\\
&=-\frac{1}{2}\kappa\bm{\pi}_{\lambda}\begin{pmatrix}m_{2}\left(v^{i}\wedge\frac{b}{L}(1-c_{\lambda}^{2})v_{i}\right)\\\
0\end{pmatrix}+O(\kappa^{2})\\\ &=-\kappa
m_{2}\left(v^{i},\frac{b}{L}\left(1-c_{\lambda}^{2}\right)v_{i}\right)+O(\kappa^{2}),\end{split}$
(2.82)

where in the last step we used the completeness relation to move
$\sqrt{1-c_{\lambda}^{2}}$ from the left to the right in the Poisson bi-vector
and the fact that $c_{\lambda}$ commutes with the Siegel propagator. This
correction is expected: in the quantum Witten theory the one-loop tadpole is
fully constructed from a Feynman diagram, but after adding stubs a part of
moduli space (far from open string degeneration) is eliminated in the Feynman
diagram and the elementary vertex $\tilde{m}_{0}$ is added to compensate. This
vertex explicitly contains the closed string degeneration region but, just
like the original Feynman diagram, it doesn’t tell us how to deal with it.
This appears to be general: adding open string stubs does not help in dealing
with closed string degeneration. This problem can be fully dealt with in open-
closed SFT, as we will explicitly see in the following.

###  2.4 Closed string field theory with stubs

In this section we will apply the prescription for adding stubs described
above to the case of closed SFT, which is a theory described by a truly
quantum BV master action [38]. The action can be written as

$S_{c}[\rho_{\rm
c}]=\sum_{g=0}^{\infty}\kappa^{2g-2}\sum_{k=0}^{\infty}\frac{1}{k!}\omega_{c}\left(\rho_{\rm
c},l_{k}^{(g)}(\rho_{\rm c}^{\wedge
k})\right)=\frac{1}{2\kappa^{2}}\omega_{c}(\rho_{\rm c},Q_{\rm c}\rho_{\rm
c})+S_{\rm int}[\rho_{\rm c}],$ (2.83)

where $\rho_{\rm c}=r^{i}_{\rm c}c_{i}\in\cal{H}_{\rm c}$ is the closed string
field, $c_{i}$ the basis vectors and $Q_{\rm c}$ the closed BRST charge. As
discussed in the previous section we introduce an auxiliary field
$\chi_{c}\in\bar{P}_{\rm c}\cal{H}_{\rm c}$, where $P_{\rm c}$ projects onto
${\rm ker}(L_{\rm c})$ (the standard choice being $L_{\rm
c}=L_{0}+\bar{L}_{0}$), as described in (2.5), obtaining

$S^{\prime}_{\rm c}[\rho_{\rm c},\chi_{\rm
c}]=\frac{1}{2\kappa^{2}}\omega_{\rm c}\left(\chi_{\rm c},Q_{\rm c}\chi_{\rm
c}\right)+\frac{1}{2\kappa^{2}}\omega_{c}(\rho_{\rm c},Q_{\rm c}\rho_{\rm
c})+S_{\rm int}[\rho_{\rm c}].$ (2.84)

Again we organize the two fields in a vector

$R_{\rm c}\coloneqq\begin{pmatrix}\rho_{\rm c}\\\ \chi_{\rm
c}\end{pmatrix}\in\mathcal{H}^{\prime}_{\rm c}\coloneqq\mathcal{H}_{\rm
c}\oplus\bar{P}_{\rm c}\mathcal{H}_{\rm c},$ (2.85)

and introduce a collective symplectic form and its corresponding Poisson bi-
vector

$\Omega_{\rm c}(R_{1},R_{2})\coloneqq{\rm Tr}\begin{pmatrix}\omega_{\rm
c}(\rho_{1},\rho_{2})&0\\\ 0&\omega_{\rm
c}\left(\chi_{1},\chi_{2}\right)\end{pmatrix},$ (2.86)

$U^{\prime}_{\rm c}=\left[\dfrac{(-)^{c^{i}}}{2}\begin{pmatrix}c_{i}\\\
0\end{pmatrix}\wedge\begin{pmatrix}c^{i}\\\
0\end{pmatrix}+\dfrac{(-)^{\bar{c}^{j}}}{2}\begin{pmatrix}0\\\
\bar{c}_{j}\end{pmatrix}\wedge\begin{pmatrix}0\\\
\bar{c}^{j}\end{pmatrix}\right]\in{\cal H}_{\rm c}^{\prime\wedge 2},$ (2.87)

such that

$\displaystyle(\Omega_{\rm c}\otimes\bm{1})(1\otimes U^{\prime}_{\rm c})$
$\displaystyle=\bm{1}_{\cal{H^{\prime}}_{\rm c}}.$ (2.88)

Let us then extend the multi string products on the new Hilbert space
$\mathcal{H}^{\prime}_{\rm c}$

$l^{\prime(g)}_{n}(R_{1},\cdots,R_{n})\coloneqq\begin{pmatrix}l^{\prime(g)}_{n}(\rho_{1},\cdots,\rho_{n})\\\
0\end{pmatrix},$ (2.89)

which can be uplifted to the co-algebras formalism as coderivations acting on
the symmetrized closed tensor algebra [18]

$\displaystyle\bm{l}^{\prime}\coloneqq\sum_{g}\kappa^{2g}\sum_{n}\bm{l}^{\prime(g)}_{n}:\mathcal{SH}^{\prime}_{\rm
c}\to\mathcal{SH}^{\prime}_{\rm c}.$ (2.90)

Therefore, the action (2.84) can be rewritten in the WZW form by introducing
the interpolated field $R_{\rm c}(t)$ such that $R_{\rm c}(0)=0$ and $R_{\rm
c}(1)=R_{\rm c}$

$\displaystyle S^{\prime}[R_{\rm c}]=\int_{0}^{1}dt\,\frac{\Omega_{\rm
c}}{\kappa^{2}}\left(\pi_{1}\bm{\partial}_{t}\,e^{\wedge R_{\rm
c}(t)}\,,\,\pi_{1}\bm{l}^{\prime}e^{\wedge R_{\rm c}(t)}\right),$ (2.91)

where $\pi_{1}$ projects onto the single closed string state sector.
Moreover, the multi-string products satisfy the quantum $L_{\infty}$ homotopy
algebra, which in this formalism reads

$(\bm{l}^{\prime}+\bm{U}^{\prime}_{\rm c})^{2}=0,$ (2.92)

and this guarantees the validity of the quantum BV master equation [39]. Let
us now perform the field redefinition (2.6) adapted to this case

$R_{\rm c}=\begin{pmatrix}\rho_{\rm c}\\\ \chi_{\rm
c}\end{pmatrix}=O_{\lambda}^{({\rm c})}\begin{pmatrix}\phi\\\ \Sigma_{\rm
c}\end{pmatrix}=\begin{pmatrix}c_{\lambda}^{({\rm
c})}&-\sqrt{1-c_{\lambda}^{({\rm c})2}}\\\ \sqrt{1-c_{\lambda}^{({\rm
c})2}}&c_{\lambda}^{({\rm c})}\end{pmatrix}\begin{pmatrix}\phi\\\ \Sigma_{\rm
c}\end{pmatrix}\coloneqq\Phi,$ (2.93)

where we can standardly choose $c_{\lambda}^{({\rm c})}=e^{\lambda L_{\rm
c}}\to e^{\lambda L_{0}^{+}}$. This rotation uplifts to a cohomomorphism
between two group-like elements

$e^{\wedge R_{\rm c}}=\bm{O}_{\lambda}^{({\rm c})}e^{\wedge\Phi}.$ (2.94)

Thus, by using the property (2.38) the action can be rewritten as follows

$\displaystyle S^{\prime}[\Phi]=\int_{0}^{1}dt\,\frac{\Omega_{\rm
c}}{\kappa^{2}}\left(\pi_{1}\bm{\partial}_{t}\,e^{\Phi(t)}\,,\,\pi_{1}\bm{L}e^{\Phi(t)}\right),$
(2.95)

where we defined

$\bm{L}\coloneqq\bm{O}_{\lambda}^{({\rm
c})-1}\bm{l}^{\prime}\bm{O}_{\lambda}^{({\rm c})},$ (2.96)

that is clearly a coderivation that satisfies the quantum $L_{\infty}$
relations, indeed

$(\bm{L}+\bm{U}^{\prime}_{\rm c})^{2}=\bm{O}_{\lambda}^{({\rm
c})-1}\left(\bm{l}^{\prime}+\bm{O}_{\lambda}^{({\rm c})}\bm{U}^{\prime}_{\rm
c}\bm{O}_{\lambda}^{({\rm c})-1}\right)^{2}\bm{O}_{\lambda}^{({\rm
c})}=\bm{O}_{\lambda}^{({\rm c})-1}(\bm{l}^{\prime}+\bm{U}^{\prime}_{\rm
c})^{2}\bm{O}_{\lambda}^{({\rm c})}=0,$ (2.97)

where in the last step we used (2.57).
Since the products satisfy the quantum homotopy relations we can consistently
apply the homological perturbation lemma to integrate out the auxiliary field
$\Sigma$ ending up on a quantum stubbed closed SFT action. We therefore
introduce the objects which define the free strong deformation retract, namely
the canonical inclusion $\iota_{\rm c}$, the canonical projector $\pi_{\rm c}$
and the propagator $h_{\rm c}$ as follows

$\displaystyle\pi_{\rm c}(\Phi)=\phi,$ (2.98) $\displaystyle\iota_{\rm
c}(\phi)=\begin{pmatrix}\phi\\\ 0\end{pmatrix},$ (2.99) $\displaystyle h_{\rm
c}(\Phi)=\begin{pmatrix}0\\\ \frac{b_{c}}{L_{c}}\Sigma_{\rm c}\end{pmatrix},$
(2.100)

which clearly satisfy the retract properties $\pi_{\rm c}\iota_{\rm
c}=1_{{\cal H}_{\rm c}}$ and $(\iota_{\rm c}\pi_{\rm c})^{2}=\iota_{\rm
c}\pi_{\rm c}$, the annihilation relations $h_{\rm c}\iota_{\rm c}=\pi_{\rm
c}h_{\rm c}=h_{\rm c}^{2}=0$ and the Hodge-Kodaira decomposition

$L_{1}^{(0)}h_{\rm c}+h_{\rm c}L_{1}^{(0)}=1_{{\cal H}^{\prime}_{\rm
c}}-\iota_{\rm c}\pi_{\rm c}.$ (2.101)

We can then promote such objects to the language of co-algebras by following
the well-known definitions of propagator and projectors on the symmetrized
tensor space, as described for example in appendix B of [18]. We can then
apply the homological perturbation lemma, which tells us that the effective
products that will describe the transferred theory take the following form

$\displaystyle\tilde{\bm{l}}$ $\displaystyle=\bm{Q}_{\rm c}+\bm{\pi}_{\rm
c}\bm{\delta L}\frac{1}{\bm{1}_{\mathcal{SH^{\prime}}_{\rm c}}+\bm{h}_{\rm
c}\left(\bm{\delta L}+\kappa^{2}\bm{U^{\prime}}_{\rm c}\right)}\bm{\iota}_{\rm
c}=$ (2.102) $\displaystyle=\bm{Q}_{\rm c}+\bm{\pi}_{\rm c}\bm{O}^{({\rm
c})-1}_{\lambda}\bm{\delta l}^{\prime}\dfrac{1}{\bm{1}_{\cal
SH^{\prime}}+\bm{O}_{\lambda}^{({\rm c})}\bm{h}_{\rm c}\bm{O}^{({\rm
c})-1}_{\lambda}\left(\bm{\delta l}^{\prime}+\kappa^{2}\bm{U}^{\prime}_{\rm
c}\right)}\bm{O}_{\lambda}^{({\rm c})}\bm{\iota}_{\rm c}$ (2.103)
$\displaystyle=\bm{Q}_{\rm c}+\bm{\pi}_{\lambda}^{({\rm c})}\bm{\delta
l}^{\prime}\dfrac{1}{\bm{1}_{\cal SH^{\prime}}+\bm{h}_{\lambda}^{({\rm
c})}\left(\bm{\delta l}^{\prime}+\kappa^{2}\bm{U}^{\prime}_{\rm
c}\right)}\bm{\iota}_{\lambda}^{({\rm c})},$ (2.104)

in which we consistently defined

$\displaystyle\bm{\pi}_{\lambda}^{({\rm c})}\coloneqq\bm{\pi}_{\rm
c}\bm{O}^{({\rm c})-1}_{\lambda},$ (2.105)
$\displaystyle\bm{\iota}_{\lambda}^{({\rm c})}\coloneqq\bm{O}_{\lambda}^{({\rm
c})}\bm{\iota}_{\rm c},$ (2.106) $\displaystyle\bm{h}_{\lambda}^{({\rm
c})}\coloneqq\bm{O}_{\lambda}^{({\rm c})}\bm{h}_{\rm c}\bm{O}^{({\rm
c})-1}_{\lambda}.$ (2.107)

Therefore, the action for the subbed closed string field (ignoring the vacuum
bubbles, which are not captured by the homotopy transfer) reads

$\displaystyle\tilde{S}_{\rm
c}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{[}}\phi{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{]}}=\int_{0}^{1}dt\,\frac{\omega_{\rm
c}}{\kappa^{2}}\left(\pi_{1}\bm{\partial}_{t}\,e^{\wedge\phi(t)}\,,\,\pi_{10}\tilde{\bm{l}}\,e^{\wedge\phi(t)}\right),$
(2.108)

where $\tilde{\bm{l}}=\sum_{g=0}^{\infty}\kappa^{2g}\tilde{\bm{l}}^{(g)}$. In
particular the truncation to $g=0$ is the classical stubbed closed SFT, which
obeys the classical master equation.

