B}\left[{\partial\over\partial u^{1}}\right]+du^{2}{\cal
B}\left[{\partial\over\partial u^{2}}\right]\right)\chi(w=0)\,.$ (5.65)

We note that (5.64) does not hold in general because $\chi$ is not an element
of ${\cal H}_{c}$ and hence $\Omega_{1}(\chi)$ is not a well defined one
form.111111Over domains where the phase of local coordinates is globally well
defined, however, the form is well defined. We shall show that if we consider
a region of the moduli space where the dilaton is close to one of the external
states, then (5.64) holds in that region up to addition of total derivatives
with respect to the moduli $v^{i}$ of the original Riemann surface, with $d$
interpreted as the exterior derivative in the space spanned by $u^{1},u^{2}$.
Since we are ignoring the total derivative terms in $v^{i}$, this allows us to
express the effect of the dilaton insertion as the insertion of

$\int_{M}\Omega_{2}(D)+\int_{\partial M}\Omega_{1}(\chi)\,,$ (5.66)

where $M$ is the region of the Riemann surface that excludes small disks
around other punctures.

  2. 2.

Next we note that the Euler number $\chi(M)$ of a two-dimensional surface $M$
with boundary $\partial M$ is given by

$\chi(M)={1\over 2\pi}\int_{M}R^{(2)}+{1\over 2\pi}\int_{\partial
M}k^{(1)}\,.$ (5.67)

Here the curvature two form $R^{(2)}$ and the geodesic curvature one form
$k^{(1)}$, computed from a conformal metric $ds^{2}=\rho^{2}\,|dz|^{2}$, are
given by

$\begin{split}R^{(2)}=&\ -2\,i\,\partial\bar{\partial}\rho\ dz\wedge
d\bar{z}\,,\\\ k^{(1)}=&\
d\theta_{\gamma}-i[dz\,\partial\log\rho-d\bar{z}\bar{\partial}\log\rho]\,,\end{split}$
(5.68)

where the term $d\theta_{\gamma}$ computes the rotation angle of the tangent
vector to the curve $\gamma\in\partial M$. The Euler number of a surface
$\Sigma_{g,n}$ of genus $g$ with $n$ boundaries is $\chi_{g,n}=2-2g-n$.

  3. 3.

Finally, we shall show that

$\Omega_{2}(D)=\ -\,{1\over 2\pi}R^{(2)}\,\ \ \hbox{and}\ \ \
\Omega_{1}(\chi)=\ -\,{1\over 2\pi}k^{(1)}\,.$ (5.69)

This, together with (5.66) and (5.67) would then establish the soft dilaton
theorem for on-shell amplitudes.

Let us begin with the first step, namely that when the dilaton is close to one
of the other vertex operators, we can replace $\Omega_{2}(D)$ by
$-d\Omega_{1}(\chi)$. In the string field theory such contributions come from
Feynman diagrams with a three-point function involving $D$, an on-shell vertex
operators $c\bar{c}V$ and an off-shell internal state in ${\cal H}_{c}$ is
connected by a propagator to the rest of the Feynman diagram. We can now write
$D=-Q\chi$ and deform the BRST contour away from the location of $\chi$ and
through $c\bar{c}V$ to act on the off-shell vertex operator and hence on the
propagator. The commutator of $Q$ with the $b_{0}^{+}$ term in the propagator
${\cal P}_{b}$ in (5.4) brings down a factor of $L_{0}^{+}$ and gives

$[Q\,,{\cal P}_{b}]\ ={1\over
2\pi}\,\,b_{0}^{-}\,\int_{0}^{2\pi}\hskip-4.0ptd\theta\,\int_{0}^{\infty}\hskip-3.0ptds\
{\partial\over\partial
s}e^{-s(L_{0}+\bar{L}_{0})}e^{i\theta(L_{0}-\bar{L}_{0})}\,.$ (5.70)

This is a total derivative term that we now argue, can be regarded as the
insertion of $-d\chi$. The boundary term from $s=\infty$ vanishes since all
the states that can contribute have $L_{0}>0$,121212There are no tachyonic
states since in the matter sector the state that flows along the propagator is
a dimension (1,1) primary $V$ or its descendent. In the ghost sector the state
has ghost number (0,2) or (2,0) and hence the state $c\bar{c}V$ also cannot
propagate. This also shows that istead of writing $D=-Q\chi$ and deforming the
BRST contour, we could use the original expression involving $D$ and place an
upper cut-off $\Lambda$ in the integral over $s$ associated with the
propagator. Then the large $\Lambda$ limit of the integral is finite and
reproduces the full integral. This would mean that the sum of all the Feynman
diagrams can be approximated by the integral of $\Omega_{2}(D)$ over the
original Riemann surface except for small regions around the original
punctures. Physically this happens because in string field theory the
punctures are represented as semi-infinite cylinders and cutting out small
regions around the punctures correspond to cutting out the regions near the
far end of the cylinders. Since the cylinders are flat, the integral of
$\Omega_{2}(D)$ from these regions vanish and there is no need to include the
$\Omega_{1}(\chi)$ boundary integrals to recover the Euler number.  whereas
the boundary term at $s=0$ is the integral of $\Omega_{1}(\chi)$ along the
part of $\partial M$ associated with the $\theta$ integral of the collapsed
propagator. The action of $Q$ on the rest of the vertex operators vanishes
since all other external states are BRST invariant and the anti-commutator of
$Q$ with the ${\cal B}[\partial/\partial v^{i}]$ insertions will generate
total derivative with respect to $v^{i}$ which will eventually integrate to
zero after integration over the $v^{i}$’s. This shows that $\Omega_{2}(D)$ can
be replaced by $-d\Omega_{1}(\chi)$ in this region of the moduli space. The
same result holds when the dilaton is close to any other other external state
vertex operators.

We now turn to the proof of (5.69), following [83]. This is somewhat non-
trivial since the dilaton vertex operator is not of the form $c\bar{c}V$ for a
dimension $(1,1)$ matter primary $V$, and hence the procedure described in the
paragraph following (3.42) does not work. In particular the form that needs to
be integrated over the moduli space now depends on the choice of local
coordinates at the puncture. We will consider some domain on the Riemann
surface with a local uniformizer $z$ used to label the points in this domain.
We write the relation between $z$ and the local coordinate $w$ around the
puncture where the dilaton is inserted as

$z=F(w;u)=F(w;u^{1},u^{2})\,,$ (5.71)

where we have parameters $u=(u^{1},u^{2})$ with $u^{1}$ and $u^{2}$ real
variables encoding the position of the puncture via the complex variable
$y(u)$ obtained by setting $w=0$ in $F$:

$y(u)=F(0;u)\,.$ (5.72)

Given this definition, we can write $z$ as a series expansion in $w$, thus
describing a general $F$ as follows:

$z=F(w;u^{1},u^{2})=y(u)+a(u)\,w+\tfrac{1}{2}b(u)w^{2}+\tfrac{1}{3!}c(u)w^{2}+{\cal
O}(w^{4})\,,$ (5.73)

with $a,b,c$ arbitrary complex-valued functions of the parameters $u^{1}$ and
$u^{2}$.

Let us now determine the antighost insertions

$\begin{split}{\cal B}\left[{\partial\over\partial u^{i}}\right]=&\ \
\ointclockwise b(z)dz{\partial F\over\partial
u^{i}}+\ointclockwise\bar{b}(\bar{z})d\bar{z}{\partial\overline{F}\over\partial
u^{i}}\\\ =&\ \ -\ointop b(w)dw\Bigl{(}{\partial F\over\partial
w}\Bigr{)}^{-1}{\partial F\over\partial
u^{i}}-\ointop\bar{b}(\bar{w})d\bar{w}\Bigl{(}{\partial\bar{F}\over\partial\bar{w}}\Bigr{)}^{-1}{\partial\overline{F}\over\partial
u^{i}}\,,\end{split}$ (5.74)

where we passed to the $w$ frame so that the insertions act directly on the
operator, itself inserted in $w$ frame. Next we use (5.73) to evaluate in a
power series in $w$ the expressions inside the above integrals. A short
calculation gives

$\Bigl{(}{\partial F\over\partial w}\Bigr{)}^{-1}{\partial F\over\partial
u^{i}}\ =\ \alpha_{i}+\beta_{i}\,w+\gamma_{i}\,w^{2}+{\cal O}(w^{3})\,,$
(5.75)

where the expansion coefficients are quickly confirmed to be

$\begin{split}\alpha_{i}=&\ {1\over a}{\partial y\over\partial
u^{i}}\,,\\\\[4.30554pt] \beta_{i}=&\ {1\over a}{\partial a\over\partial
u^{i}}-{b\over a^{2}}{\partial y\over\ \partial u^{i}}\,,\\\\[4.30554pt]
\gamma_{i}=&\ a{\partial\over\partial u^{i}}\Bigl{(}{b\over
2a^{2}}\Bigr{)}+\Bigl{(}{b^{2}\over a^{3}}-{1\over 2}{c\over
a^{2}}\Bigr{)}{\partial y\over\partial u^{i}}\,.\end{split}$ (5.76)

For the complex conjugate factor in the antighost insertion the result is the
same, with all quantities complex conjugated. This means that with the
oscillator expansion $b(w)=\sum_{n}b_{n}/w^{n+2}$, and the analogous one for
$\bar{b}(\bar{w})$, we have

${\cal B}\left[{\partial\over\partial
u^{i}}\right]=-\left(\alpha_{i}\,b_{-1}+\beta_{i}\,b_{0}+\gamma_{i}\,b_{1}\ +\
\bar{\alpha}_{i}\,\bar{b}_{-1}+\bar{\beta}_{i}\bar{b}_{0}+\bar{\gamma}_{i}\,\bar{b}_{1}+\cdots\right)\,,$
(5.77)

where the dots indicate terms with oscillators $b_{n},\bar{b}_{n}$ with $n\geq
2$, which are not needed for the dilaton computation. Now we find from (5.63),

$\begin{split}\Omega_{2}(D)=&\ \Bigl{(}-{1\over 2\pi i}\Bigr{)}\,du^{1}\wedge
du^{2}{\cal B}\left[{\partial\over\partial u^{1}}\right]{\cal
B}\left[{\partial\over\partial
u^{2}}\right](c_{1}c_{-1}-\bar{c}_{1}\bar{c}_{-1})|{0}\rangle\\\ =&\
\Bigl{(}{1\over 2\pi i}\Bigr{)}\,du^{1}\wedge
du^{2}\bigl{[}\alpha_{1}\gamma_{2}-\alpha_{2}\gamma_{1}-(\bar{\alpha}_{1}\bar{\gamma}_{2}-\bar{\alpha}_{2}\bar{\gamma}_{1})\bigr{]}|{0}\rangle\,,\end{split}$
(5.78)

where we note that within the brackets, the second term, in parentheses, is
the complex conjugate (c.c.) of the first. Moreover, both terms are
antisymmetric in the 1 and 2 labels. Using the values of the $\alpha$ and
$\gamma$ coefficients we find

$\begin{split}\Omega_{2}(D)=&\ \Bigl{(}{1\over 2\pi i}\Bigr{)}\,du^{1}\wedge
du^{2}\Bigl{[}{\partial y\over\partial u^{1}}{\partial\over\partial
u^{2}}\Bigl{(}{b\over 2a^{2}}\Bigr{)}-{\partial y\over\partial
u^{2}}{\partial\over\partial u^{1}}\Bigl{(}{b\over
2a^{2}}\Bigr{)}-(\hbox{c.c.})\bigr{]}|{0}\rangle\end{split}$ (5.79)

where terms involving the product ${\partial y\over\partial u^{1}}{\partial
y\over\partial u^{2}}$ cancel out for the $1\leftrightarrow 2$ antisymmetry.
We then find

$\begin{split}\Omega_{2}(D)=&\ {1\over 2\pi i}\,\Bigl{[}dy\wedge
d\Bigl{(}{b\over 2a^{2}}\Bigr{)}-d\bar{y}\wedge d\Bigl{(}{\bar{b}\over
2\bar{a}^{2}}\Bigr{)}\Bigr{]}|{0}\rangle\end{split}$ (5.80)

which is simply

$\begin{split}\Omega_{2}(D)=&\ {1\over 2\pi i}\,dy\wedge
d\bar{y}\,\Bigl{[}{\partial\over\partial\bar{y}}\Bigl{(}{b\over
2a^{2}}\Bigr{)}+{\partial\over\partial y}({\bar{b}\over
2\bar{a}^{2}}\Bigr{)}\Bigr{]}\,.\end{split}$ (5.81)

This is our final result for the integrated dilaton insertion two-form. Since
the vacuum state corresponds to the identity operator, we deleted
$|{0}\rangle$ and effectively this form integrates a function over the
surface.

For the $\chi$ state one form we have from (5.65) and (5.77),

$\begin{split}\Omega_{1}(\chi)=&\ -{1\over 2\pi
i}\,\Bigl{(}-\tfrac{1}{2}du^{1}{\cal B}\left[{\partial\over\partial
u^{1}}\right]-\tfrac{1}{2}du^{2}{\cal B}\left[{\partial\over\partial
u^{2}}\right]\Bigr{)}(c_{0}-\bar{c}_{0})|{0}\rangle\\\ =&\ -{1\over 2\pi
i}\,\Bigl{(}\tfrac{1}{2}du^{1}(\beta_{1}-\bar{\beta}_{1})+\tfrac{1}{2}du^{2}(\beta_{2}-\bar{\beta}_{2})\Bigr{)}|{0}\rangle\,.\end{split}$
(5.82)

Using the values of the $\beta_{i}$ one readily finds

$\Omega_{1}(\chi)=\ {1\over 2\pi
i}\,\Bigl{[}-\tfrac{1}{2}d\Bigl{(}\ln{a\over\bar{a}}\Bigr{)}+{b\over
2a^{2}}dy-{\bar{b}\over 2\bar{a}^{2}}d\bar{y}\Bigr{]}\,.$ (5.83)

This is our final result for the integrated $\chi$ one-form. One quickly
verifies that $\Omega_{2}(D)=-{\rm d}\Omega_{1}(\chi)\,.$

Note that in the analysis above we have ignored the possibility of the point
$y$ meeting the contours associated with ${\cal B}[\partial/\partial v^{i}]$
insertions. It has been shown in [83] (section 6.1) that their is no extra
contribution from these regions of $y$ integration.

At this point, to make these forms compute geometrically recognizable
quantities, we now place a condition on an a priori arbitrary conformal metric
$\rho$ on the surface. A conformal metric is one where the length element is
$ds=\rho|dz|$. We demand that the real and imaginary parts of $w$ function as
Riemann normal coordinates at the location of the puncture [85]. To deal with
the dilaton, it suffices to demand the lowest order version of the constraint;
that the metric $\rho^{w}$ in the $w$ frame satisfy
$\partial_{w}\rho^{w}|_{w=0}=\partial_{\bar{w}}\rho^{w}|_{w=0}=0$. That is, we
are setting the linear parts of the $w$ dependence of the metric to zero. To
implement this, we write the metric as $ds=\rho^{w}|dw|$ in the $w$
coordinate, with

$\begin{split}\rho^{w}(w)=&\ \rho\Bigl{|}{dz\over dw}\Bigr{|}=\
\Bigl{(}\rho(y)+(z-y){\partial\rho\over\partial
z}\Bigl{|}_{y}+(\bar{z}-\bar{y}){\partial\rho\over\partial{\bar{z}}}\bigr{|}_{y}+\cdots\Bigr{)}\,\Bigl{|}a+bw+\cdots\Bigr{|}\end{split}$
(5.84)

where dots represent terms higher order in $w$ that will not be relevant.
Noting that at the puncture, where the derivatives are evaluated, $z$ can be
traded for $y$, and to linear order in $w,\bar{w}$, we have

$\begin{split}\rho^{w}(w)=&\ \rho\Bigl{|}{dz\over dw}\Bigr{|}=\
\Bigl{(}\rho(y)+a\,w\,{\partial\rho\over\partial
y}\Bigl{|}_{y}+\bar{a}\,\bar{w}\,{\partial\rho\over\partial{\bar{y}}}\Bigr{|}_{y}+\cdots\Bigr{)}\,|a|\Bigl{(}1+{b\over
2a}w+{\bar{b}\over 2\bar{a}}\bar{w}+\cdots\Bigr{)}\\\
=&\rho(y)\,|a|\Bigl{[}1+a\,w\Bigl{(}{\partial\over\partial y}\log\rho+{b\over
2a^{2}}\Bigr{)}+\bar{a}\,\bar{w}\Bigl{(}{\partial\over\partial{\bar{y}}}\log\rho+{\bar{b}\over
2\bar{a}^{2}}\Bigr{)}+\cdots\Bigr{]}\,.\end{split}$ (5.85)

The condition that first derivatives of $\rho^{w}$ vanish at $y$ requires the
linear terms in $w$ in the above expression to vanish. This sets

${b\over 2a^{2}}=-{\partial\over\partial y}\log\rho\,,\ \ \ \ {\bar{b}\over
2\bar{a}^{2}}=-{\partial\over\partial\bar{y}}\log\rho\,.$ (5.86)

With these relations the dilaton and $\chi$ forms, become, respectively

$\begin{split}\Omega_{2}(D)=&\ \Bigl{(}{1\over 2\pi i}\Bigr{)}dy\wedge
d\bar{y}\,\Bigl{[}-2{\partial\over\partial\bar{y}}{\partial\over\partial
y}\log\rho\Bigr{]}\,,\\\ \Omega_{1}(\chi)=&\ {1\over 2\pi
i}(-i)\Bigl{[}d\theta_{a}-idy{\partial\over\partial y}\log\rho+i\,d\bar{y}\
{\partial\over\partial\bar{y}}\log\rho\Bigr{]}\,,\end{split}$ (5.87)

where we set $a=|a|e^{i\theta_{a}}$, so that $\theta_{a}$ is the phase of the
$a$ coefficient ($|a|$ is the so-called mapping radius of the local
coordinate). Comparing with the curvature two form and the geodesic curvatures
in (5.68) we have

$\Omega_{2}(D)=\ -\,{1\over 2\pi}R^{(2)}\,\ \ \hbox{and}\ \ \
\Omega_{1}(\chi)=\ -\,{1\over 2\pi}k^{(1)}\,.$ (5.88)

This result confirms the anticipated role of the dilaton $D$ and $\chi$
insertions. We integrate the dilaton form over the surface minus the
coordinate disks of the punctures, where the other external states are
inserted. This prevents the collision of the dilaton and the external states.
Having stopped at the boundaries, we must include the integration of the
$\chi$ one-form over the circles bounding the coordinate disks. Both integrals
together give the Euler number of the surface, independent of the fiducial
metric $\rho$ used to create the family of local coordinates.

