$\bar{\cal V}\equiv{\cal V}_{o}+\sum_{n=1}^{\infty}{\cal V}_{n}\,,\ \ \ \
{\cal V}_{n}\equiv\sum_{n_{o}=0}^{\infty}(g_{s})^{n+{1\over 2}n_{o}-1}\,{\cal
V}_{0,1,n,n_{o}}\,.$ (7.62)

Note that the definition of ${\cal V}_{n}$ reproduces the earlier definitions
for $n=1,2$. This time we find that the boundary of this chain mixes with the
classical closed string chain $\hat{\cal V}_{c}$:

$\partial\bar{\cal V}+\tfrac{1}{2}\\{\bar{\cal V},\bar{\cal
V}\\}_{o}+\\{\bar{\cal V},\hat{\cal V}_{c}\\}_{c}=0\,,$ (7.63)

This relation allows the formulation of a classical open string field theory
in a nontrivial closed string background. The action is written with the
familiar kinetic term for the open string field plus all the interactions
implicit in $\bar{\cal V}$ with a closed string field $\Psi_{0}$. The action
has open string gauge invariance if $\Psi_{0}$ satisfies the classical closed
string equations of motion [9].

  6. 6.

A chain $\tilde{\cal V}$ that includes all genus zero surfaces with all
numbers of boundary components ($b=0,1,\cdots$) and all allowed numbers of
open and closed string punctures:

$\tilde{\cal V}=\sum_{n_{c},b,n_{o}}(g_{s})^{n_{c}+b+{1\over 2}n_{o}-2}\,{\cal
V}_{0,b,n_{c},n_{o}}\,.$ (7.64)

This chain, all by itself, satisfies a simple recursion relation:

$\partial\tilde{\cal V}+\tfrac{1}{2}\\{\tilde{\cal V}\,,\tilde{\cal
V}\\}+\Delta^{\prime}_{o}\tilde{\cal V}=0$ (7.65)

Here the antibracket is the full one, including the closed and open
contributions. Note the action of $\Delta^{\prime}_{o}$ in which open strings
on the same boundary component are glued together. The other operator
$\Delta_{o}$ does not feature as it increases the genus.

This recursion relation has been recently shown to be relevant to the
description of the physics of $N$ D-branes in the large $N$ limit [99, 106].

The recursive approach to the construction of string vertices described in the
existence proof in this section can be implemented for practical computations
involving low-dimensional moduli spaces. One particularly useful set of
vertices arises when the starting point, the vertices with no moduli, use
local coordinates at the punctures that are related to the global coordinates
on the sphere or the upper half plane via SL$(2,\mathbb{C})$ or
SL$(2,\mathbb{R})$ maps (projective maps). In this case the construction of
$\\{{\cal V},{\cal V}\\}$ and $\Delta{\cal V}$ can be done by solving a set of
algebraic equations (as opposed to differential equations for more general
maps) and we can systematically proceed to construct higher order vertices.
Examples of such constructions and their practical applications can be found
in [107, 108, 109, 82]. As of now, however, there is no uniform description of
all the ${\cal V}$’s with projective local coordinates even at genus zero. We
shall now describe a few approaches that give uniform description of all the
${\cal V}$’s.

###  7.8 Minimal area string vertices: Witten vertex and closed string
polyhedra

We now turn to the construction of string vertices. In this subsection we
shall describe the construction based on the minimal area metric, focussing on
the bosonic string theory. The most elementary vertex is the tree-level three
point function. For open strings the vertex is a disk with three punctures on
the boundary. It is instructive to compare this with closed strings for which
the vertex is a sphere with three punctures. For the vertices that will be
constructed in this subsection, the two are related. If we view the closed
string vertex as a surface with three punctures located on the equatorial
circle, the open string vertex can be defined by cutting the surface along
that circle and keeping just, say, the northern hemisphere. This is now a disk
with three boundary punctures. The local coordinates at the punctures are
inherited from the local coordinates at the punctures of the closed string
vertex.

We begin with the simplest case: bosonic open string field theory. The vertex
constructed by Witten is a special case of the general class of the three open
string vertices that we shall consider. As remarked earlier, Witten’s open
string field theory is cubic, i.e. this theory only needs a three-string
vertex satisfying an associativity condition. We shall first describe this
construction. The surface can be built by gluing together three half disks –
the coordinate half-disks, or patches. These half disks $\\{|w_{i}|\leq
1,\hbox{Im}\,w_{i}\geq 0\\}$, with $i=1,2,3$, can be thought as the world-
sheets of the three strings and are shown in the figure below. The points
$w_{i}=0$ are the punctures.

Figure 6: Three half disks representing open string world-sheets. The
boundaries are glued as indicated by the color coding, resulting in a disk
with three boundary punctures.

The boundaries $|w_{i}|=1$ of the three disks are glued as indicated by the
coloring of the figure and according to the identifications

$\begin{split}w_{1}w_{2}=-1\,,\ \ \ \ &\hbox{for}\ |w_{1}|=1,\
\hbox{Re}\,w_{1}\leq 0\,,\\\ w_{2}w_{3}=-1\,,\ \ \ \ &\hbox{for}\ |w_{2}|=1,\
\hbox{Re}\,w_{2}\leq 0\,,\\\ w_{3}w_{1}=-1\,,\ \ \ \ &\hbox{for}\ |w_{3}|=1,\
\hbox{Re}\,w_{3}\leq 0\,.\\\ \end{split}$ (7.66)

Note that the common interaction point $Q$, at $w_{i}=i$ for all three disks,
is the midpoint of each ‘open string’ $|w_{i}|=1,\hbox{Im}\,w_{i}\geq 0$. The
gluing prescription identifies the left half of string one with the right half
of string two; the left half of string two with the right half of string
three; the left half of string three with the right half of string one,
forming a single disk with its boundary comprised by the union of the
horizontal boundaries of the three disks.

It is useful to visualize the three open string vertex as a surface with a
metric. For this note that we have, on the glued surface a Strebel quadratic
differential $\varphi$ that takes the form

$\varphi=\phi(w_{i})dw_{i}^{2}=-{1\over w_{i}^{2}}dw_{i}^{2}\,,$ (7.67)

holding on each of the three coordinate patches. It is crucial that $\varphi$
is defined globally, because the local expressions above are consistent with
the gluing conditions $w_{i}w_{i+1}=-1$. The horizontal trajectories, the
lines along which $\varphi$ is real and positive, represent open strings – one
sees that those are the sets $w_{i}=re^{i\theta}$ with $r\leq 1$ fixed, and
$\theta\in[0,\pi]$, so that $\varphi=(d\theta)^{2}$. In particular, for $r=1$
these are the open strings that are being glued together. The quadratic
differential has second order poles at the punctures $w_{i}=0$, as can be seen
directly from the above expression. The quadratic differential has a zero at
$Q$, where three half-neighborhoods of the point $i$ on the disks are glued.
Indeed, a well-defined coordinate $z$ vanishing at $Q$ takes the form
$z=(w_{i}-i)^{2/3}$. Therefore, near $w_{i}=i$, we have $\varphi\simeq
dw_{i}^{2}\sim zdz^{2}$, making the zero at $z=0$ manifest.

A quadratic differential has a transformation law
$\phi(z)dz^{2}=\phi(w)dw^{2}$ under analytic changes of coordinates that
implies the existence of a well defined conformal metric

$ds^{2}=|\phi(w_{i})|\,|dw_{i}|^{2}\,.$ (7.68)

If we write $w_{i}=e^{-T}e^{i\theta}$, the half disk corresponds to
$T\in[0,\infty]$ and $\theta\in[0,\pi]$ with flat metric
$ds^{2}=dT^{2}+d\theta^{2}$. This is a semi-infinite strip of width $\pi$. The
string vertex can therefore be viewed as the joining of three such strips,
with half-string identifications at the edges $T=0,\theta\in[0,\pi]$. This
requires a bit of folding of the 3 strips along the locus of the open string
midpoints. This is shown in Figure 7. We will later explain that this is a
minimal area metric on this surface under the condition that any open curve
with boundary endpoints, and homotopic to a puncture have length at least
$\pi$.

Figure 7: The associative open string vertex, with the half disks mapped into
three semi-infinite strips.

This picture of the three strips makes the associativity of the string vertex
manifest. One can view a product $A\star B$ of string fields as the insertion
of the state $A$ at the infinite end of one of the vertex strips and $B$ at
the infinite end of the strip next to the first. The product is read at the
last strip. For nested products we simply add more strips in the same fashion,
gluing half strings. Due to the associativity of the $\star$-product, we have
$\\{{\cal V}^{o}_{3},{\cal V}^{o}_{3}\\}=0$, implying that there is no need
for higher vertices in the classical theory. It also seems clear that
$\Delta{\cal V}^{o}_{3}\not=0$, the left-hand side being a singular surface.
This means that the ${\cal V}^{o}_{3}$ vertex is problematic at the quantum
level[56].

For any explicit calculation of open string couplings we need the vertex
described in terms of a single disk $|w|\leq 1$ with the three half-disks
embedded inside it. This can be achieved with a sequence of conformal maps,
with each half disk going into a $120^{\circ}$ wedge of the unit disk. To map
$w_{i}$ to this wedge we first use an SL($2,\mathbb{C}$) transformation

$h(u)={1+iu\over 1-iu}\,.$ (7.69)

This maps the unit upper half disk $\\{|u|<1,\hbox{Im}\,u\geq 0\\}$ to the
‘right’ half disk $\\{|h|\leq 1,\hbox{Re}\,h\geq 0\\}$. It is now simple to
apply the $h$ map to the $w_{i}$ half disks to create the correct wedges by
the power map $h^{2/3}$ and then to place the wedges properly in the $w$ disk
by global rotations. One then gets that the maps

$w(w_{1})=e^{2\pi i/3}(h(w_{1}))^{2/3},\ \ w(w_{2})=(h(w_{2}))^{2/3}\,,\ \
w(w_{3})=e^{-2\pi i/3}(h(w_{3}))^{2/3}\,,$ (7.70)

which indeed take the half disks into wedges fitting the $w$ unit disk as
shown in Figure 8 below.

Figure 8: The associative open string vertex presented globally on a $|w|\leq
1$ disk with punctures at $P_{1},P_{2},P_{3}$ and common mid-point $Q$.

A final map to the upper half plane is also quite useful. This is done by
composing the above functions with the map $z=h^{-1}(w)=-i{w-1\over w+1}$
taking the unit $w$ disk to the upper half plane, with the punctures on the
real axis. We thus get

$\begin{split}z=f_{1}(w_{1})=&\ h^{-1}(w(w_{1}))=\
\sqrt{3}+\tfrac{8}{3}w_{1}+\tfrac{16}{9}\sqrt{3}w_{1}^{2}+\tfrac{248}{81}w_{1}^{3}+{\cal
O}(w_{1}^{4})\,,\\\\[2.15277pt] z=f_{2}(w_{2})=&\ h^{-1}(w(w_{2}))=\ \ \
\tfrac{2}{3}w_{2}-\tfrac{10}{81}w_{2}^{3}+{\cal
O}(w_{2}^{5})\,,\\\\[2.15277pt] z=f_{3}(w_{3})=&\
h^{-1}(w(w_{3}))=-\sqrt{3}+\tfrac{8}{3}w_{3}-\tfrac{16}{9}\sqrt{3}w_{3}^{2}+\tfrac{248}{81}w_{3}^{3}+{\cal
O}(w_{3}^{4})\,.\end{split}$ (7.71)

We have included above the power series expansion of the local coordinates.
These are needed for explicit calculations. The picture of the vertex in the
$z$ upper-half plane is shown below. The image of the three half disks fill
the upper half plane. The string midpoint $Q$ now appears at $z=i$ (see Figure
9).

Figure 9: The associative open string vertex represented on the upper-half
$z$-plane. The punctures now lie on the real line, and the three half disks
cover the full upper-half plane.

For closed strings the three-string vertex is simply the open string vertex,
as described by the above $z$ upper-half plane, extended to the full $z$
plane. The coordinates at the punctures are the same as well. One can
visualize the closed string vertex as the result of gluing three semi-infinite
cylinders of circumference $2\pi$ across the open surfaces. The gluing
represents the doubling of the gluing of three semi-infinite strips for the
open string vertex. The resulting picture is as shown below. As before, we
have a Strebel quadratic differential on this surface, with second order poles
at the three punctures. This time there are two zeroes of the quadratic
differential, these are the two heavy dots in the figure, representing the
location where the three boundaries of the semi-infinite cylinders meet.

Figure 10: The closed string vertex in closed string field theory is obtained
by gluing the edges of three identical semi-infinite cylinders.

As mentioned before we have $\partial{\cal V}_{0,4}=-\tfrac{1}{2}\\{{\cal
V}_{0,3},{\cal V}_{0,3}\\}$, and we must now understand what is the four-
string vertex ${\cal V}_{0,4}$. Physically, the necessity for ${\cal V}_{0,4}$
arises because the three Feynman diagrams, formed using two copies of the
vertex, do not provide a full cover of the moduli space of four-punctured
spheres. These three diagrams are shown in Figure 11.

Figure 11: The three Feynman diagrams for four-string scattering, indicated as
$s,t,u$ channels.

The antibracket actually constructs the boundary of the Feynman regions
covered by the three diagrams. If we place three of the labeled punctures at
$0,1$, and $\infty$, the Feynman regions, shown in white in Figure 12, are
disks around $0,1$, and $\infty$, with boundaries ${\cal B}_{1},{\cal B}_{2},$
and ${\cal B}_{3}$. The vertex ${\cal V}_{0,4}$ represents the shaded region
of the moduli space. The darker shading shows a fundamental domain, such that
the action of the SL(2,${\mathbb{C}}$) transformations that permute the three
fixed punctures at 0, 1, $\infty$, produces the full shaded region. The local
coordinates at the punctures are known over the ${\cal B}_{i}$ sets, and the
local coordinates over ${\cal V}_{0,4}$ must match these. We shall now
describe a natural way to define the local coordinates over ${\cal V}_{0,4}$
and give a simple parameterization of the shaded region.

Figure 12: As calculated and plotted by Moeller [110], the shaded region on
the $\xi$ plane represents the string vertex ${\cal V}_{0,4}$. The boundaries
${\cal B}_{i}$ are obtained from the Feynman region as the propagator length
goes to zero.

Both of these issues can be addressed by considering a theorem of Strebel,
showing the existence and uniqueness of a certain type of quadratic
differentials. The relevant version of the theorem is easily understood [111,
112]

> Any $n$-punctured Riemann surface can be built uniquely with $n$ identical
> semi-infinite cylinders, by gluing their boundaries with some set of
> isometric identifications.

For convenience we will assume that the cylinders have circumference $2\pi$.
The resulting surface has a Strebel quadratic differential, which is equal to
$\varphi=-\tfrac{1}{w_{i}^{2}}dw_{i}^{2}$ on each semi-infinite cylinder,
described as the disk $|w_{i}|\leq 1$, punctured at $w_{i}=0$. We can choose
the $w_{i}$’s as the local coordinates as the punctures, producing a section
of $\widehat{\cal P}_{g,n}$. We shall see later that a part of of this section
for genus zero $\widehat{\cal P}_{0,n}$ can be identified as the classical
vertices ${\cal V}_{0,n}$. Note that, in fact, the vertex ${\cal V}_{0,3}$ was
constructed in this fashion. Note also that the boundaries of the cylinders,
once glued, define a critical graph; this graph has $n$ faces, one for each
cylinder, a number of vertices where $\varphi$ vanishes, and a number of
edges, each connecting a pair of vertices. One can see from Fig. 10 that for
the three string vertex the critical graph has three faces, three edges and
two vertices. If the Strebel differential is known, the local coordinates at
the punctures are clearly fixed.

For ${\cal V}_{0,4}$, by Strebel’s theorem, the surfaces are constructed with
four cylinders. The graph indicating the identifications must have four faces.
The zeroes of the quadratic differential are trivalent vertices in this graph.
These zeroes are points of localized negative curvature, elsewhere the metric
is flat, though each puncture contributes to the total accounting of curvature
in the Gauss-Bonnet formula. It turns out that the quadratic differential
generically has four zeroes, and thus the critical graph has four vertices. A
little thought reveals that the critical graph of Strebel differentials for
${\cal V}_{0,4}$ defines a tetrahedron. The graph is determined if we know the
lengths of the edges of the tetrahedron. We have six edges and four length
conditions, since the edges on each face must add up to $2\pi$. This means two
real parameters determine the unique quadratic differential – a fact
consistent with $\mathcal{M}_{0,4}$ being a space of complex dimension one. We
can work with three real parameters $a,b,c$ that, representing the lengths of
the three edges on a face, are all non-negative and add up to $2\pi$:
$a+b+c=2\pi$. It is simple to see that, all four faces have edges of length
$a,b$, and $c$. Figure 13 shows the tetrahedra, with the two possible
inequivalent ways of labeling the faces of the critical graph. For the second
configuration we use parameters $a^{\prime},b^{\prime},c^{\prime}$ satisfying
$a^{\prime}+b^{\prime}+c^{\prime}=2\pi$.