###  2.5 Open-closed string field theory with stubs

In this section we will apply our method to open-closed string field theory
[40]. We will follow the recent formulation of OC-SFT presented in [16], to
which we refer the reader for further details. The BV master action is given
by

$S_{{\rm oc}}[\rho_{\rm o},\rho_{\rm
c}]=\sum_{g,b=0}^{\infty}\kappa^{2g+b-2}\sum_{k=0}^{\infty}\sum_{\\{p_{1},...,p_{b}\\}}^{\infty}\frac{1}{b!k!(p_{1})\cdots(p_{b})}\mathcal{A}^{g,b}_{k;p_{1},...,p_{b}}\left(\rho_{\rm
c}^{\wedge k}\otimes^{\prime}\rho_{\rm o}^{\odot
p_{1}}\wedge^{\prime}\cdots\wedge^{\prime}\rho_{\rm o}^{\odot p_{b}}\right),$
(2.109)

where $\rho_{\rm o}=r^{i}_{\rm o}o_{i}\in\cal{H}_{\rm o}$ is the open string
field, $\rho_{\rm c}=r^{i}_{\rm c}c_{i}\in\cal{H}_{\rm c}$ the closed string
field, $o_{i}$ the basis vectors of $\mathcal{H}_{\rm o}$ and $c_{i}$ the
basis vectors of $\mathcal{H}_{\rm c}$. The
$\mathcal{A}^{g,b}_{k;p_{1},...,p_{b}}$ represent all the possible couplings
on all possible Riemann surfaces, including the kinetic terms for closed
strings (sphere) and open strings (disk). The action can be written in terms
of open-closed multi-string products as [16]

$\displaystyle S_{\rm oc}[\rho_{\rm c},\rho_{\rm
o}]=\sum_{g,b=0}^{\infty}\kappa^{2g+b-2}\sum_{k=0}^{\infty}\frac{1}{b!}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Bigg{[}}}}\frac{1}{(k+1)!}\omega_{\text{c}}\left(\rho_{\rm
c},l^{(g,b)}_{k,0}\left(\rho_{\rm c}^{\wedge k}\right)\right)$ (2.110)
$\displaystyle+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\sum_{\begin{subarray}{c}\\{p_{1},...,p_{b-1}\\}=0\\\
{p_{b}=1}\end{subarray}}^{\infty}}}\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{C({p_{1},\ldots,p_{b}})}}}{k!\,(p_{1})\cdots(p_{b})}\omega_{\text{o}}\left(\rho_{\rm
o},m^{(g,b)}_{k\left[p_{1},...,p_{b-1}\right]p_{b}-1}\left(\rho_{\rm
c}^{\wedge k}\otimes^{\prime}\rho_{\rm o}^{\odot p_{1}}\cdots\rho_{\rm
o}^{\odot p_{b-1}}\otimes^{\prime\prime}\rho_{\rm
o}^{\otimes(p_{b}-1)}\right)\right){{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\Bigg{]}}}}$

with the important identification

$\displaystyle\omega_{\text{o}}\left(\Psi_{b,1},m^{(g,b)}_{k\left[p_{1},...,p_{b-1}\right]p_{b}-1}\left((\Phi)_{k}\otimes^{\prime}[\Psi]_{p_{1}}\wedge^{\prime}\cdots\wedge^{\prime}[\Psi]_{p_{b-1}}\otimes^{\prime\prime}\Psi_{b,2}\otimes\cdots\otimes\Psi_{b,p_{b}}\right)\right)$
$\displaystyle\coloneqq$
$\displaystyle(-1)^{\epsilon}\omega_{\text{c}}\left(\Phi_{1},l^{(g,b)}_{k-1,[p_{1},\cdots,p_{b}]}\left(\Phi_{2}\wedge\cdots\wedge\Phi_{k}\otimes^{\prime}[\Psi]_{p_{1}}\wedge^{\prime}\cdots\wedge^{\prime}[\Psi]_{p_{b}}\right)\right)$
(2.111) $\displaystyle=$
$\displaystyle\mathcal{A}^{g,b}_{k;\\{p_{1},...,p_{b}\\}}\left((\Phi)_{k}\otimes^{\prime}[\Psi]_{p_{1}}\wedge^{\prime}...\wedge^{\prime}[\Psi]_{p_{b}}\right)$

and other more standard cyclicity relations. The constrained $l$ and $m$
products can then be upgraded to coderivation-like operators $\bm{l}$ and
$\bm{m}$ on the open-closed tensor algebra $\mathcal{SH}_{\rm
c}\otimes^{\prime}\mathcal{SCH}_{\rm o}$ (consisting of symmetrized closed
strings ($\mathcal{SH}_{\rm c}$) and cyclized open strings ($\mathcal{CH}_{\rm
o}$) on symmetrized boundaries ($\mathcal{SCH}_{\rm o}$)) in a way which is
described in detail in [16].

Following the general strategy, we isolate the two kinetic terms and we
introduce an auxiliary field for each sector

$\begin{split}S^{\prime}_{{\rm oc}}[\rho_{\rm o},\chi_{\rm o},\rho_{\rm
c},\chi_{\rm c}]&=\frac{1}{2\kappa}\omega_{\text{o}}(\rho_{\rm o},Q_{\rm
o}\rho_{\rm o})+\frac{1}{2\kappa}\omega_{\text{o}}\left(\chi_{\rm o},Q_{\rm
o}\chi_{\rm o}\right)\\\ &+\frac{1}{2\kappa^{2}}\omega_{\text{c}}(\rho_{\rm
c},Q_{\rm c}\rho_{\rm
c})+\frac{1}{2\kappa^{2}}\omega_{\text{c}}\left(\chi_{\rm c},Q_{\rm
c}\chi_{\rm c}\right)\\\ &+S_{{\rm int}}[\rho_{\rm o},\rho_{\rm
c}].\end{split}$ (2.112)

in which $\chi_{\rm o}\in\bar{P}_{\rm o}$ and $\chi_{\rm c}\in\bar{P}_{\rm c}$
where $P_{\rm o}$ and $P_{\rm c}$ project onto ${\rm ker}(L_{\rm o})$ and
${\rm ker}(L_{\rm c})$, respectively. The standard choice of stubs is $L_{\rm
c}=L_{0}^{+}$ and $L_{\rm o}=L_{0}$.
We then define

$R_{\rm c}\coloneqq\begin{pmatrix}\rho_{\rm c}\\\ \chi_{\rm
c}\end{pmatrix}\in\mathcal{H}^{\prime}_{\rm c}\coloneqq\mathcal{H}_{\rm
c}\oplus\bar{P}_{\rm c}\mathcal{H}_{\rm c},$ (2.113) $R_{\rm
o}\coloneqq\begin{pmatrix}\rho_{\rm o}\\\ \chi_{\rm
o}\end{pmatrix}\in\mathcal{H}^{\prime}_{\rm o}\coloneqq\mathcal{H}_{\rm
o}\oplus\bar{P}_{\rm o}\mathcal{H}_{\rm o},$ (2.114)

and

$\Omega_{\rm o}(R_{{\rm o}(1)},R_{{\rm o}(2)})\coloneqq{\rm
Tr}\begin{pmatrix}\omega_{\rm o}(\rho_{{\rm o}(1)},\rho_{{\rm o}(2)})&0\\\
0&\omega_{\rm o}\left(\chi_{{\rm o}(1)},\chi_{{\rm
o}(2)}\right)\end{pmatrix},$ (2.115) $\Omega_{\rm c}(R_{{\rm c}(1)},R_{{\rm
c}(2)})\coloneqq{\rm Tr}\begin{pmatrix}\omega_{\rm c}(\rho_{{\rm
c}(1)},\rho_{{\rm c}(2)})&0\\\ 0&\omega_{\rm c}\left(\chi_{{\rm
c}(1)},\chi_{{\rm c}(2)}\right)\end{pmatrix},$ (2.116) $U^{\prime}_{\rm
o}=\left[\dfrac{(-)^{o^{i}}}{2}\begin{pmatrix}o_{i}\\\
0\end{pmatrix}\wedge\begin{pmatrix}o^{i}\\\
0\end{pmatrix}+\dfrac{(-)^{\bar{o}^{j}}}{2}\begin{pmatrix}0\\\
\bar{o}_{j}\end{pmatrix}\wedge\begin{pmatrix}0\\\
\bar{o}^{j}\end{pmatrix}\right]\in{\cal H}_{\rm o}^{\prime\wedge 2},$ (2.117)
$U^{\prime}_{\rm c}=\left[\dfrac{(-)^{c^{i}}}{2}\begin{pmatrix}c_{i}\\\
0\end{pmatrix}\wedge\begin{pmatrix}c^{i}\\\
0\end{pmatrix}+\dfrac{(-)^{\bar{c}^{j}}}{2}\begin{pmatrix}0\\\
\bar{c}_{j}\end{pmatrix}\wedge\begin{pmatrix}0\\\
\bar{c}^{j}\end{pmatrix}\right]\in{\cal H}_{\rm c}^{\prime\wedge 2},$ (2.118)

such that

$\displaystyle(\Omega_{\rm o}\otimes\bm{1})(1\otimes U^{\prime}_{\rm o})$
$\displaystyle=\bm{1}_{\cal{H^{\prime}}_{\rm o}},$ (2.119)
$\displaystyle(\Omega_{\rm c}\otimes\bm{1})(1\otimes U^{\prime}_{\rm c})$
$\displaystyle=\bm{1}_{\cal{H^{\prime}}_{\rm c}}.$ (2.120)

Now, we promote the multi string products on the new Hilbert spaces

$\displaystyle l^{\prime(g,b)}_{k[p_{1},\cdots,p_{b}]}(R_{\rm c}^{\wedge
k}\otimes^{\prime}R_{\rm o}^{\odot
p_{b}}\wedge^{\prime}\cdots\wedge^{\prime}R_{\rm o}^{\odot
p_{1}})\coloneqq\begin{pmatrix}l^{(g,b)}_{k[p_{1},\cdots,p_{b}]}(\rho_{\rm
c}^{\wedge k}\otimes^{\prime}\rho_{\rm o}^{\odot
p_{b}}\wedge^{\prime}\cdots\wedge^{\prime}\rho_{\rm o}^{\odot p_{1}})\\\
0\end{pmatrix},$ (2.121) $\displaystyle
m^{\prime(g,b)}_{k[p_{1},\cdots,p_{b-1}]p_{b}}(R_{\rm c}^{\wedge
k}\otimes^{\prime}R_{\rm o}^{\odot
p_{1}}\wedge^{\prime}\cdots\wedge^{\prime}R_{\rm o}^{\odot
p_{b-1}}\otimes^{\prime\prime}R_{\rm o}^{\otimes p_{b}})\coloneqq$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\begin{pmatrix}m^{(g,b)}_{k[p_{1},\cdots,p_{b-1}]p_{b}}(\rho_{\rm
c}^{\wedge k}\otimes^{\prime}\rho_{\rm o}^{\odot
p_{1}}\wedge^{\prime}\cdots\wedge^{\prime}\rho_{\rm o}^{\odot
p_{b-1}}\otimes^{\prime\prime}\rho_{\rm o}^{\otimes p_{b}})\\\
0\end{pmatrix},$ (2.122)

which can be uplifted to the co-algebras formalism in the sense described in
[16].

$\displaystyle\bm{l}^{\prime}=\sum_{g,b}\kappa^{2g+b}\sum_{k,\left\\{p_{1},...,p_{b}\right\\}}\bm{l}^{(g,b)}_{k\left[p_{1},...,p_{b}\right]}:\mathcal{SH^{\prime}}_{\rm
c}\otimes^{\prime}\mathcal{SCH^{\prime}}_{\rm o}\to\mathcal{SH^{\prime}}_{\rm
c}\otimes^{\prime}\mathcal{SCH^{\prime}}_{\rm o}$ (2.123)
$\displaystyle\bm{m}^{\prime}=\sum_{g,b}\kappa^{2g+b-1}\sum_{k,\left\\{p_{1},...,p_{b}\right\\}}\bm{m}^{(g,b)}_{k\left[p_{1},...,p_{b-1}\right]p_{b}}:\mathcal{SH^{\prime}}_{\rm
c}\otimes^{\prime}\mathcal{SCH^{\prime}}_{\rm o}\to\mathcal{SH^{\prime}}_{\rm
c}\otimes^{\prime}\mathcal{SCH^{\prime}}_{\rm o}.$ (2.124)

Therefore, by defining an interpolation for each field, namely $R_{\rm o}(t)$
and $R_{\rm c}(t)$ such that for $t=0$ they return $0$ and for $t=1$ $R_{\rm
o}$ and $R_{\rm c}$, respectively, we can rewrite the open-closed SFT action
with free auxiliary fields in the WZW form

$\displaystyle S^{\prime}[R_{\rm c},R_{\rm o}]=\int_{0}^{1}dt\,$
$\displaystyle\frac{\Omega_{\rm
c}}{\kappa^{2}}\left(\pi_{10}\bm{\partial}_{t}\,{\cal G}(R_{\rm c}(t),R_{\rm
o}(t)))\,,\,\pi_{10}\bm{l}^{\prime}\,{\cal G}(R_{\rm c}(t),R_{\rm
o}(t))\right)+$ $\displaystyle\frac{\Omega_{\rm
o}}{\kappa}\left(\pi_{01}\,\bm{\partial}_{t}\,{\cal G}(R_{\rm c}(t),R_{\rm
o}(t))\,,\,\pi_{01}\bm{m}^{\prime}\,{\cal G}(R_{\rm c}(t),R_{\rm
o}(t))\right),$ (2.125)

where $\pi_{10}$ and $\pi_{01}$ returns a single copy of
$\mathcal{H}^{\prime}_{\rm c}$ and $\mathcal{H}^{\prime}_{\rm o}$,
respectively. In addition, the group like element is defined as in [16]

${\cal G}(R_{\rm c},R_{\rm o})=\sum_{k,b,p_{1},...,p_{b}\geq
0}\frac{1}{k!b!(p_{1})\cdots(p_{b})}\left(R_{\rm c}^{\wedge
k}\otimes^{\prime}R_{\rm o}^{\odot
p_{1}}\wedge^{\prime}\cdots\wedge^{\prime}R_{\rm o}^{\odot p_{b}}\right),$
(2.126)

where $(p)=p+\delta_{p,0}$. Furthermore, the products satisfy the quantum
open-closed homotopy algebra, namely

$\displaystyle\pi_{10}\left\\{\dfrac{1}{2}[\bm{l}^{\prime},\bm{l}^{\prime}]+\bm{l}^{\prime}\bm{m}^{\prime}+\left[\bm{l}^{\prime},\kappa\bm{U}^{\prime}_{o}+\kappa^{2}\bm{U}^{\prime}_{c}\right]\right\\}=0,$
(2.127)
$\displaystyle\kappa\pi_{01}\left\\{\dfrac{1}{2}[\bm{m}^{\prime},\bm{m}^{\prime}]+\bm{m}^{\prime}\bm{l}^{\prime}+\left[\bm{m}^{\prime},\kappa\bm{U}^{\prime}_{o}+\kappa^{2}\bm{U}^{\prime}_{c}\right]\right\\}=0,$
(2.128)

which can be succinctly written as

$(\bm{l}^{\prime}+\bm{m}^{\prime}+\kappa\bm{U}^{\prime}_{o}+\kappa^{2}\bm{U}^{\prime}_{c})^{2}=0.$

Let us now implement the field redefinition (2.6), with two independent stub
parameters $\lambda_{\rm c}$ and $\lambda_{\rm o}$

$R_{\rm c}=\begin{pmatrix}\rho_{\rm c}\\\ \chi_{\rm
c}\end{pmatrix}=O_{\lambda_{\rm c}}^{({\rm c})}\begin{pmatrix}\phi\\\
\Sigma_{\rm c}\end{pmatrix}=\begin{pmatrix}c_{\lambda_{\rm c}}^{({\rm
c})}&-\sqrt{1-c_{\lambda_{\rm c}}^{({\rm c})2}}\\\ \sqrt{1-c_{\lambda_{\rm
c}}^{({\rm c})2}}&c_{\lambda_{\rm c}}^{({\rm
c})}\end{pmatrix}\begin{pmatrix}\phi\\\ \Sigma_{\rm
c}\end{pmatrix}\coloneqq\Phi,$ (2.129) $R_{\rm o}=\begin{pmatrix}\rho_{\rm
o}\\\ \chi_{\rm o}\end{pmatrix}=O_{\lambda_{\rm o}}^{({\rm
o})}\begin{pmatrix}\psi\\\ \Sigma_{\rm
o}\end{pmatrix}=\begin{pmatrix}c_{\lambda_{\rm o}}^{({\rm
o})}&-\sqrt{1-c_{\lambda_{\rm o}}^{({\rm o})2}}\\\ \sqrt{1-c_{\lambda_{\rm
o}}^{({\rm o})2}}&c_{\lambda_{\rm o}}^{({\rm
o})}\end{pmatrix}\begin{pmatrix}\psi\\\ \Sigma_{\rm
o}\end{pmatrix}\coloneqq\Psi.$ (2.130)

As discussed in the previous sections the orthogonal matrices which define the
above field redefinitions can be upgraded to the co-algebras formalism as an
open-closed cohomomorphism-like object acting on the group-like element

${\cal G}(\Phi,\Psi)=\bm{O}_{\lambda_{\rm o},\lambda_{\rm c}}{\cal G}(R_{\rm
c},R_{\rm o}),$ (2.131)

where

$\bm{O}_{\lambda_{\rm o},\lambda_{\rm c}}=\bm{O}^{({\rm c})}_{\lambda_{\rm
c}}\otimes^{\prime}\bm{O}^{({\rm o})}_{\lambda_{\rm
o}}:\mathcal{SH^{\prime}}_{\rm c}\otimes^{\prime}\mathcal{SCH^{\prime}}_{\rm
o}\to\mathcal{SH}_{\rm c}\otimes^{\prime}\mathcal{SCH}_{\rm o}.$ (2.132)