Note that the proof of (5.88) is valid for arbitrary off-shell states
$A_{1},\cdots,A_{n}$ in the correlation function. This establishes that for a
fixed Riemann surface, integrating
$\Omega^{(g,n+1)}_{6g-6+2n+2}(D,A_{1},\cdots,A_{n})$ over the surface minus
the coordinate disks, with moduli describing the location where $D$ is
inserted, supplemented by the integral of
$\Omega^{(g,n+1)}_{6g-6+2n+1}(\chi,A_{1},\cdots,A_{n})$ over the boundaries of
the coordinate disks, with moduli describing the location where $\chi$ is
inserted, gives [83]

$(2-2g-n)\,\Omega^{(g,n)}_{6g-6+2n}(A_{1},\cdots,A_{n})\,.$ (5.89)

This is the result needed for a dilaton shift to change the coupling constant
of the theory, for (minus) the Euler number is the power of the coupling
constant appearing in the string interactions in the combination
$(g_{s})^{-\chi_{g,n}}\\{A_{1},\cdots,A_{n}\\}_{g}$ (see equation (4.11)).

The proof of the dilaton theorem in string field theory uses the above result
as a guide to construct the field redefinition that maps the combination of
the string field action and the measure with one value of the string coupling,
to the same combination with a small variation of the coupling constant. To
leading order, there is a shift of the string field by the dilaton
$|{D}\rangle$. This is the first term of a not-quite-legal gauge
transformation with gauge parameter $|{\chi}\rangle$, and the next term in the
field redefinition involves the insertion of $|{\chi}\rangle$ on the three
string vertex. But higher order terms in the redefinition need modifications,
because the illegality of $\chi$ begins to matter. The redefinition is also
different from the one in the background independence analysis, because the
dilaton is not a primary. The full redefinition is given in [83], section 8,
in the language of a ‘Hamiltonian’ that induces the field redefinition via the
antibracket.

##  6 Algebraic structures underlying string field theories

In string field theory, as presently formulated, the string field action $S$
is a functional of a string field. The string field is defined as a general
vector in the state space of a suitably chosen conformal or super-conformal
two-dimensional field theory. Most versions of string field theories share a
common algebraic structure. For classical open string field theories, the
structure on the ${\cal H}_{o}$ space where the string field lives is a cyclic
$A_{\infty}$ algebra of multilinear string field products satisfying a set of
relations and endowed with a cyclic inner product [51]. For classical closed
string field theories, the structure on ${\cal H}_{c}$ is an $L_{\infty}$
algebra of multilinear string field products satisfying a set of relations,
and a symmetric inner product [6]. Such kind of $L_{\infty}$ structures also
exist for ordinary field theories, as reviewed in detail and elaborated in
[86].

In this section we will describe such structures and show how they can be used
to construct actions that satisfy the Batalin-Vilkovisky master equation, and
are thus are guaranteed a consistent quantization. The free string field
theories require the definition of the lowest product in the algebra, a
product with one input string field, giving an output string field. This is
simply a linear operator, identified with the BRST operator of the conformal
field theory. The free string field theory also requires the inner product,
which also arises naturally from the conformal field theory. The interaction
terms in the string field theory require the definition of string vertices,
which we will discuss in the following section. For the time being, we simply
assume suitable vertices exist that allow one to construct the multilinear
string products needed for the $A_{\infty}$ and $L_{\infty}$ algebras.

For the quantum closed string field theory the $L_{\infty}$ structure is
modified. While the multilinear products of the classical theory arise from
string amplitudes on genus-zero punctured Riemann surfaces, the multilinear
products of the quantum theory require punctured Riemann surfaces of genus one
and higher. The consistency relations satisfied by the products are also
modified for the quantum theory, some authors calling the resulting structure
a ‘quantum $L_{\infty}$ algebra’. For classical open string field theories, a
tractable quantum theory requires including closed strings as part of the
spectrum. The resulting open-closed string field theories have an algebraic
structure in which $A_{\infty}$ and $L_{\infty}$ subalgebras are extended to a
larger self-consistent structure involving interactions of both open and
closed strings.

As discussed in section 4, for open superstrings, heterotic strings, and type
II strings, a complete construction in the $L_{\infty}$ framework involves
adding an extra copy of the string field, a copy that is needed to get the
action and equations of motion to work out, but that turns out to describe
free, decoupled degrees of freedom. There exist some versions of superstring
field theories that do not fit the $A_{\infty}$ or $L_{\infty}$ structures but
are rather based on Wess-Zumino-Witten (WZW) like algebraic structures; these
will be reviewed in section 8.

Homotopy algebras are a useful organizing principle in string field theory,
and their development was strongly influenced by the string constructions they
had to describe. There have also been a number of recent applications of
homotopy algebras to ordinary field theory, such as $W$-algebras, double copy
relations, and double field theory [87, 88, 89, 90, 91]. Some recent work on
homotopy algebras and BV quantization in the context of superstring field
theory can be found in [92, 69].

###  6.1 $A_{\infty}$ algebras and classical open string field theory

In this section we shall give a formal description of the $A_{\infty}$ algebra
that underlies the formulation of tree-level open string field theory. In the
following section we shall discuss the relation between this formal structure
and the formulation of open string field theory described in section 4.4.

For this algebraic structure we simply assume that we work with a vector space
$V$ with elements with a natural $\mathbb{Z}$ grading, which gives the
‘degree’ of the elements. For most aspects all that matters is the degree
modulo two; we have even elements if the degree is zero (mod 2) or odd
elements if the degree is one (mod 2). We shall denote by $d_{A}$ the degree
of the element $A$. In the analysis in this section, even degree elements will
behave as Grassmann even and odd degree elements will behave as Grassmann
odd.131313 When we apply this to open string field theory, however, this
assignment of degree is opposite to that of the Grassmannality of the vertex
operator, e.g. a vertex operator carrying odd ghost number will correspond to
an even degree element of the algebra. To avoid confusion, we have introduced
the symbol $d_{A}$ for degree of $A$, which is to be distinguished from the
Grassmannality of the vertex operators that will be denoted by $(-1)^{A}$. The
relation between the formalism developed in this section and that used in
section 4.4 will be explained in section 6.2.

We now define the products of the $A_{\infty}$ algebra. For this we define a
set of multilinear maps $b_{n}:V^{\otimes n}\to V$ with $n=1,2,\ldots~{}$.
These are products, as they take $n$ vectors in $V$ as input and the output is
a vector in $V$. All products are declared to be of degree minus one, meaning
that

$\hbox{deg}\,(b_{n}(A_{1},\cdots,A_{n}))=-1+\sum_{i=1}^{n}d_{A_{i}}\,.$ (6.1)

In order to formulate the consistency conditions satisfied by these
multilinear maps one forms the larger vector space called the ‘tensor co-
algebra’ $T(V)$

$T(V)\equiv V\oplus(V\otimes V)\oplus(V\otimes V\otimes V)\oplus\cdots$ (6.2)

Acting on such a space one can consider the linear operator ${\bf b}$ of
degree minus one which essentially is the sum of all multilinear maps. When it
acts on the $V^{\otimes n}$ subspace of $T(V)$, the operator ${\bf b}$ is
defined as follows:

${\bf b}=\sum_{i=1}^{n}\sum_{j=0}^{n-i}\,{\mathbb{1}}^{\otimes j}\otimes
b_{i}\otimes\mathbb{1}^{n-i-j}\,,\ \ \hbox{on}\ \ V^{\otimes n}\,.$ (6.3)

As one can see, in here, the product $b_{i}$ acts on all the possible
length-$i$ list of consecutive entries in $V^{\otimes n}$. It is possible to
unpack this, and to break ${\bf b}$ into a sum of linear operators:

${\bf b}=\sum_{i=1}^{\infty}{\bf b}_{i}={\bf b}_{1}+{\bf b}_{2}+\cdots\,.$
(6.4)

Then, consistent with (6.3) we have

${\bf b}_{i}=\sum_{j=0}^{n-i}\,{\mathbb{1}}^{\otimes j}\otimes
b_{i}\otimes\mathbb{1}^{n-i-j}\,,\ \ \hbox{on}\ \ V^{\otimes n}\,,\
\hbox{for}\ i\leq n\,.$ (6.5)

When $i>n$, ${\bf b}_{i}$ acting on $V^{\otimes n}$ is zero. Note that ${\bf
b}_{i}$ is an operator on $T(V)$ while $b_{i}$ is an operator on $V^{\otimes
i}$. In fact, an operator like ${\bf b}_{i}$ acting in this fashion on $T(V)$
is called a coderivation, and a sum of coderivations, such as ${\bf b}$, is
also a coderivation.141414To fully define a coderivation one must define a
coproduct $\bar{\Delta}:T(V)\to T(V)\otimes^{\prime}T(V)$, with the prime to
distinguish this product from just the tensor product within $T(V)$. The
coproduct is a linear operator acting as follows
$\bar{\Delta}(A_{1}\otimes\cdots\otimes
A_{n})=\sum_{k=1}^{n-1}(A_{1}\otimes\cdots
A_{k})\otimes^{\prime}(A_{k+1}\cdots A_{n})\,,\ \ n\geq 2,\ \ \
\bar{\Delta}(A_{1})=0\,.$ A linear operator ${\bf a}$ on $T(V)$ is a
coderivation if $\bar{\Delta}{\bf a}=({\bf a}\otimes^{\prime}{\bf I}+{\bf
I}\otimes^{\prime}{\bf a})\bar{\Delta}\,,$ with ${\bf I}$ the identity
operator on $T(V)$.  In fact, the most general coderivation can be specified
in terms of multilinear products [93], as we did for ${\bf b}$, which is
defined by the collection of $b_{i}$’s.

Acting on $V^{\otimes 3}$, for example, ${\bf b}$ becomes

$\begin{split}{\bf b}=&\ \
b_{1}\otimes\mathbb{1}\otimes\mathbb{1}+\mathbb{1}\otimes
b_{1}\otimes\mathbb{1}+\mathbb{1}\otimes\mathbb{1}\otimes
b_{1}\,+b_{2}\otimes\mathbb{1}+\mathbb{1}\otimes b_{2}\ +b_{3}\,,\ \
\hbox{on}\ \ V^{\otimes 3}\,,\end{split}$ (6.6)

and acting on $A\otimes B\otimes C\in V^{\otimes 3}$ we would get

$\begin{split}{\bf b}\,(A\otimes B\otimes C)=&\ \ b_{1}(A)\otimes B\otimes
C+(-1)^{d_{A}}A\otimes b_{1}(B)\otimes C+(-1)^{d_{A}+d_{B}}A\otimes B\otimes
b_{1}(C)\\\ &+b_{2}(A,B)\otimes C+(-1)^{d_{A}}A\otimes b_{2}(B,C)\
+b_{3}(A,B,C)\,,\end{split}$ (6.7)

with the sign factors arising when the odd degree $b$’s go across the states.
Note that the result on the right-hand side is an element of $V\oplus(V\otimes
V)\oplus(V\otimes V\otimes V)$.

The (graded) commutator of coderivations is in fact a coderivation, so ${\bf
b}^{2}=\tfrac{1}{2}\\{{\bf b}\,,{\bf b}\\}$ is a coderivation. The consistency
condition on the products is simply the requirement that the coderivation
${\bf b}^{2}$ vanishes, so that ${\bf b}$ has the property of a differential:

${\bf b}^{2}=0\,.$ (6.8)

The content of these conditions is fully seen by consideration of the action
on the various summands of $T(V)$, namely $V^{\otimes n}$ with $n\geq 1$. On
$V^{\otimes n}$ the condition can be expressed as

$\sum_{i=1}^{n}{\bf b}_{i}\,{\bf b}_{n+1-i}=0\,,\ \ \hbox{on}\ \ V^{\otimes
n}\,.$ (6.9)

The product here is composition of the operator action. The $A_{\infty}$
algebra is simply the vector space $V$ with the products $b_{n}$, that when
assembled into ${\bf b}$ as in (6.4), satisfy (6.8).

The first few cases of the conditions are

$\begin{split}\hbox{Acting on }V:\ \ &0\,={\bf b}_{1}{\bf b}_{1}\,,\\\
\hbox{Acting on }V^{\otimes 2}:\ \ &0\,={\bf b}_{1}{\bf b}_{2}+{\bf b}_{2}{\bf
b}_{1}\,,\\\ \hbox{Acting on }V^{\otimes 3}:\ \ &0\,={\bf b}_{1}{\bf
b}_{3}+{\bf b}_{2}{\bf b}_{2}+{\bf b}_{3}{\bf b}_{1}\,.\end{split}$ (6.10)

More explicitly, using the string products and the symbol $\circ$ for
composition,

$\begin{split}\hbox{Acting on }V:\ \ &0\,=b_{1}\circ b_{1}\,,\\\ \hbox{Acting
on }V^{\otimes 2}:\ \ &0\,=b_{1}\circ
b_{2}+b_{2}\circ(b_{1}\otimes\mathbb{1}+\mathbb{1}\otimes b_{1})\,,\\\
\hbox{Acting on }V^{\otimes 3}:\ \ &0\,=b_{1}\circ
b_{3}+b_{2}\circ(b_{2}\otimes\mathbb{1}+\mathbb{1}\otimes b_{2})\,,\\\ &\ \ \
\ +b_{3}(b_{1}\otimes\mathbb{1}\otimes\mathbb{1}+\mathbb{1}\otimes
b_{1}\otimes\mathbb{1}+\mathbb{1}\otimes\mathbb{1}\otimes
b_{1})\,.\end{split}$ (6.11)

We will write $b_{1}(A)=QA$, with $Q$ the BRST operator. The product $b_{2}$
of the algebra can be written as $b_{2}(A,B)=AB$. The $b_{3}$ product is the
first of an infinite series of ‘homotopies’ ($b_{3},b_{4},b_{5},\ldots$). It
is written as $b_{3}(A,B,C)=(A,B,C)\in V$. The three identities above then
give, when acting on states:

$\begin{split}\ \ &0\,=Q^{2}A\,,\\\ \ \
&0\,=Q(AB)+(QA)B+(-1)^{d_{A}}A(QB)\,,\\\ \ \
&0\,=Q(A,B,C)+(AB)C+(-1)^{d_{A}}A(BC)\,,\\\ &\ \ \ \
+(QA,B,C)+(-1)^{d_{A}}(A,QB,C)+(-1)^{d_{A}+d_{B}}(A,B,QC)\,.\end{split}$
(6.12)

The first identity is the nilpotency of the BRST operator. The second identity
states that $Q$ is an odd derivation of the product. The third identity shows
that strict associativity of the product, that is, having
$(AB)C=-(-1)^{d_{A}}A(BC)$ is not required. Here, associativity holds ‘up to
homotopy,’ that is, up to terms involving the homotopy $b_{3}$ and the BRST
operator. In the general framework of classical open string field theory we do
not require strict associativity and we have nontrivial products $b_{n\geq
3}$. Of course, if we have an associative product $b_{2}$ all the higher
homotopies vanish, and the algebraic structure of the string field theory is
simpler, with $Q$ and $b_{2}$ being the only ingredients. This is the case for
the cubic classical open string field theory formulated by Witten [5].

We shall now introduce a bilinear inner product on the vector space of the
algebra. This is a structure that, given two elements of $V$, gives us a
complex number. We thus have a bilinear inner product $V\otimes
V\to\mathbb{C}$ written as $(A,B)\in\mathbb{C}$, for $A,B\in V$. This
structure is required to have the following exchange property

$(A,B)=-(-1)^{d_{A}d_{B}}(B,A)\,.$ (6.13)

The bilinear form is symplectic in that it is antisymmetric when both vectors
are of even degree. It is only symmetric when both vectors are of odd degree.
The BRST operator $Q$ satisfies the folding over property

$(QA,B)=-(-1)^{d_{A}}(A,QB)\,.$ (6.14)

The inner product implements the cyclicity property of the products.
Intuitively, cyclicity is the natural result of open string vertex operators
being located on the boundary of a disk. In such situation the ordering of the
operators along the boundary matters, but cyclicity reflects the absence of a
special ‘first’ operator on the boundary. The axioms require that the bilinear
form and the products satisfy

$(A_{1},b_{n}(A_{2},\cdots,A_{n+1}))=(-1)^{\\#}(A_{2},b_{n}(A_{3},\cdots,A_{n+1},A_{1}))\,,$
(6.15)

with $\\#$ the sign factor necessary to rearrange the states into the final
ordering. Since $b_{n}$ is odd and degree operates like Grassmanality, we have

$\\#=d_{A_{2}}+d_{A_{1}}(1+d_{A_{2}}+d_{A_{3}}+\cdots+d_{A_{n+1}})\,.$ (6.16)

An $A_{\infty}$ algebra equipped with an inner product satisfying the
properties above is called a cyclic $A_{\infty}$ algebra. This concludes the
presentation of the algebraic structure and its axioms.

For the application to open string field theory we now note that the general
version of this theory requires a Grassmann algebra ${\cal G}$ because the
target-space theory contains anticommuting fields, such as ghosts. For
classical bosonic open string field theory, complex numbers suffice, since the
spacetime fields are valued on the complex numbers. As explained at the
beginning of section 4, the space $V$ that for open strings would naturally be
the vector space ${\cal H}_{o}$ of the BCFT, must be upgraded to a ${\cal G}$
module in the quantum theory, which with a bit of abuse of notation we still
denote by ${\cal H}_{o}$. The degree of an element is now the sum of the
degrees of the BCFT basis vector and the degree of the associated target space
field, given by its Grassmanality. The string field products thus have inputs
that are each elements of the module ${\cal H}_{o}$, and an output that is
also an element of the module ${\cal H}_{o}$. The inner product is then a map
from the tensor product of the modules to the Grassmann algebra: ${\cal
H}_{o}\otimes{\cal H}_{o}\to{\cal G}$.

The open string field $\Phi$ contains Grassmann odd open string vertex
operators multiplied by even target space fields, and Grassmann even vertex
operators times odd target space fields – thus a Grassmann odd object overall.
In this $A_{\infty}$ picture, however, this string field must be viewed as an
even degree object in the algebra: $d_{\Phi}=0$ (mod 2). The classical master
action $S(\Phi)$ for the string field is simply a sum over the various
products of the $A_{\infty}$ algebra. We claim that

$S(\Phi)\,=\,\sum_{n=1}^{\infty}{1\over n+1}(\Phi,b_{n}(\Phi,\cdots,\Phi))\,,$
(6.17)

is the master action and satisfies the classical master equation. We will show
this in the next section by relating the above action to the previously
obtained, consistent action (4.43).