Figure 13: The edges of the four semi-infinite cylinders are glued to each
other following the overap patterns of the above tetrahedra, with one cylinder
attached to each face.

One must now find the region of $\\{a,b,c\\}$ parameter space for ${\cal
V}_{0,4}$. The full parameter space
$a+b+c=a^{\prime}+b^{\prime}+c^{\prime}=2\pi$ would produce all surfaces in
$\mathcal{M}_{0,4}$, so we must determined the restricted tetrahedra that
correspond to ${\cal V}_{0,4}$. It is not difficult to track the tetrahedra
obtained from $\\{{\cal V}_{0,3},{\cal V}_{0,3}\\}$, thus determining the
boundary of ${\cal V}_{0,4}$. In fact, since that tetrahedron comes from the
gluing of two ${\cal V}_{0,3}$ and twisting, one finds that one length
parameter becomes equal to $\pi$. It then becomes clear that the regions that
define ${\cal V}_{0,4}$ are

$a,b,c\leq\pi\,,\ \ \ a^{\prime},b^{\prime},c^{\prime}\leq\pi\,.$ (7.72)

The region is an interior triangle on the $a,b,c$ parameter space, shown in
purple in Figure 14. The other three triangles represent the Feynman regions.

Figure 14: In the two-dimensional parameter space $a+b+c=2\pi$ with $a,b,c\geq
0$, the vertex region is represented by the (purple) middle triangle.

A simple change of viewpoint allows a simple characterization of the
restricted tetrahedra. Consider closed paths, not homotopic to the punctures
on the critical graph. On the left tetrahedron of Fig. 13, a path surrounding
faces 1 and 2 has length $2(b+c)$. The condition $a\leq\pi$ implies
$2\pi-(b+c)\leq\pi$, and thus $b+c\geq\pi$, or equivalently $2(b+c)\geq 2\pi$.
The length condition makes this path longer or equal to $2\pi$, the
circumference of any cylinder. The length conditions (7.72) are in fact
equivalent to the condition that all nontrivial closed paths on the critical
graph must be longer than or equal to $2\pi$.

This simple condition in fact works for all ${\cal V}_{0,n}$ with $n\geq 4$,
that is, for all classical closed string interactions. Indeed, for any fixed
$n$ consider the graph with $n$ faces, and introduce length parameters for the
gluing of the $n$ semi-infinite cylinder boundaries. We then have [112, 113,
114]

> The surfaces in ${\cal V}_{0,n}$ are those for which the critical graph has
> no nontrivial closed path shorter than $2\pi$. We call such critical graphs
> restricted polyhedra.

A relatively straightforward argument allows one to show that the geometric
master equation is satisfied for this choice of vertices. There are two
elements to that argument as one tries to establish the equality of $\\{{\cal
V},{\cal V}\\}$ and $\partial{\cal V}$. First, $\partial{\cal
V}\subset\\{{\cal V},{\cal V}\\}$ follows because the boundary of ${\cal V}$
are polyhedra with some closed path reaching the critical length of $2\pi$,
and if we cut along this path we obtain two restricted polyhedra. Second,
$\\{{\cal V},{\cal V}\\}\subset\partial{\cal V}$, because when two restricted
polyhedra are glued across a face, we get a restricted polyhedron – this is
argued by showing that any closed path that now runs over the whole glued
surface is still long enough.

While the regions ${\cal V}_{0,n}$ have a very elegant description, the
explicit construction of the quadratic differentials is quite challenging. The
cases of ${\cal V}_{0,4}$ and ${\cal V}_{0,5}$ have been considered in some
detail by Moeller [110, 115]. A new approach to obtain these quadratic
differentials based on machine learning has been recently introduced by Erbin
and Firat [116] and could eventually help carry out computations for vertices
${\cal V}_{0,n}$ for much larger $n$. An approach based on Liouville conformal
field theory to calculate quadratic differentials has been proposed in [117].

In the regions of moduli spaces not included in the ${\cal V}$ vertices, the
Strebel quadratic differential still can build the surface but the polyhedron
has nontrivial closed paths with length smaller than $2\pi$. The string field
theory does not use these quadratic differentials. Instead, these surfaces are
built using lower vertices and propagators. Consequently, the associated
quadratic differential $\varphi$ that determines the local coordinates $w_{i}$
at the punctures via $\varphi=-dw_{i}^{2}/w_{i}^{2}$ and the condition that
$|w_{i}|=1$ at the end of the semi-infinite cylinder, is different. It is a
Jenkins-Strebel quadratic differential – this means that horizontal
trajectories fill the surface. In particular, the surface can contain ring
domains, finite cylinders associated to propagators and foliated by horizontal
trajectories. Since the propagators have circumference $2\pi$, all nontrivial
closed curves will be larger than $2\pi$. Note that a Strebel differential is
just a particular class of a Jenkins-Strebel differential.

In order to have a single principle that determines the full section ${\cal
F}_{0,n}$ in $\widehat{\mathcal{P}}_{0,n}$ we turn to a minimal area problem.
In fact this minimal area problem in principle determines the quantum vertices
${\cal V}_{g,n}$ and full sections ${\cal F}_{g,n}$ in
$\widehat{\mathcal{P}}_{g,n}$ [118]:

> Minimal area problem: Given a genus $g$ Riemann surface with $n\geq 0$
> punctures ($n\geq 3$ for $g=0$, $n\geq 1$ for $g=1$) find the metric of
> minimal (reduced) area under the condition that the length of any nontrivial
> homotopy closed curve be greater than or equal to $2\pi$.

Here in this problem, a metric means a conformal metric, that is, one has a
function $\rho$ such that the length element is $ds=\rho|dz|$. Since the
problem involves all nontrivial closed curves in a surface it is ‘modular
invariant’ (the set of all nontrivial closed curves is invariant under large
diffeomorphisms) and is independent of the labeling of the punctures, thus
implementing naturally the symmetry under the exchange of punctures. It is
simple to show that any solution to a minimal area problem is unique, this
follows from the convexity of the area functional. Uniqueness is the reason we
get a section in $\widehat{\mathcal{P}}_{g,n}$. In the minimal area metric one
expects semi-infinite cylinders associated to the punctures. The term
‘reduced’ area is used because the area of the semi-infinite cylinders is
infinite and requires regularization. One uses fixed arbitrary coordinates
$\tilde{w}$ around the punctures to remove small disks
$|\tilde{w}|\leq\epsilon$ around the puncture and subtract off a logarithmic
divergence $\sim-\log\epsilon$ in the area. Under a change of the arbitrary
coordinates around the punctures, the reduced area changes by a metric
independent constant. This implies that the extremal metric is regulator
independent. The regulator also preserves the convexity of the area
functional.

We can state what the vertices ${\cal V}_{g,n}$ are based on the expected
properties of the minimal area metrics. All we need is an algorithm to decide
if any surface $\Sigma_{g,n}$ is in ${\cal V}_{g,n}$. For this we use the
minimal area metric on $\Sigma_{g,n}$ and inspect it searching for ring
domains, or annuli, revealing the existence of propagators. A ring domain, or
a cylinder of height greater or equal to $2\pi$ is declared to be a
propagator. $\Sigma_{g,n}\in{\cal V}_{g,n}$ if the minimal area metric has no
propagators.  To fix the local coordinates on $\Sigma_{g,n}$ we look at the
semi-infinite cylinders associated to the punctures. These cylinders are well
defined in that it must end; there must be a ‘last’ geodesic homotopic to the
puncture. That last geodesic must be retracted towards the puncture by a
distance $\pi$, and this retracted curve becomes the coordinate curve for this
puncture. This procedure effectively adds length $\pi$ stubs to the surface.
Equipped with such stubs, one can see that $\Delta{\cal V}$ will not produce
surfaces with closed curves shorter than $2\pi$: when gluing two coordinate
curves in a given surface, the stubs already provide $2\pi$ length to any
closed curve that goes through the created handle. In fact, we get a solution
of the master equation for any stub length greater than $\pi$. In the string
field theory the change of stub length is realized by a field redefinition.
The longer the stub length, the Feynman diagrams produce less of the moduli
space and the vertices produce more. Changing the stub length has an
interpretation as a renormalization group transformation of the action [119].

We note that as long as we only work with classical closed string field
theory, there is no need for stubs – and the coordinate curves coincide with
the faces of the polyhedron. A fair amount of work has been done with this
version of the theory, including a somewhat inconclusive effort to identify a
tachyon vacuum [58, 120]. It is the full quantum closed string field theory
that requires stubs. This is made clear by considering $\Delta{\cal V}_{0,3}$,
where two coordinate curves of the vertex are twist glued. Without stubs this
produces singular surfaces [121]. With length $\pi$ stubs all nontrivial
closed curves remain longer than $2\pi$ and one can determine the quantum
vertex ${\cal V}_{1,1}$ needed to satisfy the master equation.

While the minimal area metrics clearly exist for all genus zero surfaces and
across large parts of the moduli space of non-zero genus surfaces, there is
still no mathematical proof that they exist in all cases. Perhaps existence is
just difficult to prove and the result will be established at some point. All
minimal area metrics that arise from quadratic differentials are flat, except
for negative curvature singularities at the zeroes of the differential.
Moreover on any region of the surface there is just one band of geodesics
saturating the length condition. For general surfaces, where the minimal area
metric does not arise from a quadratic differential, it has also not been
clear if bulk curvature can exist, or lines of curvature can exist. Moreover,
it was expected that there would be regions of the surface foliated by
multiple bands of geodesics.

Progress on the minimal area problem was made by M. Headrick and Zwiebach
[122, 123], using the method of convex optimization. The original minimal-area
problem was analyzed in terms of homology (as opposed to homotopy), using
calibrations and the max flow - min cut theorem to formulate it as a local
convex program that is easily implemented numerically. Moreover, a dual
program, involving maximization of a concave functional was derived, providing
lower bounds for the minimal area. These methods were applied to find
(numerically) the minimal area metric for the once-punctured square torus. The
solution displays regions covered by intersecting bands of length-saturating
geodesics and exhibits both positive and negative bulk curvature. Further work
to understand regions with multiple intersecting bands of geodesics was
reported by Usman and Zwiebach [124]. It showed that a claim that regions with
more than four such bands are flat is simply not true. All in all, these works
provide evidence that minimal area metrics exist. The minimal area problem
fits in the area of systolic geometry, where a number of important results
have been established [125].

The minimal area problem can also be used for open string field theory and for
open-closed string field theory. If we consider diagrams with only external
open strings, we are looking at Riemann surfaces of all genus, with $b\geq 1$
boundaries and a number $m$ of boundary punctures. One can show that the
Witten classical open string field theory produces correctly all the moduli
spaces because such diagrams are solutions of a minimal area problem: the
surfaces have the minimal area under the condition that any open curve of
nontrivial homotopy (and boundary endpoints) have length bigger or equal to
$\pi$ [63]. For open-closed string field theory we need to consider all genus
surfaces with arbitrary numbers of open and closed string punctures. Here, a
natural minimal area problem requires nontrivial open curves (with boundary
endpoints) to be longer or equal to $\pi$ while nontrivial closed curves are
longer or equal to $2\pi$. In writing out this theory one must add length
$\pi$ stubs to all vertices, including those of the open string, and for the
boundary state vertex, which represents the one-point function of closed
strings in a disk. Thus, the classical open string field theory subsector of
the open-closed quantum theory is non-polynomial.

###  7.9 String vertices from hyperbolic metrics

Much is known about hyperbolic metrics on two dimensional surfaces, metrics of
Gaussian curvature $-1$. The moduli space of such metrics on an orientable
surface of genus $g$ and $n$ punctures coincides with the moduli spaces of
conformal structures on the surface, namely, with the moduli space
$\mathcal{M}_{g,n}$ of Riemann surfaces – this is a version of the
uniformization theorem [126]. It is also interesting to note that the surfaces
that must be included in the definition of ${\cal V}$, as written out in
equation (7.33), are precisely those for which hyperbolic metrics exist.
Indeed, hyperbolic metrics do not exist for spheres with less than three
punctures, and for tori without punctures. These facts motivated Pius and
Moosavian to explore the construction of closed string field theory using
hyperbolic metrics on Riemann surfaces [127, 128]. They considered surfaces
with punctures, and that led to a complication. Such metrics have cusps on the
punctures. Moreover the cutting and gluing of such metrics is not done along
geodesics, and therefore it does not give a hyperbolic metric. These
complications mean that the solution of the master equation is only
approximate and must be improved.

In a somewhat different approach, Costello and Zwiebach [34] worked with
surfaces of genus $g$ and $n$ boundary components, which allow for exact
solutions of the master equation, as we will explain. On such surfaces we
consider hyperbolic metrics in which the $n$ boundaries are geodesics, all of
length $L$. Given any such surface $\Sigma_{g,n}$ with its metric, there is an
obvious grafting map gr∞ that takes it to $\widehat{\mathcal{P}}_{g,n}$, that
is, a map to the space of Riemann surfaces of genus $g$ with $n$ punctures and
local coordinates at the punctures. The map

$\hbox{gr}_{\infty}:\Sigma_{g,n}\to\widehat{\mathcal{P}}_{g,n}\,,$ (7.73)

simply attaches semi-infinite cylinders of circumference $L$ to each of the
boundaries of the surface. The semi-infinite cylinders define the coordinate
disks for the punctures in the resulting surface, the two related by the
exponential map.

Let us call $\mathcal{M}_{g,n,L}$ the moduli space of hyperbolic metrics on a
surface of genus $g$ with $n$ geodesic boundaries, all of length $L$. The
grafting map, followed by the map $\pi$ forgetting coordinates is then a map

$\pi\circ\hbox{gr}_{\infty}:\mathcal{M}_{g,n,L}\to\mathcal{M}_{g,n}\,.$ (7.74)

It is a theorem that this map is in fact a homomorphism, a one-to-one onto
map. By considering all the hyperbolic metrics on a genus $g$ surface with $n$
geodesic boundary components we have created a cover of moduli space!

The simplest example of this is $\mathcal{M}_{0,3,L}$, the moduli space of
hyperbolic metrics on a sphere with three holes. The hyperbolic metric here is
unique, so this moduli space is just a point. This is the familiar pants
diagram shown in Figure 15, with the dotted lines representing the flat
cylinders that could be attached to the boundaries to create the surface
${\cal V}_{0,3}\in\widehat{\mathcal{P}}_{0,3}$. As we will see below this is
the closed string vertex for a hyperbolic string field theory.

Figure 15: A sketch of a hyperbolic three-closed string vertex. The boundaries
are geodesics of equal lengths.

To proceed, we need one more definition. For a surface $\Sigma$ with
boundaries and equipped with a metric, the systole of $\Sigma$, called sys
$\Sigma$, is the length of the shortest non-contractible closed geodesic which
is not a boundary component. If we have a surface with no geodesics, except
for the boundary geodesics of length $L$, the systole is declared to be $L$.
In general, for a surface in $\mathcal{M}_{g,n,L}$ the systole can be larger
than $L$, equal to $L$, if the shortest curve which is not a boundary has
length exactly equal to the boundary, or shorter than $L$.

On $\mathcal{M}_{g,n,L}$ we can now select a subset $\widetilde{\cal
V}_{g,n}(L)$ that includes those surfaces where the systole is larger than or
equal to $L$:

$\widetilde{\cal V}_{g,n}(L)\equiv\\{\Sigma\in\mathcal{M}_{g,n,L}\ \bigl{|}\
\hbox{sys}\,\Sigma\geq L\\}\,.$ (7.75)

Since the boundary geodesics are of length $L$ the above sets include all
surfaces that have no geodesic of length less than $L$. Even if there are no
boundaries ($n=0$), the definition is sensible: here
$\mathcal{M}_{g,0,L}=\mathcal{M}_{g,0}$ and the above defines a certain subset
of $\mathcal{M}_{g,0}$. For $\mathcal{M}_{0,3,L}$ the selected subset is a
single surface, the pants diagram with boundaries of length $L$.

The string vertices ${\cal V}_{g,n}(L)$ are now defined as the sets
$\widetilde{\cal V}_{g,n}(L)$ grafted with semi-infinite cylinders to turn the
boundaries into punctures with local coordinates:

${\cal V}_{g,n}(L)\equiv\hbox{gr}_{\infty}\bigl{(}\widetilde{\cal
V}_{g,n}(L)\bigr{)}\,.$ (7.76)

The main result of [34] is now simple to state:

> The sets ${\cal V}_{g,n}(L)$, with $L\leq L_{*}=2\sinh^{-1}1,$ solve the
> geometric master equation.