Therefore, thanks to (2.38) the action (2.125) can be rewritten as

$\displaystyle S^{\prime}[\Phi,\Psi]=\int_{0}^{1}dt\,$
$\displaystyle\frac{\Omega_{\rm
c}}{\kappa^{2}}\left(\pi_{10}\bm{\partial}_{t}\,{\cal
G}(\Phi(t),\Psi(t)))\,,\,\pi_{10}\bm{L}\,{\cal G}(\Phi(t),\Psi(t))\right)+$
$\displaystyle\frac{\Omega_{\rm
o}}{\kappa}\left(\pi_{01}\,\bm{\partial}_{t}\,{\cal
G}(\Phi(t),\Psi(t))\,,\,\pi_{01}\bm{M}\,{\cal G}(\Phi(t),\Psi(t))\right),$
(2.133)

where we defined

$\displaystyle\bm{L}\coloneqq\bm{O}^{-1}_{\lambda_{\rm o},\lambda_{\rm
c}}\bm{l}^{\prime}\bm{O}_{\lambda_{\rm o},\lambda_{\rm c}},$ (2.134)
$\displaystyle\bm{M}\coloneqq\bm{O}^{-1}_{\lambda_{\rm o},\lambda_{\rm
c}}\bm{m}^{\prime}\bm{O}_{\lambda_{\rm o},\lambda_{\rm c}},$ (2.135)

Moreover, the new multi-string products satisfy the quantum homotopy algebra

$\displaystyle\pi_{10}\left\\{\dfrac{1}{2}[\bm{L},\bm{L}]+\bm{L}\bm{M}+\left[\bm{L},\kappa\bm{U}^{\prime}_{o}+\kappa^{2}\bm{U}^{\prime}_{c}\right]\right\\}=0,$
(2.136)
$\displaystyle\kappa\pi_{01}\left\\{\dfrac{1}{2}[\bm{M},\bm{M}]+\bm{M}\bm{L}+\left[\bm{M},\kappa\bm{U}^{\prime}_{o}+\kappa^{2}\bm{U}^{\prime}_{c}\right]\right\\}=0,$
(2.137)

and this can be seen by following the same steps as in (2.97). Therefore,
(2.133) satisfies the quantum master equation, and we can consistently apply
the quantum homotopy transfer with the goal of integrating out the two
auxiliary fields $\Sigma_{\rm o}$ and $\Sigma_{\rm c}$. Furthermore, precisely
because the products $\bm{L}$ and $\bm{M}$ define a quantum nilpotent
structure

$(\bm{L}+\bm{M}+\kappa\bm{U}^{\prime}_{o}+\kappa^{2}\bm{U}^{\prime}_{c})^{2}=0,$

we can assert that the effective theory will also be a solution of the
corresponding quantum master equation.
Let us then define the desired strong deformation retract by introducing a
canonical projector and a canonical inclusion promoted to the co-algebras
formalism as cohomorphisms

$\displaystyle\bm{\pi}=\bm{\pi}_{\rm c}\otimes^{\prime}\bm{\pi}_{\rm o},$
(2.138) $\displaystyle\bm{\iota}=\bm{\iota}_{\rm
c}\otimes^{\prime}\bm{\iota}_{\rm o},$ (2.139)

where

$\displaystyle\pi_{\rm c}(\Phi)=\phi,$ (2.140) $\displaystyle\iota_{\rm
c}(\phi)=\begin{pmatrix}\phi\\\ 0\end{pmatrix},$ (2.141)

the same for $\pi_{\rm o}$ and $\iota_{\rm o}$. Hence, if we apply $\bm{\pi}$
to the group like element we get

$\bm{\pi}{\cal G}(\Phi,\Psi)=\sum_{k,b,p_{1},...,p_{b}\geq
0}\frac{1}{k!b!(p_{1})\cdots(p_{b})}\left(\phi^{\wedge
k}\otimes^{\prime}\psi^{\odot
p_{1}}\wedge^{\prime}\cdots\wedge^{\prime}\psi^{\odot
p_{b}}\right)=\mathcal{G}(\phi,\psi)\coloneqq\mathcal{G}.$ (2.142)

Let us now define the propagator

$\bm{h}=\bm{h}_{c}\otimes^{\prime}\bm{\iota}_{\rm o}\bm{\pi}_{\rm
o}+\bm{1}_{\mathcal{SH^{\prime}}_{\rm c}}\otimes^{\prime}\bm{h}_{\rm o},$
(2.143)

where

$\displaystyle h_{\rm c}(\Phi)=\begin{pmatrix}0\\\ \frac{b_{\rm c}}{L_{\rm
c}}\Sigma_{\rm c}\end{pmatrix},$ (2.144) $\displaystyle h_{\rm
o}(\Psi)=\begin{pmatrix}0\\\ \frac{b_{\rm o}}{L_{\rm o}}\Sigma_{\rm
o}\end{pmatrix}.$ (2.145)

Finally, by applying the quantum homological perturbation lemma we get the
following action for the stubbed open-closed theory

$\displaystyle\tilde{S}_{\rm
oc}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{[}}\phi,\psi{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{]}}=\int_{0}^{1}dt\left(\frac{\omega_{\rm
c}}{\kappa^{2}}\left(\pi_{10}\bm{\partial}_{t}\,{\cal
G}\,,\,\pi_{10}\tilde{\bm{l}}\,{\cal G}\right)+\frac{\omega_{\rm
o}}{\kappa}\left(\pi_{01}\,\bm{\partial}_{t}\,{\cal
G}\,,\,\pi_{01}\tilde{\bm{m}}\,{\cal G}\right)\right).$ (2.146)

where the effective products are

$\tilde{\bm{l}}+\tilde{\bm{m}}=\bm{Q}_{\rm c}+\bm{Q}_{\rm
o}+\bm{\pi}_{\lambda_{\rm o},\lambda_{\rm c}}(\bm{\delta
m}^{\prime}+\bm{\delta l}^{\prime})\frac{1}{\bm{1}_{\mathcal{SH}_{\rm
c}\otimes^{\prime}\mathcal{SCH}_{\rm o}}+\bm{h}_{\lambda_{\rm o},\lambda_{\rm
c}}\left(\bm{\delta m}^{\prime}+\bm{\delta
l}^{\prime}+\kappa\bm{U}^{\prime}_{\rm o}+\kappa^{2}\bm{U}^{\prime}_{\rm
c}\right)}\bm{\iota}_{\lambda_{\rm o},\lambda_{\rm c}}$ (2.147)

and we have introduced

$\displaystyle\bm{\pi}_{\lambda_{\rm o},\lambda_{\rm
c}}\coloneqq\bm{\pi}\bm{O}^{-1}_{\lambda_{\rm o},\lambda_{\rm c}},$ (2.148)
$\displaystyle\bm{\iota}_{\lambda_{\rm o},\lambda_{\rm
c}}\coloneqq\bm{O}_{\lambda_{\rm o},\lambda_{\rm c}}\bm{\iota},$ (2.149)
$\displaystyle\bm{h}_{\lambda_{\rm o},\lambda_{\rm
c}}\coloneqq\bm{O}_{\lambda_{\rm o},\lambda_{\rm
c}}\bm{h}\bm{O}^{-1}_{\lambda_{\rm o},\lambda_{\rm c}}.$ (2.150)

##  3 Equivalence of amplitudes

In this section we will show that the on-shell amplitudes for a theory with
and without stubs are equivalent at the quantum level. For the sake of
clarity, we will consider ‘quantum’ $A_{\infty}$ theories (of the kind
analyzed in section 2.3), but the proof can be straightforwardly generalized
to quantum $L_{\infty}$ theories and to open-closed SFT.
Our proof will rely on showing that the two theories give rise to the same
‘almost’ minimal model, namely the effective action for fields in the image of
$P$, properly containing the cohomology. Then amplitudes will trivially be the
same, by further restricting the external legs to the cohomology. Let’s start
with the original theory without stubs (trivially extended by the free
auxiliary field $\chi$, which certainly does not change the amplitudes). In
this case the minimal model effective products (the amplitudes) are given by
the following perturbed SDR

$\displaystyle\circlearrowright(-\bm{h}^{\prime}_{0})\left(\mathcal{TH}^{\prime},\bm{m_{1}}^{\prime}\right)$
$\displaystyle\xtofrom[\text{$\bm{\iota}^{\prime}_{0}$}]{\text{$\bm{\pi}^{\prime}_{0}$}}\left(\mathcal{T}P\mathcal{H},\bm{\pi}_{0}\bm{Q}\bm{\iota}_{0}\right)$
$\displaystyle\downarrow\quad\quad\quad$
$\displaystyle\quad\quad\quad\quad\quad\downarrow$ (3.1)
$\displaystyle\circlearrowright(-\hat{\bm{h}}^{\prime}_{0})\left(\mathcal{TH}^{\prime},\bm{m}^{\prime}+\bm{U}^{\prime}\right)$
$\displaystyle\xtofrom[\text{$\hat{\bm{\iota}}^{\prime}_{0}$}]{\text{$\hat{\bm{\pi}}^{\prime}_{0}$}}\left(\mathcal{T}P\mathcal{H},\hat{\bm{m}}+\bm{U}_{\pi_{0}}\right),$

where the (un-perturbed) projectors are defined as follows

$\displaystyle\bm{\pi}^{\prime}_{0}\coloneqq\bm{\pi}_{0}\bm{\pi}:\mathcal{TH}^{\prime}\to\mathcal{T}P\mathcal{H},$
(3.2)
$\displaystyle\bm{\iota}^{\prime}_{0}\coloneqq\bm{\iota}\bm{\iota}_{0}:\mathcal{T}P\mathcal{H}\to\mathcal{TH}^{\prime}$
(3.3)

and the propagator $\bm{h}^{\prime}_{0}$ is the co-algebraic uplifting (2.76)
of $h^{\prime}_{0}$ which acts on an element of $\mathcal{H}^{\prime}$ as

$h^{\prime}_{0}\coloneqq\begin{pmatrix}h_{0}&0\\\
0&\frac{b}{L}\end{pmatrix}\coloneqq\begin{pmatrix}\frac{b}{L}\bar{P}&0\\\
0&\frac{b}{L}\end{pmatrix}.$ (3.4)

Notice that $\bar{P}$ is not needed in the second slot because there are no
states belonging to ${\rm ker}(L)$.
Using the homotopy transfer, the effective products read

$\hat{\bm{m}}=\bm{\pi}^{\prime}_{0}\bm{m}^{\prime}\frac{1}{\bm{1}_{\mathcal{TH}^{\prime}}+\bm{h}^{\prime}_{0}(\bm{\delta
m}^{\prime}+\bm{U}^{\prime})}\bm{\iota}^{\prime}_{0}.$ (3.5)

Let’s now discuss the amplitudes of the stubbed theory, defined by the
homotopy transfer (2.78). To obtain the almost minimal model in this scenario,
we must apply a further integration-out. Therefore, computing amplitudes for
the stubbed theory corresponds to applying two consecutive homotopy transfers
to the starting theory (with the non-propagating free field $\chi$), without
stubs

$\displaystyle\circlearrowright(-\bm{h}_{\lambda})\left(\mathcal{TH}^{\prime},\bm{m_{1}}^{\prime}\right)\xtofrom[\text{$\bm{\iota}_{\lambda}$}]{\text{$\bm{\pi}_{\lambda}$}}$
$\displaystyle\circlearrowright(-\bm{h}_{0})\left(\mathcal{TH},\bm{Q}\right)$
$\displaystyle\\!\xtofrom[\text{$\bm{\iota}_{0}$}]{\text{$\bm{\pi}_{0}$}}\left(\mathcal{T}P\mathcal{H},\bm{\pi}_{0}\bm{Q}\bm{\iota}_{0}\right)$
$\displaystyle\downarrow$ $\displaystyle\downarrow$ (3.6)
$\displaystyle\circlearrowright(-\tilde{\bm{h}}_{\lambda})\left(\mathcal{TH}^{\prime},\bm{m}^{\prime}+\bm{U}^{\prime}\right)\xtofrom[\text{$\tilde{\bm{\iota}}_{\lambda}$}]{\text{$\tilde{\bm{\pi}}_{\lambda}$}}$
$\displaystyle\circlearrowright(-\hat{\bm{h}}_{0})\left(\mathcal{TH},\tilde{\bm{m}}+\bm{U}\right)$
$\displaystyle\\!\xtofrom[\text{$\hat{\bm{\iota}}_{0}$}]{\text{$\hat{\bm{\pi}}_{0}$}}\left(\mathcal{T}P\mathcal{H},\hat{\bm{m}}_{\lambda}+\bm{U}_{\pi_{0}}\right)$

This is an occurrence of the horizontal composition described in detail in
[18], which tells us that we can write

$\hat{\bm{m}}_{\lambda}=\bm{\pi}_{0}\bm{\pi}_{\lambda}\bm{m}^{\prime}\frac{1}{\bm{1}_{\mathcal{TH}^{\prime}}+\left(\bm{h}_{\lambda}+\bm{\iota}_{\lambda}\bm{h}_{0}\bm{\pi}_{\lambda}\right)\left(\bm{\delta
m}^{\prime}+\bm{U}^{\prime}\right)}\bm{\iota}_{\lambda}\bm{\iota}_{0}.$ (3.7)

This expression (3.7) has to be compared with (3.5). The two expressions
apparently differ because of the $\pi$ projectors, the $\iota$ inclusions and
the propagators. Let’s start by analyzing the apparent differences in the
propagators. First of all we have to understand that, as already pointed out
in [18], the composite co-algebraic propagator
$\bm{h}_{\lambda}+\bm{\iota}_{\lambda}\bm{h}_{0}\bm{\pi}_{\lambda}$ does not
appear to be of the typical form (2.76), indeed its action is given by

$\begin{split}&\left(\bm{h}_{\lambda}+\bm{\iota}_{\lambda}\bm{h}_{0}\bm{\pi}_{\lambda}\right)\pi_{n}=\sum_{k=0}^{n-1}\left[1_{\mathcal{H}^{\prime}}^{\otimes
k}\otimes h_{\lambda}\otimes P_{\lambda}^{\otimes(n-k-1)}+P_{\lambda}^{\otimes
k}\otimes\iota_{\lambda}h_{0}\pi_{\lambda}\otimes\left(\iota_{\lambda}P\pi_{\lambda}\right)^{\otimes(n-k-1)}\right]\pi_{n}.\end{split}$
(3.8)

Nevertheless, this composite propagator satisfies (by construction) the Hodge-
Kodaira decomposition and thus (3.8) is just an alternative equivalent way to
uplift the propagator $h_{\lambda}+\iota_{\lambda}h_{0}\pi_{\lambda}$ from
$\mathcal{H}^{\prime}$ to $\mathcal{TH}^{\prime}$. Additionally, it is
possible to show that after unpackaging (3.7) the effective products are given
in terms of $h_{\lambda}+\iota_{\lambda}h_{0}\pi_{\lambda}$ exactly as if we
had upgraded the ‘propagator’ $h_{\lambda}+\iota_{\lambda}h_{0}\pi_{\lambda}$
according to the prescription (2.76). Therefore, we can directly compare the
propagators in (3.5) and (3.7) as operators on $\mathcal{H}^{\prime}$.
Let us then compute explicitly the composite propagator using the definitions
(2.70), (2.71) and (2.72)