One surprising feature of the results given in this subsection is the apparent
mismatch between the Grassmannality of a vertex operators and its degree which
is taken to be even (odd) for vertex operators carrying odd (even) ghost
number. This is visible for example in (6.12) where every time we move $Q$
through an element $A$ of the algebra, we pick a factor of $(-1)^{d_{A}}$,
which is 1 for odd ghost number operators and $-1$ for even ghost number
operators. This can be contrasted with (4.46) where every time we move $Q$
through a Grassmann odd vertex operator inside the product $[\cdots]$, we pick
up a minus sign. Let us first note that the rules of moving Grassmann odd
elements through the vertex operators inside a product could differ if we
changed the definition of the product. In (4.46) we were implicitly assuming
that the product $[\cdots]$ followed the prescription given in section 3.5:
inside the correlators we arrange all the vertex operators first, followed by
${\cal B}$ and boundary state insertions to the right. However we could have
arranged the vertex operators interspersed with ${\cal B}$ insertions, in
which case every time we move $Q$ from one Grassmann odd vertex operator to
the next one, $Q$ also passes through a ${\cal B}$ and as a result we do not
pick up any sign. A description that could lead to the rules (6.12) is
provided by regarding the products $b_{n}(A_{1},\cdots,A_{n})$ as the result
of contraction of a surface state with the ket states
$|{A_{1}}\rangle,\cdots|{A_{n}}\rangle$. As discussed at the beginning of
section 4.4, the ket state $|{A}\rangle$ has opposite Grassmannality to that
of the vertex operator $A$ and hence an odd vertex operator will naturally
lead to even ket states. In section 6.2 we shall give the explicit relation
between the product $[A_{1},\cdots,A_{n}]$ introduced in section 4.4 and the
product $b_{n}(A_{1},\cdots,A_{n})$ introduced here for all $n$.

###  6.2 Relation between different classical open SFT formalisms

In this section we shall discuss the relation between the various quantities
defined in section 4.4 and those introduced in section 6.1. Both deal with
classical open string theory.

We begin with the definition of the bilinear inner product. We had the BPZ
product $\langle\,\cdot\,,\,\cdot\,\rangle$ used in section 4.4, and the inner
product $(\,\cdot\,,\,\cdot\,)$ used in the algebraic analysis of section 6.1.
These two, we claim, are related as follows:

$(A,B)=(-1)^{A+1}\,\langle A,B\rangle^{\prime}\,\,.$ (6.18)

Here $A$ in the exponent is the (mod 2) ghost number of $A$, which coincides
with the Grassmanality of $A$ ($A$ is a vertex operator or a string field
vertex operator). To relate this to the degree of $A$ that appeared in section
6.1, we shall make the identification151515In the mathematical literature this
kind of relation is called a ‘suspension’. It relates descriptions of the
theory in which the effective Grassmanality of all elements is altered.

$A=d_{A}+1\ (\hbox{mod}\,2)\,.$ (6.19)

Note that we are assigning opposite Grassmannality to $A$ than what was
assigned in the previous subsection. As a consistency check, the exchange
property (6.13) for $(\,\cdot\,,\,\cdot\,)$ follows immediately from the
exchange property of the BPZ inner product

$\langle B,A\rangle^{\prime}=(-1)^{AB}\langle A,B\rangle^{\prime}\,,$ (6.20)

the relation between the inner products, and the above Grassmanality/degree
relation. Indeed,

$(B,A)=(-1)^{B+1}\langle B,A\rangle^{\prime}=(-1)^{AB+B+1}\langle
A,B\rangle^{\prime}=(-1)^{AB+A+B}(A,B)=(-1)^{d_{A}d_{B}+1}(A,B)\,.$ (6.21)

Next we relate the star product $\star$ to the $A_{\infty}$ product $b_{2}$.
We have

$A\star B=(-1)^{A+1}b_{2}(A,B)=(-1)^{A+1}(AB)\,,$ (6.22)

using the definition $(AB)=b_{2}(A,B)$. This gives, using the second relation
in (6.12),

$Q(A\star B)=(-1)^{A+1}Q(AB)=(-1)^{A}((QA)B)-(A(QB))=(QA)\star
B+(-1)^{A}A\star QB\,\,.$ (6.23)

More generally, we relate the product $[A_{1}\cdots A_{n}]$ to
$b_{n}(A_{1},\cdots,A_{n})$ via:

$[A_{1}\cdots
A_{n}]=(-1)^{n(n+1)/2}\sum_{\sigma}(-1)^{s_{\sigma}}(-1)^{A_{\sigma(1)}+2A_{\sigma(2)}+\
\cdots\ +nA_{\sigma(n)}}\,b_{n}(A_{\sigma(1)},A_{\sigma(2)},\cdots
A_{\sigma(n)})\,.$ (6.24)

Here $\sigma$ is a permutation of $1,\cdots n$ and $(-1)^{s(\sigma)}$ is the
sign factor that we pick up when we rearrange the order of the $A_{i}$’s with
the rule $A_{i}A_{j}=(-1)^{A_{i}A_{j}+1}A_{j}A_{i}$.161616This rule can be
derived by examining the left hand side, defined in (4.47), whose symmetry
follows on account of (4.45). Using the $A_{\infty}$ relations (6.9) we can
now easily verify the main identity (4.46) for the $[\cdots]$ products, for
the case of Grassmann odd (i.e. even degree) $A^{\prime}_{i}$’s. For the case
$n=2$ the above identity leads to

$[A_{1},A_{2}]=A_{1}\star A_{2}-(-1)^{A_{1}A_{2}}A_{2}\star A_{1}\,.$ (6.25)

Note that we could drop the $(-1)^{n(n+1)/2}$ factor in (6.24) while replacing
$A_{\sigma(i)}$ by $d_{A_{\sigma(i)}}=A_{\sigma(i)}+1$. It also follows from
(6.24) that when all the $A_{i}$’s are Grassmann odd, i.e. of even degree, we
get

$[A_{1}\cdots A_{n}]=\sum_{\sigma}b_{n}(A_{\sigma(1)},A_{\sigma(2)},\cdots
A_{\sigma(n)})\,.$ (6.26)

Note that the $(-1)^{s_{\sigma}}$ factor disappears since for odd $A_{i}$’s we
have $A_{i}A_{j}=A_{j}A_{i}$ inside $[~{}]$.

Equation (6.24) provides an implicit definition of $b_{n}(A_{1},\cdots,A_{n})$
in terms of world-sheet correlation functions. For this note that we have
already defined $[A_{1}\cdots A_{n}]$ in terms of the world-sheet correlation
functions in section 4.4. There we summed over all cyclic permutations of the
vertex operators to define $\\{A_{0}\cdots A_{n}\\}$ and then used it to
define $[A_{1}\cdots A_{n}]$ via (4.47). To define
$b_{n}(A_{1},\cdots,A_{n})$, we proceed as follows:

  * •

Define $\\{A_{0}A_{1}\cdots A_{n}\\}^{C}$ to be that part of
$\\{A_{0}A_{1}\cdots A_{n}\\}$ where the vertex operators are arranged in the
cyclic order $0,1,\cdots,n$.

  * •

Define $[A_{1}\cdots A_{n}]^{C}$ via $\langle
A_{0}\,,\,[A_{1},\cdots,A_{n}]^{C}\,\rangle^{\prime}=\\{A_{0},A_{1},\cdots,A_{n}\\}^{C}$.

  * •

Use (6.24) with $[A_{1}\cdots A_{n}]$ replaced by $[A_{1}\cdots A_{n}]^{C}$ on
the left hand side, and pick only the identity permutation in the sum over
$\sigma$ on the right hand side.

We now argue for the consistency of the $A_{\infty}$ master action (6.17). It
follows immediately from the relation between inner products (6.18) and the
relation between products (6.26) that the $A_{\infty}$ action coincides with
the action (4.43). Furthermore, since the main identity (4.46) follows as a
consequence of the $A_{\infty}$ algebra, it is clear that the rest of the
results discussed in section 4.4 follow. In particular, the action satisfies
the classical BV master equation.

###  6.3 $L_{\infty}$ algebras and classical closed string field theory

We now turn to a discussion of the algebraic structure behind classical closed
string field theory. While for open strings the relevant algebraic structures
arise from homotopy associative algebras, for closed strings the relevant
structures are homotopy Lie algebras. For classical closed string field theory
we have an $L_{\infty}$ algebra. The $L_{\infty}$ algebra contains an infinite
number of products, all of which are graded symmetric.

To work in generality we assume we have a graded vector space $W$ with
elements with a natural $\mathbb{Z}$ grading. We write $B_{1},B_{2},\cdots\in
W$ for elements of fixed degree. We denote the degree of $B$ by $d_{B}$. The
products are viewed as brackets, such as the Lie algebra bracket with two
entries. In the situation used for the conventional formulation of string
field theory, there are products with one input, two inputs, and all numbers
of higher inputs:171717A product $b_{0}$ with no input is just a special state
in $W$. If $b_{0}$ is included in the set of products, we have a ‘curved’
$L_{\infty}$ algebra that can play a role in the formulation of string field
theory around backgrounds where the classical theory has terms linear in the
field [6, 94].

$b_{1}(B_{1})\,,\ \ b_{2}(B_{1},B_{2}),\ \ b_{3}(B_{1},B_{2},B_{3}),\ \
\cdots\,,$ (6.27)

etc. Unlike the case of open string field theory, here the relation to the
structures introduced in section 4.1 is more straightforward:

$b_{1}(B_{1})=QB_{1}\,,\ \ b_{1}(B_{1},B_{2})=[B_{1},B_{2}]_{0},\ \
b_{3}(B_{1},B_{2},B_{3})=[B_{1},B_{2},B_{3}]_{0},\ \ \cdots\,,$ (6.28)

where $Q$ is a linear operator in $W$, to be later identified as the BRST
operator of the CFT. The subscript 0 on the brackets $[~{}]$ indicates that
these will be identified with the genus zero products defined in section 4.1.
The products $b_{n}$ are graded commutative. When exchanging any two inputs in
a product, we just get a sign factor corresponding to the exchange of the two
objects according to their degree

$b_{n}(B_{1},\cdots,B_{k},B_{k+1},\cdots,B_{n})=(-1)^{d_{B_{k}}d_{B_{k+1}}}b_{n}(B_{1},\cdots,B_{k+1},B_{k},\cdots,B_{n})\,.$
(6.29)

The products are defined to be of intrinsic degree minus one, so that

$\hbox{deg}\,(b(B_{1},\cdots,B_{n}))=-1+\sum_{i=1}^{n}d_{B_{i}}\,,$ (6.30)

in accordance with the comment below (4.13).

In analogy to what we did for the $A_{\infty}$ algebra, we construct a larger
space by adding symmetrized products of the vector space $W$:

$T({W})=\sum_{n=1}^{\infty}S{W}^{\otimes n}\,.$ (6.31)

The summands in this expression are spaces $SW^{\otimes n}$, with $S$ for
symmetrized; their elements are written as

$B_{1}\wedge B_{2}\wedge\cdots\wedge B_{n}\,,$ (6.32)

where the ‘wedge’ simply means that the order of the $B$’s can be interchanged
with sign factors according to their statistics, just as in the products
above. Thus, for example, $B_{1}\wedge
B_{2}=(-1)^{d_{B_{1}}d_{B_{2}}}B_{2}\wedge B_{1}$. For a general permutation
$\sigma=\\{\sigma(1),\cdots,\sigma(n)\\}$ of the integers $\\{1,\cdots,n\\}$,
we define

$B_{1}\wedge\cdots\wedge B_{n}=\
\epsilon(\sigma;B)B_{\sigma(1)}\wedge\cdots\wedge B_{\sigma(n)}\,,$ (6.33)

where $\epsilon(\sigma;B)$ is the so called Koszul sign factor, clearly
dependent on the permutation $\sigma$ and the degree of the $B$ entries.

In order to state the main identity satisfied by the product it is convenient
to define the splitting of the set of integers $\\{1,\ldots,n\\}$ into a first
group $\\{i_{1},\ldots,i_{l}\\}$ of integers and a second group
$\\{j_{1},\ldots,j_{k}\\}$ of integers, clearly with $l+k=n$. We ask that
$l\geq 1$ but $k\geq 0$. A splitting has a sign factor $\epsilon(\sigma)$,
where the permutation is that which turns the ordered set into
$\\{i_{1},\ldots,i_{l}\,,j_{1},\ldots,j_{k}\\}$. Two splittings are said to be
equivalent if their first groups contain the same integers, regardless of
their order.

We are now able to write the identities satisfied by the products. For this we
define an odd coderivation ${\bf b}$ acting on $T(W)$ as follows:181818The
definition here is given in a form slightly different from the original one in
[6] but it is exactly equivalent to it, including all signs.

${\bf b}(B_{1}\wedge\cdots\wedge
B_{n})=\sum_{l=1}^{n}\sum_{\sigma^{\prime}}\epsilon(\sigma^{\prime},B)\,\bigl{(}\,b_{l}\bigl{(}B_{i_{1}},\cdots,B_{i_{l}}\bigr{)}\wedge
B_{j_{1}}\wedge\cdots\wedge B_{j_{n-l}}\bigr{)}\,,$ (6.34)

where the sum over $\sigma^{\prime}$ denotes the sum over all inequivalent
splittings of $\\{1,\cdots,n\\}$ into a group with $l$ integers and a group
with $n-l$ integers (the prime is to remind the reader that this is a
restricted sum). Note that a product is used to collapse the first set of
states in the splitting into a single state. Thus, for example, we have

${\bf b}(B_{1})=b_{1}(B_{1})\,,$ (6.35)

since the only allowed splitting of $\\{1\\}$ is into the sets
$\\{1\\}\,,\\{\emptyset\\}$. For the case of two inputs, we first note that
there are three splittings of $\\{1,2\\}$: into $\\{1\\},\\{2\\}$, into
$\\{2\\},\\{1\\}$, and into $\\{1,2\\},\\{\emptyset\\}$. We therefore have,
including the signs,

$\begin{split}{\bf b}(B_{1}\wedge B_{2})=&\ b_{1}(B_{1})\wedge
B_{2}+(-1)^{d_{B_{1}}d_{B_{2}}}b_{1}(B_{2})\wedge B_{1}+b_{1}(B_{1},B_{2})\\\
=&\ b_{1}(B_{1})\wedge B_{2}+(-1)^{d_{B_{1}}}B_{1}\wedge
b_{1}(B_{2})+b_{1}(B_{1},B_{2})\,.\end{split}$ (6.36)

The identity satisfied by the products is simply the condition that the
operator ${\bf b}$ squares to zero

${\bf b}^{2}=0\,.$ (6.37)

This can be shown to be equivalent to the nilpotence of the BRST operator $Q$
and the main identities (7) with the last term set to zero. We shall
illustrate this with some examples.

For the action on $B_{1}$, as in (6.35), we find the expected condition of
nilpotency of $Q$:

${\bf b}^{2}(B_{1})={\bf b}\,{\bf b}(B_{1})={\bf b}(QB_{1})=Q(QB_{1})=0\,.$
(6.38)

Acting on $B_{1}\wedge B_{2}$ and using (6.36) we find

${\bf b}^{2}(B_{1}\wedge B_{2})={\bf b}(QB_{1}\wedge
B_{2})+(-1)^{d_{B_{1}}}{\bf b}(B_{1}\wedge QB_{2})+{\bf
b}(b_{2}(B_{1},B_{2}))=0\,.$ (6.39)

Expanding by using (6.36) for the first two terms, (6.35) for the last, and
$Q^{2}=0$, we get

$\begin{split}{\bf b}^{2}(B_{1}\wedge B_{2})=\ &\
\Bigl{(}(-1)^{d_{B_{1}}+1}QB_{1}\wedge QB_{2}+b_{2}(QB_{1},B_{2})\Bigr{)}\\\ \
&\ +(-1)^{d_{B_{1}}}\Bigl{(}QB_{1}\wedge
QB_{2}+b_{2}(B_{1},QB_{2})\Bigr{)}+Q\,b_{2}(B_{1},B_{2})=0\,.\end{split}$
(6.40)

The two terms in $S{W}^{\otimes 2}$ cancel out and give no new condition. The
nontrivial condition arises from the part in $W$ on the above right-hand side:

$Q\,b_{2}(B_{1},B_{2})+b_{2}(QB_{1},B_{2})+(-1)^{d_{B_{1}}}b_{2}(B_{1},QB_{2})=0\,.$
(6.41)

This is in agreement with (7) for $N=2$. This condition states that $Q$ is a
(graded) derivation of the two-product.

Next we turn to the identity that arises from ${\bf b}^{2}(B_{1}\wedge
B_{2}\wedge B_{3})=0$. One gets, from the vanishing of the part in $W$, the
condition

$\begin{split}0\,=\,&\ Q\,b_{3}(B_{1},B_{2},B_{3})\
+b_{3}(QB_{1},B_{2},B_{3})+(-1)^{d_{B_{1}}}b_{3}(B_{1},QB_{2},B_{3})+(-1)^{d_{B_{1}}+d_{B_{2}}}b_{3}(B_{1},B_{2},QB_{3})\\\\[3.44444pt]
\ &\ \
+b_{2}(b_{2}(B_{1},B_{2}),B_{3})+(-1)^{d_{B_{2}}d_{B_{3}}}b_{2}(b_{2}(B_{1},B_{3}),B_{2})+(-1)^{d_{B_{1}}(d_{B_{2}}+d_{B_{3}})}b_{2}(b_{2}(B_{2},B_{3}),B_{1})\,.\end{split}$
(6.42)

This is in agreement with (7) for $N=3$. Note the three terms on the last
line. If these were set to zero, we would have a graded version of the Jacobi
identity for the two-product $b_{2}(B_{1},B_{2})$. But in $L_{\infty}$, this
two-product is not a strict Lie bracket, and therefore it does not satisfy a
Jacobi identity. In the $L_{\infty}$ algebra, the ‘Jacobiator’, namely the sum
of those three terms in question, is set equal to the failure of the higher
product $b_{3}(B_{1},B_{2},B_{3})$ to be a derivation of $Q$.

In general, the action of the second ${\bf b}$ in ${\bf
b}^{2}(B_{1}\wedge\ldots\wedge B_{n})$ only gives a new identity on account of
the vanishing of the terms in ${W}$. Thus, making use of (6.34), the general
conditions on the products arise by replacing the parenthesis by a bracket:

$0\ =\
\sum_{l=1}^{n}\sum_{\sigma^{\prime}}\epsilon(\sigma^{\prime},B)\,b_{n-l+1}\bigl{(}\,b_{l}\bigl{(}B_{i_{1}},\cdots,B_{i_{l}}\bigr{)}\,,B_{j_{1}},\cdots,B_{j_{n-l}}\bigr{)}\,.$
(6.43)

Since we have recovered the identity (7), with the last term discarded and all
products at genus zero, the construction of the classical master action and
the proof that it satisfies the BV master equation proceeds as in section 4.1.