This means we get a solution for boundary lengths that are bounded by the
constant $L_{*}\simeq 1.76275$.

We will not go over the details of the proof that establishes the above
result, but the intuition for it comes from familiar facts about collars in
hyperbolic metrics (see, for example, [126]). Consider a simple closed
geodesic $\gamma$ on a Riemann surface. The collar ${\cal C}(\gamma)$ of width
$w$ about $\gamma$ is the set of all points on the surface whose distance to
the curve $\gamma$ does not exceed $w/2$. For a geodesic on the interior of
the surface, the collar extends to the two sides of it. For a geodesic
boundary, the collar just extends to one side of it, so that its ‘width’ is in
fact $w/2$. Given a collection of simple closed geodesics $\gamma_{i}$ of
length $\ell_{i}$ that do not intersect, we can consider collars of width
$w_{i}$ chosen such that

$\sinh\bigl{(}\tfrac{1}{2}w_{i}\bigr{)}\sinh\bigl{(}\tfrac{1}{2}\ell_{i})=1\,.$
(7.77)

It is a remarkable result in hyperbolic geometry that such collars are
disjoint on the surface; they do not intersect! The collars are annuli that
exist around each geodesic. The shorter the geodesic, the wider the collar.
The width $w$ of the collar equals the length $\ell$ of the geodesic when both
factors in the above left-hand side are equal to one, that is for
$\ell=L_{*}=2\sinh^{-1}1$. It follows that for a geodesic of length $L<L_{*}$,
its collar will have $w_{L}>L_{*}>L$, thus:

$L\leq L_{*}\quad\to\quad w\geq L\,.$ (7.78)

This is why the above subsets ${\cal V}_{g,n}(L)$ satisfy the master equation
with $L\leq L_{*}$. In that case, the collars around the geodesic boundaries
have width greater than $L/2$, so that when they are glued together, any
nontrivial closed curve crossing the seam will have length greater than $L$,
and thus the gluing will result on a surface on the vertex region.
Furthermore, since the glued surface now has systole $L$, realized by the
gluing curve, it lies on the boundary of the vertex region. As one can see,
the collars here play the role of the stubs in the minimal area problem, whose
role was to prevent the creation of short non-contractible closed curves upon
gluing.

The above construction of the closed string vertices, is at this point, the
only fully rigorous and explicit construction available. Given the wealth of
results known about hyperbolic metrics, one would expect that in some way,
hyperbolic string field theory could be tractable analytically.

This hyperbolic construction has been extended to open-closed string field
theory by Cho [129]. The Riemann surfaces here have borders, geodesic closed
curves of length $L_{c}$ where we attach external closed string semi-infinite
cylinders, open-puncture boundaries, geodesic segments of length $L_{o}$ where
we attach external open string semi-infinite strips, and just plain boundaries
in the sense of open string theory. Moreover, where boundaries and open-
puncture boundaries meet, they are orthogonal. The string vertices for any
moduli space (with two exceptions noted below) are the surfaces for which the
hyperbolic metric has closed curve systole exceeding or equal to $L_{c}$ and
open curve systole exceeding or equal to $L_{o}$. The closed curve systole was
defined before. The open curve systole is the length of the shortest
noncontractible open curve with endpoints on boundaries that is not itself an
open-puncture boundary. For the vertices to satisfy the BV geometric master
equation the values of $L_{c}$ and $L_{o}$ must satisfy the conditions

$L_{c}\leq L_{*}^{\prime}\,,\ \ \ \ \hbox{and}\ \ \ \ \sinh L_{c}\,\sinh
L_{o}\leq 1\,.$ (7.79)

The constant $L_{*}^{\prime}$ is defined by the condition
$\sinh(L_{*}^{\prime}/2)\sinh L_{*}^{\prime}=1$ giving $L_{*}^{\prime}\simeq
1.21876$. One can see that for $L_{o}\leq L_{*}^{\prime}/2$, all $L_{c}\leq
L_{*}^{\prime}$ work. For $L_{o}>L_{*}^{\prime}/2$ the possible values of
$L_{c}$ have a smaller upper bound. The above rule for vertices does not apply
for two moduli spaces: for the disk with one bulk puncture the vertex is just
a circle of length $L_{c}$ (the limit of a cylinder of circumference $L_{c}$
and vanishing height, with a border and a boundary), and for the annulus
without punctures there is no vertex.

The explicit calculation of the local coordinates of the hyperbolic three-
string vertex has been done by Firat [130]. This allows for off-shell
computations with this vertex. In fact the result gives the coordinates for a
general hyperbolic pants diagram, that is, a sphere with three geodesic
boundaries of different lengths. This is useful because the Teichmuller moduli
space of hyperbolic surfaces has a nice decomposition in terms of the gluing
of pants. The Liouville equation for the conformal factor of the metric on the
three-holed sphere is solved for by relating it to the monodromy problem for a
second order linear ordinary differential equation, a Fuchsian equation on the
complex plane.

Hyperbolic vertices for closed string field theory have an interesting limit.
As long as we focus on the genus zero theory, the condition $L<L_{*}$ is not
really required and $L$ can be taken as large as one wishes. In fact, when
$L\to\infty$ they hyperbolic vertices approach, up to an overall constant
scale factor, the polyhedral vertices of classical string field theory [34].
This has been exploited in [117], showing that indeed the Strebel quadratic
differentials associated with the polyhedra arise from hyperbolic vertices as
a WKB approximation applied to the Fuchsian equation. This allows for the
computation of these quadratic differentials using the tools of Liouville
conformal field theory, in particular, their conformal blocks.

Finally, for useful information derived about the properties of off-shell
amplitudes in hyperbolic geometry see [127, 128].

###  7.10 String vertices for open superstring field theory

If we restrict ourselves to the NS sector of the classical open superstring
field theory, the original proposal of [131] was to use a string field in the
minus one picture, with the standard BRST operator for the kinetic term. The
cubic interaction requires a picture changing operator adding one unit of
picture number so that the total picture number in the correlator is equal to
minus two, as required. Using the associative vertex the PCO was inserted at
the string midpoint, but this leads to difficulties: associativity fails to
hold as one gets a singular collision of two picture changing operators. While
several ways to address this difficulty have been considered, a canonical way
to insert picture changing operators that does not suffer from singularities
was developed by Erler, Konopka and Sachs in [54, 28, 55]. The result is a
consistent theory for the NS sector of classical open superstrings, the NS
sector of classical heterotic strings, and the NS–NS sector of classical type
II superstrings. Our focus here will be on the NS sector of open superstrings
discussed in [54]. We will comment on the extension for heterotic strings and
for type II superstrings.

For open superstring field theory, the idea is to begin with a series of
bosonic open string products $b_{n+1}^{(0)}$, with $n\geq 0$, satisfying the
$A_{\infty}$ axioms. The superscript zero in the products indicates no
insertion of picture changing operators; the subscript is the number of states
going into the product. An explicit procedure using canonical insertions
constructs a new series of open superstring products $b_{n+1}^{(n)}$. Here,
the superscript indicates the extra picture number $n$ carried by such
insertions, which is appropriate for open superstring interactions of $n+1$ NS
fields, each of picture minus one. These products, by construction, still
define an $A_{\infty}$ algebra, and thus the open superstring action can be
written as usual. For convenience of notation we use the ‘bold’ operators
${\bf b}_{n+1}^{(0)}$ and ${\bf b}_{n+1}^{(n)}$ to denote the extension of the
products to act on the tensor co-algebra $T({\cal H}_{o})$ (see section 6.1).
We define the formal sum of superstring products, an operator of degree one,

${\bf B}^{[0]}(t)=\sum_{n=0}^{\infty}t^{n}{\bf b}_{n+1}^{(n)}\,.$ (7.80)

The use of the formal parameter $t$ as opposed to just doing the sum of the
products (as we did for open bosonic strings) will be convenient for the later
discussion. The superscript $[0]$ indicates that the number of picture
changing operators is the correct one for the products, assuming an NS field
of picture minus one. Thus the ‘zero’ deficit, denoted by the superscript.
These vertices are consistent if they satisfy the $A_{\infty}$ condition:

$\big{(}{\bf B}^{[0]}(t)\big{)}^{2}=0\,.$ (7.81)

Additionally, we require that these vertices belong to the ‘small’ Hilbert
space: acting on string fields killed by $\eta_{0}$ they give string field
killed by $\eta_{0}$. Defining the associated $\boldsymbol{\eta}_{0}$ operator
acting on the tensor algebra this condition takes the form

$\bigl{\\{}\boldsymbol{\eta}_{0}\,,{\bf B}^{[0]}(t)\bigr{\\}}=0\,.$ (7.82)

This just states that $\eta_{0}$ should be a derivation of the superstring
products. Of course, $\eta_{0}$ is also a derivation of the bosonic string
products; this means that $\bigl{\\{}\boldsymbol{\eta}_{0}\,,{\bf
b}^{(0)}_{n}\bigr{\\}}=0$, for all $n\geq 1$. In order to solve for the ${\bf
b}_{n+1}^{(n)}$ operators satisfying these conditions, one introduces a larger
set of products of various deficits, so that one can incorporate into a single
framework the bosonic products of maximal deficit. For string products with
picture number deficit $m$ we have

${\bf B}^{[m]}(t)=\sum_{n=0}^{\infty}t^{n}{\bf b}_{m+n+1}^{(n)}\
\,,m=0,1,\ldots\,.$ (7.83)

In all generality, we now put together all such products,

$\begin{split}{\bf B}(s,t)=\sum_{m=0}^{\infty}s^{m}{\bf
B}^{[m]}(t)=\sum_{m,n=0}^{\infty}s^{m}\,t^{n}\,{\bf
b}^{(n)}_{m+n+1}\,.\end{split}$ (7.84)

Note that ${\bf B}(0,t)$ is the superstring collection of products while ${\bf
B}(s,0)=\sum_{m}s^{m}{\bf b}^{(0)}_{m+1}$ is the bosonic string collection of
products. Moreover, one introduces a set of Grassmann even operators
$\boldsymbol{\mu}$ that change the picture numbers. We include variable
number, greater or equal to two, of input string states and a variable number,
greater or equal to one, of picture number insertions:

$\boldsymbol{\mu}(s,t)=\sum_{m=0}^{\infty}s^{m}\boldsymbol{\mu}^{[m]}(t)=\sum_{m,n=0}^{\infty}s^{m}\,t^{n}\,\boldsymbol{\mu}^{(n+1)}_{m+n+2}\,.$
(7.85)

At this point what is needed is guaranteed if the following equations hold:

$\bigl{\\{}{\bf B}(s,t),{\bf B}(s,t)\bigr{\\}}=0\,,\ \ \hbox{and}\ \
\bigl{\\{}\boldsymbol{\eta}_{0},{\bf B}(s,t)\bigr{\\}}=0\,.$ (7.86)

For $t=0$ these manifestly hold, because they are the constraints satisfied by
the bosonic string products. For $s=0$ these are the relations we need to hold
for the superstring products. The insight of [54] is the proposal of a set of
recursion relations expressed as a set of differential equations:

${\partial\over\partial t}{\bf B}(s,t)=\bigl{[}{\bf
B}(s,t)\,,\,\boldsymbol{\mu}(s,t)\bigr{]}\,,\ \ \ {\partial\over\partial
s}{\bf
B}(s,t)=\bigl{[}\boldsymbol{\eta}_{0}\,,\,\boldsymbol{\mu}(s,t)\bigr{]}\,.$
(7.87)

These equations allow for a recursive solution that yields, at the end, the
superstring vertices. But, more crucially, they guarantee that the required
equations (7.86) hold. Indeed, this follows by taking the $t$ derivatives of
the left-hand sides of (7.86), and showing that given that these left-hand
sides vanish for $t=0$, they must vanish for all $t$. The algorithm to find a
solution to these equations was described in [54].

We illustrate the recursive procedure with the lowest order case, the one
fixing the first product ${\bf b}_{2}^{(1)}$ of the open superstring.
Recalling that ${\bf b}_{1}^{(0)}={\bf Q}$, we write

${\bf B}(s,t)={\bf Q}+t\,{\bf b}_{2}^{(1)}+s\,{\bf b}_{2}^{(0)}+\ldots\,,\ \ \
\ \ \ \boldsymbol{\mu}(s,t)=\boldsymbol{\mu}_{2}^{(1)}+\ldots\,\,.$ (7.88)

The two differential equations above give, to leading order, two equations

${\bf b}_{2}^{(1)}=\bigl{[}{\bf Q}\,,\,\boldsymbol{\mu}_{2}^{(1)}\bigr{]}\,,\
\ \ \ \ {\bf
b}_{2}^{(0)}=\bigl{[}\boldsymbol{\eta}_{0}\,,\,\boldsymbol{\mu}_{2}^{(1)}\bigr{]}\,.$
(7.89)

With a little trial and error, the solution for $\boldsymbol{\mu}_{2}^{(1)}$
can be written in terms of $\boldsymbol{\xi}_{0}$ and the product ${\bf
b}_{2}^{(0)}$:

$\boldsymbol{\mu}_{2}^{(1)}=\tfrac{1}{3}\bigl{(}\boldsymbol{\xi}_{0}\,{\bf
b}_{2}^{(0)}-\,{\bf b}_{2}^{(0)}\boldsymbol{\xi}_{0}\bigr{)}\,.$ (7.90)

As a two-product this is

$\boldsymbol{\mu}_{2}^{(1)}(A\otimes
B)=\tfrac{1}{3}\bigl{(}\xi_{0}(A,B)-(\xi_{0}A,B)-(-1)^{A}(A,\xi_{0}B)\,\bigr{)}\,.$
(7.91)

This solution for $\boldsymbol{\mu}_{2}^{(1)}$ is easily confirmed acting on
two string fields $A,B$ in the small Hilbert space:

$\begin{split}[\boldsymbol{\eta}_{0}\,,\boldsymbol{\mu}_{2}^{(1)}](A\otimes
B)=&\ \boldsymbol{\eta}_{0}\boldsymbol{\mu}_{2}^{(1)}(A\otimes
B)\\\\[4.30554pt] =&\
\tfrac{1}{3}\,{\eta}_{0}\,\bigl{(}\xi_{0}(A,B)-(\xi_{0}A,B)-(-1)^{A}(A,\xi_{0}B)\,\bigr{)}\\\\[4.30554pt]
=&\
\tfrac{1}{3}\,\bigl{(}\eta_{0}\xi_{0}(A,B)+(\eta_{0}\xi_{0}A,B)+(A,\eta_{0}\xi_{0}B)\,\bigr{)}\\\\[4.30554pt]
=&\ \tfrac{1}{3}\,\bigl{(}(A,B)+(A,B)+(A,B)\,\bigr{)}=(A,B)={\bf
b}_{2}^{(0)}(A\otimes B)\,.\end{split}$ (7.92)

Finally, we can determine the superstring product

$\begin{split}(A,B)^{*}\equiv&\ {\bf b}_{2}^{(1)}(A\otimes B)=\bigl{[}{\bf
Q}\,,\,\boldsymbol{\mu}_{2}^{(1)}\bigr{]}(A\otimes B)\\\ =&\ {\bf
Q}\boldsymbol{\mu}_{2}^{(1)}(A\otimes B)-\boldsymbol{\mu}_{2}^{(1)}(QA\otimes
B)-(-1)^{A}\boldsymbol{\mu}_{2}^{(1)}(A\otimes QB)\\\\[4.30554pt] =&\ \
\,\tfrac{1}{3}\,Q\,\bigl{(}\xi_{0}(A,B)-(\xi_{0}A,B)-(-1)^{A}(A,\xi_{0}B)\bigr{)}\\\\[3.01385pt]
&\hskip-5.0pt-\tfrac{1}{3}\bigl{(}\xi_{0}(QA,B)-(\xi_{0}QA,B)+(-1)^{A}(QA,\xi_{0}B)\bigr{)}\\\\[3.01385pt]
&\hskip-5.0pt-\tfrac{1}{3}(-1)^{A}\bigl{(}\xi_{0}(A,QB)-(\xi_{0}A,QB)-(-1)^{A}(A,\xi_{0}QB)\bigr{)}\,.\end{split}$
(7.93)

Using the derivation property of $Q$ and $\\{Q,\xi_{0}\\}={\cal X}_{0}$ we
quickly get

$(A,B)^{*}\equiv\tfrac{1}{3}\bigl{(}{\cal X}_{0}(A,B)+({\cal
X}_{0}A,B)+(A,{\cal X}_{0}B)\bigr{)}\,.$ (7.94)

The calculation of the next product $(A,B,C)^{*}$ follows a similar line,
although it is computationally harder. The construction, however, guarantees
that all the higher superstring products exist, are in the small Hilbert
space, and satisfy the $A_{\infty}$ relations. Since the construction
guarantees that we get satisfactory vertices, the action takes the expected
form. With an open NS string field $\Phi$ of ghost number one and picture
number minus one, and in the small Hilbert space, the action is

$S(\Phi)=\tfrac{1}{2}\langle\Phi,Q\Phi\rangle^{\prime}+\sum_{n=1}^{\infty}{1\over
n+2}\langle\Phi,b_{n+1}^{(n)}(\Phi,\cdots,\Phi)\rangle^{\prime}\,.$ (7.95)

It should be noted that for open strings the use of $\boldsymbol{\xi}_{0}$ is
not mandatory. It suffices to use an operator $\boldsymbol{\xi}$ whose
anticommutator with $\boldsymbol{\eta}_{0}$ is equal to one.