$\begin{split}h_{\lambda}+\iota_{\lambda}h_{0}\pi_{\lambda}&=O_{\lambda}\left(h_{0}+\iota
h_{0}\pi\right)O_{\lambda}^{-1}\\\
&=O_{\lambda}\begin{pmatrix}\frac{b}{L}\bar{P}&0\\\
0&\frac{b}{L}\end{pmatrix}O_{\lambda}^{-1}\\\
&=\begin{pmatrix}\frac{b}{L}\bar{P}c_{\lambda}^{2}+\frac{b}{L}(1-c_{\lambda}^{2})&\frac{b}{L}\bar{P}c_{\lambda}\sqrt{1-c_{\lambda}^{2}}-\frac{b}{L}c_{\lambda}\sqrt{1-c_{\lambda}^{2}}\\\
\frac{b}{L}\bar{P}c_{\lambda}\sqrt{1-c_{\lambda}^{2}}-\frac{b}{L}c_{\lambda}\sqrt{1-c_{\lambda}^{2}}&\frac{b}{L}\bar{P}(1-c_{\lambda}^{2})+\frac{b}{L}c_{\lambda}^{2}\end{pmatrix}\\\
&=\begin{pmatrix}\frac{b}{L}\bar{P}&0\\\
0&\frac{b}{L}\end{pmatrix}=h^{\prime}_{0},\end{split}$ (3.9)

where in the last step we used the fact that
$\bar{P}(1-c_{\lambda}^{2})=(1-c_{\lambda}^{2})$ or equivalently
$\bar{P}\sqrt{1-c_{\lambda}^{2}}=\sqrt{1-c_{\lambda}^{2}}$, because
$\sqrt{1-c_{\lambda}^{2}}\rho\in\bar{P}\mathcal{H}\quad\forall\rho\in\mathcal{H}$,
as explained in section 2.1.
Let us now study the composite canonical projection $\pi_{0}\pi_{\lambda}$ by
applying it on a test state $R\in\mathcal{H}^{\prime}$

$\begin{split}\pi_{0}\pi_{\lambda}R&=\pi_{0}\pi
O_{\lambda}^{-1}\begin{pmatrix}\rho\\\ \chi\end{pmatrix}\\\
&=\pi_{0}\left(c_{\lambda}\rho+\sqrt{1-c_{\lambda}^{2}}\chi\right)\\\
&=\pi_{0}c_{\lambda}\rho\\\ &=\pi_{0}\rho,\end{split}$ (3.10)

where the third line is justified by the fact that
$\sqrt{1-c_{\lambda}^{2}}\,\chi$ does not belong to ${\rm ker}(L)$ and thus it
is annihilated by $\pi_{0}$. Moreover, in the last step we noticed that
$\pi_{0}c_{\lambda}\rho=\pi_{0}\rho$ because $\pi_{0}$ returns a non-vanishing
result when $\rho\in{\rm ker}(L)$ and clearly in such a situation
$c_{\lambda}\rho=e^{-\lambda L}\rho=\rho$. Therefore, the composite canonical
projector acts effectively as

$\pi_{0}\pi_{\lambda}=\pi_{0}\pi=\pi^{\prime}_{0}.$ (3.11)

Similarly, it is possible to show that the composite canonical inclusion
$\iota_{\lambda}\iota_{0}=\iota^{\prime}_{0}$ and thus we can conclude that

$\hat{\bm{m}}_{\lambda}=\hat{\bm{m}}.$

###  3.1 Example: open string one-loop tadpole in OC-SFT

After the general proof we have just given we can concretely test the
equivalence between amplitudes in open-closed SFT with and without stubs. To
compute an amplitude in OC-SFT, as in the previous section, we apply the
homological perturbation lemma to the following strong deformation retract

$\circlearrowright(-\bm{H}_{0})\left(\mathcal{SH}_{\rm
c}\otimes^{\prime}\mathcal{SCH}_{\rm o},\bm{Q}_{\rm c}+\bm{Q}_{\rm
o}\right)\xtofrom[\text{$\bm{\iota}_{0}$}]{\text{$\bm{\pi}_{0}$}}\left(\mathcal{S}P_{0}^{+}\mathcal{H}_{\rm
c}\otimes^{\prime}\mathcal{SC}P_{0}\mathcal{H}_{\rm
o},\bm{\pi}_{0}\left(\bm{Q}_{\rm c}+\bm{Q}_{\rm
o}\right)\bm{\iota}_{0}\right)$ (3.12)

in which the projector $\bm{\iota_{0}}\bm{\pi_{0}}$ is defined as

$\bm{\iota_{0}}\bm{\pi_{0}}=\bm{P}_{0}^{+}\otimes^{\prime}\bm{P}_{0},$ (3.13)

where $P_{0}$ and $P_{0}^{+}$ project onto the ${\rm ker}(L_{0})$ and ${\rm
ker}(L_{0}^{+})$, respectively. As discussed in [2], the open-closed co-
algebraic propagator $\bm{H}_{0}$ can be chosen as

$\bm{H}_{0}=\bm{h}_{0}^{+}\otimes^{\prime}\bm{P}_{0}+\bm{1}_{\mathcal{SH^{\prime}}_{\rm
c}}\otimes^{\prime}\bm{h}_{0},$ (3.14)

where $\bm{h}_{0}^{+}$ and $\bm{h}_{0}$ are the co-algebraic promotion of the
Siegel gauge propagators

$\displaystyle h_{0}^{+}=\frac{b_{0}^{+}}{L_{0}^{+}}\bar{P}_{0}^{+},$ (3.15)
$\displaystyle h_{0}=\frac{b_{0}}{L_{0}}\bar{P}_{0},$ (3.16)

see appendix A of [2] for more details. Therefore, the effective products of
the almost minimal model, which define the amplitudes are

$\left(\hat{\bm{l}}+\hat{\bm{m}}\right)=\bm{\pi}_{0}\left(\bm{l}+\bm{m}\right)\frac{1}{\bm{1}_{\mathcal{SH}_{\rm
c}\otimes^{\prime}\mathcal{SCH}_{\rm o}}+\bm{H}_{0}\left(\bm{\delta
m}+\bm{\delta l}+\kappa\bm{U}_{\rm o}+\kappa^{2}\bm{U}_{\rm
c}\right)}\bm{\iota}_{0}.$ (3.17)

It is particularly illuminating, as a concrete example, to specifically
discuss the open string one-loop tadpole, which is simple enough and yet non-
trivially tests the quantum structure of our construction.888Another simple
(but classical) example we have studied is the closed string two-point
amplitude on the disk, which was analyzed in detail in [41]. In this case the
stub deformation creates an explicit fundamental vertex in the middle of
moduli space, which was absent (with a specific choice of $SL(2,\mathbb{C})$
vertices) in [41]. We denote this amplitude as $A_{0;1}^{(0,2)}(\varphi_{\rm
o})$. In this particular case, the relevant homotopy relation is

$[\bm{m}^{(0,2)},\bm{m}^{(0,1)}]+\pi_{0,1}\bm{m}^{(0,1)}\bm{l}^{(0,1)}+[\bm{m}^{(0,2)},\bm{l}^{(0,0)}]+[\bm{m}^{(0,1)},\bm{U}_{\rm
o}]=0,$ (3.18)

and if we apply it to the identity we get

$m_{1,0}^{(0,1)}\left(l_{0,0}^{(01)}\right)+Q_{\rm
o}m_{0,0}^{(0,2)}+m_{0,2}^{(0,1)}(U_{\rm o})=0.$ (3.19)

Using 3.17 we can find the expression for the desired amplitudes without and
with stubs

$A_{0;1}^{0,2}(\varphi_{\rm o})=\omega_{\text{o}}\left(\varphi_{\rm
o},{m}_{0,0}^{0,2}\right)-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{1,0}^{(0,1)}\left(\frac{b_{0}^{+}}{L_{0}^{+}}\bar{P}_{0}^{+}l_{0,0}^{(0,1)}\right)\right)-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{0,2}^{(0,1)}\left(\bm{h}_{0}U_{\rm o}\right)\right),$ (3.20)

and

$A_{\lambda\,0;1}^{0,2}(\varphi_{\rm o})=\omega_{\text{o}}\left(\varphi_{\rm
o},\tilde{m}_{0,0}^{0,2}\right)-\omega_{\text{o}}\left(\varphi_{\rm
o},\tilde{m}_{1,0}^{(0,1)}\left(\frac{b_{0}^{+}}{L_{0}^{+}}\bar{P}_{0}^{+}\tilde{l}_{0,0}^{(0,1)}\right)\right)-\omega_{\text{o}}\left(\varphi_{\rm
o},\tilde{m}_{0,2}^{(0,1)}\left(\bm{h}_{0}U_{\rm o}\right)\right).$ (3.21)

As before, let us proceed by extracting the vertices with stubs that appear in
the previous expression from equation 2.147 which we list below

$\displaystyle\tilde{l}_{0,0}^{(0,1)}=c_{\lambda_{\rm c}}^{({\rm
c})}l_{0,0}^{(0,1)},$ (3.22)
$\displaystyle\tilde{m}_{1,0}^{(0,1)}(\cdot)=c_{\lambda_{\rm c}}^{({\rm
c})}m_{1,0}^{(0,1)}(c_{\lambda_{\rm o}}^{({\rm o})}\cdot),$ (3.23)
$\displaystyle\tilde{m}_{0,2}^{(0,1)}(\cdot,\cdot)=c_{\lambda_{\rm o}}^{({\rm
o})}m_{0,2}^{(0,1)}(c_{\lambda_{\rm o}}^{({\rm o})}\cdot,c_{\lambda_{\rm
o}}^{({\rm o})}\cdot),$ (3.24)
$\displaystyle\tilde{m}_{0,0}^{(0,2)}=c_{\lambda_{\rm o}}^{({\rm
o})}m_{0,0}^{(0,2)}-c_{\lambda_{\rm o}}^{({\rm o})}m_{1,0}^{(0,1)}\left(H_{\rm
c}l_{0,0}^{(0,1)}\right)-c_{\lambda_{\rm o}}^{({\rm
o})}m_{0,2}^{(0,1)}\left(\bm{H}_{\rm o}U_{\rm o}\right).$ (3.25)

Therefore, the stubbed amplitude becomes

$\begin{split}A_{\lambda\,0;1}^{0,2}(\varphi_{\rm
o})&=\omega_{\text{o}}\left(\varphi_{\rm
o},m_{0,0}^{(0,2)}\right)-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{1,0}^{(0,1)}\left(\left[H_{\rm
c}+\frac{b_{0}^{+}}{L_{0}^{+}}\bar{P}_{0}^{+}c_{\lambda_{\rm c}}^{({\rm
c})2}\right]l_{0,0}^{(0,1)}\right)\right)\\\
&\qquad-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{0,2}^{(0,1)}\left(\left[\bm{H}_{\rm o}+\bm{c}^{(\rm o)}_{\lambda_{\rm
o}}\bm{h}_{0}\right]U_{\rm o}\right)\right)\\\
&=\omega_{\text{o}}\left(\varphi_{\rm
o},\tilde{m}_{0,0}^{(0,2)}\right)-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{1,0}^{(0,1)}\left(\frac{b_{0}^{+}}{L_{0}^{+}}\bar{P}_{0}^{+}l_{0,0}^{(0,1)}\right)\right)\\\
&\qquad-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{0,2}^{(0,1)}\left(\left[\bm{H}_{\rm o}+\bm{c}^{(\rm o)}_{\lambda_{\rm
o}}\bm{h}_{0}\right]U_{\rm o}\right)\right).\end{split}$ (3.26)

Thus, following the general picture we have previously discussed, the two
amplitudes are equivalent if

$\left[\bm{H}_{\rm o}+\bm{c}^{(\rm o)}_{\lambda_{\rm
o}}\bm{h}_{0}\right]U_{\rm o}=\bm{h}_{0}U_{\rm o},$ (3.27)

which is simply demonstrated as

$\begin{split}\left[\bm{H}_{\rm o}+\bm{c}^{(\rm o)}_{\lambda_{\rm
o}}\bm{h}_{0}\right]U_{\rm o}&=\frac{1}{2}o^{i}\wedge H_{o}o_{i}+\bm{c}^{(\rm
o)}_{\lambda_{\rm o}}\left(\frac{1}{2}o^{i}\wedge h_{o}o_{i}\right)\\\
&=\frac{1}{2}o^{i}\wedge\left[H_{\rm
o}+\frac{b_{0}}{L_{0}}\bar{P}_{0}c_{\lambda_{\rm o}}^{({\rm
o})2}\right]o_{i}\\\
&=\frac{1}{2}o^{i}\wedge\frac{b_{0}}{L_{0}}\bar{P}_{0}o_{i}=\bm{h}_{0}U_{\rm
o},\end{split}$ (3.28)

where we inserted the completeness relation to move operators from the basis
vector on the left to the basis vectors on the right.
It is also instructive to provide a geometric picture of the stub deformation
in this example. This is possible by explicitly constructing the OC-SFT
vertices that realize this amplitude. We have done this is in appendix B, by
explicitly solving the homotopy relation (3.19). Using these results, we can
rewrite the stubbed and the un-stubbed amplitude in terms of surface states

$\begin{split}A_{0;1}^{0,2}(\varphi_{\rm o})=&-\frac{1}{2\pi
i}\int_{0}^{\infty}ds\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes b_{0}^{+}e^{-sL_{0}^{+}}\right)\left(\varphi_{\rm o}\otimes
l_{0,0}^{(0,1)}\right)\\\ &+\int_{t_{\rm min}}^{t_{\rm
max}}dt\langle\Sigma_{0,1}^{(0,2)}(t)|B_{t}(V_{t})\varphi_{\rm o}\\\
&-\int_{0}^{\infty}ds\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes 1_{\mathcal{H}_{\rm o}}\otimes
b_{0}e^{-sL_{0}}\right)\left(o^{i}\otimes\varphi_{\rm o}\otimes
o_{i}\right)\end{split}$ (3.29)

and

$\begin{split}A_{\lambda\,0;1}^{0,2}(\varphi_{\rm o})=&-\frac{1}{2\pi
i}\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
h_{0}^{+}c_{\lambda_{\rm c}}^{({\rm c})2}\right)\left(\varphi_{\rm o}\otimes
l_{0,0}^{(0,1)}\right)\\\ &-\frac{1}{2\pi
i}\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes H_{\rm
c}\right)\left(\varphi_{\rm o}\otimes l_{0,0}^{(0,1)}\right)\\\ &+\int_{t_{\rm
min}}^{t_{\rm max}}dt\langle\Sigma_{0,1}^{(0,2)}(t)|B_{t}(V_{t})\varphi_{\rm
o}\\\ &-\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
1_{\mathcal{H}_{\rm o}}\otimes h_{0}c_{\lambda_{\rm o}}^{({\rm
o})2}\right)\left(o^{i}\otimes\varphi_{\rm o}\otimes o_{i}\right)\\\
&-\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
1_{\mathcal{H}_{\rm o}}\otimes H_{\rm o}\right)\left(o^{i}\otimes\varphi_{\rm
o}\otimes o_{i}\right).\\\ \end{split}$ (3.30)

Now, using the standard Schwinger representation for the propagators, we get

$\begin{split}A_{\lambda\,0;1}^{0,2}(\varphi_{\rm o})=&-\frac{1}{2\pi
i}\int_{2\lambda_{\rm
c}}^{\infty}ds\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
b_{0}^{+}e^{-sL_{0}^{+}}\right)\left(\varphi_{\rm o}\otimes
l_{0,0}^{(0,1)}\right)\\\ &-\frac{1}{2\pi i}\int_{0}^{2\lambda_{\rm
c}}ds\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
b_{0}^{+}e^{-sL_{0}^{+}}\right)\left(\varphi_{\rm o}\otimes
l_{0,0}^{(0,1)}\right)\\\ &+\int_{t_{\rm min}}^{t_{\rm
max}}dt\langle\Sigma_{0,1}^{(0,2)}(t)|B_{t}(V_{t})\varphi_{\rm o}\\\
&-\int_{0}^{2\lambda_{\rm
o}}ds\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
1_{\mathcal{H}_{\rm o}}\otimes
b_{0}e^{-sL_{0}}\right)\left(o^{i}\otimes\varphi_{\rm o}\otimes
o_{i}\right)\\\ &-\int_{2\lambda_{\rm
o}}^{\infty}ds\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
1_{\mathcal{H}_{\rm o}}\otimes
b_{0}e^{-sL_{0}}\right)\left(o^{i}\otimes\varphi_{\rm o}\otimes
o_{i}\right),\end{split}$ (3.31)

that is clearly equivalent to the amplitude without stubs. Furthermore, we can
easily interpret the reorganization of the terms which build-up the amplitude
after having introduced the stubs. In particular, we have a new fundamental
vertex given by the sum of the second, third and fourth line of the above
expression (3.31) and thus now the fundamental vertex involves integration
over a bigger region of moduli space. Conversely, the portion of moduli space
covered by Feynman diagrams is reduced. Indeed the amplitude can be rewritten
as

$\begin{split}A_{\lambda\,0;1}^{0,2}(\varphi_{\rm o})=&-\int_{2\lambda_{\rm
o}}^{\infty}ds\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
1_{\mathcal{H}_{\rm o}}\otimes
b_{0}e^{-sL_{0}}\right)\left(o^{i}\otimes\varphi_{\rm o}\otimes
o_{i}\right)\\\ &+\int_{t_{\rm c}(e^{-\lambda_{\rm c}})}^{t_{\rm
o}(e^{-\lambda_{\rm
o}})}dt\langle\Sigma_{0,1}^{(0,2)}(t)|B_{t}(V_{t})\varphi_{\rm o}\\\
&-\frac{1}{2\pi i}\int_{2\lambda_{\rm
c}}^{\infty}ds\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
b_{0}^{+}e^{-sL_{0}^{+}}\right)\left(\varphi_{\rm o}\otimes
l_{0,0}^{(0,1)}\right),\end{split}$ (3.32)

where $t_{\rm o}(q)$ and $t_{\rm c}(q)$ are defined in (B.15) and (B.32). In
particular the local coordinate map associated with the surface state
$\langle\Sigma_{0,1}^{(0,2)}(t)|$ is defined as follows

$\hat{F}_{t}(w)=\begin{cases}&G_{\rm c}(w)\qquad\qquad t<t_{\rm min}\\\
&F_{t}(w)\qquad\qquad t_{\rm min}<t<t_{\rm max}\\\ &G_{\rm o}(w)\qquad\qquad
t>t_{\rm max}\end{cases}$ (3.33)

where the maps $G_{\rm c},\,G_{\rm c},\,F_{t}$ are defined in appendix B.

##  4 Conclusions

After having clarified the quantum algebraic structure of the stub deformation
in bosonic SFT, we would like to give some possible directions for future
explorations

  * •

According to our picture, adding stubs also generates constant terms in the
action, from the path integral over $\Sigma$. In this paper we have ignored
these constant terms, but it is important to understand why they are there.
The first of these terms is the one-loop determinant of the $\Sigma$ kinetic
term. Since this is the same as the one-loop determinant of the $\chi$ kinetic
term, this term has to be canceled by a counterterm in the original extended
action (2.5), if we want that integrating out $\chi$ reproduces exactly the
initial theory. The other vacuum terms are generated by the interactions of
$\Sigma$ with $\psi$ and they start at two loops. The reason why they are
present is that the stubbed and the original action should give rise to the
same vacuum energy when one reduces them to the (almost) minimal model, i.e.
after integrating out the whole $\bar{P}H$. This is also expected from
(quantum) background independence and the field-redefinition invariance of the
path-integral measure [42].999We thank A. H. Fırat for a discussion on this
point. We expect this to be true only after accounting for the extra constant
terms that are generated by integrating-out $\Sigma$ (which, after discarding
the one-loop determinant, also start at two loops). It would be worth to
systematically check this, which we expect to be a generalization of the
arguments presented in section 3. On the technical side, we would like to
understand how the vacuum bubbles that are generated by the path integral can
be formulated using the same co-algebraic homotopy transfer that generates
amplitudes with external legs.

  * •

It could be a useful exercise to repeat our analysis in the context of
superstring field theory. In particular our addition of a free non-propagating
field seems to share some similarity with Sen’s method of introducing the R
sector via a free (but propagating) field. It would be interesting to see
these two structures at work simultaneously.

  * •

Although the stub deformation is a field redefinition, it is not invertible in
a strict sense. This is geometrically clear since the inverse stub operator
$e^{+\lambda L_{0}}$ has the interpretation of ‘removing’ a strip of
worldsheet, which may not be a well-defined operation in general. Nonetheless
there could be a well-defined domain of definition for inverse stubs: for sure
we expect to be possible to remove stubs from a theory where we have just
added them, if the length of the inverse stub is smaller than the length of
the added ones. The use of inverse stubs in the kinetic operator of closed SFT
could be a way of regulating the one-loop determinant, which produces the
torus vacuum amplitude an infinite number of times on infinite equivalent
fundamental domains [38]. Similarly, the vacuum energy which is generated in
open-closed SFT after open string integration-out and closed string vacuum-
shift as discussed in [2] seems to overcount the annulus moduli space and
inverse open-string stubs could play a role in fixing this problem. More
generally, the introduction of quantum stubs could help in approaching the
elusive nature of the cosmological constant in SFT.

  * •

Open-Closed SFT has a natural ‘interpolating’ form [43] where, at the extreme
of the interpolation, the moduli space is mostly covered by open or closed
string propagators [9, 14]. It would be interesting to relate these general
interpolations with open and closed stub deformations. Again, this seems to
require the possibility of increasing the closed string stubs and reducing at
the same time the open string stubs (and viceversa).

In general, we hope the technical results in this paper will be useful for
better understanding the quantum structure of string (field) theory.

## Acknowledgments

We thank H. Erbin, A. H. Fırat and J. Vošmera for useful discussions. CM
thanks the Kavli Institute for Theoretical Physics (KITP), Santa Barbara, for
hospitality during the workshop “What is String Theory? Weaving Perspectives
Together”, where part of this work was completed. This research was supported
in part by grant NSF PHY-2309135 to the KITP. The work of CM and AR is
partially supported by the MUR PRIN contract 2020KR4KN2 “String Theory as a
bridge between Gauge Theories and Quantum Gravity”. The work of CM, AR and BV
is also partially supported by the INFN project STeFI “String Theory and
Fundamental Interactions”.

##  Appendix A Path integral and co-algebraic homotopy transfer

In this section we will prove explicitly that the homotopy transfer at the
level of co-algebras (that we heavily use in the main text) performs the path
integral over the degrees of freedom which we want to integrate out. This
amounts in showing that the transferred products given by the homotopy
transfer precisely reproduce the effective vertices obtained by standard QFT
manipulations. The equivalence of the homotopy transfer to the path integral
has been discussed in detail in [34], at the level of Batalin-Vilkovisky
formalism. Here we will translate this language to the ‘dual’ co-algebra
language, on which our SFT formalism is based. In order to be as self-
contained as possible we will start from the very beginning.

Let us then consider a generic SFT action $S[\Phi]$ that satisfies the BV
quantum master equation, where $\Phi\in\mathcal{H}$ can be decomposed as
$\Phi=\phi^{i}f_{i}$ where the $f_{i}$’s are the basis vectors of a graded
vector space ${\cal H}$, the space of string fields.
Let’s furthermore distinguish between the free kinetic term and the
interactions

$S[\Phi]=S_{\rm free}[\Phi]+S_{\rm int}[\Phi],$ (A.1)

and assume a kinetic term of the form

$S_{\rm
free}[\Phi]=\frac{1}{2}\omega\left(\Phi,Q\Phi\right)=\frac{1}{2}\omega(f_{i},Qf_{j})\phi^{i}\phi^{j}=\frac{1}{2}Q_{ij}\phi^{i}\phi^{j},$
(A.2)

where $\omega$ is a symplectic form on ${\cal H}$, $Q$ is a BRST operator (an
odd nilpotent linear operator on ${\cal H}$) and $Q_{ij}$ is defined as

$Q_{ij}\coloneqq\frac{1}{2}\omega(f_{i},Qf_{j}),$ (A.3)

with

$Q_{ij}=(-)^{f_{j}}Q_{ji},$ (A.4)

where $(-)^{v}$ accounts for the degree of the vector $v$. Let us then
introduce a projector $P$ and define $\varphi=P\Phi$ and $R=\bar{P}\Phi$. Our
aim is to find the Wilsonian effective action $\tilde{S}[\varphi]$ by
integrating out $R$ in the BV path integral, namely

$e^{-\tilde{S}[\phi]}=\int_{L}\mathcal{D}R\,e^{-S[\varphi+R]},$ (A.5)

where $L$ is a Lagrangian submanifold corresponding to some gauge fixing. To
better understand this point let us introduce a ‘propagator’ $h$ giving rise
to the Hodge-Kodaira decomposition

$\displaystyle P+hQ+Qh=1,$ (A.6)

together with $h^{2}=Ph=0$, so that $(P,hQ,Qh)$ are orthogonal projectors.
This partitions the total space ${\cal H}$ into

$\mathcal{H}=A\oplus B\oplus C,$ (A.7)

where we have defined

$\displaystyle A\coloneqq P\mathcal{H}={\rm Span}\\{a_{i}\\},$ (A.8)
$\displaystyle B\coloneqq Qh\mathcal{H}={\rm Span}\\{b_{i}\\},$ (A.9)
$\displaystyle C\coloneqq hQ\mathcal{H}={\rm Span}\\{c_{i}\\}.$ (A.10)

We also define dual basis vectors by

$\omega(a^{i},a_{j})=\omega(b^{i},c_{j})=\omega(c^{i},b_{j})=\delta^{i}_{j},$
(A.11)

from which we can write down the completeness relation

$1_{\mathcal{H}}=a_{i}\omega(a^{i},\cdot)+b_{i}\omega(c^{i},\cdot)+c_{i}\omega(b^{i},\cdot).$
(A.12)

Under this decomposition the symplectic form is given by

$\omega_{ij}=\begin{pmatrix}\omega_{ij}^{A}&0&0\\\ 0&0&\omega_{ij}^{BC}\\\
0&-\omega_{ij}^{BC}&0\end{pmatrix},$ (A.13)

where we defined

$\displaystyle\omega_{ij}^{A}\coloneqq\omega(a_{i},a_{j}),$ (A.14)
$\displaystyle\omega_{ij}^{BC}\coloneqq\omega(b_{i},c_{j}).$ (A.15)

Therefore, the string field can be written as

$\displaystyle\Phi=\varphi+R=\alpha^{i}a_{i}+\beta^{i}b_{i}+\gamma^{i}c_{i}.$
(A.16)

The distinction between fields and anti-fields in $(1-P){\cal
H}\equiv\bar{P}{\cal H}$ depends on a choice of lagrangian submanifold (a
maximal subspace of $\bar{P}{\cal H}$ where $\omega$ vanishes). In particular
it is possible to arrange all anti-fields of $\bar{P}\mathcal{H}$ into $B$.
Consequently, the Lagrangian submanifold that defines the gauge fixing will be
given by the condition $\beta^{i}=0$ or equivalently $hR=h\Phi=0$. Thus, the
path integral becomes

$\int_{L}\mathcal{D}R=\int_{L}\prod_{ij}d\beta^{i}d\gamma^{j}=\int\prod_{ij}d\beta^{i}d\gamma^{j}\delta(\beta^{i}).$
(A.17)

By defining $R^{\prime}=\gamma^{i}c_{i}$, the effective action reads

$e^{-\tilde{S}[\phi]}=\int\prod_{i}d\gamma^{i}e^{-S[\varphi+R^{\prime}]}.$
(A.18)

To perform the above integral we introduce a source term $J_{i}\gamma^{i}$,
with $d(J_{i})=d(\gamma^{i})$, and we isolate the kinetic term of the $C$
sector

$\begin{split}e^{-\tilde{S}[\phi]}&=\int\prod_{i}d\gamma^{i}e^{-\frac{1}{2}Q_{ij}\gamma^{i}\gamma^{j}-S_{\rm
free}[\varphi]-S_{\rm
int}[\varphi+R^{\prime}]+J_{i}\gamma^{i}}\bigr{|}_{J_{i}=0}\\\ &=e^{-S_{\rm
free}[\varphi]-S_{\rm
int}[\varphi+\partial_{J_{i}}c_{i}]}\int\prod_{i}d\gamma^{i}e^{-\frac{1}{2}Q_{ij}\gamma^{i}\gamma^{j}+J_{i}\gamma^{i}}\bigr{|}_{J_{i}=0}.\end{split}$
(A.19)

The problem then reduces to the computation of a Gaussian integral. Assuming
positivity of $Q_{ij}$ (also by deforming possible ‘tachyonic’ contours in the
complex plane so that, for grassmann even variables, the integral is done on a
steepest path), the integral converges when $Q_{ij}$ is non-degenerate. This
condition is satisfied if ${\rm ker}(Q)$ is contained in ${\rm Im}(P)$. Under
this assumption we can readily compute the path integral by completing the
square. To do this, we define

$h^{ij}\coloneqq(-)^{b_{i}}\omega(b^{i},hb^{j}),$ (A.20)

that satisfies the following properties

$\displaystyle h^{ij}=(-)^{b^{i}}h^{ji},$ (A.21) $\displaystyle
Q_{ji}h^{jk}=\delta_{i}^{k},$ (A.22) $\displaystyle
Q_{ij}h^{jk}=(-)^{c_{i}}\delta_{i}^{k}.$ (A.23)

Then we can easily verify

$-\frac{1}{2}Q_{ij}\gamma^{i}\gamma^{j}+J_{i}\gamma^{i}=-\frac{1}{2}\left[Q_{ij}(\gamma^{i}-h^{ik}J_{k})(\gamma^{j}-h^{jl}J_{l})\right]+\frac{1}{2}h^{ij}J_{i}J_{j},$
(A.24)

and thus, by making the change of variables $x^{i}=\gamma^{i}-h^{ik}J_{k}$ we
get

$\begin{split}e^{-\tilde{S}[\phi]}&=e^{-S_{\rm free}[\varphi]-S_{\rm
int}[\varphi+\partial_{J_{i}}c_{i}]}e^{\frac{1}{2}h^{ij}J_{i}J_{j}}\int\prod_{i}dx^{i}e^{-\frac{1}{2}Q_{ij}x^{i}x^{j}}\bigr{|}_{J_{i}=0}\\\
&={\rm sdet}(Q_{ij})^{-\frac{1}{2}}e^{-S_{\rm free}[\varphi]-S_{\rm
int}[\varphi+\partial_{J_{i}}c_{i}]}e^{\frac{1}{2}h^{ij}J_{i}J_{j}}\bigr{|}_{J_{i}=0},\end{split}$
(A.25)

Therefore, the effective action can be written as

$\tilde{S}[\varphi]={\cal A}-\log\left(e^{-S_{\rm free}[\varphi]-S_{\rm
int}[\varphi+\partial_{J_{i}}c_{i}]}e^{\frac{1}{2}h^{ij}J_{i}J_{j}}\right)\bigr{|}_{J_{i}=0}.$
(A.26)

Additionally, we can introduce a further source term $J_{i}\gamma^{i}$,
getting

$\begin{split}\tilde{S}[\varphi]&={\cal A}-\log\left(e^{-S_{\rm
free}[\varphi]-S_{\rm
int}[\varphi+\partial_{J_{i}}c_{i}]}e^{\frac{1}{2}h^{ij}J_{i}J_{j}}e^{J_{i}\gamma^{i}}\right)\bigr{|}_{J_{i}=0,\gamma^{i}=0}\\\
&={\cal A}-\log\left(e^{-S_{\rm free}[\varphi]-S_{\rm
int}[\varphi+\partial_{J_{i}}c_{i}]}e^{\frac{1}{2}h^{ij}\partial_{\gamma^{i}}\partial_{\gamma^{j}}}e^{J_{i}\gamma^{i}}\right)\bigr{|}_{J_{i}=0,\gamma^{i}=0}\\\
&={\cal
A}-\log\left(e^{\frac{1}{2}h^{ij}\partial_{\gamma^{i}}\partial_{\gamma^{j}}}e^{-S_{\rm
free}[\varphi]-S_{\rm
int}[\varphi+\partial_{J_{i}}c_{i}]}e^{J_{i}\gamma^{i}}\right)\bigr{|}_{J_{i}=0,\gamma^{i}=0}\\\
&={\cal
A}-\log\left(e^{\frac{1}{2}h^{ij}\partial_{\gamma^{i}}\partial_{\gamma^{j}}}e^{-S_{\rm
free}[\varphi]-S_{\rm
int}[\varphi+\gamma^{i}c_{i}]}\right)\bigr{|}_{\gamma^{i}=0}.\end{split}$
(A.27)

Let us then introduce a canonical projector $\Pi:{\cal F(H)}\to{\cal F}(A)$,
where ${\cal F(H)}$ is the functional space over ${\cal H}$ and ${\cal F}(A)$
the same over $A$. This projector is defined such that

$\displaystyle\Pi(\alpha^{i})=\alpha^{i},$ (A.28)
$\displaystyle\Pi(\beta^{i})=\Pi(\gamma^{i})=0,$ (A.29)

and it is generalized on a generic functional $f\in{\cal F(H)}$ by Taylor
expanding the functional in powers of $\alpha$, $\beta$ and $\gamma$. With
this understanding, the condition $\gamma^{i}=0$ in the effective action can
be implemented through the introduction of $\Pi$

$\tilde{S}[\varphi]={\cal A}-\log\left(\Pi
e^{\frac{1}{2}h^{ij}\partial_{\gamma^{i}}\partial_{\gamma^{j}}}e^{-S_{\rm
free}[\varphi]-S_{\rm int}[\varphi+\gamma^{i}c_{i}]}\right).$ (A.30)

Furthermore let us define

$\tilde{\Pi}\coloneqq\Pi
e^{\frac{1}{2}h^{ij}\partial_{\gamma^{i}}\partial_{\gamma^{j}}},$ (A.31)

thus obtaining

$\tilde{S}[\varphi]={\cal A}-\log\left(\tilde{\Pi}e^{-S_{\rm
free}[\varphi]-S_{\rm int}[\varphi+\gamma^{i}c_{i}]}\right).$ (A.32)

We introduced $\tilde{\Pi}$ to establish a connection between this standard
QFT procedure and [34], where the authors demonstrate that $\tilde{\Pi}$
represents the transferred projector derived from the homological perturbation
lemma (HPL) at the functional level, applied to the following free SDR

$\circlearrowright(-H)\left({\cal
F(H)},d\right)\xtofrom[\text{$I$}]{\text{$\Pi$}}\left({\cal F}(A),\Pi
dI\right),$ (A.33)

in which the starting differential and the propagator are defined as

$\displaystyle d\coloneqq(S_{\rm
free}[\varphi+R^{\prime}],\cdot)=\gamma^{i}\omega(c^{j},Qc_{i})\frac{\overrightarrow{\partial}}{\partial\beta^{j}}+\alpha^{i}\omega(a^{j},Qa_{i})\frac{\overrightarrow{\partial}}{\partial\alpha^{j}},$
(A.34) $\displaystyle
H\coloneqq\frac{1}{\\#_{\beta+\gamma}}\beta^{i}\omega(b^{j},hb_{i})\frac{\overrightarrow{\partial}}{\partial\gamma^{j}},$
(A.35)

where $\\#_{\beta+\gamma}$ counts the number of occurrence of the variables
$\beta$ and $\gamma$ in each monomial on which $H$ acts. The inclusion $I$
trivially embeds a function of $\alpha$ into the space of functions of
$(\alpha,\beta,\gamma)$. It is easy to see that the propagator satisfies the
Hodge-Kodaira decomposition

$dH+Hd=1-I\Pi,$ (A.36)

Next, we consider adding a deformation given by the BV Laplacian
$\delta_{1}=\Delta$

$\circlearrowright(-\tilde{H})\left({\cal
F(H)},d+\Delta\right)\xtofrom[\text{$\tilde{I}$}]{\text{$\tilde{\Pi}$}}\left({\cal
F}(A),\tilde{d}\right),$ (A.37)

then by the homological perturbation lemma the transferred canonical projector
will be given by

$\tilde{\Pi}=\Pi\frac{1}{1+\delta_{1}H}.$ (A.38)

An important result of [34] (section 4.1.1) is that A.31 and A.38 are in fact
the same projector, when applied to a generic functional $f\in{\cal F(H)}$,
namely

$\Pi
e^{\frac{1}{2}h^{ij}\partial_{\gamma^{i}}\partial_{\gamma^{j}}}f=\Pi\frac{1}{1+\delta_{1}H}f.$
(A.39)

This result allows us to draw a connection between the more standard path
integral approach and the application of the HPL to the free strong
deformation retract A.33. Moreover it ensures that $\tilde{\Pi}$ defined as
A.31 satisfies all the properties required by the HPL.
Notice that, as it is evident from A.27, this path integral will also compute
the vacuum bubbles, which will appear as constant terms in the effective
action. In particular if we set $\varphi$ to zero in the effective action
(A.32) we get

$\tilde{S}[0]=\mathcal{A}-\log\left(\tilde{\Pi}e^{-S_{\rm
int}[\gamma^{i}c_{i}]}\right),$ (A.40)

and the second term will produce the (interacting) vacuum bubbles, in addition
to the one-loop determinant $\mathcal{A}$. We can isolate the bubbles by
adding and subtracting them from A.32

$\tilde{S}[\varphi]=\mathcal{A}-\log\left(\tilde{\Pi}e^{-S_{\rm
int}[\gamma^{i}c_{i}]}\right)-\log\left(\frac{\tilde{\Pi}e^{-S_{\rm
free}[\varphi]-S_{\rm int}[\varphi+\gamma^{i}c_{i}]}}{\tilde{\Pi}e^{-S_{\rm
int}[\gamma^{i}c_{i}]}}\right).$ (A.41)

The last term

$\tilde{S}^{\prime}[\varphi]=-\log\left(\frac{\tilde{\Pi}e^{-S_{\rm
free}[\varphi]-S_{\rm int}[\varphi+\gamma^{i}c_{i}]}}{\tilde{\Pi}e^{-S_{\rm
int}[\gamma^{i}c_{i}]}}\right)$ (A.42)

is then the effective action without the vacuum bubbles, which correctly
vanishes when $\varphi=0$. In order to extract this normalized action and
concretely link it to the products defining the initial UV action, we follow
again [34], who applied the HPL to the free SDR A.33, but with a different
perturbation $\delta_{2}$ defined as

$\delta_{2}=(S_{\rm int}[\varphi+R^{\prime}],\cdot)+\Delta.$ (A.43)

This gives the following perturbed SDR

$\circlearrowright(-\tilde{H}_{2})\left({\cal
F(H)},d+\delta_{2}\right)\xtofrom[\text{$\tilde{I}_{2}$}]{\text{$\tilde{\Pi}_{2}$}}\left({\cal
F}(A),\tilde{d}_{2}\right)$ (A.44)

and the HPL dictates that

$\displaystyle\tilde{\Pi}_{2}=\Pi\frac{1}{1+\delta_{2}H}.$ (A.45)

Focusing now on the transferred differential

$\tilde{d_{2}}=\Pi dI+\Pi\delta_{2}\frac{1}{1+H\delta_{2}}I,$ (A.46)

it is possible to show (section 4.2 of [34]) that $\tilde{d}_{2}$ can be
independently written as

$\tilde{d}_{2}=(\tilde{S}^{\prime}[\varphi],\cdot)_{\alpha\alpha}+\Delta_{\alpha\alpha},$
(A.47)

where the subscript $(\cdot)_{\alpha\alpha}$ means that the BV bracket and BV
laplacian only contain derivatives in $\alpha$, namely

$\displaystyle\left(\cdot,\cdot\right)_{\alpha\alpha}=\dfrac{\overleftarrow{\partial}}{\partial\alpha^{i}}(\omega_{A})^{ij}\dfrac{\overrightarrow{\partial}}{\partial\alpha^{j}},$
(A.48)
$\displaystyle\Delta_{\alpha\alpha}=\frac{1}{2}(-)^{\alpha^{i}}(\omega_{A})^{ij}\dfrac{\overrightarrow{\partial}}{\partial\alpha^{i}}\dfrac{\overrightarrow{\partial}}{\partial\alpha^{j}}.$
(A.49)

Notice that any possible constant part of $\tilde{S}^{\prime}[\varphi]$ in
(A.47) would be eliminated by the BV bracket. We can then determine the
effective products of $\tilde{S}^{\prime}[\varphi]$ by imposing the equality
of A.46 and A.47

$\begin{split}(\tilde{S}^{\prime}[\varphi],\cdot)_{\alpha\alpha}+\Delta_{\alpha\alpha}&=\Pi
dI+\Pi\delta_{2}\frac{1}{1+H\delta_{2}}I\\\ &=\Pi
dI+\Pi\delta_{2}I-\Pi\delta_{2}\frac{1}{1+H\delta_{2}}H\delta_{2}I\\\
&=\Pi(S[\varphi+R],\cdot)I+\Pi\Delta
I-\Pi\delta_{2}\frac{1}{1+H\delta_{2}}H\delta_{2}I\\\
&=\Pi(S[\Phi],\cdot)I+\Delta_{\alpha\alpha}-\Pi\frac{1}{1+\delta_{2}H}\delta_{2}H\delta_{2}I,\end{split}$
(A.50)

where we named $\Phi=\varphi+R$. The above expression allows us to establish a
relation between the BV bracket of the effective theory and the starting
action, that can be further simplified as

$(\tilde{S}^{\prime}[\varphi],\cdot)_{\alpha\alpha}=\Pi(S[\Phi],\cdot)I-\Pi\frac{1}{1+\delta_{2}H}\delta_{2}H\delta_{2}I.$
(A.51)

By applying this expression to $\varphi$ we will now show that we precisely
recover the effective vertices as obtained by the HPL at the level of the
Hilbert space $\mathcal{H}$ and the co-algebraic description used in the main
text.
In order to find the effective products we assume that the starting action and
the effective action can be written in WZW form as follows:

$S[\Phi]=\int_{0}^{1}dt\,\omega(\dot{\Phi},\pi_{1}\bm{l}e^{\wedge\Phi}),$
(A.52)
$\tilde{S}^{\prime}[\varphi]=\int_{0}^{1}dt\,\omega(\dot{\varphi},\pi_{1}\tilde{\bm{l}}e^{\wedge\varphi}),$
(A.53)

where both $\bm{l}$ and $\tilde{\bm{l}}$ are assumed to be cyclic. Then, if we
apply A.51 to $\varphi$ we get

$(\tilde{S}^{\prime}[\varphi],\varphi)_{\alpha\alpha}=\Pi(S[\Phi],I\varphi)-\Pi\frac{1}{1+\delta_{2}H}\delta_{2}H\delta_{2}I\varphi.$
(A.54)

The left hand side can be obtained by a direct computation

$\begin{split}(\tilde{S}^{\prime}[\varphi],\varphi)_{\alpha\alpha}=\tilde{S}^{\prime}[\varphi]\frac{\mathop{\\!\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}\nolimits}{\partial\alpha^{i}}\omega_{A}^{ij}\frac{\mathop{\\!\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}\nolimits}{\partial\alpha^{j}}(\alpha^{k}a_{k})=(-)^{\alpha^{i}}\frac{\mathop{\\!\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}\nolimits}{\partial\alpha^{i}}\tilde{S}^{\prime}[\varphi]\omega_{A}^{ij}a_{j}.\end{split}$
(A.55)

The derivative acts on the action as:

$\begin{split}\frac{\mathop{\\!\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}\nolimits}{\partial\alpha^{j}}\tilde{S}^{\prime}[\varphi]&=\frac{\mathop{\\!\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}\nolimits}{\partial\alpha^{j}}\int_{0}^{1}dt\,\omega(\pi_{1}\bm{\partial}_{t}e^{\wedge\varphi},\pi_{1}\tilde{\bm{l}}e^{\wedge\varphi})\\\
&=(-)^{\alpha^{i}}\int_{0}^{1}dt\,\omega(\pi_{1}\bm{\partial}_{t}\bm{\partial}_{\alpha_{i}}e^{\wedge\varphi},\pi_{1}\tilde{\bm{l}}e^{\wedge\varphi})+\int_{0}^{1}dt\,\omega(\pi_{1}\bm{\partial}_{t}e^{\wedge\varphi},\pi_{1}\tilde{\bm{l}}\bm{\partial}_{\alpha_{i}}e^{\wedge\varphi})\end{split}$
(A.56)

where $\bm{\partial}_{\alpha_{i}}$ should be considered as a coderivation on
the same footing as $\bm{\partial}_{t}$. Then using the cyclic property of
$\tilde{\bm{l}}$ we can rewrite the second term as:

$\omega(\pi_{1}\bm{\partial}_{t}e^{\wedge\varphi},\pi_{1}\tilde{\bm{l}}\bm{\partial}_{\alpha_{i}}e^{\wedge\varphi})=(-)^{\alpha_{i}}\omega(\pi_{1}\bm{\partial}_{\alpha^{i}}e^{\wedge\varphi},\pi_{1}\bm{\partial}_{t}\tilde{\bm{l}}e^{\wedge\varphi}),$
(A.57)

so that the two terms in A.56 can be resummed using the Leibniz rule:

$\begin{split}\frac{\mathop{\\!\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}\nolimits}{\partial\alpha^{j}}\tilde{S}^{\prime}[\varphi]&=(-)^{\alpha^{i}}\int_{0}^{1}dt\,\partial_{t}\,\omega(\pi_{1}\bm{\partial}_{\alpha^{i}}e^{\wedge\varphi},\pi_{1}\tilde{\bm{l}}e^{\wedge\varphi})\\\
&=(-)^{\alpha^{i}}\omega(\pi_{1}\bm{\partial}_{\alpha^{i}}e^{\wedge\varphi},\pi_{1}\tilde{\bm{l}}e^{\wedge\varphi})=(-)^{\alpha^{i}}\omega(a_{i},\pi_{1}\tilde{\bm{l}}e^{\wedge\varphi}).\end{split}$
(A.58)

Plugging this back into A.55 we can use the completeness relation to build

$(\tilde{S}^{\prime}[\varphi],\varphi)_{\alpha\alpha}=\omega(a_{i},\pi_{1}\tilde{\bm{l}}e^{\wedge\varphi})\omega_{A}^{ij}a_{j}=\pi_{1}\tilde{\bm{l}}e^{\wedge\varphi}.$
(A.59)

Let’s now turn to the first term on the right hand side of A.54. Since the
$\beta,\gamma$ part of the Poisson bracket vanishes when acting on $\varphi$,
we are left with

$\Pi(S[\Phi],I\varphi)=\Pi\left[S[\Phi]\frac{\mathop{\\!\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}\nolimits}{\partial\alpha^{i}}\omega_{A}^{ij}\frac{\mathop{\\!\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}\nolimits}{\partial\alpha^{j}}I\varphi\right]$
(A.60)

where $I\varphi$ must be understood now as an element of $\cal F(H)$.
Following analogous steps as in A.59 we get

$\Pi(S[\Phi],I\varphi)=\Pi\left[\omega(a_{i},\pi_{1}\bm{l}e^{\wedge\Phi})\right]\omega_{A}^{ij}a_{j}.$
(A.61)

The symplectic form gives a non zero result only if
$\pi_{1}\bm{l}e^{\wedge\Phi}\in A$ so the previous expression is equivalent to

$\Pi(S[\Phi],I\varphi)=\Pi\left[\omega(a_{i},\pi_{1}\bm{\pi}\bm{l}e^{\wedge\Phi})\right]\omega_{A}^{ij}a_{j}=\Pi[\pi_{1}\bm{\pi}\bm{l}e^{\wedge\Phi}].$
(A.62)

As $\Pi$ kills all $\beta$’s and $\gamma$’s we can trade it with the
$\bm{\iota}$ immersion acting on $\varphi$ and write

$\Pi(S[\Phi],I\varphi)=\pi_{1}\bm{\pi}\bm{l}\bm{\iota}e^{\wedge\varphi}.$
(A.63)

The second term in the RHS of A.59 can be computed trough similar steps,
giving the following chain of relations

$\begin{split}\delta_{2}I\varphi&=\pi_{1}\bm{\delta l}e^{\wedge\Phi},\\\
H\delta_{2}I\varphi&=(-)^{\gamma^{j}}\beta^{j}\pi_{1}\bm{\delta
l}\bm{(}hb_{j}\bm{)}e^{\wedge\Phi},\\\
\delta_{2}H\delta_{2}I\varphi&=\pi_{1}\bm{\delta l}(\bm{h}\bm{\delta
l}+\bm{h}\bm{U})e^{\wedge\Phi},\end{split}$ (A.64)

where $\bm{(}hb_{j}\bm{)}$ is understood as the zero-coderivation associated
with the state $hb_{j}$. Additionally, we can notice that each application of
a $\delta_{2}H$ on $\delta_{2}H\delta_{2}\varphi$ consists in inserting a
$(\bm{h}\bm{\delta l}+\bm{h}\bm{U})$ in front of $e^{\wedge\Phi}$, namely

$(\delta_{2}H)^{n}\delta_{2}H\delta_{2}\varphi=\pi_{1}\bm{\delta
l}(\bm{h}\bm{\delta l}+\bm{h}\bm{U})^{n+1}e^{\wedge\Phi},$ (A.65)

Therefore A.54 becomes

$\begin{split}\pi_{1}\tilde{\bm{l}}e^{\wedge\varphi}&=\pi_{1}\bm{\pi}\bm{l}\bm{\iota}e^{\wedge\varphi}-\pi_{1}\bm{\pi}\bm{\delta
l}\frac{1}{1+\bm{h}(\bm{\delta l}+\bm{U})}\bm{h}(\bm{\delta
l}+\bm{U})\bm{\iota}e^{\wedge\varphi}\\\
&=\pi_{1}\bm{\pi}\bm{l}\frac{1}{1+\bm{h}(\bm{\delta
l}+\bm{U})}\bm{\iota}e^{\wedge\varphi},\end{split}$ (A.66)

where we have also traded the $\Pi$ in the second term of A.54 with a
$\bm{\iota}$ acting on $e^{\wedge\varphi}$, following the same reasoning
adopted in A.63. This establishes that the HPL at the co-algebraic level which
we use in the main text is the result of the path integral.

Moreover, we can remove the $\pi_{1}$ from (A.66) and add the projected
Poisson bi-vector $\bm{U}_{\pi}$ (defined on $A$) on both sides of the above
expression, obtaining

$\tilde{\bm{l}}+\bm{U}_{\pi}=\bm{\pi}\bm{l}\frac{1}{1+\bm{h}(\bm{\delta
l}+\bm{U})}\bm{\iota}+\bm{U}_{\pi}.$ (A.67)

Remarkably this is equivalent to what we would get by applying the homotopy
transfer to the nilpotent structure $(\bm{l}+\bm{U})$, that is

$\tilde{\bm{l}}+\bm{U}_{\pi}=\bm{\pi}(\bm{l}+\bm{U})\frac{1}{1+\bm{h}(\bm{\delta
l}+\bm{U})}\bm{\iota}.$ (A.68)

To see this equivalence, let us manipulate

$\begin{split}\bm{\pi}(\bm{l}+\bm{U})\frac{1}{1+\bm{h}(\bm{\delta
l}+\bm{U})}\bm{\iota}&=\bm{\pi}\bm{l}\frac{1}{1+\bm{h}(\bm{\delta
l}+\bm{U})}\bm{\iota}+\bm{\pi}\bm{U}\frac{1}{1+\bm{h}(\bm{\delta
l}+\bm{U})}\bm{\iota}\\\ &=\bm{\pi}\bm{l}\frac{1}{1+\bm{h}(\bm{\delta
l}+\bm{U})}\bm{\iota}+\bm{U}_{\pi}\bm{\pi}\frac{1}{1+\bm{h}(\bm{\delta
l}+\bm{U})}\bm{\iota}\\\ &=\bm{\pi}\bm{l}\frac{1}{1+\bm{h}(\bm{\delta
l}+\bm{U})}\bm{\iota}+\bm{U}_{\pi}\bm{\pi}\bm{\iota}-\bm{U}_{\pi}\bm{\pi}\bm{h}(\bm{\delta
l}+\bm{U})\frac{1}{1+\bm{h}(\bm{\delta l}+\bm{U})}\bm{\iota}\\\
&=\bm{\pi}\bm{l}\frac{1}{1+\bm{h}(\bm{\delta
l}+\bm{U})}\bm{\iota}+\bm{U}_{\pi}=\tilde{\bm{l}}+\bm{U}_{\pi},\end{split}$
(A.69)

where in the second line we have used (2.53) and in the third line
$\bm{\pi}\bm{\iota}=1$ and $\bm{\pi h}=0$.
So, in conclusion, we have proven the following general formula for the
Wilsonian effective action

$\tilde{S}[\varphi]={\cal
A}-\log\left(e^{\frac{1}{2}h^{ij}\partial_{\gamma^{i}}\partial_{\gamma^{j}}}e^{-S_{\rm
int}[\gamma^{i}c_{i}]}\right){\Big{|}}_{\gamma^{i}=0}+\int_{0}^{1}dt\,\omega(\dot{\varphi},\pi_{1}\tilde{\bm{l}}e^{\varphi}),$
(A.70)

which describes the transfer between the nilpotent structures

$\displaystyle(\bm{l}+\bm{U})^{2}=0\quad\to\quad(\tilde{\bm{l}}+\bm{U}_{\pi})^{2}=0,$
(A.71)

which, as expected, guarantees that the effective action is a solution to the
BV quantum master equation, provided this is true for the initial UV action.
The first two terms in (A.70) give the expression of the (connected) vacuum
bubbles, that are the additive constants in the effective action. These
constants do not enter in the BV quantum master equation, yet they are
important for the correct computation of observables. The dynamical part of
the action, on the right, is given by the homotopy transfer at the co-algebra
level. It would be interesting and useful to understand if also the constant
part of the action could be written at the co-algebra level, encoding the
vacuum amplitudes that one can construct using the interaction vertices and
the propagators.

##  Appendix B  $SL(2,\mathbb{C})$ vertices for the open string one-loop
tadpole

In this appendix, we will construct the OC-SFT vertices needed to compute the
open string one-loop tadpole. The moduli space associated with this amplitude
is one dimensional. For the sake of simplicity we will use $SL(2,\mathbb{C})$
vertices as much as we can.101010It would be also interesting to use
hyperbolic vertices [9, 14]. A similar construction has been discussed in [7],
but the closed string degeneration region was not addressed. Specifically, we
can characterize the annulus as a cylinder, $C_{(\pi,2\pi t)}$, with height
$\pi$ and circumference $2\pi t$, where $t\in[0,\infty)$ parameterizes the
moduli space. Let’s denote $z$ the complex coordinate on the cylinder, which
is subject to the identification

$z\simeq z+2\pi t.$ (B.1)

Within this parameterization, we have the open string degeneration in the
limit $t\to\infty$, which corresponds to an extremely wide and flattened
cylinder that leads to the collision of boundaries. On the other hand, the
limit $t\to 0$ is the closed string degeneration, represented by a very narrow
tube.
As anticipated in the main text, this amplitude receives contribution from
three Feynman diagrams, the first one containing the open string degeneration,
the second one containing the closed string degeneration and the third one
which is a fundamental vertex covering the interior of the moduli space

$A_{0;1}^{0,2}(\varphi_{\rm o})=-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{0,2}^{(0,1)}\left(\bm{h}_{0}U_{\rm
o}\right)\right)-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{1,0}^{(0,1)}\left(\frac{b_{0}^{+}}{L_{0}^{+}}\bar{P}_{0}^{+}l_{0,0}^{(0,1)}\right)\right)+\omega_{\text{o}}\left(\varphi_{\rm
o},m_{0,0}^{(0,2)}\right).$ (B.2)

###  B.1 Open string non-separating plumbing fixture

Let us begin by describing the open channel

$-\omega_{\text{o}}\left(\varphi_{\rm o},m_{0,2}^{(0,1)}\left(\bm{h}_{0}U_{\rm
o}\right)\right).$

To start with we give an explicit expression to the product $m_{0,2}^{(0,1)}$

$\omega_{\text{o}}\left(\Psi_{1},m_{0,2}^{(0,1)}(\Psi_{2},\Psi_{3})\right)=(-)^{\Psi_{1}+\Psi_{2}}\langle
f_{-\xi}\circ\Psi_{1}(0)f_{0}\circ\Psi_{2}(0)f_{\xi}\circ\Psi_{3}(0)\rangle_{\rm
UHP}$ (B.3)

and we define the $f$-maps as generic $SL(2,\mathbb{R})$ transformations which
insert the punctures on the real axis of the UHP in $-\xi$, $0$ and $\xi>0$
respectively, namely $f_{\xi}(0)=\xi$, $f_{0}(0)=0$ and $f_{-\xi}(0)=-\xi$. We
impose cyclicity by considering the $SL(2,\mathbb{R})$ transformation $g$ that
cyclically permutes the location of the punctures, $g(0)=-\xi$, $g(\xi)=0$ and
$g(-\xi)=\xi$ and enforcing the following conditions

$g\circ f_{-\xi}=f_{\xi},\qquad\qquad g\circ f_{0}=f_{-\xi},\qquad\qquad
g\circ f_{\xi}=f_{0}.$ (B.4)

Additionally we also impose twist symmetry $f_{\xi}(-w)=-f_{-\xi}(w)$ and
$f_{0}(-w)=-f_{0}(w)$. This gives a one-parameter ($\lambda$) family of maps

$\displaystyle f_{-\xi}(w_{1})=\frac{\xi^{2}+\lambda\xi w_{1}}{\xi-3\lambda
w_{1}},$ (B.5) $\displaystyle f_{0}(w_{2})=\lambda w_{2},$ (B.6)
$\displaystyle f_{\xi}(w_{3})=\frac{\lambda\xi w_{3}-\xi^{2}}{\xi+3\lambda
w_{3}}.$ (B.7)

The two positive real parameters $\lambda$ and $\xi$ must satisfy
$\xi>3\lambda$ in order to avoid the overlap of the local coordinate patches.
Presently, our aim is to implement the plumbing fixture procedure, identifying
the local coordinates associated with the punctures located in $\xi$ and
$-\xi$ as follows

$w_{1}\simeq-\frac{q}{w_{3}},$ (B.8)

where $q\in[0,1]$. By inverting (B.5) and (B.7) we can rewrite (B.8) getting
an identification on the UHP (with coordinate $u$) which defines an annulus

$u\simeq f_{-\xi}\circ I_{-q}\circ f^{-1}_{\xi}(u),$ (B.9)

where we have denoted $I_{-q}(w)=-q/w$. In the following, we will use the
symbol $\Sigma_{\rm open}^{\rm p.f.}(q)$ to denote this surface, which is
reported in figure 1.

Figure 1: Annulus given by the plumbing fixture procedure with boundaries in
green and orange. The red curves, identified through the equivalence relation
(B.8), delineate the contours of the local patches associated with maps
$f_{-\xi}$ and $f_{\xi}$. The $\times$ denote the open string punctures and
the purple half-circle the contour of the local patch related to $f_{0}$.

Note that identification (B.9) corresponds to composition of
$SL(2,\mathbb{R})$ maps, and therefore, following [7], we can find the
similarity transformation that ‘diagonalizes’ the previous expression. In
other words, we introduce another $SL(2,\mathbb{R})$ map

$u=p(z^{\prime})=\xi\sqrt{\frac{\xi^{2}-\lambda^{2}}{\xi^{2}-9\lambda^{2}}}\left(\frac{1-z^{\prime}}{1+z^{\prime}}\right),$
(B.10)

such that

$z^{\prime}\simeq p^{-1}\circ f_{-\xi}\circ I_{-q}\circ f^{-1}_{\xi}\circ
p(z^{\prime})=\Lambda z^{\prime},$ (B.11)

where the real number $\Lambda$ is the ratio of the two eigenvalues of the
diagonal matrix associated with

$p^{-1}\circ f_{-\xi}\circ I_{-q}\circ f^{-1}_{\xi}\circ p.$ (B.12)

Its explicit expression is

$\Lambda(\lambda,\xi,q)=\frac{\xi^{2}+3\lambda^{2}q-\sqrt{\xi^{4}+9\lambda^{4}q^{2}-10\lambda^{2}\xi^{2}q}}{\xi^{2}+3\lambda^{2}q+\sqrt{\xi^{4}+9\lambda^{4}q^{2}-10\lambda^{2}\xi^{2}q}},$
(B.13)

which is real thanks to the non-overlap condition $\xi>3\lambda$. The
resulting surface is shown in figure 2.
Now it is easy to map to the frame where the annulus is described by (B.1).
Indeed, defining

$z=-\log(z^{\prime})$

we get

$z\simeq z+\log\left(\frac{1}{\Lambda}\right).$ (B.14)

Figure 2: Annulus in the frame $z^{\prime}=p^{-1}(u)$.

Therefore, comparing with (B.1), we can read off how the modulus is related to
the plumbing fixture parameter $q$ and the SFT data $\xi$ and $\lambda$

$t_{\rm
o}(q,\lambda,\xi)=\frac{1}{2\pi}\log\left(\frac{\xi^{2}+3\lambda^{2}q+\sqrt{\xi^{4}+9\lambda^{4}q^{2}-10\lambda^{2}\xi^{2}q}}{\xi^{2}+3\lambda^{2}q-\sqrt{\xi^{4}+9\lambda^{4}q^{2}-10\lambda^{2}\xi^{2}q}}\right).$
(B.15)

Notice that by varying the plumbing fixture parameter in $[0,1]$ the modulus
changes in the interval

$t_{\rm o}(q,\lambda,\xi)\in[t_{\rm max}(\lambda,\xi),\infty),$ (B.16)

in which we defined $t_{\max}(\lambda,\xi)\coloneqq t_{\rm
o}(q=1,\lambda,\xi)$. This is the portion of moduli space covered by the non-
separating open string Feynman diagram, which depends on the data $\xi$ and
$\lambda$.
Finally, we can easily find the local coordinate map associated to the open
string puncture on $C_{(\pi,2\pi t)}$, figure 3

$\begin{split}G_{\rm o}(w)&\coloneqq g_{\rm o}\circ f_{0}(w)\\\
&=\log\left(\frac{\xi\sqrt{\xi^{4}+9\lambda^{4}q^{2}-10\lambda^{2}\xi^{2}q}+(\xi^{2}-9\lambda^{2}q)w}{\xi\sqrt{\xi^{4}+9\lambda^{4}q^{2}-10\lambda^{2}\xi^{2}q}-(\xi^{2}-9\lambda^{2}q)w}\right)\\\
&=2\tanh^{-1}\left(\frac{1}{\xi}\sqrt{\frac{\xi^{2}-9\lambda^{2}q}{\xi^{2}-\lambda^{2}q}}w\right).\end{split}$
(B.17)

where we defined $g_{\rm o}(u)\coloneqq-\log\circ\,p^{-1}(u)$.

Figure 3: Annulus in the frame $z=-\log(z^{\prime})$.

###  B.2 Closed string separating plumbing fixture

Regarding the closed channel

$-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{1,0}^{(0,1)}\left(\frac{b_{0}^{+}}{L_{0}^{+}}\bar{P}_{0}^{+}l_{0,0}^{(0,1)}\right)\right),$

this time we have to perform the plumbing fixture on the following two
fundamental vertices

$\omega_{\text{c}}\left(l_{0,0}^{(0,1)},\Phi\right)=\frac{1}{4\pi^{2}}\langle
f_{b}\circ c_{0}^{-}\Phi(0,\bar{0}\rangle_{\rm disk},$ (B.18)

and

$\omega_{\text{o}}\left(\Psi,m_{1,0}^{(0,1)}(\Phi)\right)=\frac{(-)^{\Phi+\Psi}}{2\pi
i}\langle d\circ f_{\rm o}\circ\Psi(0)d\circ f_{\rm
c}\circ\Phi(0,\bar{0})\rangle_{\rm disk},$ (B.19)

where, following [41], $f_{b}$, $f_{\rm c}$ and $f_{\rm o}$ are the
$SL(2,\mathbb{C})$ maps

$\displaystyle f_{\rm b}(w_{1})=\frac{w_{1}}{l_{\rm b}},$ (B.20)
$\displaystyle f_{\rm o}(w_{2})=\frac{w_{2}}{l_{\rm o}},$ (B.21)
$\displaystyle f_{\rm
c}(w_{3})=i\frac{1+\frac{w_{3}}{\beta_{1}}}{1+\frac{w_{3}}{\beta_{2}}},$
(B.22)

where $l_{\rm b}$ and $l_{\rm o}\in[1,\infty)$ are the fundamental closed and
open stubs, respectively, and the real parameters $\beta_{1}$ and $\beta_{2}$
are chosen to be such that $\beta_{2}>\beta_{1}>1$. Additionally, imposing the
non-overlap condition implies the following constraint

$l_{\rm o}>\frac{\beta_{1}(1-\beta_{2})}{\beta_{2}(1-\beta_{1})}.$ (B.23)

The two punctured disks are reported in figure 4.

Figure 4: On the left, the unit circle with an inserction of a closed string
at $0$ and local patches corresponding to a circle of radius $l_{\rm b}^{-1}$.
On the right, the unit disk with a closed string puncture and an open string
puncture, with local patches represented in red and purple, respectively. The
colors used are consistent with the previous section. The plumbing fixture
glues the two disks along the red curves, resulting in an annulus with orange
and green boundaries.

Furthermore we called $d$ the Cayley function, which maps the UHP into the
unitary disk in the standard way $d(0)=1,\,d(i)=0,\,d(\infty)=-1$. Therefore
the local coordinate maps on the disk are

$\displaystyle u_{1}=f_{\rm b}(w_{1}),\qquad\qquad u_{2}=d\circ f_{\rm
o}(w_{2}),\qquad\qquad u_{2}=d\circ f_{\rm c}(w_{3}),$ (B.24)

Now, let us perform the closed-string separating plumbing fixture, identifying
the local coordinates associated with the closed punctures on the two disks
using the relationship

$w_{1}\simeq\frac{q}{w_{3}},$ (B.25)

where $q\in[0,1]$.
Inverting (B.24) and inserting it into (B.25) allows us to derive the relation
between the coordinates on the two disks

$u_{2}=d\circ f_{\rm c}\circ I_{q}\circ f_{\rm b}^{-1}(u_{1}),$ (B.26)

we will call this surface as $\Sigma_{\rm closed}^{\rm p.f.}(q)$, figure 5.

Figure 5: Boundary-punctured annulus given by the separating closed plumbing
fixture

In terms of the $u_{2}$ coordinate, we are delineating an annulus with an
outer boundary of unit radius with the open puncture and an inner boundary
consisting of a circle with radius $R$, centered at $C$

$\displaystyle R(l_{\rm
b},\beta_{1},\beta_{2},q)=\frac{2q\text{$\beta_{1}$}\text{$\beta_{2}$}(\text{$\beta_{2}$}-\text{$\beta_{1}$})\text{$l_{\rm
b}$}}{4\text{$\beta_{1}$}^{2}\text{$\beta_{2}$}^{2}\text{$l_{\rm
b}$}^{2}-q^{2}(\text{$\beta_{1}$}+\text{$\beta_{2}$})^{2}},$ (B.27)
$\displaystyle C(l_{\rm
b},\beta_{1},\beta_{2},q)=\frac{q^{2}(\text{$\beta_{2}^{2}$}-\text{$\beta_{1}^{2}$})}{4\text{$\beta_{1}$}^{2}\text{$\beta_{2}$}^{2}\text{$l_{\rm
b}$}^{2}-q^{2}(\text{$\beta_{1}$}+\text{$\beta_{2}$})^{2}}.$ (B.28)

Therefore, to obtain the annulus in the frame where (B.1) holds, it is
necessary to apply an additional transformation to center both boundaries at
zero. The strategy we adopt is as follows: we start with the generic real
Möbius map that leaves the unit circle unchanged

$h_{\alpha}(u_{2})=\frac{u_{2}-\alpha}{1-\alpha u_{2}},$ (B.29)

with $\alpha\in\mathbb{R}$, and then we impose the condition

$h_{\alpha}\left(C+R\right)=-h_{\alpha}\left(C-R\right),$ (B.30)

thus getting

$\alpha_{\pm}=\frac{q^{2}\left(\text{$\beta_{1}$}^{2}+\text{$\beta_{2}$}^{2}\right)\pm
2\text{$\beta_{1}$}\text{$\beta_{2}$}\left(\sqrt{(q^{2}-\text{$\beta_{1}^{2}$}\text{$l_{b}^{2}$})(q^{2}-\text{$\beta_{2}^{2}$}\text{$l_{b}^{2}$})}\mp\text{$\beta_{1}$}\text{$\beta_{2}$}\text{$l_{\rm
b}$}^{2}\right)}{q^{2}(\text{$\beta_{1}^{2}$}-\text{$\beta_{2}^{2}$})},$
(B.31)

notice that both solutions are real because $1<\frac{\beta_{1}l_{\rm
b}}{q}<\frac{\beta_{2}l_{\rm b}}{q}$ by assumption. These two solutions
identify the two possible maps which center the boundary without the open
insertion by leaving the unit circle unchanged. In particular $h_{\alpha_{+}}$
maps the interior of the unit circle into the interior of the unit circle,
whereas $h_{\alpha_{-}}$ sends the interior of the unit circle into the
exterior of the unit circle. We will consider
$z^{\prime}=h_{\alpha_{-}}(u_{1})$ and thus the case in which the punctured
circle becomes the inner boundary, as shown in figure 6.

Figure 6: Annulus in the frame $z^{\prime}=h_{\alpha_{-}}(u_{2})$.

Finally, by applying the logarithm, we obtain the cylinder, which, however, is
rotated by $\pi/2$ and has a boundary length of $2\pi$, as reported in the
first figure of 7. To achieve the desired surface it is needed to further
apply a rotation and a dilatation, second figure of 7.

Figure 7: Annulus in the frames $z^{\prime\prime}=\log(z^{\prime})$ and
$z=it_{\rm c}z^{\prime\prime}$.

At this point, it is possible to read off the modulus by comparing with (B.1)

$t_{\rm c}(q;l_{\rm
b},\beta_{1},\beta_{2})=\pi\log\left(\frac{\text{$\beta_{1}$}\text{$\beta_{2}$}\text{$l_{b}^{2}$}-q^{2}+\sqrt{\left(q^{2}-\beta_{1}^{2}l_{b}^{2}\right)\left(q^{2}-\text{$\beta_{2}^{2}$}\text{$l_{b}^{2}$}\right)}}{ql_{\rm
b}(\text{$\beta_{2}$}-\text{$\beta_{1}$})}\right)^{-1}$ (B.32)

notice that by varying the plumbing fixture parameter in [0, 1] the modulus
changes as follows

$t_{\rm c}(q,l_{\rm b},\beta_{1},\beta_{2})\in[0,t_{\rm min}(l_{\rm
b},\beta_{1},\beta_{2})],$ (B.33)

where we defined $t_{\rm min}(l_{\rm b},\beta_{1},\beta_{2})=t_{\rm
c}(q=1,l_{\rm b},\beta_{1},\beta_{2})$.
As expected in the closed string channel we are able to cover a sub-region of
moduli space containing the closed string degeneration $t=0$.
The local coordinate associated with the open string insertion on the
canonical cylinder is given by

$\begin{split}&G_{\rm c}(w)\coloneqq g_{\rm c}\circ d\circ f_{\rm o}(w)\\\
&=it_{\rm
c}\log\left(\frac{q^{2}\left(\beta_{1}^{2}w+i\beta_{2}^{2}l_{o}\right)-\beta_{1}\beta_{2}(w+il_{o})\left(\beta_{1}\beta_{2}l_{b}^{2}+\sqrt{(q^{2}-\beta_{1}^{2}l_{b}^{2})(q^{2}-\beta_{2}^{2}l_{b}^{2})}\right)}{q^{2}\left(-\beta_{1}^{2}w+i\beta_{2}^{2}l_{o}\right)+\beta_{1}\beta_{2}(w-il_{o})\left(\beta_{1}\beta_{2}l_{b}^{2}+\sqrt{(q^{2}-\beta_{1}^{2}l_{b}^{2})(q^{2}-\beta_{2}^{2}l_{b}^{2})}\right)}\right)\\\
&=2t_{\rm c}\tan^{-1}\left(-\frac{1}{l_{\rm o}}\frac{q\beta_{1}+l_{\rm
b}e^{\pi/t_{\rm c}}}{q\beta_{2}-l_{\rm b}e^{\pi/t_{\rm
c}}}w\right).\end{split}$ (B.34)

where we have introduced

$g_{\rm c}(u_{2})\coloneqq it_{\rm c}\log\circ h_{\alpha_{-}}(u_{2}).$ (B.35)

###  B.3 Fundamental vertex and decoupling of BRST exact states

It is important to notice the following fact. The local coordinate map in the
open string channel (B.17) can never coincide with the same quantity in the
closed string channel (B.34). This remains true even when the open and closed
parameters are chosen such that $t_{\rm c}=t_{\rm o}$. Therefore, even if the
moduli space can in principle be covered with the two Feynman diagrams
containing the degenerations, this would not be sufficient off-shell.
Therefore, differently from the disk amplitude with two closed strings that
was analyzed in [41], this time we need a fundamental vertex covering the
interior of moduli space.

In order to do so we will rewrite the amplitudes (B.2) in terms of surface
states, by making explicit the integration over the moduli space. From now on,
for the sake of simplicity, we will assume that the string field $\varphi_{\rm
o}$ has even degree, which is true in the case of a physical tadpole
amplitude.
Let us start with the open channel

$\begin{split}-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{0,2}^{(0,1)}\left(\bm{h}_{0}U_{\rm
o}\right)\right)&=-(-)^{o^{i}}\omega_{\text{o}}\left(\varphi_{\rm
o},m_{0,2}^{(0,1)}\left(h_{0}o_{i},o^{i}\right)\right)\\\
&=-(-)^{o^{i}}\omega_{\text{o}}\left(o^{i},m_{0,2}^{(0,1)}\left(\varphi_{\rm
o},h_{0}o_{i}\right)\right)\\\
&=-\langle\Sigma_{0,3}^{(0,1)}|o^{i}\otimes\varphi_{\rm o}\otimes
h_{0}o_{i}\\\
&=-\int_{0}^{1}\frac{dq}{q}\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes 1_{\mathcal{H}_{\rm o}}\otimes
b_{0}q^{L_{0}}\right)\left(o^{i}\otimes\varphi_{\rm o}\otimes
o_{i}\right),\end{split}$ (B.36)

where in the second step we used cyclicity and in the third line we introduced
the surface state $\langle\Sigma_{0,3}^{(0,1)}|$ that is defined as follows

$\langle\Sigma_{0,3}^{(0,1)}|\Psi_{1}\otimes\Psi_{2}\otimes\Psi_{3}\coloneqq\langle
f_{-\xi}\circ\Psi_{1}(0)f_{0}\circ\Psi_{2}(0)f_{\xi}\circ\Psi_{3}(0)\rangle_{\rm
UHP}=(-)^{\Psi_{1}+\Psi_{2}}\omega_{\text{o}}\left(\Psi_{1},m_{0,2}^{0,1}(\Psi_{2},\Psi_{3})\right).$
(B.37)

As for the closed string channel we have

$\begin{split}-\omega_{\text{o}}\left(\varphi_{\rm
o},m_{1,0}^{(0,1)}\left(h_{0}^{+}l_{0,0}^{(0,1)}\right)\right)&=-\frac{1}{2\pi
i}\langle\Sigma_{1,1}^{(0,1)}|\varphi_{\rm o}\otimes
h_{0}^{+}l_{0,0}^{(0,1)}\\\ &=-\frac{1}{2\pi
i}\int_{0}^{1}\frac{dq}{q}\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes b_{0}^{+}q^{L_{0}^{+}}\right)\left(\varphi_{\rm o}\otimes
l_{0,0}^{(0,1)}\right),\end{split}$ (B.38)

where we have defined the surface state $\langle\Sigma_{1,1}^{(0,1)}|$ as
follows

$\langle\Sigma_{1,1}^{(0,1)}|\Psi\otimes\Phi\coloneqq\langle d\circ f_{\rm
o}\circ\Psi(0)d\circ f_{\rm c}\circ\Phi(0,\bar{0})\rangle_{\rm
disk}=(-)^{\Phi+\Psi}2\pi
i\,\omega_{\text{o}}\left(\Psi,m_{1,0}^{(0,1)}(\Phi)\right).$ (B.39)

Now, our attention turns to the fundamental vertex

$\omega_{\text{o}}\left(\varphi_{\rm o},m_{0,0}^{(0,2)}\right).$

Although it has not been explicitly constructed yet, we know that it
corresponds to an integral over a piece of a section of the fiber bundle
having the moduli space as its base and the local coordinate map as its fiber.
Clearly such subsection can be parameterized using the same variable $t$ used
to characterize the moduli space. Then the region covered by $m_{0,0}^{(0,2)}$
has to align with the interval $t\in[t_{\rm min},t_{\rm max}]$, namely the
portion that cannot be covered by the two previous Feynman diagrams. In our
specific case, based on the definitions provided in the previous section, it
is actually possible to choose the parameters such that $t_{\min}=t_{\max}$.
However it does not appear to be possible to match the local coordinates and
therefore the fundamental vertex is needed, with our choice of lower
dimensional vertices.111111When $t_{\min}=t_{\max}$ the fundamental vertex
would be associated to a vertical integration [44] changing the local
coordinate while not moving in moduli space.
In general we thus have

$\omega_{\text{o}}(\varphi_{\rm o},m_{0,0}^{(0,2)})=\int_{t_{\rm min}}^{t_{\rm