Finally, as in the case of $A_{\infty}$ algebra, we can introduce a bilinear
inner product that, given two elements $B_{1},B_{2}$ of the algebra, gives us
an element $(B_{1},B_{2})\in\mathbb{C}$, obeying the symmetry:

$(B_{1},B_{2})=(-1)^{(d_{B_{1}}+1)(d_{B_{2}}+1)}(B_{2},B_{1})\,,$ (6.44)

and the folding over property

$(QB_{1},B_{2})=(-1)^{d_{B_{1}}}\,(B_{1},QB_{2})\,.$ (6.45)

Furthermore, using the inner product to define multilinear maps to the complex
numbers:

$\\{B_{1},\cdots,B_{n}\\}_{L_{\infty}}\equiv(B_{1},b_{n-1}(B_{2},\cdots
B_{n}))\,,$ (6.46)

we also demand the graded commutativity property:

$\\{B_{1},\cdots,B_{k},B_{k+1},\cdots,B_{n}\\}_{L_{\infty}}=(-1)^{d_{B_{k}}d_{B_{k+1}}}\\{B_{1},\cdots,B_{k+1},B_{k},\cdots,B_{n}\\}_{L_{\infty}}\,.$
(6.47)

For the application to closed string field theory a few remarks are useful.
The natural choice for $W$ is the CFT vector space ${\cal H}_{c}$ upgraded to
a ${\cal G}$ module (still denoted by ${\cal H}_{c}$) to account for Grasmann
even and odd target space fields. As we mentioned before, $Q$ is the BRST
operator in ${\cal H}_{c}$ and the products are the genus zero brackets. The
inner product $(\,\cdot\,,\cdot)$, taking ${\cal H}_{c}\otimes{\cal
H}_{c}\to{\cal G}$ in fact coincides with $\langle\cdot\,,\,\cdot\,\rangle$,
defined in (3.13). For closed strings, $L_{\infty}$ degree and Grassmanality
coincide: the full string field $\Psi$ is of even degree, and as a vertex
operator multiplied by a target space field it is uniformly Grassmann even. In
terms of these quantities the action $S$ satisfying the classical BV master
equation can be written as

$S\ =\ \tfrac{1}{2}\,(\Psi,Q\,\Psi)+\sum_{n=3}^{\infty}{1\over
n!}\\{\Psi^{n}\\}_{L_{\infty}}\,,$ (6.48)

where the string field $\Psi$ is an even degree element of the $L_{\infty}$
algebra.

###  6.4 From $A_{\infty}$ to $L_{\infty}$

In section 4.4 the classical open string field theory was written in terms of
products $[\cdots]$, which, as we saw in section 6.2, can be also expressed in
terms of the $A_{\infty}$ products. We shall now explain how the action can
also be written in terms of $L_{\infty}$ products that are defined in terms of
the $A_{\infty}$ products.

Let us call $b_{n}$ the $A_{\infty}$ products and $\bar{b}_{n}$ the
$L_{\infty}$ products to be constructed from them. The vector space of states
is the same, with elements denoted $A_{1},A_{2},\cdots$. We have that the
first product needs no change:

$\bar{b}_{1}(A_{1})\equiv b_{1}(A_{1})\,.$ (6.49)

Clearly $\bar{b}_{1}\bar{b}_{1}(A_{1})=b_{1}b_{1}(A_{1})=0$, as required. For
the second product we take

$\bar{b}_{2}(A_{1},A_{2})\equiv
b_{2}(A_{1},A_{2})+(-1)^{d_{A_{1}}d_{A_{2}}}b_{2}(A_{2},A_{1})\,.$ (6.50)

This satisfies the required exchange symmetry
$\bar{b}_{2}(A_{1},A_{2})=(-1)^{d_{A_{1}}d_{A_{2}}}\bar{b}_{2}(A_{2},A_{1})$
by construction. A short computation confirms that the desired $L_{\infty}$
property (see (6.41))

$\bar{b}_{1}\bar{b}_{2}(A_{1},A_{2})+\bar{b}_{2}(\bar{b}_{1}(A_{1}),A_{2})+(-1)^{d_{A_{1}}}\bar{b}_{2}(A_{1},\bar{b}_{1}(A_{2}))=0\,,$
(6.51)

holds because the exactly same equation holds for $b_{2}$ and $b_{1}$ (see the
second equation in (6.12)). If the $A_{\infty}$ algebra is in fact
associative, there is no $b_{3}$, and we have

$b_{2}(b_{2}(A_{1},A_{2}),A_{3})+(-1)^{d_{A_{1}}}b_{2}(A_{1},b_{2}(A_{2},A_{3}))=0\,,$
(6.52)

see the third equation in (6.12). This property in fact guarantees the
identity

$\bar{b}_{2}(\bar{b}_{2}(A_{1},A_{2}),A_{3})+(-1)^{d_{A_{2}}d_{A_{3}}}\bar{b}_{2}(\bar{b}_{2}(A_{1},A_{3}),A_{2})+(-1)^{d_{A_{1}}(d_{A_{2}}+d_{A_{3}})}\bar{b}_{2}(\bar{b}_{2}(A_{2},A_{3}),A_{1})=0\,,$
(6.53)

that must hold for the $L_{\infty}$ algebra to be a Lie algebra without higher
products (see (6.42)).

More generally, if the $A_{\infty}$ algebra has higher products, the higher
products of the $L_{\infty}$ algebra are obtained as follows:

$\bar{b}_{n}(A_{1},\cdots,A_{n})\equiv\sum_{\sigma}\epsilon(\sigma,A)\,b_{n}(A_{\sigma(1)},\cdots\,,A_{\sigma(n)})\,.$
(6.54)

Here we simply sum over the complete set of permutations of the $n$ inputs,
with the Koszul sign computed with the rule
$A_{i}A_{j}=(-1)^{d_{A_{i}}d_{A_{j}}}A_{j}A_{i}$. This definition implies that
the product $\bar{b}_{n}$ is graded commutative, as required. We will not
present here a proof that the higher $L_{\infty}$ identities for the $\bar{b}$
products are satisfied on account of the full set of identities for the
$A_{\infty}$ products. Note also that since the open string field has even
degree,
$\bar{b}_{n}(\psi_{o},\cdots,\psi_{o})=n!\,b_{n}(\psi_{o},\cdots,\psi_{o})$.

Using (6.24) and (6.54) we can find the relation between the $L_{\infty}$
products $\bar{b}_{n}$ and the products $[\cdots]$ for the open string fields
constructed in (6.24). This takes the form

$[A_{1}\cdots A_{n}]=(-1)^{d_{A_{1}}+2d_{A_{2}}+\cdots+nd_{A_{n}}}\
\bar{b}_{n}(A_{1},\cdots,A_{n})\,.$ (6.55)

We can also relate the inner products in the $L_{\infty}$ and $A_{\infty}$
algebra as follows. Let $(\hskip-6.50403pt{\rm-}\ ,\
~{})\hskip-5.78172pt{\rm-}$ denote the inner product in the $L_{\infty}$
algebra, as introduced at the end of section 6.3. We equate this to the BPZ
inner product in the state space and not the inner product $(,)$ of the
$A_{\infty}$ algebra, – the two being related by (6.18). Thus we have

$(\hskip-6.50403pt{\rm-}A_{1},A_{2}\,)\hskip-5.78172pt{\rm-}=\langle
A_{1},A_{2}\rangle^{\prime}=(-1)^{d_{A_{1}}}(A_{1},A_{2})\,.$ (6.56)

The symmetry property (6.44) and (6.45) follow as consequences of (6.13) and
(6.14). Equation (6.56) gives, from (6.46),

$\begin{split}\\{A_{0}A_{1}\cdots A_{n}\\}_{L_{\infty}}\,=&\
(\hskip-6.50403pt{\rm-}\,A_{0},\bar{b}_{n}(A_{1}A_{2}\cdots A_{n})\
)\hskip-5.78172pt{\rm-}=\
(-1)^{d_{A_{0}}}\,(A_{0},\bar{b}_{n}(A_{1}A_{2}\cdots A_{n}))\\\\[3.44444pt]
=&\
(-1)^{d_{A_{0}}}\sum_{\sigma}\epsilon(\sigma,A)\,(A_{0},b_{n}(A_{\sigma(1)},\cdots\,,A_{\sigma(n)}))\,.\end{split}$
(6.57)

With some work one can show that the symmetry property (6.47) of the left hand
side follows from the symmetry properties (6.15), (6.16).

Since the open string field $\psi_{o}$ has even degree $d_{\psi_{o}}$, we can
express the action (6.17) as

$S(\psi_{o})\,=\sum_{n=1}^{\infty}{1\over
n+1}(\psi_{o},b_{n}(\psi_{o}^{n}))=\tfrac{1}{2}\,(\hskip-6.50403pt{\rm-}\psi_{o},Q\psi_{o}\,)\hskip-5.78172pt{\rm-}+\sum_{n=2}^{\infty}{1\over(n+1)!}\\{\psi_{o}^{n+1}\\}_{L_{\infty}}\,.$
(6.58)

This agrees structurally with the $L_{\infty}$ action (6.48) of closed
strings.

Equation (6.55) can be given the following interpretation. When all the
$A_{i}$’s are Grassmann odd (i.e. of even degree) we have

$[A_{1}\cdots A_{n}]=\bar{b}_{n}(A_{1},\cdots,A_{n})\,.$ (6.59)

If some of the $A_{i}$’s are Grassmann even (odd degree) then we multiply them
by Grassmann odd $c$-numbers from the left and apply (6.59). We can then move
all the Grassmann odd $c$-numbers to the left by picking up a minus sign every
time the Grassmann odd c-number passes a Grassmann odd $A_{i}$ inside
$[A_{1}\cdots A_{n}]$ and picking up a minus sign every time the Grassmann odd
c-number passes an odd degree $A_{i}$ inside $\bar{b}_{n}(A_{1},\cdots
A_{n})$. It can be easily checked that this prescription leads to (6.55). Thus
inside the argument of $\bar{b}_{n}$ the effective Grassmannality of $A_{i}$,
defined as whether or not a Grassmann odd $c$-number passing through $A_{i}$
generates a minus sign, becomes identical to the degree.

###  6.5 Quantum $L_{\infty}$ algebra

Classical closed string field theory must be supplemented by additional terms
in order to define the full quantum closed string field theory. The Feynman
rules of the classical theory do not produce covers of the higher genus moduli
spaces. This happens because the classical master action does not solve the
quantum master equation. In order to produce an action that solves the quantum
master equation one needs higher genus string products. In the language of
$L_{\infty}$ algebra, the products $b_{n}(A_{1},\cdots,A_{n})\in W$ now
include contributions from all genus, in general. We now write

$b_{n}(B_{1},\cdots,B_{n})=\sum_{g=0}^{\infty}g_{s}^{2g+n-1}\,b_{n}^{g}(B_{1},\cdots,B_{n})\,,\
\ n=0,1,\cdots\,,$ (6.60)

where with a little abuse of notation the $b_{n}$’s now denote the products
with all genus contributions, and the genus zero $b_{n}$’s of the original
$L_{\infty}$ algebra are now written as $b_{n}^{0}$. A few things should be
noted. We now have a product $b_{0}(\ )$ without an input, that maps nothing
to $W$. It receives contributions from $b_{0}^{g}$ products with $g\geq 1$
($b_{0}^{0}(\ )=0$). In closed string field theory, the inner product of
$b_{0}^{g}(\ )$ with a state $B\in{\cal H}_{c}$ is related to the one-point
function of $B$ on genus $g$ Riemann surfaces. Furthermore, the product
$b_{1}(A_{1})$ is $QA_{1}$ at genus zero, but gets additional contribution
from two-point functions on Riemann surfaces of genus $g\geq 1$.

The identity that must be satisfied by the complete set of products is a
generalization of (6.43), and it takes the form

$\begin{split}0\ =\ &\
\sum_{l=1}^{n}\sum_{\sigma^{\prime}}\epsilon(\sigma^{\prime},B)\,b_{n-l+1}\bigl{(}b_{l}\bigl{(}B_{i_{1}},\cdots,B_{i_{l}}\bigr{)},B_{j_{1}},\cdots,B_{j_{n-l}}\bigr{)}\\\\[4.30554pt]
&\hskip-5.0pt+\tfrac{1}{2}\,b_{n+2}\bigl{(}B_{1},\cdots\,,B_{n},\varphi_{s},\varphi_{r}\bigr{)}\,(\varphi_{s}^{c},\varphi_{r}^{c})\,.\end{split}$
(6.61)

This identity is equivalent to (7). It is the identity defining the quantum
$L_{\infty}$ algebra. The term in the second line is of purely quantum origin
and was not present in (6.43) . Of course, even the first term is different,
as it now includes the higher-genus contributions.

The abstract $L_{\infty}$ algebra introduced in section 6.3 is thus modified
to the quantum $L_{\infty}$ algebra. The main new ingredients are the
following.

  1. 1.

We define b as in (6.34) with the sum over $l$ starting at 0, and $b_{i}$’s
now including higher genus contributions satisfying the identity (6.61).

  2. 2.

We also introduce an operator $\boldsymbol{\theta}$ from $T(W)\to T(W)$ such
that

$\boldsymbol{\theta}\,(A_{1}\wedge\cdots\wedge
A_{n})=\tfrac{1}{2}\,A_{1}\wedge\cdots\wedge
A_{n}\wedge\varphi_{r}\wedge\varphi_{s}\,(\varphi_{r}^{c},\varphi_{s}^{c})\,.$
(6.62)

The quantum $L_{\infty}$ algebra identity (6.61) follows from the vanishing of
the following operator acting on $T(W)$

$\boldsymbol{\pi}_{1}\bigl{(}{\bf b}^{2}+{\bf
b}\,\boldsymbol{\theta}\bigr{)}=0\,,$ (6.63)

where $\boldsymbol{\pi}_{1}$ is the projection from $T(W)$ to $W$. It is shown
by Markl [95] that this condition is equivalent to the conceptually clearer
condition

$({\bf b}+\boldsymbol{\theta})^{2}=0\,.$ (6.64)

###  6.6 Open-closed string field theory

In section 4.5 we described a formulation of open-closed string field theory
where the open and closed strings appear on similar footing except for some
signs, see e.g. the $(-1)^{j-1}$ factor in the second line of
(LABEL:emainsup). The difference in sign arises because the closed string
fields are even while open string fields are odd. However we have seen in
section 6.4 that it is possible to reformulate classical open string field
theory using the products $\bar{b}_{n}(A_{1},\cdots A_{n})$ that satisfy the
same $L_{\infty}$ algebra as classical closed strings. This suggests that a
similar change in sign can be used to rewrite the open-closed string field
theory in terms of a regular quantum $L_{\infty}$ algebra without any extra
sign. To this end, we define $L_{\infty}$-type multilinear functions from the
open-closed functions:

$\\{A^{c}_{1},\cdots,A^{c}_{n_{c}};A^{o}_{1},\cdots,A^{o}_{n_{o}}\\}_{L_{\infty}}\equiv(-1)^{d_{A^{o}_{2}}+2d_{A^{o}_{3}}+\cdots\,+(n-1)d_{A^{o}_{n}}}\\{A^{c}_{1},\cdots,A^{c}_{n_{c}};A^{o}_{1},\cdots,A^{o}_{n_{o}}\\}\,,$
(6.65)

with $\\{A^{c}_{1},\cdots,A^{c}_{n_{c}};A^{o}_{1},\cdots,A^{o}_{n_{o}}\\}$ as
defined in (4.61). When restricted to classical open string field theory, this
expression reduces to $\\{A_{1}^{o},\cdots,A_{n}^{o}\\}_{L_{\infty}}$ via
(6.55)-(6.57). Then the open-closed string field theory action defined in
terms of $\\{\cdots\\}_{L_{\infty}}$ will have the same form as (4.69) since
for Grassmann odd open string states there is no difference between
$\\{\cdots\\}_{L_{\infty}}$ and $\\{\cdots\\}$. In terms of the new
$L_{\infty}$ multilinear functions, however, the $(-1)^{j-1}$ factor for open
strings will be absent in (LABEL:emainsup) and we can treat open and closed
strings on equivalent footing.

The open-closed system is a novel and fascinating structure in terms of
homotopy algebras. Much of the novelty is present for the bosonic string case.
As Kajiura and Stasheff discuss [96, 97] there are various ways of thinking of
the algebraic structure of the open-closed string theory. One can view it as a
deformation of the $A_{\infty}$ structure on the open string state space
controlled by the $L_{\infty}$ algebra on the closed string state space.
Physicists note that indeed open-closed string field theory shows how to write
classical open string field theories on general closed string backgrounds.
Mathematically, one can focus on the $L_{\infty}$ algebra acting on the open
string state space as a graded vector space. This is the homotopy version of a
Lie algebra $L$ acting on a vector space $M$, where $M$ is viewed as a
representation of $L$. There has been renewed work in formulating the open-
closed string field theory in the language of coderivations, finding
interesting limits, and the implications for constant terms in the action [98,
99, 100].

###  6.7 Homotopy transfer

We considered the Wilsonian effective action in section 5.4. The main takeaway
was that starting from a consistent string field theory one could obtain, via
a suitable projection of the string field, a subset of fields for which there
is a consistent effective field theory. In algebraic terms, and focusing here
only on the classical theories, we have the original string field theory with
an $A_{\infty}$ or $L_{\infty}$ algebra. The procedure to be discussed below,
going under the name of homotopy transfer, builds an $A_{\infty}$ or
$L_{\infty}$ algebra on the subset of fields[101, 96, 97, 65, 66]. These
algebras thus define the effective field theory of that subset of fields. The
construction clarifies the general conditions that must be satisfied in the
selection of the projector down to a subset of fields. It also provides
explicit formulae for the terms in the effective action.

Homotopy transfer for ${\bf A}_{\infty}$. Consider classical open strings
formulated in the framework of $A_{\infty}$ algebras. The $A_{\infty}$
algebras will be described by coderivations on the tensor algebra $T(V)$ as we
discussed in section 6.1. A slight change of notation is helpful here. We
wrote before ${\bf b}={\bf b}_{1}+{\bf b}_{2}+{\bf b}_{3}+\cdots$. We will set
${\bf M}={\bf b}$, write ${\bf Q}={\bf b}_{1}$, as this is the BRST operator,
and define ${\bf m}={\bf b}_{2}+{\bf b}_{3}+\cdots$. All in all, we have the
degree one operator

${\bf M}={\bf Q}+{\bf m}\,,\ \ \ {\bf m}=\sum_{n=2}^{\infty}{\bf b}_{n}\,.$
(6.66)

The main identity satisfied by the products is ${\bf M}^{2}=0$ which implies,

${\bf Q}^{2}=0\,,\ \ \ \ \hbox{and}\ \ \ \ {\bf Q}{\bf m}+{\bf m}{\bf Q}+{\bf
m}^{2}=0\,.$ (6.67)

We now follow the discussion of [67]. We introduce a projector $P$ from $V$ to
a subspace $\bar{V}$, this is the subspace of degrees of freedom we want to
write an effective field theory for,

$P\,:{V}\to\bar{V}\,,\ \ \ P^{2}=P\,.$ (6.68)

The $A_{\infty}$ inner product must be consistent with the projector:

$(A_{1}\,,PA_{2})=(PA_{1}\,,\,A_{2})\,.$ (6.69)

We also demand that ${P}$ and ${Q}$ commute

${Q\,P}={P\,Q}\,.$ (6.70)

We extend this projector to a degree zero operator ${\bf P}:T({V})\to
T(\bar{V})$ by defining the action

${\bf P}(A_{1}\otimes A_{2}\otimes\cdots\otimes A_{n})\,\equiv\,PA_{1}\otimes
PA_{2}\otimes\cdots\otimes PA_{n}\,,$ (6.71)

so that

${\bf P}^{2}={\bf P}\,,$ (6.72)

and

${\bf Q\,P}={\bf P\,Q}\,.$ (6.73)

Associated to the projector we can introduce a linear operator $h$ of degree 1
(thus odd) in ${V}$ satisfying the conditions:

$h\,P=P\,h=0\,,\ \ \ \ h^{2}=0\,,\ \ \ \ Qh+hQ=1-P\,.$ (6.74)

If $P$ projects on to a subspace containing all the $L_{0}=0$ states, for
example, then we can take $h={b_{0}\over L_{0}}(1-P)$.