For heterotic string field theory, an almost identical procedure works. The
bosonic closed string products form an $L_{\infty}$ algebra and the heterotic
closed string products, constructed with insertions on the bosonic products,
also satisfy the $L_{\infty}$ algebra while still living in the small Hilbert
space. This time it is important to use $\boldsymbol{\xi}_{0}$, since the
resulting ${\cal X}_{0}$ insertions commute with the operators $b_{0}^{-}$ and
$L_{0}^{-}$. As a result, the new products will still satisfy the required
constraints $b_{0}^{-}=0$ and $L_{0}^{-}=0$. The expression for the second
product is completely analogous to the one found above for open superstrings:

$[A,B]^{*}\equiv\tfrac{1}{3}\bigl{(}{\cal X}_{0}[A,B]+[{\cal
X}_{0}A,B]+[A,{\cal X}_{0}B]\bigr{)}\,.$ (7.96)

Here $[A,B]$ is the original bosonic string product. For type II superstrings
there are a couple of options in the way one treats left moving and right-
moving pictures. Starting with the bosonic products, an asymmetric approach
first raises by recursion the left-moving picture number, and then raises the
right-moving picture number. A more symmetric approach treats both picture
numbers as equivalent. For more details, see [28].

A relation between this approach and the vertical integration introduced in
section 3.2 has been discussed in [132]. Inclusion of Ramond sector in this
construction is possible but needs some extra ingredients. Ref. [55]
generalized the construction of open superstring field theory to describe
equations of motion of the R-sector fields. While constructing an action along
this line is also possible[133, 134], this requires inclusion of an extra free
field as in section 4.2 or using a restricted vector space for Ramond sector
fields as will be discussed in section 8.4. Generalizations of such
construction to the Ramond sector of heterotic and type II string theory have
also been discussed in [135, 136, 137].

###  7.11 Some subtleties in quantum closed superstring field theory

All constructions of ${\cal V}_{g,n}$ for bosonic string theory, as described
in sections 7.8 and 7.9, have the property that the local coordinates vary
continuously as we move in the moduli space. In other words ${\cal V}_{g,n}$
describes the piece of a submanifold of $\widehat{\cal P}_{g,n}$. This may not
be possible for open and/or closed superstring field theories due to the
existence of spurious poles. As reviewed in §3.2, the only known systematic
way of choosing the PCO locations as we move in the moduli space is to divide
the moduli space into small chambers, choose PCO locations inside a given
chamber avoiding spurious poles and then add suitable compensating terms at
the boundary between the two chambers. Therefore the choice of ${\cal
V}_{g,n}$ in string field theory should also reflect this. Given a choice of
$(6g-6+2n)$ dimensional subspace ${\cal V}_{g,n}$ of $\widehat{\cal P}_{g,n}$
for the bosonic string field theory, we divide it into small chambers, and in
each chamber choose the PCO locations such that we avoid the spurious poles.
In the $(6g-7+2n)$ dimensional boundary between two chambers we add vertical
segments that interpolate between two sets of PCO locations on two sides of
the boundary, but the vertical segments themselves must be a union of several
segments so that along each of these segments only one PCO location changes.
As discussed in §3.2, when such walls meet on a $(6g-8+2n)$ dimensional
subspace, we have to add new two dimensional vertical segments. This process
continues until we have ensured that the union of the original chambers and
the vertical segments fill a ball shaped region in $\widehat{\cal P}_{g,n}$
with no gaps. The rules for carrying out the integrals along the vertical
segments to construct the products $\\{A_{1},\cdots,A_{n}\\}$ is the same as
for the construction of the amplitudes as discussed in §3.2.

There are however two more subtleties in this construction. First, we cannot
construct ${\cal V}_{g,n}$ satisfying the geometric master equation (4.8) with
a single choice of PCO locations in each chamber, since, as discussed above
(LABEL:evertexsuper), the definition of $\Delta$ and $\\{,\\}$ in (4.8) now
involves insertion of the operator ${\cal G}$ which, in the R sector, involves
averages of PCO locations. Therefore at least close to the boundaries, ${\cal
V}_{g,n}$’s must also be chains that involve averages of subspaces of
$\widehat{\cal P}_{g,n}$. This is also necessary for the interaction vertices
${\cal V}_{g,n}$ to be symmetric under the exchange of external punctures. For
example, even for the three-point function on the sphere with NS sector
external punctures, where we need one PCO for the heterotic theory and one PCO
each in the holomorphic and anti-holomorphic sector in type II theories, it is
not possible to find a single location that remains invariant under the
$SL(2,C)$ transformations that permute the three punctures. So we must take
the averages of different choices of PCO locations.

The second subtlety arises as follows. Let us suppose that we have chosen the
${\cal V}_{g,n}$’s so that they do not contain spurious poles. Now if we
construct a Feynman diagram by joining two of them by a propagator, then it
gives a particular choice of PCO locations determined by the plumbing fixture
rules. One can now ask: is the configuration of PCOs determined this way free
from spurious singularities? Unless the answer is in the affirmative, we run
into a problem since in string field theory we do not have any freedom in
choosing the PCO locations in part of $\widehat{\cal P}_{g,n}$ that is covered
by Feynman diagrams with one or more propagators. They are fixed by the choice
of PCO locations in the construction of the interaction vertices of the
diagram. To answer this question, we need to examine the origin of spurious
singularities in a Feynman diagram, assuming that the interaction vertices
have been chosen avoiding the spurious poles. For this it will be useful to
represent the contribution from such Feynman diagrams as in quantum field
theory. In this case we need to sum over infinite number of fields that could
propagate along each internal propagator of the Feynman diagram, and this sum
could diverge. One can show that this is the source of potential spurious
poles in the Feynman diagram. This can be remedied by choosing the local
coordinates at the punctures that we use in the construction of ${\cal
V}_{g,n}$ to have long stubs in the sense described in §7.8. In this case an
external state of $L_{0}$ eigenvalue $h$ attached to a vertex will produce a
factor of $\lambda^{-h}$ for some large number $\lambda$. Since for large $h$
the number of states grow as $e^{C\sqrt{h}}$ for some constant $C$, we see
that the sum over intermediate states in a Feynman diagram can be made
convergent by taking $\lambda$ to be large, and hence the Feynman diagrams
with propagators will be free from spurious poles as long as the vertices
${\cal V}_{g,n}$ are constructed avoiding the spurious poles. For this
argument it is important to ensure that the spectrum of $L_{0}$ eigenvalues is
bounded from below and that for a given $h$ there are only a finite number of
fields so that the sum is finite. The first condition is satisfied by choosing
the states in the $-1$ picture in the NS sector and the $-1/2$ or $-3/2$
picture in the R sector. Both $-1/2$ and $-3/2$ picture R sector states have
infinite degeneracies due to the existence of bosonic zero modes $\gamma_{0}$
and $\beta_{0}$ respectively, causing apparent violation of the second
condition. However insertion of ${\cal X}_{0}$ in the R sector propagator
guarantees that the matrix element of ${\cal X}_{0}$ between a pair of states
is non-zero only for a finite number of states for any given $L_{0}$
eigenvalue [15]. This way the second condition is also satisfied.

##  8 Superstring field theories in the large Hilbert space

One of the issues we face in the construction of the interaction vertices for
superstring field theories is in the choice of the locations of the picture
changing operators. While the principle underlying the choice of picture
changing operators is known, explicit constructions are more difficult. In
this section we shall describe a way to get around this issue by working in
the large Hilbert space where we do not require the string field to be
annihilated by $\eta_{0}$. In that case we can write down the gauge invariant
string field theory action in terms of the same products that appear in the
construction of $A_{\infty}$ or $L_{\infty}$ algebras underlying the bosonic
open or closed string field theories. The price we pay is that the covering of
the moduli space by the sum over Feynman diagrams is no longer obvious, it
requires extra effort to establish this.

All our discussion here will be restricted to the tree-level string field
theories. We will consider classical open superstrings in the NS sector,
classical heterotic strings in the NS sector, and classical type II strings in
the NS-NS sector. We conclude with a brief discussion of the Ramond sector of
classical open superstrings.

###  8.1 Berkovits open superstring field theory in the NS sector

The conventional BRST quantization of the open superstring considers states
$|{\phi}\rangle$ that are killed by the BRST operator: $Q|{\phi}\rangle=0$.
The familiar physical state representatives in the NS sector take the
schematic form $|{\phi}\rangle=c\,e^{-\phi}\,V_{M}(0)|{0}\rangle$, with
$V_{M}$ a matter operator that is a dimension one-half primary, $e^{-\phi}$ a
dimension one-half primary, and $c$ a dimension minus one primary, all for a
dimension zero state. It follows that $|{\phi}\rangle$ has ghost number one
and picture number minus one, the standard assignments of ghost and picture
numbers for a NS string field. Correlators on the disk are nonvanishing unless
the total picture number of the operators adds to minus two and the ghost
numbers add up to three. This means that a kinetic term of the form
$\langle\phi,Q\phi\rangle$ is allowed, but a cubic interaction of the string
field requires the insertion of a PCO, an operator of ghost number zero and
picture number plus one, to get the picture number to work out. As discussed
in section 7.10, if one tries to use the associative open string vertex to
build the string field action, this approach runs into some complications with
collision of PCO’s but using more general non-associative vertices a classical
theory can be formulated.

Now we discuss a different approach leading to an action without any explicit
picture changing operators. This can be done by working in the ‘large’ Hilbert
space of the $\beta,\gamma$ conformal field theory. Recall that while
‘fermionizing’ the $\beta,\gamma$ system via $\beta=\partial\xi\,e^{-\phi}$
and $\gamma=\eta\,e^{\phi}$, the field $\xi$ appears under a derivative, and
thus its zero mode $\xi_{0}$ does not feature in the construction of the
$\beta,\gamma$ fields. Working on the large Hilbert space amounts to including
explicitly the anticommuting zero mode operator $\xi_{0}$ among the list of
operators, thus immediately doubling the number of basis states. In this
context, the conventional Hilbert space is called the ‘small’ Hilbert space,
to distinguish it from the ‘large’ Hilbert space that includes states obtained
by the action of $\xi_{0}$ on the small Hilbert space. We note that $\xi_{0}$
and the zero mode $\eta_{0}$ of the $\eta$ field satisfy the anticommutator
relation

$\\{\eta_{0},\xi_{0}\\}=1\,.$ (8.1)

The states in the small Hilbert space are those annihilated by $\eta_{0}$. Of
course, the large Hilbert space contains the small Hilbert space as a
nontrivial subspace. The BRST operator $Q$ and the $\eta_{0}$ operator are on
similar footing in that $QQ=0$ and $\eta_{0}\eta_{0}=0$. Moreover, they
anticommute,

$\\{Q,\eta_{0}\\}=0\,,$ (8.2)

and both have ghost number one. In the following table we indicate the
conformal dimension, ghost number, and picture number of the various fields

| dim | gh # | pic #
---|---|---|---
$\xi$ | $0$ | $-1$ | $+1$
$\eta$ | $1$ | $1$ | $-1$
$\beta$ | $3/2$ | $-1$ | $0$
$\gamma$ | $-1/2$ | $+1$ | $0$
$\exp(q\phi)$ | $-\tfrac{1}{2}q(q+2)$ | $0$ | $q$

As we will describe now, the NS open string field $\Phi$ to be used in the
large Hilbert space SFT is Grassmann even, and has both ghost number zero and
picture number zero. The construction of a fully gauge invariant interacting
action is done in analogy to the construction of a Wess-Zumino-Witten theory
of scalars on a two-dimensional compact space, nicely written when using
complex coordinates $z$ and $\bar{z}$. The role of the spatial derivatives
$\partial$ and $\bar{\partial}$ is played in the string field theory by the
operators $Q$ and $\eta_{0}$ respectively. The action is given by

$S={1\over
2g^{2}}\Bigl{\langle}\hskip-2.0pt\Bigl{\langle}(e^{-\Phi}\eta_{0}e^{\Phi})(e^{-\Phi}Qe^{\Phi})+\int_{0}^{1}dt\,\Phi\\{e^{-t\Phi}Qe^{t\Phi},e^{-t\Phi}\eta_{0}e^{t\Phi}\\}\Bigr{\rangle}\hskip-2.0pt\Bigr{\rangle}\,.$
(8.3)

This action is defined by expanding all exponentials in formal Taylor series
and preserving the order of operators, letting the multilinear function
$\langle\langle\cdots\rangle\rangle$ of an ordered set of operators be given
by

$\langle\langle A_{1}\ldots A_{n}\rangle\rangle=\Bigl{\langle}h^{-1}\circ
f_{1}^{(n)}\circ A_{1}(0)\cdots h^{-1}\circ f_{n}^{(n)}\circ
A_{n}(0)\Bigr{\rangle}^{\prime}_{L}\,,\ \ \ n\geq 2\,,$ (8.4)

where the functions $f_{k}^{(n)}(z)$, prescribing the conformal maps to insert
the operators, are

$f_{1}^{(n)}(z)=\Bigl{(}{1+iz\over 1-iz}\Bigr{)}^{2/n}\,,\ \ \ \
f_{k}^{(n)}(z)=e^{2\pi i(k-1)/n}f_{1}^{(n)}(z)\,,\ \ \ k=2,\ldots n\,,$ (8.5)

and $h^{-1}$ is the inverse of the map $h$ defined in (7.69), taking the disk
to the upper half plane. The subscript $L$ on the correlator indicates that
the upper half plane correlator in the definition of the multilinear product
is computed in the large Hilbert space, where we have

$\langle\xi\,c\partial c\partial^{2}ce^{-2\phi}\rangle_{L}^{\prime}\neq 0\,,$
(8.6)

telling us that for a nonvanishing correlator of a set of operators the total
ghost number must add up to two and the total picture number must add up to
minus one. Note that each term in the action has just one $Q$ and one
$\eta_{0}$. This requires a string field of zero picture number and zero ghost
number, so that each term has picture number minus one, due to the $\eta_{0}$,
and ghost number two, due to $Q$ and $\eta_{0}$.

The function $f_{1}^{(n)}$ maps the upper half disk $|z|\leq
1,\hbox{Im}(z)>0$, to the wedge $|\hbox{Arg}(f_{1}^{(n)})|\leq\pi/n,\
|f_{1}^{(n)}|\leq 1$, of a full unit disk, with the puncture $z=0$ mapped to
$f_{1}^{(n)}=1$. The maps with $k>1$ are obtained by a simple rotation of the
first wedge. All in all, $n$ non-overlapping wedges, touching along radial
lines fully fill the unit disk. This unit disk is further mapped to the upper
half plane by the map $h^{-1}$. For $n=2$ the multilinear function is in fact
the BPZ inner product. For $n=3$, it is the multilinear product of the
associative open string vertex (compare with (7.70)). For $n>3$, this
multilinear map can be thought as an iterated product using the associative
open string three-vertex:

$\langle\langle A_{1}\ldots A_{n}\rangle\rangle=\langle A_{1},A_{2}\star
A_{3}\star\cdots\star A_{n}\rangle_{L}^{\prime}\,.$ (8.7)

It is therefore cyclic.

One can show that the equation of motion from this nonlinear action takes a
surprisingly simple form:

$\eta_{0}\bigl{(}e^{-\Phi}Qe^{\Phi}\bigr{)}=0\,,$ (8.8)

where all products, obtained after expanding the exponentials, are to be
regarded as the usual star product. Moreover, the action is invariant under
gauge transformations with parameters $\Lambda$ and $\Omega$,

$\delta e^{\Phi}=(Q\Lambda)e^{\Phi}+e^{\Phi}(\eta_{0}\Omega)\,.$ (8.9)

The expansion of the action to cubic order (with $Q$ and $\eta_{0}$ killing
the constant term in the exponential) gives

$S={1\over
2g^{2}}\Bigl{\langle}\hskip-2.0pt\Bigl{\langle}\tfrac{1}{2}(Q\Phi)(\eta_{0}\Phi)+\
\tfrac{1}{6}(Q\Phi)\bigl{(}\Phi(\eta_{0}\Phi)-(\eta_{0}\Phi)\Phi\bigr{)}\Bigr{\rangle}\hskip-2.0pt\Bigr{\rangle}\,+{\cal
O}(\Phi^{3})\,.$ (8.10)

From the above expansion we see that the kinetic term takes the form

$S_{\rm
kin}\sim\langle\hskip-2.0pt\langle(Q\Phi)\,(\eta_{0}\Phi)\rangle\hskip-2.0pt\rangle=\langle
Q\Phi\,,\eta_{0}\Phi\rangle_{L}^{\prime}\,,$ (8.11)

with the bilinear function equal to the BPZ inner product. This means that the
linearized equation of motion is

$Q\eta_{0}|{\Phi}\rangle=0\,.$ (8.12)

Consistent with the full gauge transformations (8.9), we have two linearized
gauge invariances:

$\delta\Phi=Q|{\Lambda}\rangle\,,\ \ \ \delta\Phi=\eta_{0}|{\Omega}\rangle\,.$
(8.13)

To analyze the free field equation, we expand the string field relative to the
zero mode $\xi_{0}$:

$|{\Phi}\rangle=|{\phi^{\prime}}\rangle+\xi_{0}|{\phi}\rangle\,,\ \ \
\hbox{with}\ \ \eta_{0}|{\phi^{\prime}}\rangle=\eta_{0}|{\phi}\rangle=0\,.$
(8.14)

Here $|{\Phi}\rangle$ is in the large Hilbert space, while $|{\phi}\rangle$
and $|{\phi^{\prime}}\rangle$ are in the small Hilbert space. The $\eta_{0}$
gauge symmetry is, explicitly

$\delta|{\Phi}\rangle=\eta_{0}|{\Omega}\rangle=\eta_{0}(|{\omega}\rangle+\xi_{0}|{\omega^{\prime}}\rangle)=|{\omega^{\prime}}\rangle\,.$
(8.15)

This can be used to gauge away $|{\phi^{\prime}}\rangle$ in $|{\Phi}\rangle$
so that we have $|{\Phi}\rangle=\xi_{0}|{\phi}\rangle$. The equation of motion
then becomes

$0=Q\eta_{0}|{\Phi}\rangle=Q\eta_{0}\xi_{0}|{\phi}\rangle=Q\\{\eta_{0},\xi_{0}\\}|{\phi}\rangle=Q|{\phi}\rangle\,.$
(8.16)

Since $|{\phi}\rangle$ is in the small NS Hilbert space and satisfies the
familiar linearized field equation, we conclude that $|{\phi}\rangle$ is a
state of ghost number one and picture number minus one. Given that the string
field $|{\Phi}\rangle$ is obtained by acting with $\xi_{0}$ on
$|{\phi}\rangle$ we have, as expected, that the large Hilbert space NS open
string field $\Phi$ has ghost and picture number zero.

The NS sector open string field theory described here can be shown to be
equivalent to the one described in section 7.10 after partial gauge fixing and
appropriate field redefinition[138, 139, 140, 141]. BRST quantization of this
theory has been discussed in [142].

###  8.2 Heterotic string field theory in the NS sector

In this theory the holomorphic sector is the NS sector of open superstrings
while the anti-holomorphic sector is that of a bosonic open string. We are
thus effectively tensoring a string field that is Grassmann even and of ghost
and picture number zero (NS large Hilbert space string field) with a string
field that is Grassmann odd and of ghost number one (bosonic open string
field). As a result, the heterotic string field $V$ will be a Grassmann odd
string field of ghost number one and picture number zero, anniihilated by
$b_{0}^{-}$ and $L_{0}^{-}$. In vertex operator language, $V$ for physical
states would be represented by $\xi c\bar{c}V_{M}e^{-\phi}$, where $V_{M}$ is
a matter sector primary operator of dimension $(1,1/2)$. This is clearly an
operator carrying zero picture number and ghost number one.

The basic nonvanishing correlator in the large Hilbert space is now

$\langle\xi\,c\partial
c\partial^{2}c\,\bar{c}\partial\bar{c}\partial^{2}\bar{c}\,e^{-2\phi}\rangle_{L}\neq
0\,,$ (8.17)

showing that operators with a non-vanishing correlator must have total ghost
number five and total picture minus one. Since the closed string inner product
$\langle A,B\rangle_{L}$ includes a $c_{0}^{-}$ factor, a non-vanishing inner
product requires the ghost numbers of $A$ and $B$ to add up to four, and the
picture numbers to add up to minus one.

The tree-level action to cubic order in the string field $V$ takes the form
[25]:

$S\,=\,{2}\,\Bigl{[}\,\frac{1}{2}\langle\eta_{0}V,QV\rangle_{L}+\frac{\kappa}{3!}\langle\eta_{0}V,[V,QV]_{0}\rangle_{L}\Bigr{]}+{\cal
O}(V^{3})\,,$ (8.18)

where $\kappa$ is related to $g_{s}$ by a constant of proportionality that
depends on the normalization in (8.17). In the above, the bracket $[\ ,\
]_{0}$ is the genus zero contribution to the product. Together with the higher
genus zero products appearing below, they represent any consistent solution of
the $L_{\infty}$ axioms in closed bosonic string theory, e.g. the products
constructed using the minimal area metric or hyperbolic metric. The kinetic
term implies a linearized equation of motion $\eta_{0}QV=0$ with linearized
gauge invariances

$\delta V=Q\Lambda+\eta_{0}\Omega\,,$ (8.19)

The construction of the full nonlinear action and its gauge transformations is
somewhat involved. A key ingredient is the generalization of the structure
$A_{Q}\equiv e^{-\Phi}Qe^{\Phi}$ appearing in the open superstring action. The
insight comes from the observation that $A_{Q}$ is a pure gauge configuration
in bosonic open string theory: it is of ghost number one, to leading order it
is $Q\Phi$ and it satisfies the cubic equation of motion
$QA_{Q}+A_{Q}A_{Q}=0$. The heterotic string field $V$ of ghost number one and
picture zero can play the role of a gauge parameter in bosonic closed string
theory. Thus $QV$ is the leading term of a pure gauge configuration in closed
string field theory. To find the full nonlinear expression $G(V)$ of the pure
gauge we introduce an integration parameter $\tau$ and the function $G(\tau
V)$ that gives the desired $G(V)$ for $\tau=1$ and vanishes for $\tau=0$.
Moreover, we require that $G(\tau V+d\tau V)$ and $G(\tau V)$ differ by a
gauge transformation with parameter $d\tau V$ applied to $G(\tau V)$.
Therefore,

$G(\tau V+d\tau V)=G(\tau V)+Q(d\tau V)+\sum_{n=1}^{\infty}{\kappa^{n}\over
n!}[G(\tau V)^{n}\,,d\tau V]_{0}+{\cal O}(d\tau^{2})\,.$ (8.20)

This gives the differential equation

$\partial_{\tau}G(\tau V)=QV+\sum_{n=1}^{\infty}{\kappa^{n}\over n!}[G(\tau
V)^{n}\,,V]_{0}\,.$ (8.21)

This equation, written more schematically as

$\partial_{\tau}G=QV+\kappa[G,V]_{0}+{\kappa^{2}\over 2}[G,G,V]_{0}+{\cal
O}(\kappa^{3})$ (8.22)

can be solved in a power series expansion

$G=G^{(0)}+\kappa G^{(1)}+\kappa^{2}G^{(2)}+{\cal O}(\kappa^{3})\,.$ (8.23)

The first few terms are quickly calculated

$G(V)=QV+{\kappa\over 2}[V,QV]_{0}+{\kappa^{2}\over
3!}\bigl{(}[V,QV,QV]_{0}+[V,[V,QV]_{0}]_{0}\bigr{)}+{\cal O}(\kappa^{3})\,.$
(8.24)

Since $G(V)$ is obtained from 0 by a series of infinitesimal gauge
transformations, it should be a pure gauge configuration. This can be
confirmed by showing that $G(V)$ is a solution of the bosonic closed string
field theory equation of motion $F(\Psi)=0$. This is done by forming a first
order linear differential equation for $F(G(\tau V))$, taking the form
$\partial_{\tau}F(G(\tau V))\sim F(G(\tau V))$. Since $G(\tau V)$ vanishes for
$\tau=0$ we also have $F(G(0))=0$ and by uniqueness, the solution of the
differential equation is $F(G(\tau V))=0$. For $\tau=1$ this is the claimed
property.

The full nonlinear action can be written in a number of equivalent ways,
useful for simple derivations of key properties. Perhaps the simplest form is
as follows ([26], section 5)252525A similarly simple form of the open string
field theory action also exists, and is given in [26], section 2.2.

$S={2}\,\int_{0}^{1}dt\,\langle\eta_{0}V\,,G(tV)\rangle_{L}\,.$ (8.25)

One can also determine the full nonlinear extensions of the linearized gauge
transformations (8.19); for details see [26].

###  8.3 Type II string field theory in the NS–NS sector

For type II closed string field theory in the NS–NS sector, physical states in
the large Hilbert space can be represented by operators of the form
$\xi\bar{\xi}c\bar{c}e^{-\phi}e^{-\bar{\phi}}V_{M}$ that have left-moving and
right moving ghost numbers equal to zero as well as left-moving and right
moving picture numbers equal to zero. Experience with bosonic closed string
theory suggests that the string field must have simply zero ghost number,
without imposing two separate conditions; the inner product, for example
contains the factor $c_{0}^{-}$ of indefinite left and right ghost numbers. On
the other hand, it seems natural to fix separately the left and right picture
numbers. Thus, the Grassmann even string field $\Psi$ will have ghost number
zero and left and right picture numbers zero.

The basic nonvanishing correlator for type II theory in the large Hilbert
space is

$\langle\,\xi\bar{\xi}\,c\partial
c\partial^{2}c\,\,\bar{c}\bar{\partial}\bar{c}\bar{\partial}^{2}\bar{c}\,e^{-2\phi}e^{-2\bar{\phi}}\rangle_{L}\neq
0\,,$ (8.26)

which means that for a nonvanishing correlator of several operators ghost
numbers must add up to 4, and both left and right picture numbers must add up
to minus one. Therefore, in the bilinear form $\langle A,B\rangle$, the ghost
numbers of $A$ and $B$ must add up to three, and the left and right picture
numbers must add up to minus one. The kinetic term for the theory consistent
with these requirements is

$S_{2}=\tfrac{1}{2}\langle\eta_{0}\Psi\,,\,Q\bar{\eta}_{0}\Psi\rangle_{L}\,.$
(8.27)

The linearized equation of motion is $\eta_{0}\bar{\eta}_{0}Q\Psi=0$ with
linearized gauge invariances

$\delta\Psi=Q\Lambda+\eta_{0}\Omega+\bar{\eta}_{0}\bar{\Omega}\,.$ (8.28)

We can expand the string field
$|{\Psi}\rangle=|{\psi_{0}}\rangle+\xi_{0}|{\psi_{1}}\rangle+\bar{\xi}_{0}|{\psi_{2}}\rangle+\xi_{0}\bar{\xi}_{0}|{\psi_{3}}\rangle$
with $|{\psi_{i}}\rangle$, with $i=0,\ldots,3$, in the small Hilbert space and
thus annihilated by both $\eta_{0}$ and $\bar{\eta}_{0}$. The $\Omega$ gauge
invariance can be used to set to zero $|{\psi_{0}}\rangle$ and
$|{\psi_{2}}\rangle$. Then, the $\bar{\Omega}$ gauge invariance can be used to
set $|{\psi_{1}}\rangle$ to zero. Thus we have
$|{\Psi}\rangle=\xi_{0}\bar{\xi}_{0}|{\psi_{3}}\rangle$, with
$|{\psi_{3}}\rangle$ in the small Hilbert space and in the minus one, minus
one picture. The linearized equation of motion then implies
$Q|{\psi_{3}}\rangle=0$, as required.

The cubic term of this theory begins to show some of the complexity. One
finds:

$S_{3}=\tfrac{1}{3!}\,\langle\eta_{0}\Psi\,,[Q\bar{\eta}_{0}\Psi,\bar{\eta}_{0}\Psi\,]^{*}\,\rangle_{L}\,,$
(8.29)

where the starred two-product must have an insertion of the PCO zero mode
$\bar{\cal X}_{0}=\\{Q,\bar{\xi}_{0}\\}$ to compensate for the presence of the
two $\bar{\eta}_{0}$ factors. The $\bar{\cal X}_{0}$ is inserted on the three
legs of the product:

$[A,B]^{*}\equiv\tfrac{1}{3}\bigl{(}\bar{\cal X}_{0}[A,B]_{0}+[\bar{\cal
X}_{0}A,B]_{0}+[A,\bar{\cal X}_{0}B]_{0}\bigr{)}\,.$ (8.30)

Since $\\{Q,\bar{\cal X}_{0}\\}=0$, we see that $Q$ is a derivation of the
starred product. Since $\\{\eta_{0},\bar{\cal
X}_{0}\\}=\\{\bar{\eta}_{0},\bar{\cal X}_{0}\\}=0$, both $\eta_{0}$ and
$\bar{\eta}_{0}$ are derivations of the starred product. A full type II action
was built by Matsunaga [27] and takes the form

$S=\int_{0}^{1}dt\,\langle\eta_{0}\Psi_{t}\,,\,{\cal
G}^{*}(\Psi(t))\rangle_{L}\,.$ (8.31)

Here, with $\Psi(t)=t\Psi$, we have that the NS–NS
$\Psi_{t}=\Psi+\bar{\eta}_{0}(\cdots)$

$\Psi_{t}=\Psi+\tfrac{\kappa}{2}\,t\,[\bar{\eta}_{0}\Psi,\Psi]^{*}\,+{\cal
O}(\kappa^{2})\,,$ (8.32)

and

${\cal
G}^{*}(\Psi)=Q\bar{\eta}_{0}\Psi+\tfrac{\kappa}{2}\,t\,[Q\bar{\eta}_{0}\Psi,\bar{\eta}_{0}\Psi]^{*}\,+{\cal
O}(\kappa^{2})\,.$ (8.33)

The complete definition of ${\cal G}^{*}(\Psi)$ for the choice $\Psi(t)=t\Psi$
is as follows. Under an infinitesimal change $t\to t+\delta t$, ${\cal
G}^{*}(\Psi(t+\delta t))$ is obtained from ${\cal G}^{*}(\Psi(t))$ by an
infinitesimal gauge transformation with parameter $\delta
t\,\bar{\eta}_{0}\Psi$ in the heterotic string field theory formulated in the
small Hilbert space in the anti-holomorphic superstring sector. Recall that
the string field for the small Hilbert space heterotic string is of ghost
number two and picture number minus one. Indeed $(\bar{\eta}_{0}\Psi)$,
playing the role of the gauge parameter in the above equation for ${\cal
G}^{*}$, has ghost number one and picture number minus one. The computation of
${\cal G}^{*}$ uses the $L_{\infty}$-type heterotic action discussed in [28],
with starred products bearing $\bar{\cal X}_{0}$ insertions in rather
intricate combinatoric patters and satisfying the $L_{\infty}$ constraint
equations. The complete definition of $\Psi_{t}$ is a bit more complicated but
can be found in [27].

###  8.4 Open superstring field theory in the Ramond sector

So far the construction described in this section deals only with the NS
sector of string field theory. At the classical level this is a consistent
truncation of the full string field theory. Nevertheless even at the classical
level, a complete construction of string field theory must include the Ramond
sector. In this section we shall describe an extension of the open superstring
field theory described in section 8.1 that includes the Ramond sector[29, 30].

The NS sector of this string field theory is identical to the one described in
section 8.1. For introducing the Ramond sector fields, we shall work in the
small Hilbert space. Let us recall some basic facts about the R-sector. The
$(\beta,\gamma)$ ghosts admit mode expansions

$\beta(z)=\sum_{n\in{\mathbb{Z}}}\beta_{n}z^{-n-{3\over
2}},\qquad\gamma(z)=\sum_{n\in{\mathbb{Z}}}\gamma_{n}z^{-n+{1\over 2}}\,,$
(8.34)

with the vacua $|-1/2\rangle=e^{-\phi/2}(0)|0\rangle$,
$|-3/2\rangle=e^{-3\phi/2}(0)|0\rangle$ satisfying,

$\beta_{n}|-1/2\rangle=0\ \hbox{for $n\geq 0$},\qquad\gamma_{n}|-1/2\rangle=0\
\hbox{for $n\geq 1$}\,.$ (8.35)

We denote by $G_{0}^{m}$ the zero mode of the matter sector superconformal
generator and define,

$X=-\delta(\beta_{0})G^{m}_{0}+\delta^{\prime}(\beta_{0})b_{0},\qquad
Y=-c_{0}\delta^{\prime}(\gamma_{0})\,.$ (8.36)

The R-sector string field $\Psi$ is taken to be a state of ghost number 1 and
picture number $-1/2$ subject to the restriction that $\Psi$ must be of the
form

$\Psi=\phi-(\gamma_{0}+2c_{0}b_{0}\gamma_{0}+c_{0}G_{0}^{m})\psi\,,$ (8.37)

with $\phi$, $\psi$ satisfying the constraints:

$b_{0}\phi=0,\quad\beta_{0}\phi=0,\quad\eta_{0}\phi=0,\quad
b_{0}\psi=0,\quad\beta_{0}\psi=0,\quad\eta_{0}\psi=0\,.$ (8.38)

This condition on $\Psi$ may also be expressed as

$XY\Psi=\Psi,\qquad\eta_{0}\Psi=0\,,$ (8.39)

where the second condition just expresses the fact that $\Psi$ is in the small
Hilbert space. We have written this condition explicitly since later we shall
couple the R-sector fields to the NS sector fields and the latter belong to
the large Hilbert space.

The quadratic term in the string field theory action in the R-sector is given
by

$S^{(0)}_{R}=-\tfrac{1}{2}\langle\Psi,YQ\Psi\rangle^{\prime}\,,$ (8.40)

where the absence of the subscript $L$ stands for the fact that the BPZ inner
product is computed in the small Hilbert space. The action is invariant under
the linearized gauge transformation,

$\delta|\Psi\rangle=Q|\lambda\rangle\,,$ (8.41)

where $|\lambda\rangle$ is a ghost number 1, picture number $-1/2$ operator,
satisfying.

$XY\lambda=\lambda,\qquad\eta_{0}\lambda=0\,.$ (8.42)

The full interacting tree-level open string field theory involves the R-sector
string field $\Psi$ of ghost number one and picture number $-1/2$ in the small
Hilbert space satisfying (8.39) and the NS sector string field $\Phi$ of ghost
number zero and picture number zero in the large Hilbert space. The action is
given by

$S=-\tfrac{1}{2}\langle\Psi,YQ\Psi\rangle^{\prime}-\int_{0}^{1}dt\,\langle
A_{t}(t),\ QA_{\eta}(t)+(F(t)\Psi)^{2}\rangle_{L}^{\prime}\,,$ (8.43)

where

$A_{\eta}(t)=\eta e^{\Phi(t)}e^{-\Phi(t)},\qquad
A_{t}(t)=\partial_{t}e^{\Phi(t)}e^{-\Phi(t)}\,,$ (8.44)

and

$F(t)\Psi=\Theta(\beta_{0})\\{A_{\eta}(t),\Theta(\beta_{0})\\{A_{\eta}(t),\cdots\Theta(\beta_{0})\\{A_{\eta}(t),\Psi\\}\cdots\\}\\}\,,$
(8.45)

$\Theta$ denotes the Heaviside function and $\Phi(t)$ is subject to the
boundary condition that $\Phi(1)=\Phi$. The products of the string fields
appearing in this expression (including those obtained by expanding the
exponentials) are to be interpreted as star products. One can show that the
action (8.43) depends only on $\Psi$ and the boundary value $\Phi$ of
$\Phi(t)$. The full non-linear gauge symmetries of this action can be found in
[29].

Note that the construction described above has two main ingredients: use of
the Ramond sector string field in the restricted Hilbert space (8.39) and the
construction of the gauge invariant coupling between the R-sector states and
the NS sector states in the large Hilbert space. One could choose to use one
and not the other and combine this with some other construction to write down
gauge invariant string field theory action. This is precisely what was done in
[133, 134] where the authors used the R-sector string fields in the restricted
Hilbert space and the NS sector string fields in the small Hilbert space to
write down a gauge invariant string field theory action with cyclic
$A_{\infty}$ structure. Ref.[143] has formulated a different version of open
superstring field theory based on integrals over supermoduli space. We shall
not discuss these developments in this review in any further detail.

##  9 Applications of string field theory

String field theory has had a number of applications that go beyond what the
world-sheet formulation of string theory can achieve. In this section we shall
review some of these applications. Tachyon condensation, for example involves
non-perturbative physics. We will review this subject here, including the
construction of the tachyon vacuum solution. There are a number of other
important solutions of open string field theory that we will not be able to
discuss here [144, 145, 146, 147, 148, 149, 150].

We will also consider a number of topics in string perturbation theory that
require string field theory to resolve a number of ambiguities in the world-
sheet approach to the theory. Indeed, we will consider mass renormalization
and vacuum shift. We will also review recent work on D-instanton contributions
to closed string amplitudes. Finally, we will sketch the arguments for
unitarity, crossing symmetry of string amplitudes, as well as for the
ultraviolet finiteness of the theory.

There have been recent efforts to use string field theory to carry out
conformal perturbation theory calculations that beyond leading orders become
extremely subtle and possibly ambiguous. This is an important area of future
research that we will not be able to cover here [151, 152, 153, 82].

With encouraging progress to date, it remains a goal to understand or improve
string field theory to a degree that it gives a fully non-perturbative
formulation of string theory.

###  9.1 Tachyon condensation

Tachyon condensation is one of the first applications of string theory to
problems that cannot be addressed using the standard world-sheet description
of scattering amplitudes. Since by now there are plenty of reviews on this
subject (see e.g. [12] for analytic approaches and [14] for numerical
approaches), we shall keep our discussion brief.

Open strings describe the excitations on the D-branes. Many D-branes in string
theory have tachyonic excitation in the open string sector – excitations with
negative mass-squared. This includes not only the D-branes in the bosonic
string theory, but also unstable D-brane systems in superstring theory, e.g. a
coincident pair of a BPS brane and its anti-brane, or non-BPS D-branes[154].

In perturbative quantum field theory, the existence of a tachyonic excitation
is associated with the presence of a scalar field whose potential has a
maximum at the background value of the field around which we carry out the
perturbation expansion. If we study the theory by assuming that the
fluctuations of the fields around this background remain small, we would
conclude that there are tachyons in the spectrum. This, however, is not
correct because the assumption that the fluctuations of the fields around the
maximum of the potential can remain small is clearly wrong. The correct way to
study such theories is to first identify a (local) minimum of the potential
and then expand the potential around the minimum and quantize the theory. This
will describe excitations with non-negative mass-squared. Therefore, in
reality there are no tachyonic modes in the spectrum.

It is generally expected that the open string tachyons on D-branes have
similar origin, namely that they are the result of attempting to quantize the
system by expanding the potential around a maximum. However in the world-sheet
description of string theory there is no systematic way to find the minimum of
the potential and quantize the system by expanding the potential around the
minimum. Nevertheless, using indirect arguments, one could make a guess about
various properties of the minimum. These claims are the content of the
following conjectures [155, 52]:

  1. 1.

The minimum of the tachyon potential on an unstable brane system, not
protected by any conservation laws, describes a configuration where the brane
disappears altogether, leaving just the closed string vacuum. Examples of such
systems are D-branes in bosonic string theory and unstable D-branes or brane-
anti-brane systems in superstring theory. One immediate consequence of this is
that the difference in the height of the potential between the perturbative
vacuum describing the original unstable brane system and the minimum of the
potential must be equal to the tension of the original brane system.

  2. 2.

Since the minimum describes a configuration without any D-branes, there are no
open string excitations around the minimum.

  3. 3.

There are also non-trivial soliton solutions where the tachyon approaches the
minimum of the potential asymptotically, but takes a complicated form in the
interior. These are expected to describe lower dimensional D-branes. Examples
of such solutions are lump solutions in bosonic string D-branes describing
lower dimensional D-branes, a kink solution on a type II string non-BPS
D-brane describing a BPS D-brane of one lower dimension, vortex solution on a
brane-antibrane pair describing a BPS D-brane of two lower dimensions, etc.

In special cases the third conjecture may be studied using conformal field
theory (CFT) techniques, since both the initial configuration and the final
configuration, being D-branes, have a description in CFT. Studying the first
two conjectures is harder since there is no good description of the tachyon
vacuum in conformal field theory.262626Some understanding of this may be
obtained using the boundary string field theory[156, 157, 158] formulated in
[159, 160]. We shall not discuss this here but refer the reader to the
original literature. Open string field theory is well suited for studying all
the conjectures since we can simply study classical solutions and their
properties in string field theory.

The main difficulty in studying the conjectures arises because the tachyon
field is coupled to all the other fields, and so a systematic study requires
studying classical equations of motion of a field theory with infinite number
of fields. The initial study of this problem [161] was done using the ‘level
truncation approach’ a method first tried by Kostelecky and Samuel in open
string field theory [162, 163] before the tachyon conjectures had been
formulated. In level truncation, we truncate the string field to a finite set
of fields based on the $L_{0}$ eigenvalue of the state that they multiply. We
shall illustrate this with the example of unstable D-branes in bosonic string
theory. At the leading order we keep just the tachyon field
$\phi\,c_{1}|0\rangle$ in the expansion of the string field since
$c_{1}|0\rangle$ is the state of lowest $L_{0}$ eigenvalue. Also since we are
looking for a translationally invariant solution, we keep only the zero
momentum state. Substituting this into the cubic open string field theory
action one gets,

$-S=V\,g_{o}^{-2}\Bigl{[}-\tfrac{1}{2}\phi^{2}+\tfrac{1}{3}\left(\tfrac{3\sqrt{3}}{4}\right)^{3}\phi^{3}\Bigr{]}\,,$
(9.1)

where $V=(2\pi)^{D}\delta^{(p+1)}(0)$ is the space-time volume of the D-brane.
The open string fields appearing in the cubic version of (4.43) have been
scaled by a factor of $g_{o}^{-1}$ to bring out an explicit factor of
$g_{o}^{-1}$ multiplying a $g_{o}$ independent action. The
$\left({3\sqrt{3}/4}\right)^{3}$ factor comes from the conformal
transformations needed to construct the cubic interaction vertex – these have
been given in (4.54), (4.55). On the other hand, as mentioned in (4.44) and
proved in (5.35), the open string coupling $g_{o}$ can be shown to be related
to the D-brane tension ${\cal T}$ via the relation

${\cal T}={1\over 2\pi^{2}g_{o}^{2}}\,.$ (9.2)

The potential in (9.1) has a local minimum at $\phi=(4/3\sqrt{3})^{3}$, where
it takes value

$-S=-V\,g_{o}^{-2}\,\tfrac{1}{6}\,\left(\tfrac{16}{27}\right)^{3}=-V\,{\cal
T}\,2\pi^{2}\,\tfrac{1}{6}\,\left(\tfrac{16}{27}\right)^{3}\simeq-0.68\,V\,{\cal
T}\,.$ (9.3)

According to the first conjecture, the expected result is $-V\,{\cal T}$. So
we are not very far from the expected result.

The analysis at higher order is facilitated using the observation that
classical open string field theory admits a consistent truncation in which we
set to zero all open string fields associated with excitations on non-trivial
matter primary. Such states enter the open string field theory action in pairs
and triplets but not singly, and it is therefore consistent to set them to
zero. The truncated string field, containing excitations by matter Virasoro
generators $L^{m}_{-n}$ and ghost oscillators acting on the vacuum state, is
known as the universal sector. Since the zero momentum tachyon belongs to the
universal sector, we can try to extend the solution to higher levels remaining
within the universal sector. A further simplification occurs by the use of
‘twist symmetry’ that allows us to restrict to string fields multiplying
states with odd $L_{0}$ eigenvalue, the tachyon being the one that multiplies
a state of $L_{0}$ eigenvalue $-1$. At the next order, working in the Siegel
gauge, we include two more fields, describing the coefficient of
$c_{1}L^{m}_{-2}|0\rangle$ and $c_{-1}|0\rangle$. One can construct the open
string field theory action with this truncated string field and find the
extremum of the action. The result improves dramatically, yielding about 95%
of the expected answer[161]. One can also check that even though we construct
the solution in the Siegel gauge, it also satisfies the equations of motion of
the out of Siegel gauge fields to very good accuracy. This procedure has now
been extended to very high level, yielding results very close to the expected
result[164, 165]. This procedure has also been generalized to describe tachyon
vacuum solution on unstable D-branes of type II string theory using Berkovits
formulation of open superstring field theory[166, 167]. One surprising feature
of the analysis based on level truncation is that there is no small parameter
in which we carry out the expansion, – a priori the contribution due to the
higher level fields could be as large as that from the lower level fields. Yet
the level truncation seems to converge rapidly.

One can also use a slightly modified version of this procedure to describe
soliton solutions in open (super-) string field theory describing lower
dimensional D-branes[168]. Instead of working with a non-compact world-volume
of the parent D-brane, we compactify certain directions that would eventually
be transverse to the soliton. Then the momenta carried by the string field in
these directions is quantized and the expansion of string fields carrying
momenta in these directions also admit a discrete level expansion, with higher
level fields multiplying states of higher $L_{0}$ eigenvalues. We can now
proceed as before to construct classical solutions in level expansion that
describe lower dimensional D-branes and check if the action associated with
the solution accounts for the difference in the value of the action for the
original D-brane and the lower dimension D-branes that we want to construct.
In all the cases that have been studied, the results come close to the
expected values.

Except for the leading order results, most of the analysis in the level
truncation method has been carried out numerically, since analytical solution
of algebraic equations of many variables is hard to find. The situation
changed dramatically in 2005 when Martin Schnabl [169] wrote down an analytic
solution in bosonic open string field theory describing the tachyon vacuum,
and showed analytically that the energy density of this solution exactly
cancels the tension of the original D-brane. The solution was written down not
in the Siegel gauge but in a different gauge that has come to be known as the
Schnabl gauge. Since then different versions of the solution, related by gauge
transformation to the original solution, have been constructed and the
algebraic structure underlying the construction of the solution has been
understood [170, 171]. This has been reviewed in detail in [12]. We now also
have a systematic procedure for describing any configuration of arbitrary
number of static D-branes as a classical solution in the open string field
theory on a specific D-brane as long as all of them share the same closed
string background[147, 174]. The tools used are surface states, including the
so-called wedge states, and the $K,b,c$ algebra of string fields [170], both
of which will be discussed below.

At present we do not have a complete understanding of the relation between the
Siegel gauge solution using level truncation and Schnabl’s analytic solution.
It will definitely be helpful to understand this relation better.

###  9.2 Tachyon vacuum solution and the ${\cal K}$bc algebra.

We give here some of the basic ideas that go into the construction of
classical open string field theory solutions, focusing on the tachyon vacuum
solution. As mentioned above, this is just one of many solutions known at this
point. For more details, consult the references above.

Solutions of OSFT are visualized as surface states dressed with some
insertions of CFT operators. Surface states are the basic ingredient, they
simply associate a surface with a puncture and a chosen local coordinate at
the puncture to a state of the string. In our case, we have a surface $\Sigma$
which is topologically a disk, a chosen point $P$ on the boundary of $\Sigma$,
and a local coordinate at $P$, namely, a map $f$ from a canonical half disk
$\hbox{Im}(w)>0,|w|\leq 1$, to the surface $\Sigma$ equipped with a coordinate
$z$. We have $z(P)=f(w=0)$ and the real boundary of the half disk is mapped to
(part of) the boundary of $\Sigma$ (see Fig. 16). The surface state
$\langle{\Sigma}|$ is defined through overlaps

$\langle{\Sigma}|\phi\rangle^{\prime}=\langle\phi(w=0)\rangle^{\prime}_{\Sigma}=\langle
f\circ\phi(0)\rangle^{\prime}_{\Sigma}\,.$ (9.4)

The state $|{\Sigma}\rangle$ just follows from BPZ conjugation. The surface in
the surface state is best visualized as $\Sigma^{\prime}$, the surface defined
as $\Sigma$ minus the image of the coordinate half disk. This removal induces
a parameterized boundary, ‘the’ open string. Of course $\Sigma^{\prime}$ still
has part of the standard boundary of $\Sigma$, where the open string boundary
conditions hold. The left (L) and right (R) parts of the open string, viewed
from $\Sigma^{\prime}$, are the images under $f$ of $|w|=1$ with
$\hbox{arg}(w)\in[0,\pi/2]$ and $\hbox{arg}(w)\in[\pi/2,\pi]$, respectively.
The open string midpoint is $Q$ (Fig. 16).

Figure 16: A surface state $|{\Sigma}\rangle$ associated to the surface
$\Sigma$ which has a puncture $P$ at the boundary and a local coordinate
around it. This local coordinate is a map from the $\xi$ upper-half disk to
$\Sigma$. Shown below is the surface $\Sigma^{\prime}$, with the coordinate
patch removed, introducing a new boundary composed by the left and right parts
($L$ and $R$) of an open string.

The star multiplication of two surface states $|{\Sigma_{1}}\rangle$ and
$|{\Sigma_{2}}\rangle$ is also a surface state
$|{\Sigma_{1}\star\Sigma_{2}}\rangle$. This surface
$(\Sigma_{1}\star\Sigma_{2})^{\prime}$, is obtained by gluing the $R$ half-
string in $\Sigma_{1}^{\prime}$ to the $L$ half-string of
$\Sigma_{2}^{\prime}$. In the surface $(\Sigma_{1}\star\Sigma_{2})^{\prime}$
the $L$ part of the string is that of $\Sigma_{1}^{\prime}$ and the $R$ part
of the string is that of $\Sigma_{2}^{\prime}$. With this rule, it is simple
to visualize the star product of surface states. If one wishes, one can add to
$(\Sigma_{1}\star\Sigma_{2})^{\prime}$ a coordinate half-disk consistent with
its $L$ and $R$ parts, in this way getting the full surface
$\Sigma_{1}\star\Sigma_{2}$.

There is plenty of flexibility in defining surface states here: we must choose
how to present the disk $\Sigma$ and how to specify the local coordinates.
This can be illustrated with the example of ‘wedge states’, surface states
characterized by a real parameter $\alpha\geq 0$. The full surface, called
$C_{\alpha+1}$, for a cylinder of circumference $\alpha+1$, is the region in
the $z$ UHP with $-\tfrac{1}{2}\leq\hbox{Re}(z)\leq\alpha+\tfrac{1}{2}$ (see
Fig. 17). The vertical lines at $-\tfrac{1}{2}$ and $\alpha+\tfrac{1}{2}$ are
identified via $z\sim z+\alpha+1$. Viewed as a disk, the ‘center’ is the point
at $i\infty$ and the boundary is the real segment
$z\in[-\tfrac{1}{2},\tfrac{1}{2}+\alpha]$. The surface is, of course, a semi-
infinite cylinder, or just a cylinder. The puncture is at the origin $z=0$,
and the local coordinate patch is the region
$-\tfrac{1}{2}\leq\hbox{Re}(z)\leq\tfrac{1}{2}$, a unit width vertical strip,
where we can identify the $L$ and $R$ edges of the surface state as the
vertical lines $\hbox{Re}(z)=\tfrac{1}{2}$ and $\hbox{Re}(z)=-\tfrac{1}{2}$,
respectively. The local coordinate $w$ on the strip is related to $z$ as
follows

$z=\,\tfrac{2}{\pi}\,\hbox{arctan}\,w\,.$ (9.5)

This indeed maps the $w$ upper-half disk, to the unit width vertical strip,
also sending $w=0$ to $z=0$, the position of the puncture, and $w=i$ to
$z=i\infty$, the position of the open string mid-point. This unit-strip
picture of the half disk is sometimes called the sliver frame. The surface
$\Omega_{\alpha}^{\prime}$ is then just the region of width $\alpha$ to the
right of the coordinate strip. Its left boundary is the half-string $L$, and
by the identification, its right boundary is the half string $R$. This is the
wedge state $\Omega_{\alpha}^{\prime}$ or sometimes simply called
$\Omega_{\alpha}$ (Fig. 17).

Figure 17: The wedge state $\Omega_{\alpha}$. The coordinate patch is shown
shaded and the left-most and right-most boundaries are identified to form the
cylinder $C_{\alpha+1}$. The state $\Omega^{\prime}_{\alpha}$ has the
coordinate patch removed and has boundaries $L$ and $R$, for the left- and
right halves of the open string.

It should now be clear that we have the star product[175]:

$|{\Omega_{\alpha}}\rangle\star|{\Omega_{\beta}}\rangle=|{\Omega_{\alpha+\beta}}\rangle\,,$
(9.6)

since gluing a region of width $\alpha$ to a region of width $\beta$ gives a
region $\Sigma^{\prime}$ of width $\alpha+\beta$. Note that the full surface
$\Sigma$, obtained by adding the unit-width coordinate strip, is a cylinder of
circumference $\alpha+1$. A few wedge states are easily recognized. The wedge
state with $\alpha=0$ is the identity string field ${\cal I}$, this is clear
because the surface $\Omega_{0}^{\prime}$ is just a strip of vanishing length,
which changes no state by star multiplication. The wedge state for $\alpha=1$
is actually the SL(2,$\mathbb{R}$) vacuum $|{\Omega_{1}}\rangle=|{0}\rangle$;
this can be seen by mapping the region $-1/2\leq{\rm Re}(z)\leq 3/2$, with the
identification $z\equiv z+2$, to the full upper half $w$ plane via the map
(9.5). Finally the state obtained in the limit $\alpha=\infty$ turns out to be
well defined and it is called the sliver state $|{\Omega_{\infty}}\rangle$.
The sliver is a rank one projector of the star algebra, satisfying the
expected property
$|{\Omega_{\infty}}\rangle\star|{\Omega_{\infty}}\rangle=|{\Omega_{\infty}}\rangle$.
A few references that discuss manipulations involving wedge states and related
surface states, with special emphasis on projectors, are [176, 177, 178, 179,
180].

The next step in this construction is to understand how to change the width of
a wedge state via the action of an operator. For this consider an
arbitrary272727We follow the discussion in [12], with a small modification
that does not require the operator $\phi$ to carry definite conformal weight.
local boundary operator $\phi$ and the overlap

$\langle\phi,\Omega_{\alpha}\rangle^{\prime}=\langle
f\circ\phi(0)\rangle^{\prime}_{C_{\alpha+1}}\,,$ (9.7)

where $f(w)$ is the map appearing on the right hand side of (9.5). We evaluate
the right-hand side by first doing a conformal scaling transformation $z\to
f_{\alpha}(z)\equiv z/(\alpha+1)$ that turns the surface into a cylinder of
unit circumference

$\langle\phi,\Omega_{\alpha}\rangle^{\prime}=\langle f_{\alpha}\circ
f\circ\phi(0)\rangle^{\prime}_{C_{1}}\,.$ (9.8)

Now note that $f_{\alpha+\delta\alpha}$, a scaling by
$1/(1+\alpha+\delta\alpha)$, can be represented as the composition

$f_{\alpha+\delta\alpha}=f_{\delta\alpha/(1+\alpha)}\circ f_{\alpha}\,.$ (9.9)

Applying (9.8) with $\alpha$ replaced by $\alpha+\delta\alpha$ we find

$\langle\phi,\Omega_{\alpha+\delta\alpha}\rangle^{\prime}=\Big{\langle}f_{\delta\alpha/(1+\alpha)}\circ
f_{\alpha}\circ f\circ\phi(0)\Big{\rangle}^{\prime}_{C_{1}}\,.$ (9.10)

The effect of the infinitesimal scaling $f_{\delta\alpha/(1+\alpha)}$ can be
reproduced by the insertion of the operator $I\
-(1+\alpha)^{-1}\delta\alpha\,\oint_{0}{dz\over 2\pi i}\,zT(z)$, with $I$ the
identity operator. Thus we have, using (9.8)

$\begin{split}\langle\phi,\Omega_{\alpha+\delta\alpha}\rangle^{\prime}=&\
\langle\phi,\Omega_{\alpha}\rangle^{\prime}-{\delta\alpha\over
1+\alpha}\Big{\langle}\oint_{0}{dz\over 2\pi i}\,zT(z)f_{\alpha}\circ
f\circ\phi(0)\Big{\rangle}^{\prime}_{C_{1}}\\\ =&\
\langle\phi,\Omega_{\alpha}\rangle^{\prime}\,-{\delta\alpha\over
1+\alpha}\Big{\langle}\oint_{0}{dz\over 2\pi
i}\,zT(z)f\circ\phi(0)\Big{\rangle}^{\prime}_{C_{\alpha+1}}\,,\end{split}$
(9.11)

where in the last step we have rescaled the cylinder width back to
$(1+\alpha)$, noting that $\oint_{0}{dz\over 2\pi i}\,zT(z)$ is invariant
under this rescaling. We thus conclude that

$\Bigl{\langle}\phi,{d\over
d\alpha}\Omega_{\alpha}\Bigr{\rangle}^{\prime}=-{1\over\alpha+1}\Bigl{\langle}\
\oint_{0}{dz\over 2\pi
i}zT(z)\,f\circ\phi(0)\Bigr{\rangle}^{\prime}_{C_{\alpha+1}}\,.$ (9.12)

At this point, by contour deformation, the integral around the vertex operator
is deformed into the sum of two integrals, one on the boundary
$\hbox{Re}\,z=\alpha+\tfrac{1}{2}$ going up and one on
$\hbox{Re}\,z=-\tfrac{1}{2}$ going down. Of course these edges are identified,
so the integrals can be combined into a single integral, say, on the $R$
boundary $\hbox{Re}\,z=\alpha+\tfrac{1}{2}$. A short calculation gives

$\Bigl{\langle}\phi,{d\over
d\alpha}\Omega_{\alpha}\Bigr{\rangle}^{\prime}=-\Bigl{\langle}\
\int_{R}{dz\over 2\pi
i}T(z)\,f\circ\phi(0)\Bigr{\rangle}^{\prime}_{C_{\alpha+1}}\,.$ (9.13)

The right-hand side is in fact the overlap of $\langle{\phi}|$ with a wedge
state acted by the contour integral of $T$. We write this as

${d\over d\alpha}\Omega_{\alpha}=-\,\Omega_{\alpha}\,{\cal K}=-{\cal
K}\,\Omega_{\alpha}\,,$ (9.14)

where the last equality follows because the insertion could be contour-
deformed within the wedge to the left line $\hbox{Re}\,z=\tfrac{1}{2}$. Here
${\cal K}$ is a string field, a state, and the products in the above equation
are star products. The state ${\cal K}$ is the surface state of an
infinitesimally thin wedge with the insertion of the stress tensor or,
equivalently, the identity string field ${\cal I}$ acted by the contour
integral $\int{dz\over 2\pi i}T(z)$ over half the string, written
schematically as

${\cal K}=\left(\int_{L}T\right)\,{\cal I}\,.$ (9.15)

Since $\Omega_{0}={\cal I}$, the above equation is integrated to find

$\Omega_{\alpha}=e^{-\alpha{\cal K}}\,.$ (9.16)

The exponential here is defined throught string products ($e^{X}={\cal
I}+X+\tfrac{1}{2}X\star X+\cdots$). This representation of the wedge states
allows us to manipulate them effectively, as we shall see below.

Completely analogous to ${\cal K}$ we have the state $B$, defined using the
antighost field $b(z)$ instead of $T(z)$:

$B=\left(\int_{L}b\right)\,{\cal I}\,.$ (9.17)

Finally we have the state $c$, defined an infinitesimal strip with a simple
pointwise insertion of the operator $c(z)$ at the boundary (the base of the
wedge). The ${\cal K}$bc algebra is the set of identities satisfied by the
string fields $c,B$, and ${\cal K}$. Including also the action of the BRST
operator $Q$ we have the relations

$\begin{split}&\ \ c^{2}=0\,,\ \ \ B^{2}=0\,,\ \ \ \\{c,B\\}=1\,,\\\ &\ \
[{\cal K}\,,B\,]=0\,,\ \ \ [\,{\cal K}\,,c\,]=\partial c\,,\\\ &\ \ Q{\cal
K}=0\,,\ \ \ QB={\cal K}\,,\ \ \ Qc=c{\cal K}c\,.\end{split}$ (9.18)

It turns out it is relatively easy to write solutions of the open string field
theory using these states. It is harder, however, to tell what the physics of
the solution is. Take, for example, the identity-based string field
$\Psi=c(1-{\cal K})$ [172, 173]. We see that $Q\Psi=c{\cal K}c-c{\cal K}c{\cal
K}=-(c-c{\cal K})(c-c{\cal K})=-\Psi^{2}$, thus $\Psi$ is a solution. The BRST
operator at the solution, $Q_{\Psi}=Q+[\Psi,\cdot]$, actually satisfies
$Q_{\Psi}B=QB+\\{c(1-{\cal K}),B\\}={\cal K}+\\{c,B\\}-\\{c,B\\}{\cal K}={\cal
I}$. An operator $A$ satisfying $Q_{\Psi}A={\cal I}$ is called a homotopy
operator. Its existence implies that any BRST closed state $\chi$ is trivial:
if $Q_{\Psi}\chi=0$, we have $\chi=Q_{\Psi}(A\star\chi)$. For this solution
$B$ is a homotopy operator and therefore there are no physical states at the
resulting configuration. The solution represents the tachyon vacuum but
regrettably, it is quite singular; the evaluation of the action for this
solution is ill-defined.

A general solution found by Okawa[170] is written for general $F=F({\cal K})$:

$\Psi=\ Fc\,{{\cal K}\over 1-F^{2}}\,BcF\,,\ \ \ \hbox{with}\ \ \ F=F({\cal
K})\,.$ (9.19)

This can be checked with simple computations. First one notices that

$(1-FBcF)^{-1}=1+{F\over 1-F^{2}}\,BcF\,,$ (9.20)

which is verified by checking that the right-hand side multiplied by
$(1-FBcF)$ is indeed the identity (this requires using $BcBc=Bc$ and
$BcFBc=FBc$). Then one checks that

$\Psi=(1-FBcF)Q(1-FBcF)^{-1}\,.$ (9.21)

In this form, where $\Psi=\xi Q\xi^{-1}$, with $\xi=1-FBcF$, it is essentially
obvious that the equation of motion is satisfied:
$Q\Psi=(Q\xi)(Q\xi^{-1})=-\Psi^{2}$ because $\Psi^{2}=(\xi Q\xi^{-1})(\xi
Q\xi^{-1})=-\xi[(Q\xi^{-1})\xi]Q\xi^{-1}=\xi[-\xi^{-1}Q\xi]Q\xi^{-1}=(Q\xi)(Q\xi^{-1})$.
With a bit more work one can check that there is a homotopy operator

$A={1-F^{2}\over{\cal K}}\,B\,,\ \ \ \ Q_{\Psi}A={\cal I}\,.$ (9.22)

Thus, formally, all these solutions are the tachyon vacuum. The question
becomes: What are the allowed $F({\cal K})$ for which the solution and the
homotopy operator are both well defined? There are two options that are
particularly interesting [181]:

  1. 1.

$F({\cal K})=e^{-{\cal K}/2}=\Omega_{{}_{1/2}}$.

This corresponds to the original solution by Schnabl and gives

$\begin{split}\Psi=&\ \Omega_{{}_{1/2}}c\ {{\cal K}\over 1-\Omega_{{}_{1}}}\
Bc\,\Omega_{{}_{1/2}}=\Omega_{{}_{1/2}}c\
\bigl{(}1+\Omega_{{}_{1}}+\Omega_{{}_{2}}+\cdots\bigr{)}\ {\cal
K}Bc\,\Omega_{{}_{1/2}}\,,\\\ =&\ \Omega_{{}_{1/2}}c\ {\cal K}Bc\
\Omega_{{}_{1/2}}+\Omega_{{}_{1/2}}c\
\Bigl{(}\sum_{n=1}^{\infty}\Omega_{n}\Bigr{)}{\cal
K}Bc\,\Omega_{{}_{1/2}}\,,\end{split}$ (9.23)

the last expression representing the solution as a leading ghost term
supplemented by an infinite sum of wedge states of growing width, each with
two insertions of $c$ and one of $B$ (Fig. 18, left). In this solution the
homotopy operator is believed to be well defined. The solution itself,
however, has a subtlety: the infinite sum cannot be truncated to evaluate the
energy, and, as a result a so-called phantom term has to be added to the
solution.

  2. 2.

$F({\cal K})={1\over\sqrt{1+{\cal K}}}$.

Written by Erler and Schnabl, this is the simplest known regular solution, and
it seems to have no subtlety [171]. It is given by

$\Psi={1\over\sqrt{1+{\cal K}}}\,c\,(1+{\cal K})\,Bc\,\,{1\over\sqrt{1+{\cal
K}}}\,.$ (9.24)

Recalling that $\Omega_{t}=e^{-t{\cal K}}$, we have the representation

${1\over\sqrt{1+{\cal
K}}}={1\over\sqrt{\pi}}\int_{0}^{\infty}{dt\over\sqrt{t}}e^{-t}\,\Omega_{t}\,,$
(9.25)

and this can be used to rewrite the solution as

$\Psi={1\over\pi}\int_{0}^{\infty}\int_{0}^{\infty}{dt_{1}\over\sqrt{t_{1}}}{dt_{2}\over\sqrt{t_{2}}}e^{-(t_{1}+t_{2})}\,\Omega_{t_{1}}\,c(1+{\cal
K})Bc\,\,\Omega_{t_{2}}\,.$ (9.26)

We see that this time we have a continuous sum over wedge states that goes
from the identity to the sliver. The solution has a width $t_{1}$ wedge, with
insertions on its right side (two ghosts sandwiching $B$ and $(1+{\cal K})$
operators) glued to a width $t_{2}$ wedge (Fig. 18, right). Verifying that
this solution has the right energy is a relatively short calculation using
tools of conformal field theory.

Figure 18: Left: The $n$-th term in the tachyon vacuum solution by Schnabl, a
wedge state of width $n+1$ with two $c$ insertions and a $B$ insertion. Right:
The integrand in the simple tachyon vacuum solution, with left and right
wedges of widths $t_{1}$ and $t_{2}$, respectively, and insertions in between
them.

Similar analytical solutions in the open superstring field theory discussed in
section 8.1 were found in [182]. Refs. [183, 184, 185] use similar techniques
for describing solutions associated with marginal deformations of the boundary
conformal field theory. Some of these numerical and analytical methods have
also been extended to a discussion of time dependent solutions in open string
field theory[186, 187] but we shall not discuss them in this review.

An important question in this area is: Given a particular solution in open
string field theory, what is its physical interpretation? For this purpose one
can use gauge-invariant observables of open string field theory. The idea for
such observables originated from an open-closed vertex arising in the
factorization of open string amplitudes in the associative open SFT [188,
189]. In this vertex, the open string is viewed as a semi-infinite strip, with
the two halves of the final open string glued together and the closed string
state inserted on the conical singularity at the string midpoint.

Figure 19: The open-closed vertex used for gauge-invariant observables in
OSFT. The open string state is inserted at the infinitely far past of the open
string. The closed string state is inserted at the midpoint of the final open
string (heavy dot), which has its two halves glued.

Because of the conical singularity, only on-shell closed string states are
naturally inserted at this singular open string midpoint, and it was shown in
[8] that if the associative OSFT is supplemented by just this interaction, the
Feynman rules of the theory provides a cover of all the moduli spaces of
surfaces with boundaries and both open and closed string punctures. This
covering property suggested the open-string gauge invariance of the open-
closed vertex coupling to on-shell closed strings. Such property was noted
independently by Hashimoto and Itzhaki [190] and Gaiotto et.al. [191], thus
providing an observable of the open string field theory.

A key step was made by Ellwood [192] who conjectured and gave evidence that
the above gauge invariant observable (now called Ellwood invariant) is related
to the closed string boundary state $|B\rangle$. Recall that the boundary
state (for bosonic strings) is a ghost-number-three closed string state
reproducing the one-point functions of arbitrary closed string operators on a
disk as follows: $\langle c_{0}^{-}{\cal O}\rangle_{\rm disk}=\langle
B|c_{0}^{-}|{\cal O}\rangle$. Let $\Phi=0$ represent the open string
background used to formulate the OSFT, and $\hat{\Phi}$ be the open string
field classical solution that represents a different open string background.
Moreover, let $|B_{0}\rangle$ and $|B_{\hat{\Phi}}\rangle$ denote the boundary
state for the original $\Phi=0$ background and the boundary state for the
background represented by $\hat{\Phi}$, respectively. With $\langle{E[{\cal
O}_{c}]}|$ denoting the ghost number two open string state such that
$\langle{E[{\cal O}_{c}]}|\phi_{o}\rangle$ denotes the disk two-point function
of any open string state $\phi_{o}$ and on-shell closed string state ${\cal
O}_{c}$ ($Q_{c}{\cal O}_{c}=0$) in the open-closed vertex, we have Ellwood’s
relation:

$-4\pi i\,\langle E[{\cal O}_{c}]|\,\hat{\Phi}\rangle=\langle{{\cal
O}_{c}}|c_{0}^{-}|B_{\hat{\Phi}}\rangle-\langle{{\cal
O}_{c}}|c_{0}^{-}|B_{0}\rangle\,.$ (9.27)

The ${\cal O}_{c}$-based Ellwood invariant on the left-hand side computes the
difference of one-point functions of ${\cal O}_{c}$ in the final and original
open string backgrounds. For additional details on normalization of the
various quantities in this relation see, for example, [14]. An extension of
Ellwood’s construction to deal with general closed string states and thus get
all the information of the boundary state was given in [193]. An explicit
procedure for constructing the boundary state corresponding to a given
solution in open string field theory was given in [194].

###  9.3 Mass renormalization and vacuum shift

In this section we shall first briefly review where the world-sheet formalism
breaks down in dealing with mass renormalization and vacuum shift and then
describe how string field theory can be used to resolve these problems.

####  9.3.1 Issue with mass renormalization

It is commonly believed that the world-sheet formalism for string theory
allows us to calculate on-shell amplitudes, i.e. S-matrix elements. This is
not strictly true, however. In the world-sheet formalism we define the on-
shell states as appropriate representatives of BRST cohomology, normalized
according to certain rules, and then compute the S-matrix element as integrals
of correlation function of the corresponding vertex operators on the Riemann
surfaces. If we translate this to the language of string field theory, it will
amount to defining the on-shell states as appropriately normalized solutions
to the linearized classical equations of motion and computing the S-matrix
element as the convolution of these wave-functions with the off-shell Green’s
functions. However, we know from our study of quantum field theory that except
in very special circumstances, this is not the correct procedure for computing
the S-matrix. There are two ways this procedure fails:

  1. 1.

We know that in quantum field theory the masses of various fields can get
renormalized. Therefore a state that is on-shell at tree level is no longer
on-shell at the loop level. If we ignore this effect and try to compute the
convolution of the tree-level wave-function of the state with the Green’s
function we shall encounter divergences. In perturbation theory these
divergences show up as self-energy diagrams on external legs. If we try to
evaluate them by setting the external momenta to satisfy the tree-level on-
shell condition, then we shall get divergent results since these diagrams will
contain on-shell internal propagators. An example of this has been shown in
Fig.20 (a) where the internal line carrying mometum $k$ is forced to be on-
shell if we require the external state to be on-shell.

  2. 2.

Even in special cases where the mass renormalization effect is absent, it e.g.
for massless gauge particles, Goldstone bosons etc., there may be wave-
function renormalization effects that tell us that a correctly normalized
state for computing tree-level S-matrix elements may not be correctly
normalized at the loop level. If we do not take this into account, we do not
get a unitary S-matrix.

We encounter similar issues in the naive world-sheet approach to the
computation of the string S-matrix. For example if we take the vertex
operators to be elements of the BRST cohomology satisfying the tree-level on-
shell condition and try to evaluate a one loop amplitude we shall get
divergent result from degenerate Riemann surfaces of the kind shown in Fig. 20
(b). The relation to the diagram in Fig. 20 (a) follows from the Schwinger
parameter representation of the Siegel gauge propagator that is needed to
express string theory Feynman diagrams as world-sheet contribution:

${1\over L_{0}+\bar{L}_{0}}\delta_{L_{0},\bar{L}_{0}}={1\over
2\pi}\int_{|q|\leq 1}{d^{2}q\over|q|^{2}}q^{L_{0}}\bar{q}^{\bar{L}_{0}}\,.$
(9.28)

For $L_{0}>0$ this is an identity but for $L_{0}<0$ the right hand side
diverges from the $q\simeq 0$ region even though the left hand side is finite.
In this case the left hand side can be regarded as the analytic continuation
of the right hand side. For $L_{0}=0$, however, both sides are divergent. On
the left-hand side such divergences occur from the on-shell internal
propagator of string field theory as in the case of Fig.20 (a). On the right
hand side such divergences, coming from the $q\simeq 0$ region, are associated
with degenerate Riemann surfaces of the kind shown in Fig.20 (b) once we
identify $q$ with the plumbing fixture variable.

$k$$k$(a)

$\times$$\times$$\times$$\times$(b)

Figure 20: (a) Mass renormalization diagram in quantum field theory. (b)
Divergent contribution to string theory amplitudes associated with one loop
mass renormalization. The crosses denote vertex operators of external states.

In the world-sheet theory there is no systematic procedure for dealing with
such divergences282828When mass renormalization is absent and hence there are
no divergences, the amplitude in the world-sheet theory may be still ambiguous
since the integrals become only conditionally convergent[195]. Working within
the conventional world-sheet formalism, [195] showed that these are associated
to possible ambiguities in the finite part of the wave-function
renormalization. We need to ensure, however, that when the finite part of the
wave-function renormalization is fixed in one amplitude, we use the same
renormalization prescription for all other amplitudes. To do this one needs
string field theory; in the conventional world-sheet description one cannot
relate the regulator in one amplitude to a regulator in a different amplitude.
A similar remark holds for the situation where tadpoles of massless moduli
fields vanish but there is an ambiguity related to redefinition of the moduli
fields. but in string field theory we can use the usual insights from quantum
field theory. In particular, the LSZ prescription tells us that instead of
dealing with diagrams of the type shown in Fig. 20 (a) in isolation, we need
to first compute the off-shell propagator by summing over all self-energy
diagrams, locate the poles of the propagator to compute renormalized masses
and the residues at the poles to compute the wave-function renormalizations,
and then evaluate the amputated Green’s function at the (loop corrected) on-
shell external momenta, renormalized by the wave-function renormalization
factors. Use of amputated Green’s function ensures that there are no self-
energy insertions on external legs as in Fig.20 (a), and we get a finite
result.

Since string field theory can be regarded as a regular quantum field theory,
we can follow the same procedure to get finite string amplitudes. An extra
complication arises because string theory is a gauge theory; not all the poles
in the gauge-fixed propagator are physical and we have to carefully sort out
the physical poles from the others. This can be done [196, 197] but we shall
later describe a way to get around this using the 1PI effective action [72,
198].

$k=0$(a)

$\times$$\times$$\times$$\times$(b)

Figure 21: (a) Massless tadpole diagram in quantum field theory. (b) Divergent
contribution to string theory amplitudes associated with massless tadpoles.
The crosses denote vertex operators of external states.

####  9.3.2 Issues with vacuum shift

Consider a massless scalar field theory with action

$S=-\int d^{D}x\left[{1\over
2}\partial_{\mu}\phi\partial^{\mu}\phi+{g_{s}\over 3!}\phi^{3}\right]\,.$
(9.29)

At one loop order this theory develops a tadpole of the $\phi$ field. If we
ignore its presence and try to compute amplitudes using the usual Feynman
rules, we run into divergences from internal $\phi$ propagators that carry
zero momentum. An example of this has been shown in Fig.21 (a) where we see an
intermediate massless propagator carrying strictly zero momentum due to
momentum conservation. As we discussed above, the analogous divergences arise
in string theory via degenerate Riemann surfaces of the type shown in Fig.21
(b).

In quantum field theory there is a well defined procedure for dealing with
this problem. Tadpole diagrams may be regarded as a contribution to the
effective action that is linear in the fields. Therefore the effective action
takes the form:

$S=-\int d^{D}x\left[{1\over
2}\partial_{\mu}\phi\partial^{\mu}\phi+{g_{s}\over
3!}\phi^{3}+c\,g_{s}\phi+{\cal O}(\phi^{2})\right]\,,$ (9.30)

where $c$ is some constant. Therefore the true extrema of the effective action
are at

$\phi=\pm\sqrt{-2c}+{\cal O}(g_{s})\,.$ (9.31)

If $c<0$ then there are two real solutions, one of which is a minimum and the
other one is a maximum of the effective potential. In this case we can define
perturbative amplitudes by expanding the fields around the minimum. This
minimum at non-zero $\phi$ represents the vacuum shift from $\phi=0$. Of
course the theory is non-perturbatively ill defined since the potential is
unbounded from below, but that will not be relevant for our discussion (and
can be avoided by adding higher order terms in the classical potential). If
$c$ is positive then there is no extremum at real values of $\phi$ and the
theory is ill defined, at least perturbatively.

In string theory there is no known systematic way to solve the tadpole problem
in the world-sheet theory although some suggestions have been made in [199].
The main idea of [199] is similar in spirit to that in quantum field theory,
i.e. cancel the loop generated tadpoles against a tadpole generated at tree
level by going away from the conformal background. However, both the loop
induced tadpole and the tree-level tadpoles involve regulators and a priori
there is no systematic way of extracting an unambiguous finite answer after
cancelling these divergences. In string field theory, on the other hand, we
can follow the same procedure as in a regular quantum field theory and arrive
at an unambiguous result.292929A hybrid approach, that does not use the full
power of string field theory, but uses enough ingredients involving off-shell
amplitudes, was used in [200] to address the tadpole problem. We shall
illustrate this below.

####  9.3.3 Solution of the vacuum shift problem using string field theory

In the following we shall consider the case of closed superstring theory but
the results also hold in other string theories with trivial changes.

The equations of motion derived from the 1PI effective action (5.47) takes the
form:

$\displaystyle Q(|\Psi\rangle-{\cal G}|\widetilde{\Psi}\rangle)=0\,,$
$\displaystyle Q|\widetilde{\Psi}\rangle+\sum_{n=0}^{\infty}{1\over
n!}[\Psi^{n}]_{\rm 1PI}=0\,.$ (9.32)

Our goal will be to look for solution to this equation in perturbation theory,
with perturbation parameter the coupling constant $g_{s}$. Recall that $g_{s}$
appears in the above equations through the definition of the product
$[\Psi^{n}]$ where $g_{s}$ is the expansion parameter for the sum over genus
(see, for example, (4.13)).

For a translationally invariant vacuum solution we can work at zero momentum.
For simplicity we set the RR fields to zero so that we have only the NS sector
fields. In this case ${\cal G}=1$ and we can set

$|\widetilde{\Psi}\rangle=|\Psi\rangle\,,$ (9.33)

so that the equation takes the form:

$Q|\Psi\rangle+\sum_{n=0}^{\infty}{1\over n!}[\Psi^{n}]_{\rm 1PI}=0\,.$ (9.34)

We can try to solve this equation iteratively in a power series in $g_{s}$,
starting with the leading order solution $|\Psi_{0}\rangle=0$. We shall assume
that the solution can be expanded in integer powers of $g_{s}$, but the
analysis can be easily extended to the situation where the solution has
fractional powers of $g_{s}$. If $|\Psi_{k}\rangle$ denotes the solution with
all powers of $g_{s}$ up to and including $g_{s}^{k}$, then a formal iterative
solution is given by:

$|\Psi_{k+1}\rangle=-{b_{0}^{+}\over L_{0}^{+}}\sum_{n=0}^{k+1}{1\over
n!}[\Psi_{k}^{n}]_{\rm 1PI}+{\cal O}(g_{s}^{k+2}).$ (9.35)

Since $|\Psi_{k+1}\rangle$ must be a solution up to and including order
$g_{s}^{k+1}$, terms of ${\cal O}(g_{s}^{k+2})$ can be ignored. This means
that the ${\cal O}(g_{s}^{k+2})$ terms arising from the $[\Psi_{k}^{n}]$
products (due to the infinite sum over genus) are to be ignored. The state
$|\Psi_{k+1}\rangle$ is a solution because one can easily check that the lower
order equations imply that the state $\sum_{n=0}^{k+1}{1\over
n!}[\Psi_{k}^{n}]_{\rm 1PI}$ is BRST invariant up to corrections of order
$g_{s}^{k+2}$.

The problem with this formal solution, however, is that
$\sum_{n=0}^{k+1}{1\over n!}[\Psi_{k}^{n}]_{\rm 1PI}$ may have components
along states with $L_{0}^{+}=0$ and hence $1/L_{0}^{+}$ acting on these states
is ill defined. To address this we introduce a projection operator ${\bf P}$
into the $L_{0}^{+}=0$ states and write:

$\begin{split}&|\Psi_{k+1}\rangle=-{b_{0}^{+}\over L_{0}^{+}}(1-{\bf
P})\sum_{n=0}^{k+1}{1\over n!}[\Psi_{k}^{n}]_{\rm
1PI}+\,|\psi_{k+1}\rangle\,+{\cal O}(g_{s}^{k+2})\,,\ \ \ {\bf
P}|\psi_{k+1}\rangle=|\psi_{k+1}\rangle,\\\ &Q|\psi_{k+1}\rangle=-{\bf
P}\sum_{n=0}^{k+1}{1\over n!}[\Psi_{k}^{n}]_{\rm 1PI}+{\cal
O}(g_{s}^{k+2})\,.\end{split}$ (9.36)

The first line is an ansatz for $|\Psi_{k+1}\rangle$, which introduces an