The next definition tells us how $h$ is promoted to an operator ${\bf h}$
acting on $T({V})$. In fact, ${\bf h}$ maps ${V}^{\otimes n}$ to itself as
follows:

$\begin{split}h(A_{1}\otimes A_{2}\otimes\cdots\otimes A_{n})\equiv\ &\ \
(hA_{1})\otimes PA_{2}\otimes\cdots\otimes PA_{n}\\\
&+(-1)^{d_{A_{1}}}A_{1}\otimes hA_{2}\otimes\cdots\otimes PA_{n}\\\ &\ \ \
\vdots\ \ \ \ \ \ \ \ \ \vdots\\\
&+(-1)^{d_{A_{1}}+d_{A_{2}}+\cdots+d_{A_{n-1}}}\ A_{1}\otimes
A_{2}\otimes\cdots\otimes hA_{n}\,.\end{split}$ (6.75)

The surprising fact of this definition is that projectors are only inserted to
the right of the insertion of $h$. The sign factors simply indicate that $h$
is odd. The definition is such that the properties (6.74) holding on ${V})$
get upgraded to identities in $T({V})$:

${\bf h}\,{\bf P}={\bf P\,h}=0\,,\ \ \ \ {\bf h}^{2}=0\,,\ \ \ \ {\bf
Q\,h}+{\bf h\,Q}={\bf 1}-{\bf P}\,.$ (6.76)

The first three equations are almost obvious, the last one takes more work. It
is straightforward to see just by acting on $A_{1}\otimes A_{2}$ that the
unusual placement of projectors is needed for the last equation to work out.

We can now state that we have an $A_{\infty}$ algebra on $\bar{V}$, although
we will describe it as one in ${V}$ in such a way that any action on the
kernel of $P$ is manifestly zero. The homotopy transfer result for
$A_{\infty}$ states that the operator $\overline{\bf M}$ defining the new
products is given by [66, 67]

$\overline{\bf M}={\bf P\,Q\,P}+{\bf P\,m}\,{{\bf 1}\over{\bf 1+h\,m}}\,{\bf
P}\,,$ (6.77)

and satisfies $\overline{\bf M}^{2}=0$, as required to define an $A_{\infty}$
algebra. Additionally, one must show that $\overline{\bf M}$ is a coderivation
in the tensor coalgebra (see, for example [67], section 8.1.3). Finally, the
new string products must be cyclic, just as the original ones. The ${\bf P}$
to the right of each term on the right-hand side ensures that $\overline{\bf
M}$ acts nontrivially only on $T(\bar{V})$. The ${\bf P}$ to the left of each
term shows that the range of $\overline{\bf M}$ is in $T(\bar{V})$. The
fraction is defined by its power series expansion

${{\bf 1}\over{\bf 1+h\,m}}={\bf 1}-{\bf h\,m}+{\bf h\,m}\,{\bf
h\,m}-\cdots\,.$ (6.78)

Let us verify that $\overline{\bf M}^{2}=0.$ The square of the first term
vanishes: ${\bf P\,Q\,P}\,{\bf P\,Q\,P}={\bf P\,Q\,}{\bf P\,Q\,P}={\bf
P\,Q\,}{\bf\,Q\,P}=0$ since ${\bf Q}$ and ${\bf P}$ commute. Given this we
have

$\overline{\bf M}^{2}={\bf P}\Bigl{\\{}{\bf Q}\,,{\bf m}\,{{\bf 1}\over{\bf
1+h\,m}}\Bigr{\\}}\,{\bf P}+\ {\bf P\,m}\ {{\bf 1}\over{\bf 1+h\,m}}\ {\bf
P\,m}\ {{\bf 1}\over{\bf 1+h\,m}}\,{\bf P}\,.$ (6.79)

With a little work the first term is seen to give

$\begin{split}{\bf P}\Bigl{\\{}{\bf Q}\,,{\bf m}\,{{\bf 1}\over{\bf
1+h\,m}}\Bigr{\\}}\,{\bf P}=\ &{\bf P}\,\bigl{\\{}{\bf Q\,,m}\bigr{\\}}{{\bf
1}\over{\bf 1+h\,m}}\,{\bf P}+{\bf P\,m}\ {{\bf 1}\over{\bf
1+h\,m}}\,\bigl{[}{\bf Q\,,1+h\,m}\bigr{]}\ {{\bf 1}\over{\bf 1+h\,m}}\,{\bf
P}\\\ =\ &\hskip-5.0pt-{\bf P\,m^{2}{{\bf 1}\over{\bf 1+h\,m}}\ P}+{\bf P\,m}\
{{\bf 1}\over{\bf 1+h\,m}}\bigl{(}({\bf 1-P}){\bf m+h\,m^{2}}\bigr{)}\ {{\bf
1}\over{\bf 1+h\,m}}{\bf P}\,.\end{split}$

Substituting this back into (6.79) one finds that, indeed $\overline{\bf
M}^{2}=0$.

It is instructive to expand the result (6.77) to write the new products
$\overline{\bf M}=\overline{\bf Q}+\overline{\bf m}_{2}+\overline{\bf
m}_{3}+\cdots$. One finds:

$\begin{split}\overline{\bf Q}\ =\ &{\bf P\,Q\,P}\,,\\\ \overline{\bf m}_{2}\
=\ &{\bf P\,m_{2}\,P}\,,\\\ \overline{\bf m}_{3}\ =\ &{\bf P}\bigl{[}{\bf
m_{3}-m_{2}\,h\,m_{2}}\bigr{]}\,{\bf P}\,,\\\ \overline{\bf m}_{4}\ =\ &{\bf
P}\bigl{[}{\bf
m_{4}-m_{2}\,h\,m_{3}-m_{3}\,h\,m_{2}+m_{2}\,h\,m_{2}\,h\,m_{2}}\bigr{]}\,{\bf
P}\,.\end{split}$ (6.80)

Acting on string fields $A_{i}$, defining $\bar{A}_{i}=PA_{i}$, and using
$m_{2}(A\otimes B)=A\star B$, we have for the first three products the
explicit action:

$\begin{split}\overline{Q}A_{1}\ =\ &PQ\bar{A}_{1}\,,\\\
\overline{m}_{2}(A_{1}\otimes A_{2})\ =\ &P(\bar{A}_{1}\star\bar{A}_{2})\,,\\\
\overline{m}_{3}(A_{1}\otimes A_{2}\otimes A_{3})\ =\
&Pm_{3}(\bar{A}_{1},\bar{A}_{2},\bar{A}_{3})-P\bigl{(}h(\bar{A}_{1}\star\bar{A}_{2})\star\bar{A}_{3}+\bar{A}_{1}\star
h(\bar{A}_{2}\star\bar{A}_{3})\bigr{)}\,.\end{split}$ (6.81)

We can connect to the results of section 5.4 by identifying $h$ as the
propagator of the heavy fields that are being integrated out.

Homotopy transfer for ${\bf L_{\infty}}$. This case has some new features
although the end result takes a similar form; here we will follow [68] with
some changes of notation. Again we work with the tensor co-algebra $T({W})$
defined by the sum of symmetrized tensor products $S{W}^{\otimes n}$. We had
before ${\bf b}={\bf b}_{1}+{\bf b}_{2}+{\bf b}_{3}+\cdots$. We will set ${\bf
L}={\bf b}$, write ${\bf Q}={\bf b}_{1}$, and define $\boldsymbol{\ell}={\bf
b}_{2}+{\bf b}_{3}+\cdots$. All in all, we have the degree one operator ${\bf
L}$ given by

${\bf L}={\bf Q}+\boldsymbol{\ell}\,,\ \ \
\boldsymbol{\ell}=\sum_{n=2}^{\infty}{\bf b}_{n}\ .$ (6.82)

The main identity satisfied by the products is ${\bf L}^{2}=0$ which implies,

${\bf Q}^{2}=0\,,\ \ \ \ \hbox{and}\ \ \ \ {\bf
Q}\boldsymbol{\ell}+\boldsymbol{\ell}{\bf Q}+\boldsymbol{\ell}^{2}=0\,.$
(6.83)

We introduce the projector $P:\,{W}\to\bar{W}$, taking ${W}$ to the subspace
$\bar{W}$ of the effective field theory. We demand that

$P^{2}=P\,,\ \ \ PQ=QP\,,\ \ \ (B_{1},PB_{2})=(PB_{1},B_{2})\,.$ (6.84)

We extend this projector to a degree zero operator ${\bf P}:T({W})\to
T(\bar{W})$. We write:

${\bf P}(B_{1}\wedge B_{2}\wedge\cdots\wedge B_{n})\,\equiv\,PB_{1}\wedge
PB_{2}\wedge\cdots\wedge PB_{n}\,.$ (6.85)

so that ${\bf P}^{2}={\bf P}$. Moreover it follows from (6.84) that ${\bf P}$
and ${\bf Q}$ commute ${\bf Q\,P}={\bf P\,Q}$. As before, associated to the
projector there is a linear operator $h$ of degree 1 satisfying191919We have
made a sign change on the right-hand side of the last equation, relative to
the one used in [68]. This is in order to have $L_{\infty}$ results analogous
to the $A_{\infty}$ results.

$h\,P=P\,h=0\,,\ \ \ \ h^{2}=0\,,\ \ \ \ Qh+hQ=1-P\,.$ (6.86)

The only new challenge here is defining ${\bf h}$, the associated odd operator
on the symmetrized tensor algebra. This is done recursively. Using the
notation $\bar{A}=PA$, we have

$\begin{split}h(A_{1}\wedge\ldots\wedge A_{n})\equiv\ &{1\over
n!}\sum_{\sigma\in
S_{n}}\epsilon(\sigma;A)\Bigl{(}h(A_{\sigma(1)}\wedge\ldots\wedge
A_{\sigma(n-1)})\wedge A_{\sigma(n)}\\\\[2.15277pt] &\ \ \
+(-1)^{A_{\sigma(1)}+\ldots+A_{\sigma(n-1)}}\bar{A}_{\sigma(1)}\wedge\cdots\wedge\bar{A}_{\sigma(n-1)}\wedge
hA_{\sigma(n)}\Bigr{)}\,.\end{split}$ (6.87)

The reader can check that this gives, for example,

$h(A_{1}\wedge A_{2})=\tfrac{1}{2}\bigl{(}hA_{1}\wedge
A_{2}+hA_{1}\wedge\bar{A}_{2}+(-1)^{A_{1}}(\bar{A}_{1}\wedge
hA_{2}+A_{1}\wedge hA_{2})\bigr{)}\,.$ (6.88)

The definition is such that the properties (6.86) holding on ${W}$ get
upgraded to identities in $T({W})$:

${\bf h}\,{\bf P}={\bf P\,h}=0\,,\ \ \ \ {\bf h}^{2}=0\,,\ \ \ \ {\bf
Q\,h}+{\bf h\,Q}={\bf 1}-{\bf P}\,.$ (6.89)

Since we have here the same identities on the tensor algebra as in the
$A_{\infty}$ case, the operator $\overline{\bf L}$ defining the new products
is given by the analog of (6.77):

$\overline{\bf L}={\bf P\,Q\,P}+{\bf P}\,\boldsymbol{\ell}\,{{\bf 1}\over{\bf
1+h\,}\boldsymbol{\ell}}\,{\bf P}\,,$ (6.90)

and satisfies $\overline{\bf L}^{2}=0$, as required to be define $L_{\infty}$
algebra. Moreover, $\overline{\bf L}$ is a coderivation in the tensor
coalgebra and the new string products are graded symmetric [68].

The analogs of (6.80) hold, in particular, $\bar{\boldsymbol{\ell}}_{3}={\bf
P}[\boldsymbol{\ell}_{3}-\boldsymbol{\ell}_{2}{\bf
h}\boldsymbol{\ell}_{2}]{\bf P}$. Acting on string fields $A_{i}$, defining
$\bar{A}_{i}=PA_{i}$, and $[A_{1},A_{2}]=\ell_{2}(A_{1}\wedge A_{2})$, we have

$\begin{split}\overline{Q}A_{1}\ =\ &Q\bar{A}_{1}\,,\\\
\overline{\ell}_{2}(A_{1}\wedge A_{2})\ =\ &P\,[\bar{A}_{1},\bar{A}_{2}]\,,\\\
\overline{\ell}_{3}(A_{1}\wedge A_{2}\wedge A_{3})=\
&P\ell_{3}(\bar{A}_{1}\wedge\bar{A}_{2}\wedge\bar{A}_{3})\\\\[2.58334pt]
&\hskip-5.0pt-P\,\bigl{(}\,\bigl{[}\,h[\bar{A}_{1}\,,\,\bar{A}_{2}]\,,\bar{A}_{3}\bigr{]}+(-1)^{A_{2}A_{3}}\bigl{[}\,h[\bar{A}_{1}\,,\,\bar{A}_{3}]\,,\bar{A}_{2}\bigr{]}+\bigl{[}\,\bar{A}_{1}\,,\,h[\bar{A}_{2}\,,\,\bar{A}_{3}]\bigr{]}\,\bigr{)}\,.\end{split}$
(6.91)

The expression for $\bar{\ell}_{3}$ is relevant to the computation of four-
point functions. The last three terms are the contributions from three Feynman
diagrams of the original field theory, where, due to $h$, only the states
outside $\bar{W}$ propagate.

##  7 String vertices

We have studied so far the algebraic structure of the various string field
theories. These structures required a number of string products. These string
products are defined, in general, using subsets of moduli spaces of Riemann
surfaces with punctures and local coordinates at the punctures.

In this section we will discuss in detail how these subsets of moduli spaces
are selected, and how we can introduce coordinates around the punctures. This
is the information about the string vertices. The consistency conditions on
these choices is an equation quite analogous to the Batalin Vilkovisky master
equation. Equipped with such string vertices we then discuss how such
information allows us to write the string field theory interactions.

At the basis of all of this lies the concept of a BV algebra. It begins by
defining a complex ${\cal C}$, usually an infinite dimensional vector space,
whose elements form a graded-commutative, associative algebra under a simple
multiplication called a dot product. Given two elements $X,Y\in{\cal C}$ we
write the dot product as $X\cdot Y\equiv XY$ belonging to the same complex.
Graded commutativity of the dot product means $XY=(-1)^{XY}\,YX$ where $X,Y$
in the sign factor refer to the Grassmannality of $X$ and $Y$. On the space
${\cal C}$ one introduces an odd operator $\Delta$ that squares to zero:

$\Delta(\Delta X)=\Delta^{2}X=0\,,\ \ \forall X\in{\cal C}\,.$ (7.1)

Moreover, the operator $\Delta$ must be a second order (super) derivation of
the dot product:

$\begin{split}\Delta(XYZ)=&\ \ \
\Delta(XY)Z+(-1)^{X}X\Delta(YZ)+(-1)^{XY+Y}Y\Delta(XZ)\\\\[4.30554pt] &-\Delta
X(YZ)-(-1)^{X}X(\Delta Y)Z-(-1)^{X+Y}XY\Delta Z\,,\end{split}$ (7.2)

for $X,Y,$ and $Z$ arbitrary elements in ${\cal C}$ with definite
Grassmanality. These properties define the BV algebra $({\cal C},\Delta)$.
Still, one extra object, the antibracket $\\{X,Y\\}$ of two elements, can be
defined from the above structure as the failure of $\Delta$ to be a first
order derivation:

$\\{X,Y\\}\equiv(-1)^{X}\Delta(XY)-(-1)^{X}(\Delta X)Y-X\Delta Y\,.$ (7.3)

There are three properties of the antibracket that follow from the above
definition: the exchange property, the Jacobi-like property and the
interaction with the dot product. These are quick to verify and read:

$\begin{split}\\{X,Y\\}=\ &\ -(-1)^{(X+1)(Y+1)}\\{Y,X\\}\,,\\\ 0\ \ =&\ \
(-1)^{(X+1)(Z+1)}\\{\,\\{X,Y\\}\,,Z\\}+\hbox{cyclic}\,,\\\ \\{X\,,YZ\\}=\ &\
\\{X,Y\\}Z+(-1)^{XY+Y}Y\\{X,Z\\}\,.\end{split}$ (7.4)

We will now explain how this BV structure is defined when ${\cal C}$ is built
from Riemann surfaces with punctures, carrying a choice of local coordinates
at each puncture. Then we will consider how this BV structure exists on
functions defined on the vector space of a CFT. Finally, the relation between
these structures will be established. This allows one to find an equation for
string vertices that implies that the string field theory satisfies the BV
master equation.

For simplicity we shall restrict most of our analysis to closed bosonic string
field theory, and will describe the results for open closed string field
theory at the end. Generalization of this analysis to superstring field theory
is straightforward.

###  7.1 BV structures on moduli spaces

We first introduce the key definitions. We let ${\cal M}_{g,n}$ be the moduli
space of Riemann surfaces of genus $g$ and $n$ punctures. The punctures are
labeled and are thus distinguishable. This moduli space is finite dimensional.
The space ${\cal P}_{g,n}$ is the moduli space of genus $g$ surfaces and $n$
punctures with a chosen analytic coordinate at each puncture. This space is
infinite dimensional for $n>0$, because one requires an infinite number of
parameters to define an analytic coordinate around a puncture. The space
${\cal P}_{g,n}$ is a fibering over ${\cal M}_{g,n}$, with a projection that
simply forgets about the analytic coordinates at the punctures. For closed
string field theory, however, it is necessary to introduce a space
$\widehat{\cal P}_{g,n}$, also fibered over ${\cal M}_{g,n}$. This space is
obtained from ${\cal P}_{g,n}$ by a projection that forgets the phase of the
local coordinate at each puncture.

It is worth noting that local coordinate around a puncture is a map from a
unit disk $|w|\leq 1$ to the surface, with $w=0$ mapped to the puncture. The
image of $|w|=1$ under the map is a coordinate curve, a simple closed Jordan
curve surrounding the puncture and homotopic to it. Specifying the phase of
the local coordinate is equivalent to singling out a special point in the
coordinate curve, say the image of $w=1$. The coordinate curve with this
special point marked is in fact enough, by the Riemann mapping theorem, to
specify the local coordinate at the puncture. The coordinate curve, without
any marked point, specifies the local coordinate up to a phase. This phase can
be viewed as the marking of a special point on a closed string. We can
therefore imagine any element of $\widehat{{\cal P}}_{g,n}$ as a Riemann
surface of genus $g$, with $n$ punctures and a coordinate curve around each
puncture. A coordinate curve defines a coordinate disk – the disk which is the
image of $|w|\leq 1$. The various coordinate disks should not have regions of
overlaps. It is possible, however, for their boundaries to touch.

To build the complex ${\cal C}$ we consider finite dimensional subspaces
${\cal A}_{g,n}$ of $\widehat{\cal P}_{g,n}$. Note that these subspaces can
have any dimension from zero to infinity when $n>0$. Next we consider spaces
of disconnected ‘generalized’ surfaces as follows:

${\cal A}_{g_{1},n_{1}}\times\ldots\times{\cal A}_{g_{r},n_{r}}\,.$ (7.5)

A ‘surface’ in this term is a disconnected one of the form
$(\Sigma_{g_{1},n_{1}},\cdots\Sigma_{g_{r},n_{r}})$, in fact a collection of
$r$ surfaces, one from each subspace. Here, the disconnected surface has a
total of $N=\sum_{i=1}^{r}n_{i}$ punctures. We assign a degree to the
subspaces ${\cal A}_{g,n}$ by setting it equal to its dimensionality

$\hbox{deg}\,({\cal A}_{g,n})=\hbox{dim}({\cal A}_{g,n})\,(\hbox{mod}\,2)\,.$
(7.6)

The order of the factors in the above product matters only up to signs. Thus,
for example

${\cal A}_{g_{1},n_{1}}\times{\cal A}_{g_{2},n_{2}}=(-1)^{{\cal
A}_{g_{1},n_{1}}\cdot{\cal A}_{g_{2},n_{2}}}\,{\cal
A}_{g_{2},n_{2}}\times{\cal A}_{g_{1},n_{1}}\,.$ (7.7)

An important technical point has to do with labeling of punctures. In an
element of ${\cal C}$ we need the punctures to be labeled from $1$ to $N$.
This is easily done by labeling the punctures successively from left to right;
those in the first space from $1$ to $n_{1}$, those in the second space from
$n_{1}+1$ to $n_{1}+n_{2}$ and so on. Moreover we need the element of ${\cal
C}$ to be invariant under the exchange of labels –we can call this a symmetric
space. This can be implemented above by applying to the surfaces in (7.5) a
permutation operator ${\bf P}$ in the symmetric group $S_{N}$ that permutes
the $N$ labels of the punctures, and a normalization factor. The result is
finally an element of the complex ${\cal C}$, denoted with double brackets
$\left[\left[\cdots\right]\right]$:

$\left[\left[{\cal A}_{g_{1},n_{1}}\,,\ldots\,,{\cal
A}_{g_{r},n_{r}}\right]\right]\equiv{1\over n_{1}!\cdots
n_{r}!}\sum_{\sigma\in S_{N}}{\bf P}_{\sigma}({\cal
A}_{g_{1},n_{1}}\times\ldots\times{\cal A}_{g_{r},n_{r}})\in{\cal C}\,.$ (7.8)

The normalization factor is such that if the ${\cal A}$ spaces are themselves
symmetric, the only effect of the permutations is to move labels across the
disconnected surfaces. The $\left[\left[\cdots\right]\right]$ are graded
symmetric under the exchange of the ${\cal A}$ spaces within the brackets. The
orientation of these $\left[\left[\cdots\right]\right]$ spaces is induced by
the orientation of the ${\cal A}$ subspaces. The general element of ${\cal C}$
consists of linear superpositions of the above elements multiplied by real
numbers.

Having defined ${\cal C}$ we must now define the dot product. This is actually
easy, we take

$\Big{[}\Big{[}{\cal A}_{g_{1},n_{1}}\,,\ldots\,,{\cal
A}_{g_{r},n_{r}}\Big{]}\Big{]}\cdot\left[\left[{\cal
B}_{g^{\prime}_{1},n^{\prime}_{1}}\,,\ldots\,,{\cal
B}_{g^{\prime}_{p},n^{\prime}_{p}}\right]\right]=\left[\left[{\cal
A}_{g_{1},n_{1}}\,,\ldots\,,{\cal A}_{g_{r},n_{r}},{\cal
B}_{g^{\prime}_{1},n^{\prime}_{1}}\,,\ldots\,,{\cal
B}_{g^{\prime}_{p},n^{\prime}_{p}}\right]\right]\,.$ (7.9)

This product is graded commutative, as expected, and is also manifestly
associative. The unit element of this algebra are surfaces without any
connected components, and thus also without any punctures.

To complete the definition of the BV structure we need to define the $\Delta$
operator. For this we must make use of twist gluing introduced in (4.10).
Consider two punctures $P_{i}$ and $P_{j}$ on a generalized surface; the
punctures may lie on the same surface or on different surfaces. Let $w_{i}$
and $w_{j}$ denote the local coordinates around the punctures. Twist gluing of
these two punctures means gluing the two punctures via the relation
$w_{i}w_{j}=e^{i\theta}$, with $\theta\in[0,2\pi]$. This operation reduces the
number of punctures by two and increases the dimensionality by one, by adding
the circle associated with the various values of $\theta$. Since we consider
the collection of all surfaces labelled by $\theta$, this definition is
insensitive to the choice of the phases of the local coordinates $w_{i}$,
$w_{j}$ and makes sense as operations on the elements of $\widehat{\cal
P}_{g,n}$.

We now define an operator $\Delta_{ij}$ with $i\not=j$ that acts on a product
space ${\cal A}={\cal A}_{1}\times\cdots\times{\cal A}_{r}$. We let
$\Delta_{ij}{\cal A}$ equal $\tfrac{1}{2}$ times the set of surfaces obtained
by twist gluing the punctures $P_{i}$ and $P_{j}$ for all the surfaces in
${\cal A}$. When the punctures lie on the same surface, the number of
connected components of the generalized surface does not change, but the genus
of the surface in question increases by one unit. When the punctures lie on
two different surfaces, these are joined by the gluing, and the number of
connected components decreases by one. In either case the dimensionality of
the space increases by one unit, with the twist angle parametrizing the new
dimension. If the number of punctures in ${\cal A}$ is one or less, the action
of $\Delta_{ij}$ gives zero. The orientation of $\Delta_{ij}A$ is defined by
the ordered set of tangent vectors $[{\partial\over\partial\theta},\\{{\cal
A}\\}]$, where $\\{{\cal A}\\}$ is the ordered collection of tangent vectors
that define the orientation of ${\cal A}$.

More generally, we can now define the $\Delta$ operator as follows using the
definition of the elements of ${\cal C}$. For any $X=\left[\left[{\cal
A}_{n_{1},g_{1}},\cdots,{\cal A}_{g_{r},n_{r}}\right]\right]$ we
define202020The $\Delta$ operator here is what we called $\Delta_{c}$ in
section 4.1. Since at this stage we are dealing exclusively with closed
strings, we drop the subscript $c$ to avoid cluttering. Later when we consider
open closed string field theory, we shall have both $\Delta_{c}$ and
$\Delta_{o}$.

$\Delta X\equiv\Delta_{ij}X\equiv{1\over n_{1}!\cdots n_{r}!}\sum_{\sigma\in
S_{N}}\Delta_{ij}{\bf P}_{\sigma}({\cal
A}_{n_{1},g_{1}}\times\cdots\times{\cal A}_{g_{r},n_{r}})\,,$ (7.10)

with $i\not=j$ and $1\leq i,j\leq N$. This definition is independent of the
choice of $i$ and $j$ due to the symmetry of the space $X$.

Let us now understand why the above-defined $\Delta$ operation satisfies
$\Delta(\Delta X)=0$. This is actually a matter of signs, and can be
understood by considering a generalized surface $X$ with four or more
punctures. Indeed, for three or less punctures the first $\Delta$ reduces the
number of punctures to one or less, and the second $\Delta$ simply gives zero.
For four or more punctures, we will evaluate $\Delta^{2}X$ in two ways, as
$\Delta_{12}\Delta_{12}X$ and as $\Delta_{12}\Delta_{34}X$. The independence
of $\Delta$ on the choice of punctures implies that the two evaluations should
be the same. But we will show that they actually differ by a sign.

In calculating with the first option $\Delta_{12}\Delta_{12}X$, consider
acting on an element $\Sigma_{X}\in X$. We first twist glue punctures $P_{1}$
and $P_{2}$ of $\Sigma_{X}$, then relabel the punctures $(P_{3}\ldots P_{N})$
as $(P_{1}\ldots P_{N-2})$ and twist glue the new $P_{1}$ and $P_{2}$
punctures. Effectively the second gluing operation is joining the original
$P_{3}$ and $P_{4}$ punctures. Therefore, the orientation of the space
$\Delta_{12}\Delta_{12}X$ at the subspace $\Delta_{12}(\Delta_{12}\Sigma_{X})$
will contain the tangent vectors

$\Delta_{12}(\Delta_{12}\Sigma_{X})\ \ \hbox{orientation}\ \
\Bigl{[}{\partial\over\partial\theta_{34}}\,,{\partial\over\partial\theta_{12}},\\{X\\}\Bigr{]}\,,$
(7.11)

where ${\partial\over\partial\theta_{12}}$ is the tangent vector associated
with the gluing of punctures one and two and
${\partial\over\partial\theta_{34}}$ is the tangent vector associated with the
gluing of punctures three and four. The ordering of the tangent vectors in the
above expression is fixed as shown.

In calculating with the second option $\Delta_{12}\Delta_{34}X$, we again
focus on the element $\Sigma_{X}\in X$. We first twist glue punctures $P_{3}$
and $P_{4}$ of $\Sigma_{X}$, and then twist glue the punctures $P_{1}$ and
$P_{2}$, which need no relabeling. This time we have

$\Delta_{12}(\Delta_{34}\Sigma_{X})\ \ \hbox{orientation}\ \
\Bigl{[}{\partial\over\partial\theta_{12}}\,,{\partial\over\partial\theta_{34}},\\{X\\}\Bigr{]}\,.$
(7.12)

We see that with this second option we have the opposite orientation as in the
first option. Given that the two ways of calculation $\Delta^{2}X$ give
answers that differ by a minus sign, we conclude that $\Delta^{2}X=0$.

The final property of $\Delta$ that must be shown is that it is a second-order
derivation of the dot product, as indicated in (7.2). The verification of this
relation is a matter of keeping track of the types of surfaces produced by the
various terms. For example, we write
$\Delta(XYZ)=R_{XX}+R_{YY}+R_{ZZ}+R_{XY}+R_{XZ}+R_{YZ}$, where $R_{XX}$, for
example denotes surfaces where the two punctures glued are in $X$, and
$R_{XY}$ denotes surfaces where one puncture is in $X$ and one is in $Y$. By
tracking surfaces for each term on the right-hand side one can prove that the
equation holds. This requires some care with normalization factors and signs
(see [102])

We now turn to the antibracket that as written in (7.3) is obtained as the
failure of $\Delta$ to be a derivation of the dot product.

$\\{X,Y\\}\equiv(-1)^{X}\Delta(XY)-(-1)^{X}(\Delta X)Y-X\Delta Y\,.$ (7.13)

For the first term on the right-hand side we write

$(-1)^{X}\Delta(XY)=S_{XX}+S_{YY}+S_{XY}\,,$ (7.14)

where $S_{XX}$ are the surfaces where two punctures in $X$ are glued, $S_{YY}$
are the surfaces where two punctures in $Y$ are glued, and $S_{XY}$ are the
surfaces where the punctures glued are one in $X$ and one in $Y$. All these
surfaces have the orientation
$(-1)^{X}\\{\partial/\partial\theta,[X],[Y]\\}=\\{[X],\partial/\partial\theta,[Y]\\}$.
It also becomes clear that for the other two terms on the right-hand side

$(-1)^{X}(\Delta X)Y=S_{XX}\,,\ \ \ X\Delta Y=S_{YY}\,,$ (7.15)

with the sign factors manifestly coming from the orientations. We thus see
that $\\{X,Y\\}=S_{XY}$ with orientation
$\\{[X],\partial/\partial\theta,[Y]\\}$; the antibracket glues two punctures,
one on the first entry and one on the second entry. In summary, if
$\Sigma_{X}$ denotes an element of $X$ and $\Sigma_{Y}$ denotes an element of
$Y$, then $\\{X,Y\\}$ consists of surfaces where one puncture of $\Sigma_{X}$
is glued to one puncture of $\Sigma_{Y}$, and the final punctured surface is
symmetrized on all the remaining punctures. It will often be the case that $X$
and $Y$ are just spaces with one connected component, and thus $\Sigma_{X}$
and $\Sigma_{Y}$ are just each a Riemann surface with punctures, as opposed to
a generalized surface.

Beyond the dot product and the $\Delta$ operator, we also have a boundary
operator $\partial$ acting on ${\cal C}$. For any $X\in{\cal C}$ we write
$\partial X$ for the boundary of $X$. The boundary operator acting on a space
of surfaces simply takes the boundary of the space, giving us another space of
surfaces, with an orientation induced by the orientation of the original
space212121Given a point $p\in{\cal A}$, a set of basis vectors
$[v_{1},\ldots,v_{k}]$ in $T_{p}(\partial A)$ defines the orientation of
$\partial A$ if $[n,v_{1},\ldots,v_{k}]$, with $n$ an outward pointing basis
vector of $T_{p}{\cal A}$, is the orientation of ${\cal A}$ at $p$. . Since
$\partial$ decreases the dimensionality of a space by one unit, it changes
degree. The operator $\partial$ is an odd derivation of the dot product

$\partial(XY)=(\partial X)Y+(-1)^{X}X\partial Y\,.$ (7.16)

Moreover $\partial$ anticommutes with $\Delta$, as one can see by tracking the
ordering of vectors defining the orientations;

$\Delta\partial X=-\partial\Delta X\,.$ (7.17)

Finally, using the definition of the antibracket from the $\Delta$ operator,
one can quickly check that $\partial$ is an odd derivation of the antibracket:

$\partial\\{X,Y\\}=\\{\partial X,Y\\}+(-1)^{X+1}\\{X,\partial Y\\}\,.$ (7.18)

The sign factor in the second term a consequence of the orientation
$\\{[X],\partial/\partial\theta,[Y]\\}$ of $\\{X,Y\\}$, and to reach $Y$,
$\partial$ goes ‘across’ the $X$ space and the gluing operation.

###  7.2 BV structures on CFT

The central point to be discussed here, focused on bosonic closed string field
theory, is that for any matter CFT of $c=26$ coupled to the reparameterization
ghosts CFT, we can represent the BV structures discussed above on the space of
functions of string fields. We call ${\cal H}$ the state space of this
complete CFT.

As discussed in section 3.2 this conformal field theory supplies string field
valued forms that can be integrated over moduli spaces of Riemann surfaces.
More precisely we have objects

$\langle\Omega^{(g,n)}_{k}|\in({\cal H}^{*})^{\otimes n}$ (7.19)

related to $\Omega^{(g,n)}_{k}(A_{1},\cdots,A_{n})$ introduced in (3.42) via

$\langle\Omega^{(g,n)}_{k}|A_{1}\rangle\otimes\cdots\otimes|A_{n}\rangle=\Omega^{(g,n)}_{k}(A_{1},\cdots,A_{n})\,,$
(7.20)

that are differential forms of degree $k$ in $\widehat{\cal P}_{g,n}$. These
are suitable for integration over $k$ dimensional subspaces ${\cal
A}^{(k)}_{g,n}$ of $\widehat{\cal P}_{g,n}$. The forms above satisfy a very
nontrivial property involving the BRST operator and the boundary operator on
moduli spaces. We have

$\int_{{\cal
A}_{g,n}}\langle{\Omega^{(g,n}_{k}}|\sum_{i=1}^{n}Q^{(i)}=(-1)^{k}\int_{\partial{\cal
A}_{g,n}}\langle{\Omega^{(g,n)}_{k-1}}|\,.$ (7.21)

Suppose we have a symmetric space ${\cal A}^{(k)}_{g,n}$, we then define

$f\bigl{(}{\cal A}^{(k)}_{g,n}\bigr{)}\equiv{1\over n!}\int_{{\cal
A}^{(k)}_{g,n}}\langle\Omega^{(g,n)}_{k}|\Psi\rangle_{1}\cdots|\Psi\rangle_{n}\,.$
(7.22)

More generally we have

$f\bigl{(}\left[\left[{\cal A}^{(k_{1})}_{g_{1},n_{1}}\,\cdots\,{\cal
A}^{(k_{r})}_{g_{r},n_{r}}\right]\right]\bigr{)}\equiv\prod_{i=1}^{r}{1\over
n_{i}!}\int_{{\cal
A}^{(k_{i})}_{g_{i},n_{i}}}\langle\Omega^{(g_{i},n_{i})}_{k_{i}}|\Psi\rangle_{1}\cdots|\Psi\rangle_{n_{i}}\,.$
(7.23)

This definition makes it clear that

$f(XY)=f(X)f(Y)\,,\ \ \ X,Y\in{\cal C}\,.$ (7.24)

The operations $\Delta$ and $\\{\cdot\,,\cdot\\}$ on string functionals are
defined as follows. We first define BPZ-implementing states $|R_{12}\rangle$
and $\langle R_{12}|$ via the relations

$~{}_{1}\langle A|~{}_{2}\langle B|R_{12}\rangle=\langle
A|B\rangle,\qquad\langle R_{12}|A\rangle_{1}|B\rangle_{2}=\langle
A|B\rangle\,,$ (7.25)

for any pair of states $|{A}\rangle$, $|{B}\rangle$ in the full CFT state
space ${\cal H}^{\prime}$. We than define symplectic forms $\langle{\omega}|$
and $|{S}\rangle$ as follows222222In [6, 75, 79], the states
$|R^{\prime}_{12}\rangle$ and $\langle R^{\prime}_{12}|$ defined as
projections of $|R_{12}\rangle$ and $\langle R_{12}|$ to the kernel of
$L_{0}^{-}$ were used in (7.26). This is not needed when the symplecting forms
act on states that are already projected, as is the case here.

$\begin{split}\langle{\omega_{12}}|=&\
\langle{R_{12}}|c_{0}^{-(2)}\,,\\\\[2.15277pt] |{S_{12}}\rangle=&\
b_{0}^{-(1)}|{R_{12}}\rangle\,.\end{split}$ (7.26)

With these structures we can now express the antibracket of two string
functions $F$ and $G$, defined in (4.20) and (4.21), as

$\\{F,G\\}=(-1)^{G+1}{\partial F\over\partial|{\Psi}\rangle}{\partial
G\over\partial|{\Psi}\rangle}|{S}\rangle$ (7.27)

Here the ket $|{S}\rangle$ glues the two state spaces left open by the
differentiation with respect to the string field. They are $\langle
F_{R}|c_{0}^{-}$ and $\langle G_{R}|c_{0}^{-}$ in the notation of (4.20).
There is no need to specify left and right derivatives because the string
field is even. The symplectic form can be used to write the kinetic term
$S_{0,2}$ of the string field theory.

$S_{0,2}=\tfrac{1}{2}\langle{\omega_{12}}|Q^{(2)}|{\Psi}\rangle_{1}|{\Psi}\rangle_{2}\,.$
(7.28)

For the $\Delta$ action on functions we have

$\Delta
F=\tfrac{1}{2}(-1)^{F+1}\Bigl{(}{\partial\over\partial|{\Psi}\rangle}{\partial\over\partial|{\Psi}\rangle}F\Bigr{)}|{S}\rangle\,.$
(7.29)

We now have a wonderful interplay in which[102]

$f(\Delta X)=-\Delta f(X)\,,$ (7.30)

where the $\Delta$ operation on the left-hand side acts on a space of surfaces
and the $\Delta$ on the right-hand side is that acting on the space of string
field functions. It now follows from the definition of the antibracket in
terms of the $\Delta$ operators that

$f\bigl{(}\\{X,Y\\}\bigr{)}=-\\{f(X),f(Y)\\}\,,$ (7.31)

where the antibracket on the right-hand side is that of string functionals.

We also have that the boundary operator is implemented by the antibracket with
the kinetic term of the string field theory:

$\\{S_{0,2},f(X)\\}=-f(\partial X)\,.$ (7.32)

###  7.3 Geometric BV master equation and string field theory master equation

The string field theory vertices are, geometrically, collections of Riemann
surfaces. The three-string vertex ${\cal V}_{0,3}$ of the classical theory is
a Riemann sphere with three punctures and local coordinates, up to phases, at
each of the punctures. The four-string vertex ${\cal V}_{0,4}$ consists of a
subset of $\widehat{\cal P}_{0,4}$, namely a collection of four punctured
spheres, each with local coordinates, up to phases, at each of the punctures.
In general, for the full quantum theory we have vertices ${\cal
V}_{g,n}\subset\widehat{\cal P}_{g,n}$, for various values of the genus $g$
and the number of punctures $n$. We formally sum these sets of surfaces to
form the full string vertex ${\cal V}$ defined as follows:

${\cal V}=\sum_{g,n}\,g_{s}^{2g+n-2}\,{\cal
V}_{g,n}\,,\quad\hbox{with}\quad\begin{cases}n\geq 3\,,\ \hbox{for}\
\,g=0\,,\\\ n\geq 1\,,\ \hbox{for}\ \,g=1\,,\\\ n\geq 0\,,\ \hbox{for}\
\,g\geq 2\,.\end{cases}$ (7.33)

The $g=0$ vertices, belonging to the classical theory, begin with three
punctures. The string vertex does not contain Riemann spheres with two or less
punctures. The $g=1$ vertices, associated with the leading quantum correction
to the action, does not contain the surfaces with zero punctures (the plain
un-punctured torus). For $g\geq 2$ surfaces with all numbers of punctures,
including zero, appear, although the surfaces with no punctures give a
constant contributions to the action. As such, they are not relevant in the
verification that the action satisfies the master equation. These constant
terms, however, are required for the string action to be background
independent. Recalling that the Euler number of genus $g$ surfaces with $n$
punctures (viewed as boundaries) is $\chi_{g,n}=2-2g-n$, we see that the
moduli spaces enter ${\cal V}$ weighted by the factor $(g_{s})^{-\chi_{g,n}}$,
just as multilinear functions were built in (4.11). Interestingly, the list of
vertices above is precisely that for which the surfaces have negative Euler
number and thus admit hyperbolic metrics of constant negative curvature. The
string vertices are symmetric under the exchange of punctures, a simple
condition discussed in the context of our discussion of the BV algebra on
moduli spaces.

The ${\cal V}_{g,n}$ are often taken to be pieces of sections on the bundle
$\widehat{\cal P}_{g,n}$ over $\mathcal{M}_{g,n}$. This condition, if
satisfied, makes the construction more canonical, as each underlying Riemann
surface in the vertex is constructed once and only once. Examples where this
is achieved are the constructions of vertices from minimal area metrics, and
the construction using hyperbolic metrics. It is clear, however, that the
consistency of the string field theory is satisfied by vertices that are more
general and are not strict sections. All that is needed is that ${\cal
V}_{g,n}$ maps onto its image under the projection $\pi:\widehat{\cal
P}_{g,n}\to\mathcal{M}_{g,n}$ with degree one. As explained in section 3.2 for
the case of the full subspace ${\cal F}_{g,n}$, this means that a generic
surface is counted once _with multiplicity_. On the left of Fig. 4 we show a
string vertex ${\cal V}_{g,n}$ that is a piece of a section over
$\mathcal{M}_{g,n}$. On the right we have a string vertex ${\cal V}_{g,n}$
that is not a piece of a section over $\mathcal{M}_{g,n}$. In both cases the
projection $\pi$ maps ${\cal V}_{g,n}$ to its image with degree one. As
discussed in section 4, ${\cal V}_{g,n}$ is part of ${\cal F}_{g,n}$, it
consists of Riemann surfaces that are not produced by the Feynman diagrams
with one or more propagators.

Figure 4: A string vertex ${\cal V}_{g,n}$ that is a piece of a section over
$\mathcal{M}_{g,n}$ (left), and a string vertex ${\cal V}_{g,n}$ that is not a
piece of a section over $\mathcal{M}_{g,n}$ (right). In both cases the
projection $\pi$ maps ${\cal V}_{g,n}$ to its image in $\mathcal{M}_{g,n}$
with degree one.

The string vertices, or just simply ${\cal V}$ must satisfy a consistency
condition for the resulting string field theory to be consistent. This is a
geometric version of the BV master equation and it takes the form

$\partial{\cal V}+\Delta{\cal V}+\tfrac{1}{2}\\{{\cal V}\,,{\cal V}\\}=0\,,$
(7.34)

We have discussed the objects in this equation before. We have an antibracket
and a $\Delta$ operator in the space of surfaces, and the boundary operator
$\partial$.

We have all the ingredients to show that the above geometric BV equation
implies that the string field theory action $S$ constructed using the vertex
${\cal V}$ satisfies the BV master equation in the space of string fields. The
action is easily written down using the function $f$ of string field
constructed in the previous section using spaces of Riemann surfaces. We take

$S=S_{0,2}+f({\cal V})\,.$ (7.35)

To verify the master equation we first note that the kinetic term $S_{0,2}$
satisfies a couple of simple properties:

$\\{S_{0,2},S_{0,2}\\}=0\,,\ \ \ \Delta S_{0,2}=0\,.$ (7.36)

The first follows by direct computation (see, [6]) and the second because the
kinetic term couples vertex operators whose ghost numbers add up to four, and
those cannot be paired to a field and its associated antifield – in such pairs
the ghost numbers of the vertex operators add up to five. With these results,
the verification of the master equation reduces to

$\begin{split}0=\ &\ \tfrac{1}{2}\\{S,S\\}+\Delta S\\\ =\ &\
\\{S_{0,2},f({\cal V})\\}+\tfrac{1}{2}\\{f({\cal V})\,,f({\cal V})\\}+\Delta
f({\cal V})\\\ =\ &-f(\partial{\cal V})-\tfrac{1}{2}f\bigl{(}\\{{\cal
V}\,,{\cal V}\\}\bigr{)}-f(\Delta{\cal V})\\\ =\ &-f\Bigl{(}\partial{\cal
V}+\tfrac{1}{2}\\{{\cal V}\,,{\cal V}\\}+\Delta{\cal
V}\Bigr{)}=0\,,\end{split}$ (7.37)

which is thus shown to hold on account of the geometrical BV equation
satisfied by ${\cal V}$.

While having the string field action satisfy the BV master equation is the
main reason for the geometrical BV equation, there are other aspects to this
geometrical equation that are also quite important. In particular, focusing on
the boundary $\partial{\cal V}_{g,n}$ we have the pictorial representation of
the master equation

The first term on the right-hand side shows the gluing pattern of the
antibracket, joining two lower vertices whose surfaces, upon gluing, give
surfaces of genus $g$ and with $n$ punctures, the same as the surface on the
left-hand side. The second term shows the gluing pattern of the $\Delta$
operator acting on single surfaces. Since this operation increases the genus
and reduces the number of punctures by two, the surfaces it acts upon belong
to ${\cal V}_{g-1,n+2}$. The sets of surfaces on both sides of the equation
must be the same as elements of $\widehat{\cal P}_{g,n}$. As an equation the
above reads

$\partial{\cal V}_{g,n}=-\Delta{\cal V}_{g-1,n+2}\ -\
\tfrac{1}{2}\hskip-10.0pt\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\
n_{1}+n_{2}=n+2\end{subarray}}\hskip-5.0pt\\{{\cal V}_{g_{1},n_{1}},{\cal
V}_{g_{2},n_{2}}\\}\ \equiv{\cal O}_{g,n}\,.$ (7.38)

Note that the twist gluing operation of local coordinates $w_{1},w_{2}$ in
$w_{1}w_{2}=t$, with $|t|=1$ is a boundary of the gluing operation
$w_{1}w_{2}=t$ with $|t|\leq 1$. The latter creates a set of surfaces that is
associated to a propagator joining the legs of the vertex. In this picture,
the twist-gluing set represents the collapsed propagator, a propagator that
has zero length. The $t=0$ surface is a degenerate surface, and corresponds to
the limit of an infinitely long propagator. The relation displayed above tells
us that the boundary of the vertex coincides with the collapsed propagator
boundary of the Feynman diagrams built just with one propagator. We write this
as

$\partial{\cal V}_{g,n}=-\partial_{p}R_{1}\,,$ (7.39)

here $\partial_{p}$ denotes collapsed propagator boundary, and $R_{1}$ is the
set of surfaces built with lower vertices and just one propagator.

###  7.4 Deligne-Mumford compactification, noded surfaces and Feynman diagrams

It is useful to pause to describe explicitly the degenerate surfaces and their
role in the theory. Such surfaces, called noded surfaces, are the key element
in the Deligne-Mumford compactification $\overline{{\cal M}}_{g,n}$ of the
moduli space ${\cal M}_{g,n}$. A noded surface is a connected complex space
where points have neighborhoods complex isomorphic to $\\{|z|<1\\}$, in which
case they are regular points, or complex isomorphic to the set $\\{zw=0;\
|z|<1,\ |w|<1\\}$ in $\mathbb{C}^{2}$, in which case they are nodes. We
imagine a node as the single common point where two separate pieces of
surfaces touch. Each component of the complement of the nodes, called a part
of the noded surface, has negative Euler characteristic. A part can be a
three-punctured sphere (counting the node as a puncture) but not, for example,
a one or two punctured sphere. Depending on the number of nodes and their
configuration the surface can have one or more parts. If a surface has a
single node, it may have one or two parts, depending whether the degeneration
is separating or non-separating. It is also useful to describe the
neighborhoods of the noded surfaces within the compactified space, and so
specify the topology of the compactification. For this, we consider the case
of a noded surface $\Sigma_{0}$ with a single node, and explain what it means
to open up the node. Fix a complex number $t$ with $|t|<\epsilon$ small to
remove a neighborhood ${\cal U}$ of the node, ${\cal
U}=\\{|z|<|t|,\,|w|<|t|\\}$ and form the identification space
$\Sigma_{t}=(\Sigma_{0}-{\cal U})/(zw=t)$. This removes the node and connects
locally the pieces of surfaces that before just touched via a plumbing fixture
(note that the coordinates $z$ and $w$ must be defined all the way to $|z|=1$
and $|w|=1$). The surfaces near $\Sigma_{0}$ in the moduli space are: (a)
noded surfaces in which the parts are quasi-conformally close to the parts of
$\Sigma_{0}$, and (b) smooth surfaces with small $t\not=0$ obtained by opening
up the node in the surfaces in (a). With this compactification,
$\overline{{\cal M}}_{g,n}$ is now compact and boundaryless. Note that the
string field theory construction of surfaces with propagators and plumbing
fixtures $zw=t$ precisely describes the neigborhoods of the noded surfaces,
with the limit $t\to 0$ as the noded surface. The noded surfaces are viewed as
divisors in the moduli space.

The vertices in ${\cal V}$ must be sufficient to construct, with Feynman
rules, all the Riemann surfaces in the various moduli spaces – this is, after
all, the way string amplitudes must be generated. In general, after using
string vertices together with a propagator to form Feynman graphs of string
field theory, the result is some submanifold $\mathcal{F}_{g,n}$ in
$\widehat{\cal P}_{g,n}$. This is clear, because gluing of surfaces in
$\widehat{\cal P}_{g,n}$ gives surfaces in $\widehat{\cal P}_{g,n}$. It is
traditionally required that $\mathcal{F}_{g,n}$ should be a full section of
the bundle $\widehat{\cal P}_{g,n}$, so that the map
$\mathcal{F}_{g,n}\to\mathcal{M}_{g,n}$ is a homeomorphism. As will be
discussed in the following sections, this can actually be achieved with
minimal area string vertices, and almost surely with hyperbolic string
vertices. It is not necessary, however. All that is required is that the map
$\mathcal{F}_{g,n}\to\mathcal{M}_{g,n}$ be of degree one. Such a map is
surjective. This is what colloquially would be referred as “covering the
moduli space”. This is enough to imply that the integral over
$\mathcal{F}_{g,n}$ of any differential form pulled back from
$\mathcal{M}_{g,n}$ will be the same as the integral over $\mathcal{M}_{g,n}$.
Since the integrand for on-shell string states is a top-form on
$\mathcal{M}_{g,n}$, this implies that the amplitude for on-shell string
states computed by integrating over $\mathcal{M}_{g,n}$ coincides with that
computed using string field theory Feynman rules. More precisely, and for
clarity, we work with the compactified moduli space $\overline{{\cal
M}}_{g,n}$ and we also view $\mathcal{F}_{g,n}$ as a space without boundary.
However, as will be discussed shortly, and in more detail in §9, we shall
encounter several situations where the degenerate surfaces require special
treatment.

In string field theory we build $\mathcal{F}_{g,n}$ as follows

$\mathcal{F}_{g,n}={\cal V}_{g,n}\oplus R_{1}\oplus\cdots R_{3g-3+n}\,.$
(7.40)

Here $R_{I}$ is a the set of surfaces built using $I$ propagators. As
indicated above, the maximum number of propagators is $3g-3+n$. Since
$\mathcal{F}_{g,n}$ is boundaryless, a closed chain in mathematical language,
the space to the right must be boundaryless. There are two types of boundaries
that appear in this construction: (i) the boundaries of the vertices and, (ii)
the boundaries from surfaces where the length of a propagator is zero. When
the length of the propagators is infinite we approach the divisors; the noded
surfaces that are not counted as boundaries. The two kinds of boundaries
cancel each other by the geometric master equation. In fact, one can show that
the vertex boundaries of $R_{I}$ cancel against the propagator boundary of
$R_{I+1}$, this being an extension of (7.39). Note also that $R_{3g-3+n}$ has
no vertex boundary, as it is a region built by gluing a set of ${\cal
V}_{0,3}$ that have no moduli.

The infinite length propagators, leading to the noded surfaces require special
attention. If the momentum propagating along the propagator is generic and can
be varied continuously, then by choosing the momentum appropriately we can
make the forms $\Omega^{(g,n)}_{k}$ fall off sufficiently rapidly near these
regions. The amplitudes for other values of momenta for which
$\Omega^{(g,n)}_{k}$ does not fall off near these regions can be defined by
analytic continuation. A systematic procedure of carrying out this analytic
continuation was described in [37]. There are however cases where the momentum
along the propagator is not generic, e.g. when it is forced to be either zero
or on-shell due to momentum conservation. In such cases the contribution from
these regions requires special attention. This will be the subject of our
discussion in section 9. In this section we shall proceed by assuming that
that $\Omega^{(g,n)}_{k}$ falls off sufficiently fast near these regions.

It is useful to consider string vertices ${\cal V}_{g,n}$ that are not
submanifolds $\widehat{\cal P}_{g,n}$, but rather, more general singular
chains of degree $6g-6+2n$. By using singular chains, the space ${\cal
V}_{g,n}$ could be the formal sum of several disjoint spaces, or could have
self-intersections, for example. Since the purpose of introducing string
vertices is to integrate over them, and there is a well-behaved theory of
integration of differential forms on singular chains, these kinds of vertices
are allowed. As we will see, due to complications with picture-changing
operators in superstring field theory, the string vertices are often singular
chains. In such cases $\mathcal{F}_{g,n}$ is also a singular chain. Because of
the symmetry under permutation of the punctures, it is an $S_{n}$ invariant
chain.

The precise statement of covering of moduli space that leads to correct on-
shell string amplitudes is that the chain $\mathcal{F}_{g,n}$, constructed as
Feynman diagrams using ${\cal V}$ and a propagator, when projected via the
forgetting of the coordinates, must represent the fundamental homology class
of $\overline{\mathcal{M}}_{g,n}$. As discussed in [34], this constraint will
be satisfied if the chain ${\cal V}$ satisfies the geometric master equation
(7.34).

In solving the geometrical master equations to find ${\cal V}$ the starting
point is the vertex ${\cal V}_{0,3}$, the three-punctured sphere. Since all
three punctured spheres are conformally equivalent, we just need an assignment
of local coordinates at the punctures respecting the symmetry under exchange
of punctures. The simplest cases from the geometrical master equation arise
from its constraints on vertices of complex dimension one: these are ${\cal
V}_{0,4}$ and ${\cal V}_{1,1}$ We have

$\partial{\cal V}_{0,4}=-\tfrac{1}{2}\\{{\cal V}_{0,3},{\cal V}_{0,3}\\}\,,\ \
\ \hbox{and}\ \ \ \partial{\cal V}_{1,1}=-\Delta{\cal V}_{0,3}\,.$ (7.41)

The next step is to find the sets ${\cal V}_{0,4}$ and ${\cal V}_{1,1}$ that
satisfy the above conditions. As we will discuss concretely, this is always
possible. One includes in the sets all the surfaces that lie in the interior
of the moduli spaces up to the prescribed boundaries. On these surfaces the
local coordinates must be chosen as known on the boundaries and continuously
within the set.

###  7.5 Existence of string vertices

While we need explicit solutions of the geometric BV equations in order to do
off-shell computations in string field theory, a more basic question is
whether these exists a solution to these equations. This was answered in the
affirmative by Costello[74], with the full details of this argument given in
[34]. We briefly sketch the main points of the inductive argument.

Suppose we have found all the vertices ${\cal V}_{g^{\prime},n^{\prime}}$ such
that $3g^{\prime}-3+n^{\prime}<3g-3+n$. At this point we need to find ${\cal
V}_{g,n}$ satisfying

$\partial{\cal V}_{g,n}={\cal O}_{g,n}\,.$ (7.42)

The right hand side ${\cal O}_{g,n}$ of this equation was defined in (7.38),
and it is $(6g-6+2n)-1$ chain on $\widehat{\cal P}_{g,n}$. It is built
entirely from ${\cal V}_{g^{\prime},n^{\prime}}$ which we have already
constructed. By construction, ${\cal O}_{g,n}$ is an $S_{n}$ invariant chain,
we require ${\cal V}_{g,n}$ to be $S_{n}$ invariant as well. It is a simple
calculation to show that $\partial{\cal O}_{g,n}=0$; it only requires using
the master equation for the spaces we have already constructed. This is a
consistency condition for the above equation: the chain ${\cal O}_{g,n}$ must
be closed. But this does not suffice, the chain should be trivial so that we
can solve for ${\cal V}_{g,n}$. In other words the appropriate homology class
of ${\cal O}_{g,n}$ must vanish:

$[{\cal O}_{g,n}]\in H_{6g-6+2n-1}(\widehat{\cal P}_{g,n})^{S_{n}}\ \
\hbox{must vanish}.$ (7.43)

The first step in proving this is to show that $\widehat{\cal P}_{g,n}$ is
homotopy equivalent to $\mathcal{M}_{g,n}$. This implies that the homology
groups of $\widehat{\cal P}_{g,n}$ and of $\mathcal{M}_{g,n}$ are isomorphic.
This step is somewhat technical. One first shows that $\widehat{\cal P}_{g,n}$
is isomorphic to a space $\widehat{\cal P}^{\prime}_{g,n}$ of Riemann surfaces
of genus $g$ with $n$ boundary components and an angular coordinate, defined
up to rotation, on each boundary. This latter space is actually known to be
homotopy equivalent of $\mathcal{M}_{g,n}$, establishing the result.

The second step consists in showing that
$H_{6g-6+2n-1}((\mathcal{M}_{g,n})^{S_{n}})$ is zero. In fact, a stronger
result is true: the homology $H_{6g-6+2n-1}(\mathcal{M}_{g,n})$ vanishes for
all spaces except for $(g,n)=(0,4)$. This is shown by identifying
$H_{6g-6+2n-1}(\mathcal{M}_{g,n})$, via Poincare duality, with the first
cohomology class
$H^{1}(\overline{\mathcal{M}}_{g,n},\overline{\mathcal{M}}_{g,n}/\mathcal{M}_{g,n})$
of the compactified moduli spaces, relative to the nodal surfaces in the
compactification. An exact sequence of relative cohomology groups leads to the
desired result. This means that the homology
$H_{6g-6+2n-1}((\mathcal{M}_{g,n})^{S_{n}})$ vanishes as well for these
spaces.

In the case of the four-punctured sphere $(g,n)=(0,4)$, the unrestricted one-
dimensional homology group $H_{1}(\mathcal{M}_{0,4})$ does not vanish, but the
$S_{4}$-invariant subgroup does. This is intuitively clear: the first homology
group of $\mathcal{M}_{0,4}$ is two dimensional, with representative cycles
where the last marked point $z_{4}$ moves around $0$ (the location of $z_{1}$)
or moves around $1$ (the location of $z_{2}$). This is clear: for each of
these cycles there is no region of the sphere without a puncture for which the
cycle is a boundary. There is no independent cycle where $z_{4}$ moves around
the point at $\infty$ (the location of $z_{3}$). On the other hand a
symmetrized cycle would have to be symmetric also under the $S_{3}$ subgroup
of $S_{4}$ that permutes the first three punctures. This cycle would require
three closed curves: one around $0$, one around $1$, and one around $\infty$.
They can be chosen not to intersect. It is clear then that there is a region
for which this cycle is a boundary, showing that the symmetric homology group
vanishes.

Similar results for open-closed bosonic string theory have been discussed in
[103] and in heterotic string theory and superstring theory have been
discussed in [104].

###  7.6 Uniqueness of string vertices up to canonical transformations

Solutions of the master equation $\Delta S+\tfrac{1}{2}\\{S,S\\}=0$ are not
unique. The infinitesimal canonical transformation

$\delta S=\Delta\epsilon+\\{S,\epsilon\\}\,,$ (7.44)

leaves the master equation unchanged to order $\epsilon$. This new action is
equivalent to the original one. We now show that solutions of the geometric BV
master equation are also not unique in a similar way. There is a notion of a
canonical transformation of the vertex ${\cal V}$ that generates a new vertex
that also satisfies the geometric master equation. This is relatively simple
to verify for infinitesimal canonical transformations, as we do now.

If we have some collection of $(6g-6+2n+1)$-chains $\mathcal{W}_{g,n}$ of
$\widehat{\cal P}_{g,n}$, then we can vary the string vertices ${\cal
V}_{g,n}$ by

$\delta_{\mathcal{W}}{\cal V}=\\{{\cal
V}\,,\mathcal{W}\\}+\Delta\mathcal{W}+\partial\mathcal{W}\,,$ (7.45)

where, as before, ${\cal V}=\sum g_{s}^{2g+n-2}{\cal V}_{g,n}$,
$\mathcal{W}=\sum g_{s}^{2g-n+2}\mathcal{W}_{g,n}$. We claim that the new
vertices, obtained by the canonical transformation,

${\cal V}+\delta_{\mathcal{W}}{\cal V}$ (7.46)

satisfy the quantum master equation, to leading order in
$\delta_{\mathcal{W}}{\cal V}$:232323Useful identities $\
\Delta\\{X,Y\\}=\\{\Delta X,Y\\}+(-1)^{X+1}\\{X,\Delta Y\\}$ and
$(-1)^{(X_{1}+1)(X_{3}+1)}\bigl{\\{}\\{X_{1},X_{2}\\},X_{3}\bigr{\\}}+\hbox{cyclic}=0$.
All $\partial,\Delta$, and $\\{\cdot,\cdot\\}$ change degree by one unit.

$\begin{split}&\tfrac{1}{2}\\{{\cal V}+\delta_{\mathcal{W}}{\cal V},{\cal
V}+\delta_{\mathcal{W}}{\cal V}\\}+\Delta({\cal V}+\delta_{\mathcal{W}}{\cal
V})+\partial({\cal V}+\delta_{\mathcal{W}}{\cal V})\\\
&=\bigl{\\{}\partial{\cal V}+\Delta{\cal V}+\tfrac{1}{2}\\{{\cal V}\,,{\cal
V}\\}\,,\mathcal{W}\bigr{\\}}+O((\delta_{\mathcal{W}}{\cal
V})^{2})=0\,.\end{split}$ (7.47)

Two sets of string vertices which are related by geometric canonical
transformations like this yield string field actions related by canonical
transformation. Indeed, recalling that $S_{\cal V}=S_{0,2}+f({\cal V})$ , it
now follows from that

$\begin{split}S_{{\cal V}+\delta_{\mathcal{W}}{\cal V}}=&\ S_{\cal
V}+f({\delta_{\mathcal{W}}{\cal V}})=S_{\cal V}+f\bigl{(}\\{{\cal
V}\,,\mathcal{W}\\}+\Delta\mathcal{W}+\partial\mathcal{W}\bigr{)}\\\ =&\
S_{\cal V}-\\{f({\cal V})\,,f(\mathcal{W})\\}-\Delta
f(\mathcal{W})-\\{S_{0,2},\mathcal{W}\\}\\\ =&\ S_{\cal V}-\\{S_{\cal
V}\,,f(\mathcal{W})\\}-\Delta f(\mathcal{W})\,.\end{split}$ (7.48)

We see that the variation of the action fits the structure shown in (7.44),
with $\epsilon=-f(\mathcal{W})$, showing that the action for the new vertices
is obtained by an infinitesimal canonical transformation of the original
action.

The converse of the above results are important. We showed that an
infinitesimal canonical transformation of ${\cal V}$ yields a vertex that also
satisfies the master equation. It is possible to show that given any two
solutions of ${\cal V}$ and ${\cal V}^{\prime}$ of the BV master equation, one
can construct a large canonical transformation that relates the two [34]. The
setup for the proof is as follows.

As a first step, we define ‘large’ canonical transformations as some kind of
exponential of small ones. Let ${\cal V}$ be a set of string vertices, and
$\mathcal{W}$ a collection $\mathcal{W}_{g,n}$ of $(6g-6+2n+1)$-dimensional
$S_{n}$-invariant singular chains in $\widehat{\cal P}_{g,n}$ which define an
infinitesimal canonical transformation. We define a family ${\cal V}(t)$ of
string vertices by requiring ${\cal V}(0)={\cal V}$ and demanding that they
satisfy the differential equation

$\frac{\mathrm{d}}{\mathrm{d}t}{\cal V}(t)=\delta_{\mathcal{W}}{\cal V}(t)\,,$
(7.49)

where $\delta_{\mathcal{W}}{\cal V}$ was defined above. If ${\cal V}$
satisfies the master equation, so does ${\cal V}(t)$ for all $t$. To show
this, we write the master equation for ${\cal V}(t)$ in the form $M_{\cal
V}(t)=0$ by defining

$M_{\cal V}(t)\equiv\partial{\cal V}(t)+\Delta{\cal
V}(t)+\tfrac{1}{2}\left\\{{\cal V}(t),{\cal V}(t)\right\\}\,.$ (7.50)

Clearly $M_{\cal V}(0)=0$ since ${\cal V}$ satisfies the master equation. A
short calculation using the differential equation (7.49) shows that

$\frac{\mathrm{d}M_{{\cal V}}}{\mathrm{d}t}=\\{M_{{\cal
V}}(t)\,,\mathcal{W}\\}\,.$ (7.51)

This equation implies that if $M_{\cal V}(0)=0$ all derivatives of $M_{\cal
V}(t)$ will vanish at $t=0$. This shows $M_{\cal V}(t)$ vanishes at all $t$.
We write the solution of (7.49) for the instantaneous vertices by taking
multiple derivatives, evaluating at $t=0$, and writing the Taylor series. One
finds

${\cal V}(t)={\cal V}+t\delta_{\mathcal{W}}{\cal
V}+\tfrac{1}{2}t^{2}\\{\delta_{\mathcal{W}}{\cal
V},\mathcal{W}\\}+\tfrac{1}{3!}t^{3}\bigl{\\{}\\{\delta_{\mathcal{W}}{\cal
V},\mathcal{W}\\},\mathcal{W}\bigr{\\}}+\cdots\ .$ (7.52)

We define the exponential of the canonical transformation via
$\exp(t\delta_{\mathcal{W}}){\cal V}\equiv{\cal V}(t)$. Note that the series
solution does not fit the naive expansion of the exponential; we do not
encounter nor define iterated variations
$\delta_{\mathcal{W}}\delta_{\mathcal{W}}{\cal V}$. The definition implies
that

$\exp(\delta_{\mathcal{W}}){\cal V}\equiv{\cal V}+\delta_{\mathcal{W}}{\cal
V}+\tfrac{1}{2}\\{\delta_{\mathcal{W}}{\cal
V},\mathcal{W}\\}+\tfrac{1}{3!}\bigl{\\{}\\{\delta_{\mathcal{W}}{\cal
V},\mathcal{W}\\},\mathcal{W}\bigr{\\}}+\cdots\ .$ (7.53)

Having defined large canonical transformations, the uniqueness of vertices is
argued as follows. Suppose that ${\cal V}_{g,n}$ and ${\cal V}^{\prime}_{g,n}$
are two sets of string vertices, both of which satisfy the master equation.
The goal is to show that there exists a sequence of $S_{n}$-invariant singular
chains $\mathcal{W}_{g,n}$ such that their exponentiation –a large canonical
transformation– relates the two sets of vertices

$\exp(\delta_{\mathcal{W}}){\cal V}={\cal V}^{\prime}.$ (7.54)

The construction of $\mathcal{W}$ is done by induction (for details see [34],
section 2.3). The initial step deals with the lowest genus zero vertex, the
three punctured sphere. Here ${\cal V}_{0,3}$ and ${\cal V}^{\prime}_{0,3}$
are both $S_{3}$ invariant points in $\widehat{\cal P}_{0,3}$. Since this
space is connected, we let $\mathcal{W}_{0,3}$ be a $S_{3}$-invariant path
connecting ${\cal V}_{0,3}$ to ${\cal V}^{\prime}_{0,3}$. Viewing
$\mathcal{W}_{0,3}$ as a one-chain, we have $\partial\mathcal{W}_{0,3}={\cal
V}^{\prime}_{0,3}-{\cal V}_{0,3}\,.$ This implies that we have satisfied
(7.54) to leading order. The full induction argument is a bit intricate and
uses the vanishing of the homology groups $H_{6g-6+2n}(\widehat{\cal
P}_{g,n})$ for all $(g,n)$ except $(0,3)$.

Figure 5: The boundary of a general open-closed vertex coincides with a closed
string $\Delta$ operator acting on a vertex, two types of open string $\Delta$
operators acting on vertices, and closed and open antibrackets of vertices.
Wavy lines represent gluing of closed string punctures, continuous lines
represent gluing of open string punctures. The heavy dots represent open
string boundaries, and external open and closed states are not shown.

###  7.7 Open-closed vertices and their main identity

The main geometric identity for open-closed string field theory was given in
(4.60). The pictorial representation is given in Figure 5, where, on the first
line we show the three ways in which two punctures on the same surface can be
glued together via the complete $\Delta$ operator. This includes $\Delta_{c}$
for closed strings, $\Delta_{o}^{\prime}$ for open strings on the same
boundary component, and $\Delta_{o}$ for open string on different boundary
components. The closed string anti-bracket and the open string anti-bracket
are on the second and third lines, respectively.

The various open-closed vertices are combined to form the open closed chain
${\cal V}$ as follows

${\cal V}=\sum_{g,b,n_{c},n_{o}}(g_{s})^{-\chi_{g,b,n_{c},n_{o}}}\,{\cal
V}_{g,b,n_{c},n_{o}}\,,$ (7.55)

with $\chi_{g,b,n_{c},n_{o}}=2-2g-n_{c}-b-\tfrac{1}{2}n_{o}$, the Euler number
of the surfaces in the associated vertex. The surfaces appearing in the above
sum are those for which the Euler number is negative, with the exception of
the disk with one closed string puncture and the annulus without punctures,
both of which have Euler number zero.242424As in the case of purely closed
string theory, the surfaces without punctures give constant contribution to
the string field theory action and do not affect any physical quantity
computed from the theory. Not included in ${\cal V}$ are therefore the spheres
$(g=b=0)$ with $n_{c}\leq 2$, the torus ($g=1,b=0$) with $n_{c}=0$, both of
which are not included in the pure closed string, and the disks
$(g=0,b=1,n_{c}=0)$ with $n_{o}\leq 2$. The above open-closed chain satisfies
the familiar geometric master equation,

$\partial{\cal V}+\Delta{\cal V}+\tfrac{1}{2}\\{{\cal V}\,,{\cal V}\\}=0\,,$
(7.56)

which is formally identical to the one considered for closed string theory.
Here, of course, $\Delta=\Delta_{c}+\Delta^{\prime}_{o}+\Delta_{o}$, as
discussed above, and for the anti-bracket we have both closed and open sector
contributions:
$\\{\cdot,\,\cdot\\}=\\{\cdot,\,\cdot\\}_{c}+\\{\cdot,\,\cdot\\}_{o}$.

In this open-closed main identity, we can identify subsets of vertices that
satisfy consistent sub-identities under the possible operations [9]. The first
two below are ‘obvious’, the others less so.

  1. 1.

Pure closed string theory. Setting $b=n_{o}=0$ in (7.55) we get a chain ${\cal
V}_{c}$ containing all closed string vertices that satisfies

$\partial{\cal V}_{c}+\Delta{\cal V}_{c}+\tfrac{1}{2}\\{{\cal V}_{c}\,,{\cal
V}_{c}\\}_{c}=0\,.$ (7.57)

Note that no anti-bracket or delta operation on surfaces with boundaries can
produce a surface without boundaries. The chain of classical closed string
vertices $\hat{\cal V}_{c}=\sum_{n=3}^{\infty}(g_{s})^{n-2}{\cal V}_{0,n,0,0}$
satisfies

$\partial\hat{\cal V}_{c}+\tfrac{1}{2}\\{\hat{\cal V}_{c}\,,\hat{\cal
V}_{c}\\}_{c}=0\,.$ (7.58)

  2. 2.

Pure classical open string theory. Setting $g=0,n_{c}=0$ and $b=1$ in (7.55)
we get a chain ${\cal V}_{o}$ that includes disks with three or more open
string punctures. Note that the $\Delta$ operator in the open string sector
cannot produce surfaces in this class: $\Delta_{o}^{\prime}$ increases the
number of boundaries and we cannot begin with $b=0$, and $\Delta_{o}$
increases genus, so the genus cannot remain zero. It follows that we have

$\partial{\cal V}_{o}+\tfrac{1}{2}\\{{\cal V}_{o}\,,{\cal V}_{o}\\}_{o}=0\,.$
(7.59)

  3. 3.

${\cal V}_{1}$ consisting of disks with $n_{c}=1$ and $n_{o}\geq 0$. Here
${\cal V}_{1}=\sum_{n_{o}=0}^{\infty}(g_{s})^{{1\over 2}n_{o}}\,{\cal
V}_{0,1,1,n_{o}}\,.$ Inspection of Figure 5 shows that the boundary of this
chain interacts with the classical open string chain ${\cal V}_{o}$ as
follows:

$\partial{\cal V}_{1}+\\{{\cal V}_{o}\,,{\cal V}_{1}\\}_{o}=0\,.$ (7.60)

  4. 4.

${\cal V}_{2}$ consisting of disks with $n_{c}=2$ and $n_{o}\geq 0$. Here
${\cal V}_{2}=\sum_{n_{o}=0}^{\infty}(g_{s})^{1+{1\over 2}n_{o}}\,{\cal
V}_{0,1,2,n_{o}}\,.$ Inspection of Figure 5 shows that the boundary of this
chain interacts with the classical open string chain ${\cal V}_{o}$, the chain
${\cal V}_{1}$ and the classical closed string three-vertex ${\cal V}_{0,3}$
as follows:

$\partial{\cal V}_{2}+\\{{\cal V}_{o}\,,{\cal
V}_{2}\\}_{o}+\tfrac{1}{2}\\{{\cal V}_{1}\,,{\cal V}_{1}\\}_{o}+\\{{\cal
V}_{1}\,,{\cal V}_{0,3}\\}_{c}\,=0\,.$ (7.61)

The recursion relations (7.60) and (7.61) can be used to discuss global
symmetries of classical open string field theory generated by closed string
states in the BRST cohomology [9]. An example is Poincare transformations.
Such symmetries were first noted and explored by Hata and Nojiri in
covariantized light-cone string field theories [105].

  5. 5.

Define now a chain $\bar{\cal V}$ of disks with all allowed numbers of open
and closed punctures: