#  Correlators of arbitrary untwisted operators and excited twist operators
for $N$ branes at angles.

Igor Pesando1

1Dipartimento di Fisica, Università di Torino
and I.N.F.N. - sezione di Torino
Via P. Giuria 1, I-10125 Torino, Italy

<EMAIL_ADDRESS>
###### Abstract

We compute the generic correlator with $L$ untwisted operators and $N$
(excited) twist fields for branes at angles on $T^{2}$ and show that it is
given by a generalization of the Wick theorem. We give also the recipe to
compute efficiently the generic OPE between an untwisted operator and an
excited twisted state.

keywords: D-branes, Conformal Field Theory

##  1 Introduction and conclusions

Since their introduction, D-branes have been very important in the formal
development of string theory as well as in attempts to apply string theory to
particle phenomenology and cosmology. However, the requirement of chirality in
any physically realistic model leads to a somewhat restricted number of
possible D-brane set-ups. An important class is intersecting brane models
where chiral fermions can arise at the intersection of two branes at angles.
An important issue for these models is the computation of Yukawa couplings and
flavour changing neutral currents.

Besides the previous computations many other computations often involve
correlators of twist fields and excited twist fields. It is therefore
important and interesting in its own to be able to compute these correlators
also because it is annoying to be able to compute, at least theoretically, all
possible correlators involving all kinds of excited spin fields while not
being able to do so with twist fields. As known in the literature [1] and
explicitly shown in [2] for the case of magnetized branes these computations
boil down to the knowledge of the Green function in presence of twist fields
and of the correlators of the plain twist fields. In many previous papers
correlators with excited twisted fields have been computed on a case by case
basis without a clear global picture, see for example ([3], [4], [5])

In this technical paper we have analyzed the $N$ excited twist fields
amplitudes with $L$ boundary vertices at tree level for open strings localized
at $D$-branes intersections on $R^{2}$ (or $T^{2}$) using the classical path
integral approach ([1],[6]) which is more efficient than the also classical
sewing approach ([7], [8]). This approach has been explored in many papers in
the branes at angles setup as well as the T dual magnetic branes setup see for
example ([9], [11], [12], [13], [14], [15], [16], [17], [18], [19],[20], [21],
[22]). We will nevertheless follow a slightly different approach, the so
called Reggeon vertex [23], which allows to compute the generating function of
all correlators, in particular we will use the formulation put forward in
[24].

This paper is organized as follows. In section 2 we review the geometrical
framework of branes at angles and fix our conventions. In the same section we
discuss carefully how to make use of the doubling trick in presence of
multiple cuts and the existence of local and global constraints. In section 3
we compute the OPE of chiral and boundary vertex operators with an arbitrary
excited twist operator by relying on the operator to state correspondence. We
propose also a better notation for excited twist operators than that usually
used which requires a new symbol for any excited twist operator. In this same
section and for use in the fourth we establish also which chiral operators are
best suited to obtain excited twist operators in the easiest way. Finally in
section 4 we compute the generating function of correlators of $N$ excited
twist operators with $L$ boundary operators. We do this in steps by first
computing the interaction of boundary and chiral vertices with twist field
operators and then computing the desired correlators by letting appropriate
combinations of chiral vertex operators collide with twist fields. Our main
result is the generating function given in eq. (123) which shows that all
correlators can be computed once the $N$ plain twist operators correlator
together with the Green function in presence of these $N$ twists are known.
This expression remains nevertheless quite formal since it requires the
precise knowledge of the Green function 111 Note that the Green functions used
in this paper are dimensionful and normalized as
$\partial_{u}\bar{\partial}_{\bar{u}}G^{IJ}(u,{\bar{u}};v,{\bar{v}};\\{\epsilon_{t}\\})=-\frac{\alpha^{\prime}}{2}\delta^{IJ}\delta^{2}(u-v)$.
and its regularized versions. Therefore somewhat explicit expressions of these
quantities are given in appendix E (see also appendixes C and D) and
completely explicit expressions for all involved quantities in the $N=3$ case
are given in appendix F. From these expressions it is clear that the
computation of amplitudes, i.e. moduli integrated correlators, with
(untwisted) states carrying momenta are very unwieldy because Green functions
can at best be expressed as sum of product of type D Lauricella functions.
This should however not be a complete surprise since in [25] it was shown that
twist fields correlators in orbifold setup are connected to loop amplitudes
which, up to now, have not been expressed in term of simpler functions.

##  2 Review of branes at angles

The Euclidean action for a string configuration is given by

$S=\frac{1}{4\pi\alpha^{\prime}}\int
d\tau_{E}\int_{0}^{\pi}d\sigma~{}(\partial_{\alpha}X^{I})^{2}=\frac{1}{4\pi\alpha^{\prime}}\int_{H}d^{2}u~{}(\partial_{u}X{\bar{\partial}_{\bar{u}}}{\bar{X}}+{\bar{\partial}_{\bar{u}}}X\partial_{u}{\bar{X}})$
(1)

where $u\in H$, the upper half plane,
$d^{2}u=e^{2\tau_{E}}d\tau_{E}d\sigma=\frac{du~{}d\bar{u}}{2i}$ and $I=1,2$ or
$z,{\bar{z}}$ so that $X=X^{z}=\frac{1}{\sqrt{2}}(X^{1}+iX^{2})$,
${\bar{X}}=X^{\bar{z}}=X^{*}$. The complex string coordinate is a map from the
upper half plane to a closed polygon $\Sigma$ in $\mathbb{C}$, i.e.
$X:H\rightarrow\Sigma\subset\mathbb{C}$. For example in fig. 1 we have
pictured the interaction of $N=4$ branes at angles $D_{t}$ with $t=1,\dots N$.
The interaction between brane $D_{t}$ and $D_{t+1}$ is in
$f_{t}\in\mathbb{C}$. We use the rule that index $t$ is defined modulo $N$.

$D_{2}$$D_{1}$$D_{4}$$f_{4}$$f_{3}$$D_{3}$$f_{2}$$\Sigma$$f_{1}$$D_{1}$$D_{1}$$D_{4}$$D_{3}$$D_{2}$$\tau_{1}$$\tau_{2}$$\tau_{3}$$\tau_{4}$$\sigma=\pi$$\sigma=0$$X(\sigma,\tau)$
Figure 1: Map from the Minkowskian worldsheet to the target polygon $\Sigma$.

As shown in [2] given the number of twist fields $N$ there are $N-2$ different
sectors which correspond to the number of reflex angles (the interior angles
bigger than $\pi$) as shown in figure 2 in the case $N=6$.

$a)$$b)$$c)$$d)$ Figure 2: The four different cases with $N=6$. $a)$
$M_{ccw}=2$ and $M=4$ where $M_{ccw}$ is measured counterclockwise and
$M_{cw}$ clockwise. $b)$ $M_{ccw}=3$ and $M=3$. $c)$ $M_{ccw}=4$ and $M=2$.
$d)$ $M_{ccw}=5$ and $M=1$.

The intuitive reason why they are different is that we need go through the
straight line, i.e. no twist, if we want to go from a reflex angles to a more
usual convex one.

They are labeled by an integer $1\leq M\leq N-2$ given by

$M=\sum_{t=1}^{N}\epsilon_{t}$ (2)

where $0<\epsilon_{t}<1$ (we define also $\bar{\epsilon}_{t}=1-\epsilon_{t}$
for simplifying the expressions) are the twists defined as in eq. (8) and
correspond to the angles measured from brane $D_{t}$ to brane $D_{t+1}$ when
they are labeled clockwise as in figure 4. It is also possible to have the
very same geometrical configuration where branes are simply labeled
counterclockwise as shown in figure 4 for which we have $M_{ccw}=N-M$ when we
still measure angles from brane $D_{t}$ to brane $D_{t+1}$ .

$f_{1}$$D_{2}$$D_{1}$$D_{4}$$f_{4}$$f_{3}$$D_{3}$$f_{2}$$\pi\epsilon_{4}$$\pi\epsilon_{3}$$\pi\epsilon_{2}$$\pi\epsilon_{1}$$\Sigma$
Figure 3: A polygon $\Sigma$ with an reflex angle and branes labeled clockwise
with $N=4$ and $M=1$.

$f_{4}$$f_{4}$$D_{4}$$D_{1}$$D_{2}$$f_{1}$$f_{2}$$D_{3}$$f_{3}$$\pi\epsilon_{1}$$\pi\epsilon_{2}$$\pi\epsilon_{3}$$\pi\epsilon_{4}$$\Sigma$
Figure 4: A polygon $\Sigma$ with an reflex angle and branes labeled
counterclockwise with $N=4$ and $M_{ccw}=3$.

To understand the meaning we notice that if we relabel the branes as in the
clockwise case, i.e. we consider the branes labeled as in figure 4 but with
the angles measured as in figure 4 and we use the conventions we use for the
local description, described in the next section, the interactions on
worldsheet would take place in the reversed world sheet time order or in the
same order but on the other boundary as shown in figure 5. Since the physics
must be connected and actually the two correlators are connected by complex
conjugation and exchange $\epsilon\leftrightarrow 1-\epsilon$ we have chosen
to measure them clockwise.

$\pi\epsilon_{1}$$D_{2}$$D_{1}$$D_{4}$$f_{4}$$f_{3}$$D_{3}$$f_{2}$$\Sigma$$f_{1}$$D_{1}$$D_{1}$$D_{4}$$D_{3}$$D_{2}$$\tau_{1}$$\tau_{2}$$\tau_{3}$$\tau_{4}$$\sigma=\pi$$\sigma=0$$X(\sigma,\tau)$
Figure 5: Map from the worldsheet to the target polygon $\Sigma$ with
complementary angles.

###  2.1 The local description

Locally at the interaction point $f_{t}\in\partial\Sigma\subset\mathbb{C}$ the
boundary conditions for the brane $D_{t}$ are given by

$\displaystyle
Re(e^{-i\pi\alpha_{t}}X^{\prime}_{loc}|_{\sigma=0})=Im(e^{-i\pi\alpha_{t}}X_{loc}|_{\sigma=0})-g_{t}=0$
(3)

where $g_{t}\in\mathbb{R}$ is the distance from the line parallel to the brane
going through the origin. Similarly the boundary conditions for the brane
$D_{t+1}$ is given by

$\displaystyle
Re(e^{-i\pi\alpha_{t+1}}X^{\prime}_{loc}|_{\sigma=\pi})=Im(e^{-i\pi\alpha_{t+1}}X_{loc}|_{\sigma=\pi})-g_{t+1}=0$
(4)

The interaction point is then

$f_{t}=\frac{e^{i\pi\alpha_{t+1}}g_{t}-e^{i\pi\alpha_{t}}g_{t+1}}{\sin~{}\pi(\alpha_{t+1}-\alpha_{t})}$
(5)

When we write the Minkowskian string expansion as
$X(\sigma,\tau)=X_{L}(\tau+\sigma)+X_{R}(\tau-\sigma)$ the previous boundary
conditions imply (and not become since they are not completely equivalent
because of zero modes)

$\displaystyle
X^{\prime}_{L~{}loc}(\xi)=e^{i2\pi\alpha_{t}}X^{\prime}_{R~{}loc}(\xi),~{}~{}~{}~{}X^{\prime}_{L~{}loc}(\xi+\pi)=e^{i2\pi\alpha_{t+1}}X^{\prime}_{R~{}loc}(\xi-\pi)$
(6)

or in a more useful way in order to explicitly compute the mode expansion

$\displaystyle
X^{\prime}_{L~{}loc}(\xi+2\pi)=e^{i2\pi\epsilon_{t}}X^{\prime}_{L~{}loc}(\xi),~{}~{}~{}~{}X^{\prime}_{R~{}loc}(\xi+2\pi)=e^{-i2\pi\epsilon_{t}}X^{\prime}_{R~{}loc}(\xi)$
(7)

where we have defined

$\epsilon_{t}=\left\\{\begin{array}[]{c
c}(\alpha_{t+1}-\alpha_{t})&\alpha_{t+1}>\alpha_{t}\\\
1+(\alpha_{t+1}-\alpha_{t})&\alpha_{t+1}<\alpha_{t}\end{array}\right.$ (8)

so that $0<\epsilon_{t}<1$ and there is no ambiguity in the phase
$e^{i2\pi\epsilon_{t}}$ entering the boundary conditions. The quantity
$\pi\epsilon_{t}$ is the angle between the two branes $D_{t}$ and $D_{t+1}$
measured counterclockwise as shown in fig. 6.

$D_{t}$$D_{t+1}$$\pi\alpha_{t+1}$$\pi\alpha_{t}$$\pi\epsilon_{t}$$D_{t}$$D_{t+1}$$\pi\alpha_{t+1}$$\pi\alpha_{t}$$\pi\epsilon_{t}$
Figure 6: The connection between $\epsilon$ and the geometrical angles
$\alpha$s defining the branes.

A consequence of this definition is that $\epsilon$ becomes $1-\epsilon$ when
we flip the order of two branes. For example the angles in fig. 4 become those
in fig. 4 when we reverse the order we count the branes, i.e. when we follow
the boundary clockwise instead of counterclockwise.

We introduce as usual the Euclidean fields
$X_{loc}({u_{loc}},{{\bar{u}}_{loc}})$,
$\bar{X}_{loc}({u_{loc}},{{\bar{u}}_{loc}})$ by a worldsheet Wick rotation in
such a way they are defined on the upper half plane by
${u_{loc}}=e^{\tau_{E}+i\sigma}\in H$. The previous choice of having brane
$D_{t}$ at $\sigma=0$ (3) and brane $D_{t+1}$ at $\sigma=\pi$ (4) implies that
in the local description where the interaction point is at $x=0$ $D_{t}$ is
mapped into $x>0$ and $D_{t+1}$ into $x<0$. The boundary conditions (6) can
then immediately be written as

$\displaystyle\partial X_{loc}(x_{loc}+i0^{+})$
$\displaystyle=e^{i2\pi\alpha_{t}}\bar{\partial}\bar{X}_{loc}(x_{loc}-i0^{+})~{}~{}~{}~{}~{}~{}0<x_{loc},$
$\displaystyle\partial X_{loc}(x_{loc}+i0^{+})$
$\displaystyle=e^{i2\pi\alpha_{t+1}}\bar{\partial}\bar{X}_{loc}(x_{loc}-i0^{+})~{}~{}~{}~{}x_{loc}<0$
(9)

and similarly relations for ${\bar{X}}$ which can be obtained by complex
conjugation. When we add to the previous conditions the further constraints

$Im(e^{-i\pi\alpha_{t}}X(x,x))=g_{t}~{}~{}~{}x>0$ (10)

we obtain a system of conditions which are equivalent to the original ones (3,
4).

In order to express the boundary conditions (7) in the Euclidean formulation
it is better to introduce the local fields defined on the whole complex plane
by the doubling trick as

$\displaystyle\partial{\cal X}_{loc}({z_{loc}})$
$\displaystyle=\left\\{\begin{array}[]{cc}\partial_{u}X_{loc}({u_{loc}})&{z_{loc}}={u_{loc}}\mbox{
with }{Im~{}}{z_{loc}}>0\mbox{ or }{z_{loc}}\in\mathbb{R}^{+}\\\
e^{i2\pi\alpha_{t}}{\bar{\partial}_{\bar{u}}}{\bar{X}}_{loc}({{\bar{u}}_{loc}})&{z_{loc}}={{\bar{u}}_{loc}}\mbox{
with }{Im~{}}{z_{loc}}<0\mbox{ or
}{z_{loc}}\in\mathbb{R}^{+}\end{array}\right.$ (13)
$\displaystyle\partial{\bar{\cal X}}_{loc}({z_{loc}})$
$\displaystyle=\left\\{\begin{array}[]{cc}\partial_{u}{\bar{X}}_{loc}({u_{loc}})&{z_{loc}}={u_{loc}}\mbox{
with }{Im~{}}{z_{loc}}>0\mbox{ or }{z_{loc}}\in\mathbb{R}^{+}\\\
e^{-i2\pi\alpha_{t}}{\bar{\partial}_{\bar{u}}}X_{loc}({{\bar{u}}_{loc}})&{z_{loc}}={{\bar{u}}_{loc}}\mbox{
with }{Im~{}}{z_{loc}}<0\mbox{ or
}{z_{loc}}\in\mathbb{R}^{+}\end{array}\right.$ (16)

In this way we can write eq.s (7) as

$\displaystyle\partial{\cal
X}_{loc}(e^{i2\pi}\delta)=e^{i2\pi\epsilon_{t}}\partial{\cal
X}_{loc}(\delta),~{}~{}~{}~{}\partial{\bar{\cal
X}}_{loc}(e^{i2\pi}\delta)=e^{-i2\pi\epsilon_{t}}\partial{\bar{\cal
X}}_{loc}(\delta)$ (17)

Notice that the two Minkowskian boundary conditions in eq.s (7) can be mapped
one into the other by complex conjugation while the corresponding ones in the
Euclidean version given in eq.s (17) are independent and each is mapped into
itself by complex conjugation therefore the Euclidean classical solutions for
${\cal X}_{loc}$ and ${\bar{\cal X}}_{loc}$ are independent.

The quantization of the string with given boundary conditions yields

$\displaystyle X_{loc}({u_{loc}},{{\bar{u}}_{loc}})$ $\displaystyle=f_{t}$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{i\pi\alpha_{t}}\sum_{n=0}^{\infty}\left[\frac{\bar{\alpha}_{(t)n+\bar{\epsilon}_{t}}}{n+\bar{\epsilon}_{t}}{u_{loc}}^{-(n+\bar{\epsilon}_{t})}-\frac{\alpha_{(t)n+\epsilon_{t}}^{\dagger}}{n+\epsilon_{t}}{u_{loc}}^{n+\epsilon_{t}}\right]$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{i\pi\alpha_{t}}\sum_{n=0}^{\infty}\left[-\frac{\bar{\alpha}_{(t)n+\bar{\epsilon}_{t}}^{\dagger}}{n+\bar{\epsilon}_{t}}{{\bar{u}}_{loc}}^{n+\bar{\epsilon}_{t}}+\frac{\alpha_{(t)n+\epsilon_{t}}}{n+\epsilon_{t}}{{\bar{u}}_{loc}}^{-(n+\epsilon_{t})}\right]$
$\displaystyle\bar{X}_{loc}({u_{loc}},{{\bar{u}}_{loc}})$
$\displaystyle=f_{t}^{*}$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{-i\pi\alpha_{t}}\sum_{n=0}^{\infty}\left[-\frac{\bar{\alpha}_{(t)n+\bar{\epsilon}_{t}}^{\dagger}}{n+\bar{\epsilon}_{t}}{u_{loc}}^{n+\bar{\epsilon}_{t}}+\frac{\alpha_{(t)n+\epsilon_{t}}}{n+\epsilon_{t}}{u_{loc}}^{-(n+\epsilon_{t})}\right]$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{-i\pi\alpha_{t}}\sum_{n=0}^{\infty}\left[\frac{\bar{\alpha}_{(t)n+\bar{\epsilon}_{t}}}{n+\bar{\epsilon}_{t}}{{\bar{u}}_{loc}}^{-(n+\bar{\epsilon}_{t})}-\frac{\alpha_{(t)n+\epsilon_{t}}^{\dagger}}{n+\epsilon_{i}}{{\bar{u}}_{loc}}^{n+\epsilon_{t}}\right]$
(18)

with $\bar{\epsilon}_{t}=1-\epsilon_{t}$ and we can interpret $f_{t}$ as the
classical solution

$X_{loc,cl}=f_{t},~{}~{}~{}~{}\bar{X}_{loc,cl}=\bar{f}_{t}$ (19)

since it is the only solution to the equations of motion with finite euclidean
action.

We find also the non trivial commutation relations ($n,m\geq 0$)

$[\alpha_{(t)n+\epsilon_{t}},\alpha_{(t)m+\epsilon_{t}}^{\dagger}]=(n+\epsilon_{t})\delta_{m,n},~{}~{}~{}~{}[\bar{\alpha}_{(t)n+\bar{\epsilon}_{t}},\bar{\alpha}_{(t)m+\bar{\epsilon}_{t}}^{\dagger}]=(n+\bar{\epsilon}_{t})\delta_{m,n}$
(20)

and the vacuum is defined in the usual way by

$\alpha_{(t)n+\epsilon_{t}}|T_{t}\rangle=\bar{\alpha}_{(t)n+\bar{\epsilon}_{t}}|T_{t}\rangle=0~{}~{}~{}~{}n\geq
0$ (21)

At first sight it may seem that the vacuum encodes the $\epsilon_{t}$
information only but as we show in eq. (67) it contains also information about
$f_{t}$ and $\alpha_{t}$ and $\alpha_{t+1}$ which can be extracted from the
proper OPEs.

###  2.2 Global description

In the local description, where the interaction point is at ${x_{loc}}=0$,
$D_{t}$ is mapped into ${x_{loc}}>0$ and $D_{t+1}$ into ${x_{loc}}<0$ this
means that in the global description the world sheet interaction points are
mapped on the boundary of the upper half plane so that $x_{t+1}<x_{t}$. The
global equivalent of the local boundary conditions given in eq.s (9) which is
useful for the path integral formulation where we cannot use the equations of
motion is

$\displaystyle\partial_{u}X(u,{\bar{u}})|_{u=x+i0^{+}}$
$\displaystyle=e^{i2\pi\alpha_{t}}\bar{\partial}_{\bar{u}}\bar{X}(u,{\bar{u}})|_{u=x+i0^{+}}~{}~{}~{}~{}x_{t}<x<x_{t-1}$
$\displaystyle\partial_{u}{\bar{X}}(u,{\bar{u}})|_{u=x+i0^{+}}$
$\displaystyle=e^{-i2\pi\alpha_{t}}\bar{\partial}_{\bar{u}}X(u,{\bar{u}})|_{u=x+i0^{+}}~{}~{}~{}~{}x_{t}<x<x_{t-1}$
(22)

To the previous constraints one must also add the global equivalent to eq.
(10)

$X(x_{t},\bar{x}_{t})=f_{t},~{}~{}~{}~{}\bar{X}(x_{t},\bar{x}_{t})=f_{t}^{*}$
(23)

in order to get a system of boundary conditions equivalent to the original
ones. When using an operatorial approach the global equivalent of eq.s (9)
become

$\displaystyle\partial X_{L}(x+i0^{+})$
$\displaystyle=e^{i2\pi\alpha_{t}}\bar{\partial}\bar{X}_{R}(x-i0^{+})~{}~{}~{}~{}x_{t}<x<x_{t-1}$
$\displaystyle\partial{\bar{X}}_{L}(x+i0^{+})$
$\displaystyle=e^{-i2\pi\alpha_{t}}\bar{\partial}X_{R}(x-i0^{+})~{}~{}~{}~{}x_{t}<x<x_{t-1}$
(24)

If we introduce the global fields defined on the whole complex plane by the
doubling trick as

$\displaystyle\partial{\cal X}(z)$
$\displaystyle=\left\\{\begin{array}[]{cc}\partial_{u}X(u)&z=u\mbox{ with
}{Im~{}}z>0\mbox{ or }z\in\mathbb{R}-[x_{N},x_{1}]\\\
e^{i2\pi\alpha_{1}}{\bar{\partial}_{\bar{u}}}{\bar{X}}(\bar{u})&z=\bar{u}\mbox{
with }{Im~{}}z<0\mbox{ or }z\in\mathbb{R}-[x_{N},x_{1}]\end{array}\right.$
(27) $\displaystyle\partial{\bar{\cal X}}(z)$
$\displaystyle=\left\\{\begin{array}[]{cc}\partial_{u}{\bar{X}}(u)&z=u\mbox{
with }{Im~{}}z>0\mbox{ or }z\in\mathbb{R}-[x_{N},x_{1}]\\\
e^{-i2\pi\alpha_{1}}{\bar{\partial}_{\bar{u}}}X(\bar{u})&z=\bar{u}\mbox{ with
}{Im~{}}z<0\mbox{ or }z\in\mathbb{R}-[x_{N},x_{1}]\end{array}\right.$ (30)

the local boundary conditions (7) can be written in the global formulation as

$\displaystyle\partial{\cal X}(x_{t}+e^{i2\pi}\delta)$
$\displaystyle=e^{i2\pi\epsilon_{t}}\partial{\cal X}(x_{t}+\delta)$
$\displaystyle\partial{\bar{\cal X}}(x_{t}+e^{i2\pi}\delta)$
$\displaystyle=e^{-i2\pi\epsilon_{t}}\partial{\bar{\cal X}}(x_{t}+\delta).$
(31)

Finally it is worth noticing the behavior of the previously introduced fields
under complex conjugation when $z$ is restricted to
$z\in\mathbb{C}-[-\infty,x_{1}]$

$\displaystyle[\partial{\cal X}(z)]^{*}$
$\displaystyle=e^{-i2\pi\alpha_{1}}\partial{\cal
X}(z\rightarrow\bar{z})={\bar{\partial}}{\bar{\cal
X}}(\bar{z})=\left\\{\begin{array}[]{c
c}{\bar{\partial}_{\bar{u}}}{\bar{X}}({\bar{u}})&{\bar{z}}={\bar{u}}\\\
e^{-i2\pi\alpha_{1}}\partial_{u}X(u)&{\bar{z}}=u\end{array}\right.$ (34)
$\displaystyle[\partial{\bar{\cal X}}(z)]^{*}$
$\displaystyle=e^{-i2\pi\alpha_{1}}\partial{\bar{\cal
X}}(z\rightarrow\bar{z})={\bar{\partial}}{\cal
X}(\bar{z})=\left\\{\begin{array}[]{c
c}{\bar{\partial}_{\bar{u}}}X({\bar{u}})&{\bar{z}}={\bar{u}}\\\
e^{i2\pi\alpha_{1}}\partial_{u}{\bar{X}}(u)&{\bar{z}}=u\end{array}\right.$
(37)

where $\partial{\cal X}(z\rightarrow\bar{z})$ means that the holomorphic
$\partial{\cal X}(z)$ is evaluated at ${\bar{z}}$. The previous expressions
also show that it is not necessary to introduce the antiholomorphic fields
${\bar{\partial}}{\cal X}(\bar{z})$ and ${\bar{\partial}}{\bar{\cal
X}}(\bar{z})$ which it is possible to construct applying the doubling trick on
${\bar{\partial}_{\bar{u}}}X(\bar{u})$ and
${\bar{\partial}_{\bar{u}}}{\bar{X}}(\bar{u})$ respectively.

##  3 Twisted Fock Space and OPEs

Given the vacuum $|T\rangle$ 222 In this section we have dropped the
dependence on $t$ as much as possible to simplify the notation e.g.
$\epsilon_{t}\rightarrow\epsilon$.  defined as usual in eq. (21) and the
expansions (18) we can immediately write a normalized basis element of the
Fock space as

$\prod_{n=0}^{\infty}\left[\frac{1}{N_{n}!}\left(\frac{\alpha^{\dagger}_{n+\epsilon}}{\sqrt{n+\epsilon}}\right)^{N_{n}}\frac{1}{\bar{N}_{n}!}\left(\frac{\bar{\alpha}^{\dagger}_{n+\bar{\epsilon}}}{\sqrt{n+\bar{\epsilon}}}\right)^{\bar{N}_{n}}\right]|T\rangle$
(38)

We want now to explore the state to operator correspondence. To the vacuum
$|T\rangle$ we associate the twist field $\sigma_{\epsilon,f}(x)$ which
depends both on the twist $\epsilon$ and on the position $f\in\mathbb{C}$ so
that

$|T\rangle=\lim_{x\rightarrow 0}\sigma_{\epsilon,f}(x)|0\rangle_{SL(2)}$ (39)

with normalization

$\langle T|T\rangle=1$ (40)

For the other states in Fock space it is however better to introduce the non
normalized states

$\prod_{n=0}^{\infty}\left({k_{\epsilon}}n!\alpha^{\dagger}_{n+\epsilon}\right)^{N_{n}}\left(k_{\bar{\epsilon}}n!\bar{\alpha}^{\dagger}_{n+\bar{\epsilon}}\right)^{\bar{N}_{n}}|T\rangle$
(41)

with ${k_{\epsilon}}=-i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{i\pi\alpha_{t}}$
and
$k_{\bar{\epsilon}}=-i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{-i\pi\alpha_{t}}$
to which we make correspond the (generically non primary) chiral operators

$\left[\prod_{n=0}^{\infty}\left(\partial^{n+1}_{u}X\right)^{N_{n}}\left(\partial^{n+1}_{u}\bar{X}\right)^{\bar{N}_{n}}\sigma_{\epsilon,f}\right](x)$
(42)

This notation can be partially misleading since, for example, it is not true
that (see eq. (52) for the right OPE) 333 Notice that on shell
$\partial^{2}_{u}X(u,{\bar{u}})=\frac{1}{2}\partial^{2}_{u}X_{L}(u)$ ,

$\partial^{2}_{u}X(u,{\bar{u}})~{}\sigma_{\epsilon,f}(x)\sim\frac{1}{(u-x)^{\\#}}(\partial^{2}_{u}X\sigma_{\epsilon,f})(x)+\dots$
(43)

and therefore the operator (42) is just a way to write the operator which
corresponds to the state (41) under the operator to state correspondence. The
advantage of this notation is that it clearly shows which state corresponds to
which operator and that it is consistent with the usual untwisted state to
operator correspondence for which

$\prod_{n=1}^{\infty}\left[\left({k_{\epsilon}}n!\alpha^{\dagger}_{n}\right)^{N_{n}}\left(k_{\bar{\epsilon}}n!\bar{\alpha}^{\dagger}_{n}\right)^{\bar{N}_{n}}\right]|0\rangle_{SL(2)}\leftrightarrow\left[\prod_{n=1}^{\infty}\left(\partial^{n}_{u}X\right)^{N_{n}}\left(\partial^{n}_{u}\bar{X}\right)^{\bar{N}_{n}}\right](u,{\bar{u}})$
(44)

We have in particular, as we soon show, that

$\displaystyle\partial_{u}X(u,{\bar{u}})\sigma_{\epsilon,f}(x)$
$\displaystyle\sim(u-x)^{\epsilon-1}(\partial_{u}X\sigma_{\epsilon,f})(x)$
$\displaystyle\partial_{u}{\bar{X}}(u,{\bar{u}})\sigma_{\epsilon,f}(x)$
$\displaystyle\sim(u-x)^{\bar{\epsilon}-1}(\partial_{u}{\bar{X}}\sigma_{\epsilon,f})(x)$
(45)

so that the excited twist operators $(\partial_{u}X\sigma_{\epsilon,f})(x)$
and $(\partial_{u}{\bar{X}}\sigma_{\epsilon,f})(x)$ are what usually written
as $\tau(x)$ and $\bar{\tau}(x)$. This observation also hints to another
reason why to use the notation (42) which is to avoid the proliferation of new
symbols, one for each excited twist operator.

Finally notice that almost all boundary operators can be recovered from chiral
ones, for example when $x_{t}<x<x_{t-1}$ and with the help of the boundary
conditions (22) we get

$\partial_{x}(X(x,x))=\partial_{u}X(u,{\bar{u}})|_{u=x}+\bar{\partial}_{\bar{u}}X(u,{\bar{u}})|_{u=x}=\partial_{u}X(u,{\bar{u}})|_{u=x}+e^{i2\pi\alpha_{t}}\partial_{u}\bar{X}(u,{\bar{u}})|_{u=x}$
(46)

where both $\partial_{u}X(u,{\bar{u}})|_{u=x}$ and
$\partial_{u}\bar{X}(u,{\bar{u}})|_{u=x}$ are chiral operators computed on the
boundary. A notable exception is however $e^{i{\bar{k}}X(x,x)+ik\bar{X}(x,x)}$
which is intrinsically non chiral since it cannot be expressed off shell using
chiral operators. In particular from the previous eq. (46) it follows that
boundary operators have more complex OPEs, such as

$\displaystyle\partial_{x}X(u,{\bar{u}})|_{u={x_{1}}}\sigma_{\epsilon,f}({x_{2}})\sim({x_{1}}-{x_{2}})^{\epsilon-1}(\partial_{u}X\sigma_{\epsilon,f})({x_{2}})+({x_{1}}-{x_{2}})^{\bar{\epsilon}-1}e^{i2\pi\alpha_{t}}(\partial_{u}{\bar{X}}\sigma_{\epsilon,f})({x_{2}}).$
(47)

###  3.1 Chiral OPEs from local Fock space

Let us now see how to compute OPEs of a chiral operator with an excited twist
field using the local operatorial formalism. We will make several examples to
make clear the way to proceed.

We start considering the simplest example, i.e. the OPE
$\partial_{u}X(u,{\bar{u}})\sigma_{\epsilon,f}(x)$. In order to find this OPE
we compute444 One could wonder whether eq. (48) should actually written using
the quantum fluctuation, i.e. $\lim_{x\rightarrow
0}\partial_{u}X_{q}(u,{\bar{u}})\sigma_{\epsilon,f}(x)|0\rangle_{SL(2)}=\partial_{u}X_{loc,q}({u_{loc}})|T\rangle$.
The answer is no because CFT has no meaningful way of splitting $X$ into the
classical and quantum part, moreover if we assume the previous expression we
naively get $\lim_{x\rightarrow
0}\partial_{u}X(u,{\bar{u}})\sigma_{\epsilon,f}(x)|0\rangle_{SL(2)}=\lim_{x\rightarrow
0}\partial_{u}(X_{cl}+X_{q})(u,{\bar{u}})\sigma_{\epsilon,f}(x)|0\rangle_{SL(2)}=\partial_{u}(X_{cl}+X_{loc,q})({u_{loc}})|T\rangle$
and while $\partial_{u}X_{cl}$ has the expected singularity for $u\rightarrow
x_{t}$ it contains also information on the location of the other twist fields,
explicitly
$\partial_{u}X_{cl}\sim(u-x_{t})^{\epsilon-1}R(u,\\{x_{{\hat{t}}\neq t}\\})$
which would imply that full excited twists would contain information on these
locations, e.g.
$(\partial_{u}X\sigma_{\epsilon_{t},f_{t}})(x_{t})=\left(R(x_{t},\\{x_{{\hat{t}}\neq
t}\\})\sigma_{\epsilon_{t},f_{t}}\right)(x_{t})+(\partial_{u}X_{q}\sigma_{\epsilon_{t},f_{t}})(x_{t})$.

$\lim_{x\rightarrow
0}\partial_{u}X(u,{\bar{u}})\sigma_{\epsilon,f}(x)|0\rangle_{SL(2)}=\partial_{u_{loc}}X_{loc}({u_{loc}})|T\rangle$
(48)

where $\partial_{u}X_{loc}({u_{loc}})$ is the operator in the twisted Fock
space which corresponds to the abstract operator $\partial_{u}X(u,{\bar{u}})$
and we can identify ${u_{loc}}=u-x$. Now using the explicit expansion (18) we
get

$\partial_{u_{loc}}X_{loc}({u_{loc}})|T\rangle={u_{loc}}^{\epsilon-1}~{}{k_{\epsilon}}\alpha_{\epsilon}^{\dagger}|T\rangle+{u_{loc}}^{\epsilon}~{}{k_{\epsilon}}\alpha_{\epsilon+1}^{\dagger}|T\rangle+\dots$
(49)

from which we deduce not only the leading order of the OPE (45) but also the
higher order terms

$\partial_{u}X(u,{\bar{u}})\sigma_{\epsilon,f}(x)=(u-x)^{\epsilon-1}~{}(\partial_{u}X\sigma_{\epsilon,f})(x)+(u-x)^{\epsilon}~{}(\partial_{u}^{2}X\sigma_{\epsilon,f})(x)+\dots$
(50)

As a second example we consider the OPE
$\partial_{u}^{2}X(u,{\bar{u}})\sigma_{\epsilon,f}(x)$. Proceeding as before
and using the fact that the local Fock space operator which corresponds to
$\partial_{u}^{2}X(u,{\bar{u}})$ is $\partial^{2}_{u_{loc}}X_{loc}({u_{loc}})$
we can find

$\partial^{2}_{u_{loc}}X_{loc}({u_{loc}})|T\rangle={u_{loc}}^{\epsilon-2}~{}(\epsilon-1)~{}{k_{\epsilon}}\alpha_{\epsilon}^{\dagger}|T\rangle+{u_{loc}}^{\epsilon-1}~{}\epsilon~{}{k_{\epsilon}}\alpha_{\epsilon+1}^{\dagger}|T\rangle+\dots$
(51)

so that we can deduce the OPE

$\partial_{u}^{2}X(u,{\bar{u}})\sigma_{\epsilon,f}(x)=(u-x)^{\epsilon-2}~{}(\epsilon-1)~{}(\partial_{u}X\sigma_{\epsilon,f})(x)+(u-x)^{\epsilon-1}~{}\epsilon~{}(\partial_{u}^{2}X\sigma_{\epsilon,f})(x)+\dots$
(52)

which shows clearly what stated before about the wrongness of eq. (43).
Obviously eq. (52) is compatible with eq. (51) since the former can be
obtained from the latter by taking the derivative $\partial_{u}$.

In the previous examples the local Fock space operator has the same functional
form of the abstract one but as discussed in [26] for the T dual configuration
of branes with magnetic field this is not always the case. The correct
mapping, which is rederived in section 3.3, is given by

$\displaystyle\left[\prod_{n=1}^{\infty}\left(\partial^{n}_{u}X\right)^{N_{n}}\left(\partial^{n}_{u}\bar{X}\right)^{\bar{N}_{n}}\right](u,{\bar{u}})\leftrightarrow\left.\prod_{n=1}^{\infty}\frac{\partial^{N_{n}}}{\partial\bar{c}_{n}^{N_{n}}}\frac{\partial^{\bar{N}_{n}}}{\partial
c_{n}^{\bar{N}_{n}}}{\cal S}_{c}(c,{\bar{c}})\right|_{\bar{c}_{n\geq
1}=c_{n\geq 1}=0}$ (53)

where the chiral generating function ${\cal S}_{c}(c,{\bar{c}})$ is given by

$\displaystyle{\cal S}_{c}(c,{\bar{c}})=$
$\displaystyle:\exp\left\\{\sum_{n=1}^{\infty}\left[\bar{c}_{n}\partial^{n}_{u_{loc}}X_{loc}({u_{loc}},{{\bar{u}}_{loc}})+c_{n}\partial^{n}_{u_{loc}}\bar{X}_{loc}({u_{loc}},{{\bar{u}}_{loc}})\right]\right\\}:$
$\displaystyle\exp\left\\{\sum_{n,m=1}^{\infty}\bar{c}_{n}~{}c_{m}~{}\partial^{n}_{u_{loc}}~{}\partial^{m}_{v_{loc}}|_{{v_{loc}}={u_{loc}}}\Delta^{z{\bar{z}}}_{c}({u_{loc}},{{\bar{u}}_{loc}};{v_{loc}},{{\bar{v}}_{loc}};\epsilon)\right\\}$
(54)

where $:\dots:$ is the normal ordering and

$\Delta^{z{\bar{z}}}_{c}({u_{loc}},{{\bar{u}}_{loc}};{v_{loc}},{{\bar{v}}_{loc}};\epsilon)=G^{z{\bar{z}}}_{N=2}({u_{loc}},{{\bar{u}}_{loc}};{v_{loc}},{{\bar{v}}_{loc}};\\{0,\epsilon;\infty,\bar{\epsilon}\\})-G^{z{\bar{z}}}_{U(t)}({u_{loc}},{{\bar{u}}_{loc}};{v_{loc}},{{\bar{v}}_{loc}})$
(55)

is the regularized Green function with the twist $\sigma_{\epsilon,f}$ at
${x_{loc}}=0$ and the anti-twist $\sigma_{\bar{\epsilon},f}$ in
${x_{loc}}=\infty$ and
$G^{z{\bar{z}}}_{U(t)}({u_{loc}},{{\bar{u}}_{loc}},{v_{loc}},{{\bar{v}}_{loc}})$
is the Green function for the untwisted string with boundary conditions
corresponding to $D_{t}$ (see appendix A for more details)555 For the chiral
correlators there is no difference in using the untwisted Green function for
$D_{t}$ or $D_{t+1}$ since the $G^{z{\bar{z}}}_{U(t)}$ is the only piece of
the untwisted Green function $G^{IJ}_{U}$ which contributes and it is
insensitive to the angle at which the brane is rotated, this is not anymore
true for the boundary correlators as we discuss later. . It is worth
discussing how $\Delta_{c}$ has to be interpreted either as a kind of
generating function or as a difference of Green functions, one of which
associated to a couple of twist fields. This point of view is important in
order to avoid confusions which could arise when considering the role of
$\Delta_{c}$ in correlators with many twist fields. The answer is given in the
derivation of eq. (95) which shows that $\Delta_{c}$ is a difference of Green
functions.

As an example of the consequences of the previous expression (54) we can
compute the local operator which corresponds to the energy-momentum tensor
$T(u)=-\frac{2}{\alpha^{\prime}}\partial_{u}X\partial_{u}\bar{X}$. We find the
local operator

$\displaystyle T_{loc}({u_{loc}})$
$\displaystyle=-\frac{2}{\alpha^{\prime}}:\partial_{u_{loc}}X_{loc}\partial_{u_{loc}}\bar{X}_{loc}:-\frac{2}{\alpha^{\prime}}\partial_{u_{loc}}\partial_{v_{loc}}\Delta^{z{\bar{z}}}_{c}|_{{v_{loc}}={u_{loc}}}$
$\displaystyle=-\frac{2}{\alpha^{\prime}}:\partial_{u_{loc}}X_{loc}\partial_{u_{loc}}\bar{X}_{loc}:-\frac{\epsilon\bar{\epsilon}}{2}\frac{1}{{u_{loc}}^{2}}$
(56)

Then we can compute the OPE $T(u)\sigma_{\epsilon,f}(x)$ from

$\left[-\frac{2}{\alpha^{\prime}}:\partial_{u_{loc}}X_{loc}\partial_{u_{loc}}\bar{X}_{loc}:-\frac{\epsilon\bar{\epsilon}}{2}\frac{1}{{u_{loc}}^{2}}\right]|T\rangle=-\frac{\epsilon\bar{\epsilon}}{2}\frac{1}{{u_{loc}}^{2}}|T\rangle+\frac{1}{{u_{loc}}}\alpha^{\dagger}_{\epsilon}\bar{\alpha}^{\dagger}_{\bar{\epsilon}}|T\rangle+O(1)$
(57)

to be

$T(u)\sigma_{\epsilon,f}(x)\sim\frac{\epsilon\bar{\epsilon}}{2}\frac{1}{(u-x)^{2}}\sigma_{\epsilon,f}(x)+\frac{1}{(u-x)^{2}}\frac{1}{{k_{\epsilon}}k_{\bar{\epsilon}}}\left(\partial_{u}X\partial_{u}\bar{X}\sigma_{\epsilon,f}\right)(x)+O(1)$
(58)

from which we read both the conformal dimension of $\sigma_{\epsilon,f}$ and
that
${k_{\epsilon}}k_{\bar{\epsilon}}\partial_{x}\sigma_{\epsilon,f}=\left(\partial_{u}X\partial_{u}\bar{X}\sigma_{\epsilon,f}\right)(x)$.
It is noteworthy that the double pole comes from the extra piece
$\partial\partial\Delta_{c}$ in eq. (56) which would not be present in a naive
state to operator correspondence.

As a last example to make clear the algorithm we consider the more complex OPE
$(\partial^{2}_{u}X\,\partial_{u}X\,\partial_{u}\bar{X})(u)\left(\partial^{2}_{u}\bar{X}\sigma_{\epsilon,f}\right)(x)$
at the leading order. First we compute the operatorial realization of
$(\partial^{2}_{u}X\,\partial_{u}X\,\partial_{u}\bar{X})(u)$ to be

$\displaystyle:(\partial^{2}_{u_{loc}}X_{loc}\partial_{u_{loc}}X_{loc}\partial_{u_{loc}}\bar{X}_{loc})(u):$
$\displaystyle+\partial^{2}_{u_{loc}}\partial_{v_{loc}}\Delta^{z{\bar{z}}}_{c}|_{{v_{loc}}={u_{loc}}}\partial_{u_{loc}}X_{loc}+\partial_{u_{loc}}\partial_{v_{loc}}\Delta^{z{\bar{z}}}_{c}|_{{v_{loc}}={u_{loc}}}\partial^{2}_{u_{loc}}X_{loc}$
$\displaystyle=:(\partial^{2}_{u_{loc}}X_{loc}\partial_{u_{loc}}X_{loc}\partial_{u_{loc}}\bar{X}_{loc})(u):-\frac{{k_{\epsilon}}k_{\bar{\epsilon}}\epsilon(1-\epsilon)(2-\epsilon)}{2{u_{loc}}^{3}}\partial_{u_{loc}}X_{loc}-\frac{{k_{\epsilon}}k_{\bar{\epsilon}}\epsilon(1-\epsilon)}{2{u_{loc}}^{2}}\partial^{2}_{u_{loc}}X_{loc}$
(59)

then we associate the state
$k_{\bar{\epsilon}}\bar{\alpha}^{\dagger}_{\bar{\epsilon}+2}|T\rangle$ to the
excited twist $\left(\partial^{2}_{u}\bar{X}\sigma_{\epsilon,f}\right)(x)$
then an easy computation gives

$(\partial^{2}_{u}X\,\partial_{u}X\,\partial_{u}\bar{X})(u)\left(\partial^{2}_{u}\bar{X}\sigma_{\epsilon,f}\right)(x)\sim({k_{\epsilon}}k_{\bar{\epsilon}})^{2}\frac{\epsilon(1-\epsilon)^{2}}{2(u-x)^{4-\epsilon}}\sigma_{\epsilon,f}(x)$
(60)

where the leading order contribution comes from the terms linear in $X_{loc}$.

###  3.2 Boundary OPEs from local Fock space

In the computation of the interaction of twisted states with untwisted ones
quite often we are not interested in chiral operators but in boundary
operators such as $e^{i{\bar{k}}X(x,x)+ik\bar{X}(x,x)}$. For this case we must
extend the analysis given in the previous section. The correct mapping is then
given by

$\displaystyle\left[e^{i{\bar{k}}X+ik\bar{X}}\prod_{n=1}^{\infty}\left(\partial^{n}_{x}X\right)^{N_{n}}\left(\partial^{n}_{x}\bar{X}\right)^{\bar{N}_{n}}\right](x+i0^{+},x-i0^{+})\leftrightarrow\left.\prod_{n=1}^{\infty}\frac{\partial^{N_{n}}}{\partial\bar{c}_{n}^{N_{n}}}\frac{\partial^{\bar{N}_{n}}}{\partial
c_{n}^{\bar{N}_{n}}}{\cal S}(c,{\bar{c}})\right|_{c_{0}=i\,k,c_{n\geq 1}=0}$
(61)

where the generating function ${\cal S}(c,{\bar{c}})$ is given by

$\displaystyle{\cal S}(c,{\bar{c}})=$
$\displaystyle:\exp\left\\{\sum_{n=0}^{\infty}\left[\bar{c}_{n}\partial^{n}_{x_{loc}}X_{loc}({x_{loc}}+i0^{+},{x_{loc}}-i0^{+})+c_{n}\partial^{n}_{x_{loc}}\bar{X}_{loc}({x_{loc}}+i0^{+},{x_{loc}}-i0^{+})\right]\right\\}:$
$\displaystyle\exp\left\\{\sum_{n,m=0}^{\infty}\bar{c}_{n}~{}c_{m}~{}\partial^{n}_{x_{1}}~{}\partial^{m}_{x_{2}}\Delta_{bou}^{z{\bar{z}}}({x_{1}};{x_{2}};\epsilon)|_{{x_{1}}={x_{2}}={x_{loc}}}\right\\}$
$\displaystyle\exp\left\\{\frac{1}{2}\sum_{n,m=0}^{\infty}c_{n}~{}c_{m}~{}\partial^{n}_{x_{1}}~{}\partial^{m}_{x_{2}}\Delta_{bou}^{{\bar{z}}{\bar{z}}}({x_{1}};{x_{2}};\epsilon)|_{{x_{1}}={x_{2}}={x_{loc}}}\right\\}$
$\displaystyle\exp\left\\{\frac{1}{2}\sum_{n,m=0}^{\infty}\bar{c}_{n}~{}\bar{c}_{m}~{}\partial^{n}_{x_{1}}~{}\partial^{m}_{x_{2}}\Delta_{bou}^{zz}({x_{1}};{x_{2}};\epsilon)|_{{x_{1}}={x_{2}}={x_{loc}}}\right\\}$
(62)

where $:\dots:$ is the normal ordering and we defined

$\Delta_{bou}^{IJ}({x_{1}};{x_{2}};\epsilon)=\left\\{\begin{array}[]{r
l}\scriptstyle
G^{IJ}({x_{1}}+i0^{+},{x_{1}}-i0^{+};{x_{2}}+i0^{+},{x_{2}}-i0^{+};\\{0,\epsilon;\infty,\bar{\epsilon}\\})&\scriptstyle-G^{IJ}_{U(t)}({x_{1}}+i0^{+},{x_{1}}-i0^{+};{x_{2}}+i0^{+},{x_{2}}-i0^{+})\\\
&{x_{1}},{x_{2}}>0\\\ \scriptstyle
G^{IJ}({x_{1}}+i0^{+},{x_{1}}-i0^{+};{x_{2}}+i0^{+},{x_{2}}-i0^{+};\\{0,\epsilon;\infty,\bar{\epsilon}\\})&\scriptstyle-G^{IJ}_{U(t+1)}({x_{1}}+i0^{+},{x_{1}}-i0^{+};{x_{2}}+i0^{+},{x_{2}}-i0^{+})\\\
&{x_{1}},{x_{2}}<0\end{array}\right.$ (63)

with
$G^{IJ}({u_{loc}},{{\bar{u}}_{loc}},{v_{loc}},{{\bar{v}}_{loc}};\\{0,\epsilon;\infty,\bar{\epsilon}\\})$
the Green function666We normalize the Green function such that
$\partial\bar{\partial}G({u_{loc}},{{\bar{u}}_{loc}},{v_{loc}},{{\bar{v}}_{loc}})=\frac{-\alpha^{\prime}}{2}\delta^{2}({u_{loc}}-{v_{loc}})$.
with the twist $\sigma_{\epsilon,f}$ at ${x_{loc}}=0$ and the anti-twist
$\sigma_{\bar{\epsilon},f}$ in ${x_{loc}}=\infty$ and
$G^{IJ}_{U(t)}({u_{loc}},{{\bar{u}}_{loc}},{v_{loc}},{{\bar{v}}_{loc}})$ is
the Green function of the untwisted string with both ends on $D_{t}$ and the
need to distinguish $x>0$ from $x<0$ is due to the different boundary
conditions of the twisted string in this ranges As shown in eq. (147) in
appendix A all $\Delta^{IJ}_{bou}$ are equal to a real symmetric function
$\Delta_{bou}(x_{1},x_{2})$ up to phases which combine to allow the previous
generating function to be written as

$\displaystyle{\cal S}(c,{\bar{c}})=$
$\displaystyle:\exp\left\\{\sum_{n=0}^{\infty}\left[\bar{c}_{n}\partial^{n}_{x_{loc}}X_{loc}({x_{loc}}+i0^{+},{x_{loc}}-i0^{+})+c_{n}\partial^{n}_{x_{loc}}\bar{X}_{loc}({x_{loc}}+i0^{+},{x_{loc}}-i0^{+})\right]\right\\}:$
$\displaystyle\exp\left\\{\sum_{n,m=0}^{\infty}c_{n\parallel
D_{t}}~{}c_{m\parallel
D_{t}}~{}\partial^{n}_{x_{1}}~{}\partial^{m}_{x_{2}}\Delta_{bou}({x_{1}};{x_{2}};\epsilon)|_{{x_{1}}={x_{2}}={x_{loc}}}\right\\}$
(64)

when $x_{loc}>0$ with

$c_{n\parallel
D_{t}}=\frac{e^{-i\pi\alpha_{t}}c_{n}+e^{i\pi\alpha_{t}}\bar{c}_{n}}{\sqrt{2}}.$
(65)

When $x_{loc}<0$ we get a very similar expression but with the substitution
$c_{\parallel D_{t}}\rightarrow c_{\parallel D_{t+1}}$ since in this case the
vertex is on the brane $D_{t+1}$.

Given the previous results we can now compute the operatorial realization of
the the vertex $e^{ik\cdot X(x,x)}$ to be, similarly to the results
([26],[27])

$\left\\{\begin{array}[]{c c}|{x_{loc}}|^{-\alpha^{\prime}k^{2}_{\parallel
D_{t}}}e^{-\frac{1}{2}R^{2}(\epsilon_{t})\alpha^{\prime}k^{2}_{\parallel
D_{t}}}~{}:e^{i(\bar{k}X_{loc}({x_{loc}},{x_{loc}})+k\bar{X}_{loc}({x_{loc}},{x_{loc}}))}:&{x_{loc}}>0~{}[{x_{t}}<x<{x_{t-1}}]\\\
|{x_{loc}}|^{-\alpha^{\prime}k^{2}_{\parallel
D_{t+1}}}e^{-\frac{1}{2}R^{2}(\epsilon_{t})\alpha^{\prime}k^{2}_{\parallel
D_{t+1}}}~{}:e^{i(\bar{k}X_{loc}({x_{loc}},{x_{loc}})+k\bar{X}_{loc}({x_{loc}},{x_{loc}}))}:&{x_{loc}}<0~{}[{x_{t+1}}<x<{x_{t}}]\end{array}\right.$
(66)

where
$R^{2}(\epsilon_{t})=2\psi(1)-\psi(\epsilon_{t})-\psi(\bar{\epsilon}_{t})>0$,
$\psi(z)=\frac{d\ln\Gamma(z)}{dz}$ being the digamma function. Notice that we
have not required the momentum $k$ to be tangent to the brane since the normal
part gives simply a phase due to the boundary conditions nevertheless, as
commented before, the explicit expression for $\Delta^{IJ}$ implies that only
the momentum parallel to the $D_{t}$ brane $k_{\parallel
D_{t}}=\frac{e^{-i\pi\alpha_{t}}k+e^{i\pi\alpha_{t}}\bar{k}}{\sqrt{2}}$ enters
the form factor
$e^{-\frac{1}{2}R^{2}(\epsilon_{t})\alpha^{\prime}k^{2}_{\parallel D_{t}}}$.

Using the previous operatorial realization we can then obtain the OPEs

$\displaystyle
e^{i(\bar{k}X(x,x)+k\bar{X}(x,x))}\sigma_{\epsilon_{t},f_{t}}({x_{t}})\sim$
$\displaystyle(x-{x_{t}})^{-\alpha^{\prime}k^{2}_{\parallel
D_{t}}}e^{-\frac{1}{2}R^{2}(\epsilon_{t})\alpha^{\prime}k^{2}_{\parallel
D_{t}}}e^{i(k\bar{f}_{t}+\bar{k}f_{t})}$
$\displaystyle\Big{[}\sigma_{\epsilon_{t},f_{t}}({x_{t}})-(x-{x_{t}})^{\epsilon_{t}}\frac{\sqrt{2}e^{-i\pi\alpha_{t}}k_{\parallel
D_{t}}}{\epsilon_{t}}(\partial_{u}X\sigma_{\epsilon_{t},f_{t}})({x_{t}})$
$\displaystyle~{}-(x-{x_{t}})^{\bar{\epsilon}_{t}}\frac{\sqrt{2}e^{-i\pi\alpha_{t}}k_{\parallel
D_{t}}}{\bar{\epsilon}_{t}}(\partial_{u}\bar{X}\sigma_{\epsilon_{t},f_{t}})({x_{t}})+\dots\Big{]}~{}~{}~{}~{}{x_{t}}<x<{x_{t-1}}$
(67)

and similarly for the $x_{loc}<0$ case which corresponds to
${x_{t+1}}<x<{x_{t}}$ with the substitution $k_{\parallel D_{t}}\rightarrow
k_{\parallel D_{t+1}}$ and
$(x-{x_{t}})^{\epsilon_{t}}\rightarrow({x_{t}}-x)^{\epsilon_{t}}$ The previous
OPE justify writing not only $\sigma_{\epsilon_{t}}({x_{t}})$ but
$\sigma_{\epsilon_{t},f_{t}}({x_{t}})$ since the $f_{t}$ and $\bar{f}_{t}$ can
be computed using the phases of the leading order terms with different
momenta. Moreover the phases $e^{-i\pi\alpha_{t}}$ and $e^{-i\pi\alpha_{t+1}}$
can be extracted from the projections $k_{\parallel D_{t}},k_{\parallel
D_{t+1}}$ of momentum $k$.

In a similar way we can compute the operator associated to
$\partial_{x}X(x,x)~{}e^{ik\cdot X(x,x)}$ to be

$|{x_{loc}}|^{-\alpha^{\prime}k^{2}_{\parallel
D_{t}}}e^{-\frac{1}{2}R^{2}(\epsilon_{t})\alpha^{\prime}k^{2}_{\parallel
D_{t}}}:\left(\partial_{x_{loc}}X_{loc}({x_{loc}},{x_{loc}})-\frac{ie^{i\pi\alpha_{t}}\,\alpha^{\prime}k_{\parallel
D_{t}}}{\sqrt{2}{x_{loc}}}\right)e^{i(\bar{k}X_{loc}({x_{loc}},{x_{loc}})+k\bar{X}_{loc}({x_{loc}},{x_{loc}}))}:$
(68)

when ${x_{loc}}>0~{}[{x_{t}}<x<{x_{t-1}}]$ and similarly for
${x_{loc}}<0~{}[{x_{t+1}}<x<{x_{t}}]$. It is worth stressing that the term
proportional to $\frac{i\alpha^{\prime}k_{\parallel D_{t}}}{{x_{loc}}}$ is
fundamental in getting the right interaction among a gluon and twisted matter
([28],[26], [27]) when the amplitude is computed in operatorial formalism.

###  3.3 Short derivation of the generating function ${\cal S}$

We will now quickly review the motivations to write down the generating vertex
(62) and why it works.

Our aim is to get some hints on how to regularize the contact divergences that
appear in the path integral computation of amplitudes in presence of twist
fields. We consider the boundary case since all the others can be treated in
an analogous manner. In the untwisted case the operatorial generating function
is simply the Sciuto-Della Selva- Saito vertex [23] 777 Because of the
boundary conditions not all terms at the exponent are independent but this
does not change the arguments in this section.

$\displaystyle{\cal S}_{U}(c,{\bar{c}})=$
$\displaystyle:e^{\sum_{n=0}^{\infty}\left[\bar{c}_{n}\partial^{n}_{x_{loc}}X_{U,loc}({x_{loc}}+i0^{+},{x_{loc}}-i0^{+})+c_{n}\partial^{n}_{x_{loc}}\bar{X}_{U,loc}({x_{loc}}+i0^{+},{x_{loc}}-i0^{+})\right]}:$
(69)

Dropping the loc indication we can rewrite it as

${\cal S}_{U}[J]=:e^{\int dx_{a}J_{I}(x_{a})X^{I}_{U}(x_{a},x_{a})}:$ (70)

where we have introduced the current $J_{I}(x_{a})$ which must be set to
$J_{I}(x_{a})=\sum_{n=0}^{\infty}c_{In}\partial^{n}_{x}\delta(x_{a}-x)$ when
we want to reproduce the original vertex but can also be taken more general as
we do in the following.

We can now compute the OPE of two such generating functions (vertices)

$\displaystyle{\cal S}_{U}[J_{1}]{\cal S}_{U}[J_{2}]$ $\displaystyle=e^{\int
d{x_{a}}\int
d{x_{b}}J_{1I}({x_{a}})J_{2J}({x_{b}})G^{IJ}_{U,bou}({x_{a}},{x_{b}})}:{\cal
S}_{U}[J_{1}]{\cal S}_{U}[J_{2}]:$ $\displaystyle=e^{\int d{x_{a}}\int
d{x_{b}}J_{1I}({x_{a}})J_{2J}({x_{b}})G^{IJ}_{U,bou}({x_{a}},{x_{b}})}{\cal
S}_{U}[J_{1}+J_{2}]$ (71)

which is valid for generic currents as long as they have compact support and
the points in the support of $J_{1}$ have bigger absolute values than those in
the support of $J_{2}$ so that the operatorial product is radial ordered. Now
the non operatorial term $e^{\int d{x_{a}}\int
d{x_{b}}J_{1I}({x_{a}})J_{2J}({x_{b}})G^{IJ}_{U,bou}({x_{a}},{x_{b}})}$ can be
roughly understood as the generating function for the OPE coefficients and we
want to check that it is reproduced when using the generating function (62)
which is defined in the twisted Fock space. In particular we consider the
generating function with a more general current given by

${\cal S}_{T}[J]=e^{\frac{1}{2}\int d{x_{a}}\int
d{x_{b}}J_{I}({x_{a}})J_{J}({x_{b}})\Delta^{IJ}_{bou}({x_{a}},{x_{b}})}:e^{\int
dx_{a}J_{I}(x_{a})X^{I}(x_{a},x_{a})}:$ (72)

It is then immediate to compute the product

$\displaystyle{\cal S}_{T}[J_{1}]{\cal S}_{T}[J_{2}]$ $\displaystyle=e^{\int
d{x_{a}}\int
d{x_{b}}J_{1I}({x_{a}})J_{2J}({x_{b}})G^{IJ}_{N=2,bou}({x_{a}},{x_{b}};\\{0,\epsilon;\infty;\bar{\epsilon}\\})}:{\cal
S}_{T}[J_{1}]{\cal S}_{T}[J_{2}]:$ $\displaystyle=e^{\int d{x_{a}}\int
d{x_{b}}J_{1I}({x_{a}})J_{2J}({x_{b}})G^{IJ}_{U,bou}({x_{a}},{x_{b}})}{\cal
S}_{T}[J_{1}+J_{2}]$ (73)

where we have added and subtracted $\int d{x_{a}}\int
d{x_{b}}J_{1I}({x_{a}})J_{2J}({x_{b}})G^{IJ}_{U,bou}({x_{a}},{x_{b}})$ to the
exponent in order to complete the prefactor in eq. (72) in the case where
$J=J_{1}+J_{2}$ and used the symmetry
$\Delta^{IJ}{{}_{bou}}({x_{a}},{x_{b}})=\Delta^{JI}{{}_{bou}}({x_{b}},{x_{a}})$.
Having verified that the generating function (62) gives vertices with the same
OPEs as the untwisted ones we can exam the reason which leads to it ([25],
[29], [26]). In operatorial formalism the normal ordered vertex (70) for the
untwisted case can be obtained from a regularized non normal ordered
generating function by a multiplicative renormalization as

${\cal S}_{U}[J]=\lim_{\eta\rightarrow 0^{+}}{\cal N}(\eta)~{}{\cal
S}_{U,reg}[J,\eta]$ (74)

where the regularized generating function is defined by a point splitting as

${\cal S}_{U,reg}[J,\eta]=e^{\int
dx_{a}J_{I}(x_{a})[X^{I(+)}_{U}(x_{a},x_{a})+X^{I(-)}_{U}(x_{a}e^{-\eta},x_{a}e^{-\eta})]}$
(75)

without normal ordering and the multiplicative renormalization is given by the
inverse of the factor we get by normal ordering the regularized generating
function

${\cal N}(\eta)=e^{-\frac{1}{2}\int d{x_{a}}\int
d{x_{b}}J_{I}({x_{a}})J_{J}({x_{b}})G^{IJ}_{U,bou}({x_{a}},{x_{b}}e^{-\eta})}$
(76)

Now the generating function for the twisted string is defined in an analogous
way by regularizing the non normal ordered generating function and then
renormalizing in a minimal way using the same renormalization factor as in the
untwisted case (76), explicitly

${\cal S}_{T}[J]=\lim_{\eta\rightarrow 0^{+}}{\cal N}(\eta)~{}{\cal
S}_{T,reg}[J,\eta]$ (77)

with

${\cal S}_{T,reg}[J,\eta]=e^{\int
dx_{a}J_{I}(x_{a})[X^{I(+)}(x_{a},x_{a})+X^{I(-)}(x_{a}e^{-\eta},x_{a}e^{-\eta})]}$
(78)

###  3.4 Getting excited twists

We are interested in excited twist states hence we would now write a kind of
SDS vertex which generates these states (41). The main observation is then
that

$\partial_{u_{loc}}^{n-1}\left[{u_{loc}}^{\bar{\epsilon}}\partial_{u_{loc}}X_{loc}({u_{loc}},{{\bar{u}}_{loc}})\right]=(n-1)!~{}{k_{\epsilon}}\alpha_{n-1+\epsilon}^{\dagger}+O({u_{loc}})$
(79)

therefore a normal ordered products of these operators gives directly an
excited twist state, e.g.

$\displaystyle\lim_{{u_{loc}}\rightarrow 0}$
$\displaystyle:\partial_{u_{loc}}^{n-1}\left[{u_{loc}}^{\bar{\epsilon}}\partial_{u_{loc}}X_{loc}({u_{loc}},{{\bar{u}}_{loc}})\right]\partial_{u_{loc}}^{m-1}\left[{u_{loc}}^{\epsilon}\partial_{u_{loc}}{\bar{X}}_{loc}({u_{loc}},{{\bar{u}}_{loc}})\right]:|T\rangle$
$\displaystyle={k_{\epsilon}}k_{\bar{\epsilon}}(n-1)!(m-1)!\alpha_{n-1+\epsilon}^{\dagger}\bar{\alpha}_{m-1+\epsilon}^{\dagger}|T\rangle=\left(\partial^{n}X\partial^{m}{\bar{X}}\sigma_{\epsilon,f}\right)(0)|0\rangle_{SL(2)}$
(80)

The generating function for products of these operators is obviously

$\displaystyle{\cal T}(d,\bar{d})=$ $\displaystyle\lim_{{u_{loc}}\rightarrow
0}:\exp\left\\{\sum_{n=1}^{\infty}\left[\bar{d}_{n}\partial_{u_{loc}}^{n-1}\left[{u_{loc}}^{\bar{\epsilon}}\partial_{u_{loc}}X_{loc}({u_{loc}},{{\bar{u}}_{loc}})\right]+d_{n}\partial_{u_{loc}}^{n-1}\left[{u_{loc}}^{\epsilon}\partial_{u_{loc}}{\bar{X}}_{loc}({u_{loc}},{{\bar{u}}_{loc}})\right]\right]\right\\}:$
(81)

since

$\displaystyle\left[\prod_{n=1}^{\infty}\left(\partial^{n}_{u}X\right)^{N_{n}}\left(\partial^{n}_{u}\bar{X}\right)^{\bar{N}_{n}}\sigma_{\epsilon,f}\right](0)|0\rangle_{SL(2)}\leftrightarrow\lim_{{u_{loc}}\rightarrow
0}\left.\prod_{n=1}^{\infty}\frac{\partial^{N_{n}}}{\partial\bar{d}_{n}^{N_{n}}}\frac{\partial^{\bar{N}_{n}}}{\partial
d_{n}^{\bar{N}_{n}}}{\cal T}(d,\bar{d})\right|_{\bar{d}=d=0}|T\rangle$ (82)

Comparing with eq. (54) we realize that there is not the exponent quadratic in
$d$, this means that the abstract operator corresponding to e.g. eq. (80) is
not simply $\lim_{u\rightarrow
x}\partial_{u}^{n-1}\left[(u-x)^{\bar{\epsilon}}\partial_{u}X\right]\partial_{u}^{m-1}\left[(u-x)^{\epsilon}\partial_{u}{\bar{X}}\right]$
but it is

$\displaystyle\lim_{u\rightarrow x}\Big{(}$
$\displaystyle\partial_{u}^{n-1}\left[(u-x)^{\bar{\epsilon}}\partial_{u}X\right]\partial_{u}^{m-1}\left[(u-x)^{\epsilon}\partial_{u}{\bar{X}}\right]$
$\displaystyle-\partial_{u}^{n-1}\partial_{v}^{m-1}\left[(u-x)^{\bar{\epsilon}}(v-x)^{\epsilon}\partial_{u}\partial_{v}\Delta^{z{\bar{z}}}_{c}(u-x,{\bar{u}}-x;v-x,{\bar{v}}-x;\epsilon)|_{v=u}\right]\Big{)}$
(83)

in fact computing its OPE with the twist field $\sigma_{\epsilon,f}(x)$ as in
section 3.1 we get

$\displaystyle\Big{(}\partial_{u}^{n-1}\left[(u-x)^{\bar{\epsilon}}\partial_{u}X\right]\partial_{u}^{m-1}\left[(u-x)^{\epsilon}\partial_{u}{\bar{X}}\right]$
$\displaystyle-\partial_{u}^{n-1}\partial_{v}^{m-1}\left[(u-x)^{\bar{\epsilon}}(v-x)^{\epsilon}\partial_{u}\partial_{v}\Delta^{z{\bar{z}}}_{c}\right]\Big{)}\sigma_{\epsilon,f}(x)$
$\displaystyle\sim\left(\partial^{n}X\partial^{m}{\bar{X}}\sigma_{\epsilon,f}\right)(x)+O(u-x)$
(84)

Notice that this OPE explains why we can use $u-x$ as argument of
$\Delta^{z{\bar{z}}}$ and not another function which behaves as
$u-x+O(u-x)^{2}$. We conclude therefore that the generating function for the
abstract operators which give excited twists is given by

$\displaystyle{\cal T}_{abs}(d,\bar{d})=$
$\displaystyle\exp\left\\{\sum_{n=1}^{\infty}\left[\bar{d}_{n}\partial_{u}^{n-1}\left[(u-x)^{\bar{\epsilon}}\partial_{u}X(u,{\bar{u}})\right]+d_{n}\partial_{u}^{n-1}\left[(u-x)^{\epsilon}\partial_{u}{\bar{X}}(u,{\bar{u}})\right]\right]\right\\}$
$\displaystyle\exp$
$\displaystyle\left\\{-\sum_{n,m=1}^{\infty}\bar{d}_{n}~{}d_{m}~{}\partial^{n-1}_{u}~{}\partial^{m-1}_{v}|_{v=u}\left[(u-x)^{\bar{\epsilon}}(v-x)^{\epsilon}\partial_{u}\partial_{v}\Delta^{z{\bar{z}}}_{c}(u-x,{\bar{u}}-x;v-x,{\bar{v}}-x;\epsilon)\right]\right\\}$
(85)

Explicitly this means that

$\displaystyle\left[\prod_{n=1}^{\infty}\left(\partial^{n}_{u}X\right)^{N_{n}}\left(\partial^{n}_{u}\bar{X}\right)^{\bar{N}_{n}}\sigma_{\epsilon,f}\right](x)=\lim_{u\rightarrow
x}\left.\prod_{n=1}^{\infty}\frac{\partial^{N_{n}}}{\partial\bar{d}_{n}^{N_{n}}}\frac{\partial^{\bar{N}_{n}}}{\partial
d_{n}^{\bar{N}_{n}}}{\cal
T}_{abs}(d,\bar{d})\right|_{\bar{d}=d=0}\sigma_{\epsilon,f}(x)$ (86)

##  4 The path integral approach

The classic method [1] to compute twists correlators is by the path integral

$\displaystyle\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle=\int_{{\cal
M}(\\{x_{t},\epsilon_{t},f_{t}\\})}{\cal D}Xe^{-S_{E}}$ (87)

where ${\cal M}(\\{x_{t},\epsilon_{t},f_{t}\\})$ is the space of string
configurations satisfying the boundary conditions (24) and (23). Since the
integral is quadratic we can then efficiently separate the classical fields
from the quantum fluctuations as

$X(u,\bar{u})=X_{cl}(u,\bar{u};\\{x_{t},\epsilon_{t},f_{t}\\})+X_{q}(u,\bar{u};\\{x_{t},\epsilon_{t}\\})$
(88)

where $X_{cl}$ satisfies the previous boundary conditions (22, 23) while
$X_{q}$ satisfies the same boundary conditions but with all $f_{t}=0$. After
this splitting we obtain

$\displaystyle\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle={\cal
N}(x_{t},\epsilon_{t})e^{-S_{E,cl}(x_{t},\epsilon_{t},f_{t})}$ (89)

where the factor ${\cal N}(x_{t},\epsilon_{t})$ is the quantum contribution.
In particular when all $f_{t}$ are equal, i.e. $f_{t}=f$ the classical action
$S_{E,cl}(x_{t},\epsilon_{t},f_{t}=f)$ is zero since the branes are the
boundary of a zero area polygon therefore we can identify

${\cal
N}(x_{t},\epsilon_{t})=\langle\sigma_{\epsilon_{1},f_{1}=f}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}=f}(x_{N})\rangle$
(90)

which is also true for the quantum fluctuation for which $f=0$. Our aim is now
to compute correlators with both (excited) twist field operators and untwisted
operators $V_{\xi_{i}}(x_{i})$ ($i=1\dots L$) associated with the untwisted
state $\xi_{i}$ of which the ones in eq. (45) are a particular case. We start
with correlators with plain twist field operators which can be computed as

$\displaystyle\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\prod_{i=1}^{L}V_{\psi_{i}}(x_{i})\rangle=\int_{{\cal
M}(\\{x_{t},\epsilon_{t},f_{t}\\})}{\cal
D}X~{}\prod_{i=1}^{L}V_{\psi_{i}}(x_{i})~{}e^{-S_{E}}$ (91)

To do so we notice that it is by far easier not to compute the previous
correlator but to compute the generating function of all the correlators, i.e.
the Reggeon vertex, in the form of the previous path integral (87) plus linear
sources

$\displaystyle V_{N+L}(J_{i})=\int_{{\cal
M}(\\{x_{t},\epsilon_{t},f_{t}\\})}{\cal D}X~{}e^{-S_{E}+\sum_{i=1}^{L}\int
dx~{}J_{iI}(x)X^{I}(x,x)}$ (92)

where
$J_{iI}(x)=\sum_{n=0}^{\infty}c_{(i)nI}\partial^{n}_{x_{i}}\delta(x-x_{i})$
since all untwisted operators can be obtained by taking derivatives with
respect to the coefficients $c_{(i)nI}$ as explained in the previous section.
This starting point is very similar to ([30],[24]) where it was recognized
that the generator for all closed (super)string amplitudes is a quadratic path
integrals. The idea in the previous papers is that the appropriate boundary
condition for R and/or NS sector can be obtained simply by inserting linear
sources with the desired boundary conditions. Because of this assumption the
quantum fluctuations are the same for all the amplitudes: from the purely NS
to the mixed ones. It was later realized that this prescription misses a
proper treatment of quantum fluctuations [31] and that when this part is
considered the amplitudes factorize correctly [8].

###  4.1 Boundary correlators with non excited twists on $\mathbb{R}^{2}$

Our strategy is therefore to compute the path integral (92) by properly
defining it in order to regularize the divergences which arise as usual
because of current self interactions. Taking inspiration from what is done for
deriving the generating function we first regularize the $\delta$ functions in
the currents, a step which corresponds to the point splitting in operatorial
formalism and then subtract the self interaction of the untwisted string. We
are therefore led to consider the path integral

$\displaystyle V_{N+L}(J_{i})=\lim_{\\{\eta_{i}\\}\rightarrow 0}$
$\displaystyle\int_{{\cal M}(\\{x_{t},\epsilon_{t},f_{t}\\})}{\cal
D}X~{}e^{-S_{E}}$ $\displaystyle\times$
$\displaystyle\prod_{i=1}^{L}\left[e^{-\frac{1}{2}\int d{x_{a}}\int
d{x_{b}}J_{iI}({x_{a}},\eta_{i})J_{iJ}({x_{b}},\eta_{i})G^{IJ}_{U(t_{i}),bou}({x_{a}},{x_{b}})}e^{\int
dx~{}J_{iI}(x,\eta_{i})X^{I}(x,x)}\right]$ (93)

where the regularized currents are defined as

$J_{iI}(x,\eta_{i})=\sum_{n=0}^{\infty}c_{(i)nI}\partial^{n}_{x_{i}}\delta(x-x_{i};\eta_{i})$
(94)

with $\delta(x-x_{i};\eta_{i})$ a regularization of the $\delta$ such that
$\lim_{\eta_{i}\rightarrow 0}\delta(x-x_{i};\eta_{i})=\delta(x-x_{i})$ and
$G^{IJ}_{U(t_{i}),bou}({x_{a}},{x_{b}})$ is the untwisted Green function with
boundary conditions which depends on the brane on which the point $x_{i}$ is.
This dependence is the reason why we have written $U(t_{i})$.

It is then immediate to compute the previous path integral by using the
splitting of $X$ into quantum and classical part (88) and get

$\displaystyle V_{N+L}(J_{i})$
$\displaystyle=\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle$
$\displaystyle\times$
$\displaystyle\prod_{i=1}^{L}e^{\sum_{n=0}^{\infty}c_{(i)nI}\partial^{n}_{x_{i}}X^{I}_{cl}(x_{i},x_{i})}$
$\displaystyle\times$
$\displaystyle\prod_{i=1}^{L}e^{\frac{1}{2}\sum_{n=0}^{\infty}c_{(i)nI}\sum_{m=0}^{\infty}c_{(i)mJ}\partial^{n}_{x_{i}}\partial^{m}_{\hat{x}_{i}}\Delta^{IJ}_{(N,M),bou(i)}(x_{i},\hat{x}_{i};\\{x_{t},\epsilon_{t}\\})|_{\hat{x}_{i}=x_{i}}}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq i<j\leq
L}e^{\sum_{n=0}^{\infty}c_{(i)nI}\sum_{m=0}^{\infty}c_{(j)mJ}\partial^{n}_{x_{i}}\partial^{m}_{x_{j}}G^{IJ}_{(N,M),bou}(x_{i},x_{j};\\{x_{t},\epsilon_{t}\\})}$
(95)

where we have introduced the boundary Green function

$G^{IJ}_{(N,M),bou}(x_{i},x_{j};\\{x_{t},\epsilon_{t}\\})=G^{IJ}_{(N,M)}(x_{i}+i0^{+},x_{i}-i0^{+};x_{j}+i0^{+},x_{j}-i0^{+};\\{x_{t},\epsilon_{t}\\})$
(96)

in presence of $N$ twist fields $\sigma_{\epsilon_{t},f_{t}=0}(x_{t})$ (with
$f_{t}=0$ since these terms come from the quantum fluctuations) in the sector
$M=\sum_{t=1}^{N}\epsilon_{t}$ and the regularized Green function

$\Delta^{IJ}_{(N,M),bou(i)}(x_{i},\hat{x}_{i};\\{x_{t},\epsilon_{t}\\})=G^{IJ}_{(N,M),bou}(x_{i},\hat{x}_{i};\\{x_{t},\epsilon_{t}\\})-G^{IJ}_{U(t_{i}),bou}(x_{i},\hat{x}_{i})$
(97)

As it happened for the $N=2$ boundary Green function and its regularized
version all the components of the boundary Green function are proportional to
real symmetric functions $G_{(N,M),bou}(x_{i},x_{j})$ and
$\Delta_{(N,M),bou(i)}(x_{i},x_{j})$ respectively up to phases. Using the
result from the appendix (D) where the explicit expressions for
$G_{(N,M),bou}(x_{i},x_{j})$ and $\Delta_{(N,M),bou}(x_{i},x_{j})$ are given
we can then write

$\displaystyle V_{N+L}(J_{i})$
$\displaystyle=\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle$
$\displaystyle\times$
$\displaystyle\prod_{i=1}^{L}e^{\sum_{n=0}^{\infty}c_{(i)nI}\partial^{n}_{x_{i}}X^{I}_{cl}(x_{i},x_{i})}$
$\displaystyle\times$
$\displaystyle\prod_{i=1}^{L}e^{\sum_{n=0}^{\infty}c_{(i)n\parallel
D_{t_{i}}}\sum_{m=0}^{\infty}c_{(i)m\parallel
D_{t_{i}}}\partial^{n}_{x_{i}}\partial^{m}_{\hat{x}_{i}}\Delta_{(N,M),bou(i)}(x_{i},\hat{x}_{i};\\{x_{t},\epsilon_{t}\\})|_{\hat{x}_{i}=x_{i}}}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq i<j\leq
L}e^{2\sum_{n=0}^{\infty}c_{(i)n\parallel
D_{t_{i}}}\sum_{m=0}^{\infty}c_{(j)m\parallel
D_{t_{i}}}\partial^{n}_{x_{i}}\partial^{m}_{x_{j}}G_{(N,M),bou}(x_{i},x_{j};\\{x_{t},\epsilon_{t}\\})}$
(98)

where $c_{n\parallel D_{t_{i}}}$ are defined as in eq. (65) and $t_{i}$ is
chosen by the brane on which $x_{i}$ lies, i.e.
${x_{t_{i}+1}}<x_{i}<{x_{t_{i}}}$.

####  4.1.1 Some consequences

There are now two immediate consequences. The first and more trivial is that
all correlators with just one derivative vertex have contribution only from
the classical solution $X_{cl}$ [5], i.e

$\displaystyle\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\partial_{x}^{n}X^{I}(x,x)\rangle=\partial_{x}^{n}X^{I}_{cl}(x,x)\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle$
(99)

The second and more interesting is that all correlators can be essentially
computed with Wick theorem plus classical contributions plus self interactions
which are absent in Wick theorem. This implies that a string in presence of
not excited twist fields (defects) is free and can be quantized almost in the
usual manner.

####  4.1.2 Some examples

As a first example we want to compute the correlator of a tachyon with $N$
twist fields

$\displaystyle\langle
e^{ik_{I}X^{I}(x,x)}\prod_{t=1}^{N}\sigma_{\epsilon_{t},f_{t}}(x_{t})\rangle=V_{N+1}|_{c_{1}=ik,\bar{c}_{1}=i\bar{k}}$
$\displaystyle=\langle\prod_{t=1}^{N}\sigma_{\epsilon_{t},f_{t}}(x_{t})\rangle~{}e^{ik\bar{X}_{cl}(x,x)+i\bar{k}X_{cl}(x,x)}e^{-k_{\parallel
D_{t}}^{2}\Delta_{(N,M),bou(i)}(x,x)}$ (100)

which shows that untwisted matter sees a kind of form factor of the
interacting twisted matter in accord with the result from OPE and what
discussed in [26] for the case of a stringy instanton. Again a priori we can
take $k^{I}$ not parallel to the brane $D_{t}$ but the explicit form of the
Green function implies a projection in the direction parallel to $D_{t}$. It
is interesting to compare the previous result with what we can get using the
OPE (67) in the previous correlator. Using the OPE at the leading order we get

$\displaystyle\langle
e^{ik_{I}X^{I}(x,x)}\prod_{t=1}^{N}\sigma_{\epsilon_{t},f_{t}}(x_{t})\rangle\sim(x-{x_{t}})^{-\alpha^{\prime}k^{2}_{\parallel
D_{t}}}e^{-\frac{1}{2}R^{2}(\epsilon_{t})\alpha^{\prime}k^{2}_{\parallel
D_{t}}}e^{i(k\bar{f}_{t}+\bar{k}f_{t})}\langle\prod_{t=1}^{N}\sigma_{\epsilon_{t},f_{t}}(x_{t})\rangle$
(101)

Now using the obvious behavior

$X_{cl}(u,{\bar{u}})\sim_{u\rightarrow
x_{t}}f_{t}+O((u-x_{t})^{\bar{\epsilon}_{t}})+O(({\bar{u}}-x_{t})^{\epsilon_{t}})$
(102)

and the result shown in app. D

$\Delta_{(N,M),bou(i)}(x,\hat{x})\sim_{x\rightarrow
x_{t}}\frac{\alpha^{\prime}}{2}\ln|x-x_{t}|^{2}+R^{2}(\epsilon_{t})+O(x-\hat{x})+O(x-x_{t})$
(103)

in the exact expression we find the expected consistency with the OPE result.

Secondly let us consider the correlator of a gauge boson with twisted matter

$\displaystyle\langle\epsilon_{I}\partial_{x}X^{I}e^{ik_{I}X^{I}(x,x)}$
$\displaystyle\prod_{t=1}^{N}\sigma_{\epsilon_{t},f_{t}}(x_{t})\rangle=\langle\prod_{t=1}^{N}\sigma_{\epsilon_{t},f_{t}}(x_{t})\rangle~{}e^{ik\bar{X}_{cl}(x,x)+i\bar{k}X_{cl}(x,x)}e^{-k_{\parallel
D_{t}}^{2}\Delta_{(N,M),bou(i)}(x,x)}$
$\displaystyle\sqrt{2}e^{-i\pi\alpha_{t}}\epsilon_{\parallel
D_{t}}\left(\partial_{x}X_{cl}(x,x)+\sqrt{2}e^{i\pi\alpha_{t}}ik_{\parallel
D_{t}}\partial_{x}\Delta_{(N,M),bou(i)}(x,\hat{x})|_{\hat{x}=x}\right)$ (104)

which exhibits the same structure as the vertex (68) and the OPE it can be
computed using it.

Next we consider the interaction of two tachyons with the twisted matter in
order to show the Wick-like expression in a simple case

$\displaystyle\langle$ $\displaystyle
e^{ik_{1I}X^{I}(x_{1},x_{1})}e^{ik_{2I}X^{I}(x_{2},x_{1})}\prod_{t=1}^{N}\sigma_{\epsilon_{t},f_{t}}(x_{t})\rangle=V_{N+2}|_{c_{j}=ik_{j},\bar{c}_{j}=i\bar{k}_{j}}$
$\displaystyle=$
$\displaystyle\langle\prod_{t=1}^{N}\sigma_{\epsilon_{t},f_{t}}(x_{t})\rangle\prod_{j=1}^{2}\left[e^{ik_{j}\bar{X}_{cl}(x_{j},x_{j})+i\bar{k}_{j}X_{cl}(x_{j},x_{j})}e^{-k_{\parallel
D_{t_{j}}}^{2}\Delta_{(N,M),bou(i)}(x_{j},x_{j})}\right]$ $\displaystyle
e^{-k_{\parallel D_{t_{1}}}k_{\parallel D_{t_{2}}}G_{(N,M),bou}(x_{1},x_{2})}$
(105)

where $G_{(N,M),bou}(x_{1},x_{2})$ is the common factor of all the components
of the boundary Green function $G_{(N,M),bou}^{IJ}(x_{1},x_{2})$ and is given
in eq. (167) whose explicit expression (166) implies that only the momenta
parallel to the brane on which the vertex lies contributes.

Finally, we consider a more lengthy example

$\displaystyle\langle$
$\displaystyle(\partial_{x}^{2}X\partial_{x}X\partial_{x}\bar{X})(x_{1})(\partial_{x}^{2}\bar{X})(x_{2})\prod_{t=1}^{N}\sigma_{\epsilon_{t},f_{t}}(x_{t})\rangle=\frac{\partial^{3}}{\partial\bar{c}_{(1)2}\partial\bar{c}_{(1)1}\partial
c_{(1)1}}\frac{\partial}{\partial c_{(2)2}}V_{N+2}\Big{|}_{c_{(j)n}=0}$
$\displaystyle=$
$\displaystyle\langle\prod_{t=1}^{N}\sigma_{\epsilon_{t},f_{t}}(x_{t})\rangle\Bigg{[}(\partial_{x}^{2}X_{cl}\partial_{x}X_{cl}\partial_{x}\bar{X}_{cl})(x_{1})(\partial_{x}^{2}\bar{X}_{cl})(x_{2})$
$\displaystyle+e^{i\pi(-\alpha_{t_{1}}-\alpha_{t_{2}})}~{}(\partial_{x}^{2}X_{cl}\partial_{x}X_{cl})(x_{1})~{}\partial_{x_{1}}\partial_{x_{2}}^{2}G_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle+e^{i\pi(\alpha_{t_{1}}-\alpha_{t_{2}})}~{}(\partial_{x}^{2}X_{cl}\partial_{x}\bar{X}_{cl})(x_{1})~{}\partial_{x_{1}}\partial_{x_{2}}^{2}G_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle+e^{i\pi(\alpha_{t_{1}}-\alpha_{t_{2}})}~{}(\partial_{x}X_{cl}\partial_{x}\bar{X}_{cl})(x_{1})~{}\partial_{x_{1}}^{2}\partial_{x_{2}}^{2}G_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle+\frac{1}{2}~{}\partial_{x}^{2}X_{cl}(x_{1})\partial_{x}\bar{X}_{cl}(x_{2})~{}\partial_{x_{1}}\partial_{\hat{x}_{1}}\Delta_{(N,M),bou}(x_{1},\hat{x}_{1})|_{\hat{x}_{1}=x_{1}}$
$\displaystyle+\frac{1}{2}~{}\partial_{x}X_{cl}(x_{1})\partial_{x}\bar{X}_{cl}(x_{2})~{}\partial_{x_{1}}^{2}\partial_{\hat{x}_{1}}\Delta_{(N,M),bou}(x_{1},\hat{x}_{1})|_{\hat{x}_{1}=x_{1}}$
$\displaystyle+e^{2i\pi\alpha_{t_{1}}}~{}\partial_{x}\bar{X}_{cl}(x_{1})\partial_{x}\bar{X}_{cl}(x_{2})~{}\partial_{x_{1}}^{2}\partial_{\hat{x}_{1}}\Delta_{(N,M),bou}(x_{1},\hat{x}_{1})|_{\hat{x}_{1}=x_{1}}$
$\displaystyle+\frac{1}{2}e^{i\pi(\alpha_{t_{1}}-\alpha_{t_{2}})}~{}\partial_{x_{1}}\partial_{\hat{x}_{1}}\Delta_{(N,M),bou}(x_{1},\hat{x}_{1})|_{\hat{x}_{1}=x_{1}}~{}\partial_{x_{1}}^{2}\partial_{x_{2}}^{2}G_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle+\frac{1}{2}e^{i\pi(\alpha_{t_{1}}-\alpha_{t_{2}})}~{}\partial_{x_{1}}^{2}\partial_{\hat{x}_{1}}\Delta_{(N,M),bou}(x_{1},\hat{x}_{1})|_{\hat{x}_{1}=x_{1}}~{}\partial_{x_{1}}\partial_{x_{2}}^{2}G_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle+\frac{1}{2}e^{i\pi(\alpha_{t_{1}}-\alpha_{t_{2}})}~{}\partial_{x_{1}}^{2}\partial_{\hat{x}_{1}}\Delta_{(N,M),bou}(x_{1},\hat{x}_{1})|_{\hat{x}_{1}=x_{1}}~{}\partial_{x_{1}}\partial_{x_{2}}^{2}G_{(N,M),bou}(x_{1},x_{2})\Bigg{]}$
(106)

###  4.2 Boundary correlators with non excited twists on $T^{2}$

The wrapping contributions have been studied for the pure twist field
correlators in [32] for the N=3 case and in [14] for the case $M=N-2$ and
there is not any difference among the different $M$ values therefore the
results obtained there are valid. Let us anyhow quickly review them. Given a
minimal $N$-polygon in $T^{2}$ with vertices $\\{f_{t}\\}$, i.e. with all
vertices in the fundamental cell, we can consider non minimal polygons which
wrap the $T^{2}$. These can be easierly described as polygons which have
vertices $\\{\tilde{f}_{t}\\}$ in the covering $\mathbb{R}^{2}$ where
$T^{2}\equiv R^{2}/\Lambda$ with the lattice defined as
$\Lambda=\\{n_{1}e_{1}+n_{2}e_{2}|n_{1},n_{2}\in\mathbb{Z}\\}$. These
configurations give an additive contribution to the classical path integral as

$\displaystyle V_{N+M}^{(T^{2})}(J_{i})$
$\displaystyle=\sum_{\\{\tilde{f}_{t}\\}}\left[\langle\sigma_{\epsilon_{1},\tilde{f}_{1}}(x_{1})\dots\sigma_{\epsilon_{N},\tilde{f}_{N}}(x_{N})\rangle\prod_{i=1}^{M}e^{\sum_{n=0}^{\infty}c_{(i)nI}\partial^{n}_{x_{i}}X^{I}_{cl}(x_{i},x_{i};\\{x_{t},\epsilon_{t},\tilde{f}_{t}\\})}\right]$
$\displaystyle\times$
$\displaystyle\prod_{i=1}^{M}e^{\frac{1}{2}\sum_{n=0}^{\infty}c_{(i)nI}\sum_{m=0}^{\infty}c_{(i)mJ}\partial^{n}_{x_{i}}\partial^{m}_{\hat{x}_{i}}\Delta^{IJ}_{(N,M),bou(i)}(x_{i},\hat{x}_{i};\\{x_{t},\epsilon_{t}\\})|_{\hat{x}_{i}=x_{i}}}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq i<j\leq
M}e^{\sum_{n=0}^{\infty}c_{(i)nI}\sum_{m=0}^{\infty}c_{(j)mJ}\partial^{n}_{x_{i}}\partial^{m}_{x_{j}}G^{IJ}_{(N,M),bou}(x_{i},x_{j};\\{x_{t},\epsilon_{t}\\})}$
(107)

In order to determine the possible vertices $\\{\tilde{f}_{t}\\}$ without
redundancy it is necessary to keep a vertex fixed and then expand the polygon.
For definiteness we keep fixed the vertex $\tilde{f}_{1}=f_{1}$ which lies at
the intersection between $D_{N}$ and $D_{1}$. We then move the next vertex
$f_{2}$ along the $D_{1}$ brane. Explicitly we write
$\tilde{f}_{2}=\tilde{f}_{1}+(f_{2}-f_{1})+n_{1}t_{1}=f_{2}+n_{1}t_{1}$ with
$n_{1}\in\mathbb{Z}$ and $t_{1}$ the shortest tangent vector to $D_{1}$ which
is in $\Lambda$. We can now continue for all the other vertices for which we
have
$\tilde{f}_{t}=\tilde{f}_{t-1}+(f_{t}-f_{t-1})+n_{t-1}t_{t-1}=f_{t}+\sum_{k=1}^{t-1}n_{k}t_{k}$.
For consistency we need requiring $\tilde{f}_{N+1}\equiv\tilde{f}_{1}=f_{1}$,
therefore the possible wrapped polygons are obtained from the solution of the
Diophantine equation

$\displaystyle\sum_{{\hat{t}}=1}^{N}n_{\hat{t}}t_{\hat{t}}=0$ (108)

which cannot be solved in general terms but only on a case by case basis as
discussed in [14].

###  4.3 Chiral correlators with non excited twists.

As a warming up for the computation of excited twist fields which we perform
in the next section we consider the interaction of chiral vertices with plain
twists.

We can now follow the same strategy we used in section 4.1 and compute the
path integral with the insertion of an arbitrary number $L_{c}$ of currents
which act as generating functions for the chiral vertex operators. As done for
the boundary correlators in section 4.1 we first regularize the currents and
then subtract the self interaction of the untwisted string. We are therefore
led to consider the path integral

$\displaystyle V_{N+L_{c}}(J_{c})=\lim_{\\{\eta_{c}\\}\rightarrow 0}$
$\displaystyle\int_{{\cal M}(\\{x_{t},\epsilon_{t},f_{t}\\})}{\cal
D}X~{}e^{-S_{E}}$ $\displaystyle\times$
$\displaystyle\prod_{c=1}^{L_{c}}\Big{[}e^{-\frac{1}{2}\int d^{2}u_{a}\int
d^{2}u_{b}J_{cI}(u_{a},\eta_{c})J_{cJ}(u_{b},\eta_{c})\partial_{u_{a}}\partial_{u_{b}}G^{IJ}_{U(t_{c})}(u_{a},{\bar{u}}_{a};u_{b},{\bar{u}}_{b})}$
$\displaystyle e^{\int
d^{2}u~{}\partial_{u}J_{cI}(u,\eta_{c})\partial_{u}X^{I}(u,{\bar{u}})}\Big{]}$
(109)

where in the second line we have written the regularization factor analogous
to the one used in section 4.1 which regularizes the currents in the third
line. In the previous expression the regularized currents are defined as

$J_{cI}(u,{\bar{u}},\eta_{c})=\sum_{n=1}^{\infty}c_{(c)nI}\partial^{n-1}_{u}\delta^{2}(u-u_{c};\eta_{c})$
(110)

with $\delta^{2}(u-u_{c};\eta_{c})$ a regularization of the $\delta^{2}()$
such that $\lim_{\eta_{c}\rightarrow 0}\delta^{2}(u;\eta_{c})=\delta^{2}(u)$.
Notice that we need using a $\delta^{2}()$ in the previous expression since we
use “directional” derivatives along $u$. To subtract the self interaction of
the untwisted string we used
$G^{IJ}_{U(t_{c})}(u_{a},{\bar{u}}_{a};u_{b},{\bar{u}}_{b})$, the untwisted
Green function computed for an arbitrary $t_{c}$ since we chose to consider
currents with at least one derivative $\partial_{u}$ so that only
$G^{z{\bar{z}}}_{U}$ give with a non vanishing contribution which is also
independent on the brane $D_{t}$.

Finally performing the path integral we get

$\displaystyle V_{N+L_{c}}(J_{c})$
$\displaystyle=\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle$
$\displaystyle\times$
$\displaystyle\prod_{c=1}^{L_{c}}\Bigg{\\{}e^{\sum_{n=1}^{\infty}c_{(c)nI}\partial^{n-1}_{u_{c}}[\partial_{u}X^{I}_{cl}(u_{c},{\bar{u}}_{c})]}$
$\displaystyle\times$ $\displaystyle
e^{\frac{1}{2}\sum_{n,m=1}^{\infty}c_{(c)nI}c_{(c)mJ}\partial^{n}_{u_{d}}\partial^{m}_{v_{\hat{c}}}\Delta_{(N,M)(c)}^{IJ}(u_{c},{\bar{u}}_{c};v_{c},{\bar{v}}_{c};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})|_{v_{c}=u_{c}}]}\Bigg{\\}}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq c<\hat{c}\leq
N}e^{\sum_{n,m=1}^{\infty}d_{(c)nI}d_{(\hat{c})mJ}\partial^{n}_{u_{t}}\partial^{m}_{v_{\hat{t}}}G_{(N,M)}^{IJ}(u_{c},{\bar{u}}_{c};v_{\hat{c}},{\bar{v}}_{\hat{c}};\\{x_{t},\epsilon_{t}\\})}$
(111)

where we have written the dependence on the complex conjugate variables such
as ${\bar{u}}$ even if the derivatives are independent in order to be
consistent with the notation used in the boundary case. The regularized chiral
Green function is defined as expected as

$\displaystyle\partial_{u}\partial_{v}\Delta_{(N,M)(c)}^{IJ}(u,{\bar{u}};v,{\bar{v}}\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})$
$\displaystyle=\partial_{u}\partial_{v}G_{(N,M)}^{IJ}(u,{\bar{u}};v,{\bar{v}};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})-\partial_{u}\partial_{v}G^{IJ}_{U(t)}(u,{\bar{u}};v,{\bar{v}};\epsilon_{t})$
(112)

and we notice that the subtraction term is different from zero only when
$IJ=z{\bar{z}}$ or $IJ={\bar{z}}z$ because of the derivatives.

###  4.4 Correlators of excited twists on $\mathbb{R}^{2}$

Finally we can compute the correlators of excited twist fields by letting the
appropriate chiral currents collide with the twist fields. We follow the same
strategy we used in section 4.1 and in the previous section 4.3 and compute
the path integral with the insertion of one generating function (85) for each
twist field. As done for the boundary correlators in section 4.1 we first
regularize the $\delta^{2}()$ functions in the currents and then subtract the
self interaction of the untwisted string. We are therefore led to consider the
path integral

$\displaystyle V_{N}(K_{t})=$
$\displaystyle\lim_{\\{u_{t}\\}\rightarrow\\{x_{t}\\}}\lim_{\eta_{t}\rightarrow
0}\int_{{\cal M}(\\{x_{t},\epsilon_{t},f_{t}\\})}{\cal D}X~{}e^{-S_{E}}$
$\displaystyle\times$ $\displaystyle\prod_{t=1}^{N}\Big{[}e^{-\frac{1}{2}\int
d^{2}u_{a}\int
d^{2}u_{b}K_{tI}(u_{a},\eta_{t})K_{tJ}(u_{b},\eta_{t})(u_{a}-x_{t})^{\epsilon_{tI}}(u_{b}-x_{t})^{\epsilon_{tJ}}\partial_{u_{a}}\partial_{u_{b}}G^{IJ}_{U(t)}(u_{a},{\bar{u}}_{a};u_{b},{\bar{u}}_{b})}$
$\displaystyle e^{\int
d^{2}u~{}K_{tI}(u,\eta_{t})(u-x_{t})^{\epsilon_{tI}}\partial_{u}X^{I}(u,{\bar{u}})}$
$\displaystyle e^{-\int d^{2}u_{a}\int
d^{2}u_{b}K_{tz}(u_{a},\eta_{t})K_{t{\bar{z}}}(u_{b},\eta_{t})\left[(u_{a}-x_{t})^{\bar{\epsilon}_{t}}(u_{b}-x_{t})^{\epsilon_{t}}\partial_{u_{a}}\partial_{u_{b}}\Delta^{z{\bar{z}}}_{c}(u_{a}-x_{t},{\bar{u}}_{a}-x_{t};u_{b}-x_{t},{\bar{u}}_{b}-x_{t};\epsilon_{t})\right]}\Big{]}$
(113)

where in the second line we have written the regularization factor analogous
to the one used in section 4.1 which regularizes the third line and finally in
the last line we have the necessary subtraction term which is in eq. (85). In
the previous expression the regularized currents are defined as

$K_{tI}(u,{\bar{u}},\eta_{t})=\sum_{n=1}^{\infty}d_{(t)nI}\partial^{n-1}_{u}\delta^{2}(u-u_{t};\eta_{t})$
(114)

with $\delta^{2}(u-u_{t};\eta_{t})$ a regularization of the $\delta^{2}()$ as
in previous section. For writing a more compact expression we have also used
$\epsilon_{tI}$ defined as

$\epsilon_{tz}=\bar{\epsilon}_{t},~{}~{}~{}\epsilon_{t{\bar{z}}}=\epsilon_{t}$
(115)

Finally performing the path integral we get

$\displaystyle V_{N}(K_{t})$
$\displaystyle=\lim_{\\{u_{t}\\}\rightarrow\\{x_{t}\\}}\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle$
$\displaystyle\times$
$\displaystyle\prod_{t=1}^{N}\Bigg{\\{}e^{\sum_{n=1}^{\infty}d_{(t)nI}\partial^{n-1}_{u_{t}}[(u_{t}-x_{t})^{\epsilon_{tI}}\partial_{u}X^{I}_{cl}(u_{t},{\bar{u}}_{t})]}$
$\displaystyle\times$ $\displaystyle
e^{\frac{1}{2}\sum_{n,m=1}^{\infty}d_{(t)nI}d_{(t)mJ}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{t}}[(u_{t}-x_{t})^{\epsilon_{tI}}(v_{t}-x_{t})^{\epsilon_{tJ}}\partial_{u}\partial_{v}\Delta_{(N,M)(t)}^{IJ}(u_{t},{\bar{u}}_{t};v_{t},{\bar{v}}_{t};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})]|_{v_{t}=u_{t}}}\Bigg{\\}}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq t<{\hat{t}}\leq
N}e^{\sum_{n,m=1}^{\infty}d_{(t)nI}d_{({\hat{t}})mJ}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{\hat{t}}}[(u_{t}-x_{t})^{\epsilon_{tI}}(v_{\hat{t}}-x_{\hat{t}})^{\epsilon_{{\hat{t}}J}}\partial_{u}\partial_{v}G_{(N,M)}^{IJ}(u_{t},{\bar{u}}_{t};v_{\hat{t}},{\bar{v}}_{\hat{t}};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})]}$
(116)

where we have defined the regularized Green function at the twist fields $t$
to be

$\displaystyle\partial_{u}\partial_{v}$
$\displaystyle\Delta_{(N,M)(t)}^{IJ}(u,{\bar{u}};v,{\bar{v}};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})=$
$\displaystyle=$
$\displaystyle\partial_{u}\partial_{v}\Delta_{(N,M)(c)}^{IJ}(u,{\bar{u}};v,{\bar{v}};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})-\partial_{u}\partial_{v}\Delta^{IJ}_{c}(u-x_{t},{\bar{u}}-x_{t};v-x_{t},{\bar{v}}-x_{t};\epsilon_{t})$
$\displaystyle=\partial_{u}\partial_{v}G_{(N,M)}^{IJ}(u,{\bar{u}};v,{\bar{v}};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})-\partial_{u}\partial_{v}G_{N=2}^{IJ}(u-x_{t},{\bar{u}}-x_{t};v-x_{t},{\bar{v}}-x_{t};\epsilon_{t})$
(117)

and we used $G_{U(t)}^{IJ}(u-x_{t},v-x_{t})=G_{U(t)}^{IJ}(u,v)$ to write the
last line and again we have written the dependence on ${\bar{u}}$ and
${\bar{v}}$ even if the derivatives are independent on them for having a
consistent notation. Actually because of the chiral derivatives the previous
expression simplifies in two cases to

$\displaystyle\partial_{u}\partial_{v}\Delta_{(N,M)(t)}^{zz}(u,{\bar{u}};v,{\bar{v}};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})$
$\displaystyle=\partial_{u}\partial_{v}G_{(N,M)}^{zz}(u,{\bar{u}};v,{\bar{v}};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})$
$\displaystyle\partial_{u}\partial_{v}\Delta_{(N,M)(t)}^{{\bar{z}}{\bar{z}}}(u,{\bar{u}};v,{\bar{v}};;\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})$
$\displaystyle=\partial_{u}\partial_{v}G_{(N,M)}^{{\bar{z}}{\bar{z}}}(u,{\bar{u}};v,{\bar{v}};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})$
(118)

It is interesting to notice that regularized Green functions are got by
subtracting the divergent part with the proper monodromy at the point of
regularization which at the points where a twist field is located means
$G_{N=2}$ while in all other points means $G_{U}$. In particular both
$(u-x_{t})^{\epsilon_{tI}}(v-x_{t})^{\epsilon_{tJ}}\partial_{u}\partial_{v}\Delta_{(N,M)(t)}^{IJ}$
and
$(u-x_{t})^{\epsilon_{tI}}(v-x_{t})^{\epsilon_{tJ}}\partial_{u}\partial_{v}G_{(N,M)}^{IJ}$
are analytic functions at $u=x_{t}$ whose explicit expression is given in
appendix E.

More explicitly the previous generating function can be written as

$\displaystyle V_{N}(K_{t})$
$\displaystyle=\lim_{\\{u_{t}\\}\rightarrow\\{x_{t}\\}}\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle$
$\displaystyle\times$
$\displaystyle\prod_{t=1}^{N}\Bigg{\\{}e^{\sum_{n=1}^{\infty}\left[\bar{d}_{(t)n}\partial^{n-1}_{u_{t}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}\partial_{u}X_{cl}(u_{t},{\bar{u}}_{t})]+d_{(t)n}\partial^{n-1}_{u_{t}}[(u_{t}-x_{t})^{\epsilon_{t}}\partial_{u}{\bar{X}}_{cl}(u_{t},{\bar{u}}_{t})]\right]}$
$\displaystyle\times$ $\displaystyle
e^{\frac{1}{2}\sum_{n,m=1}^{\infty}\bar{d}_{(t)n}\bar{d}_{(t)m}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{t}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}(v_{t}-x_{t})^{\bar{\epsilon}_{t}}\partial_{u}\partial_{v}G_{(N,M)(t)}^{zz}(u_{t},{\bar{u}}_{t};v_{t},{\bar{v}}_{t};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})]|_{v_{t}=u_{t}}}$
$\displaystyle\times$ $\displaystyle
e^{\frac{1}{2}\sum_{n,m=1}^{\infty}d_{(t)n}d_{(t)m}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{t}}[(u_{t}-x_{t})^{\epsilon_{t}}(v_{t}-x_{t})^{\epsilon_{t}}\partial_{u}\partial_{v}G_{(N,M)(t)}^{{\bar{z}}{\bar{z}}}(u_{t},{\bar{u}}_{t};v_{t},{\bar{v}}_{t};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})]|_{v_{t}=u_{t}}}$
$\displaystyle\times$ $\displaystyle
e^{\sum_{n,m=1}^{\infty}\bar{d}_{(t)n}d_{(t)m}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{t}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}(v_{t}-x_{t})^{\epsilon_{t}}\partial_{u}\partial_{v}\Delta_{(N,M)(t)}^{z{\bar{z}}}(u_{t},{\bar{u}}_{t};v_{t},{\bar{v}}_{t};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})]|_{v_{t}=u_{t}}}\Bigg{\\}}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq t<{\hat{t}}\leq
N}\Bigg{\\{}e^{\sum_{n,m=1}^{\infty}\bar{d}_{(t)n}\bar{d}_{({\hat{t}})m}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{\hat{t}}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}(v_{\hat{t}}-x_{\hat{t}})^{\bar{\epsilon}_{{\hat{t}}}}\partial_{u}\partial_{v}G_{(N,M)}^{zz}(u_{t},{\bar{u}}_{t};v_{\hat{t}},{\bar{v}}_{\hat{t}};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})]}$
$\displaystyle\times
e^{\sum_{n,m=1}^{\infty}d_{(t)n}d_{({\hat{t}})m}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{\hat{t}}}[(u_{t}-x_{t})^{\epsilon_{t}}(v_{\hat{t}}-x_{\hat{t}})^{\epsilon_{{\hat{t}}}}\partial_{u}\partial_{v}G_{(N,M)}^{{\bar{z}}{\bar{z}}}(u_{t},{\bar{u}}_{t};v_{\hat{t}},{\bar{v}}_{\hat{t}};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})]}$
$\displaystyle\times
e^{\sum_{n,m=1}^{\infty}\bar{d}_{(t)n}d_{({\hat{t}})m}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{\hat{t}}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}(v_{\hat{t}}-x_{\hat{t}})^{\epsilon_{{\hat{t}}}}\partial_{u}\partial_{v}G_{(N,M)}^{z{\bar{z}}}(u_{t},{\bar{u}}_{t};v_{\hat{t}},{\bar{v}}_{\hat{t}};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})]}$
$\displaystyle\times
e^{\sum_{n,m=1}^{\infty}d_{(t)n}\bar{d}_{({\hat{t}})m}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{\hat{t}}}[(u_{t}-x_{t})^{\epsilon_{t}}(v_{\hat{t}}-x_{\hat{t}})^{\bar{\epsilon}_{{\hat{t}}}}\partial_{u}\partial_{v}G_{(N,M)}^{{\bar{z}}z}(u_{t},{\bar{u}}_{t};v_{\hat{t}},{\bar{v}}_{\hat{t}};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})]}\Bigg{\\}}$
(119)

####  4.4.1 Some examples

Let us consider the simplest non trivial correlator with one excited twist
field

$\displaystyle\langle\left(\partial^{n}X\sigma_{\epsilon_{t},f_{t}}\right)\prod_{{\bar{t}}\neq
t}\sigma_{\epsilon_{\bar{t}},f_{\bar{t}}}\rangle=\frac{\partial}{\partial\bar{d}_{(t)nz}}V_{N}|_{d=\bar{d}=0}$
$\displaystyle=\langle\prod_{{\bar{t}}}\sigma_{\epsilon_{\bar{t}},f_{\bar{t}}}\rangle\lim_{u_{t}\rightarrow
x_{t}}\partial^{n-1}_{u_{t}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}\partial_{u}X_{cl}(u_{t})]$
(120)

where the limit is not strictly necessary since the expression
$(u_{t}-x_{t})^{\bar{\epsilon}_{t}}\partial_{u}X_{cl}(u_{t})$ is regular in
$x_{t}$. The explicit expression can be easily computed and it is given in
appendix F for some cases. Nevertheless correlators with only one excited
twist can be less trivial as

$\displaystyle\langle$
$\displaystyle\left(\partial^{n}X\partial^{m}\bar{X}\sigma_{\epsilon_{t},f_{t}}\right)\prod_{{\bar{t}}\neq
t}\sigma_{\epsilon_{\bar{t}},f_{\bar{t}}}\rangle=\frac{\partial^{2}}{\partial\bar{d}_{(t)nz}\partial
d_{(t)mz}}V_{N}|_{d=\bar{d}=0}$
$\displaystyle=\langle\prod_{{\bar{t}}}\sigma_{\epsilon_{\bar{t}},f_{\bar{t}}}\rangle\lim_{u_{t}\rightarrow
x_{t}}\Big{[}\partial^{n-1}_{u_{t}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}\partial_{u}X_{cl}(u_{t})]\partial^{m-1}_{u_{t}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}\partial_{u}X_{cl}(u_{t})]$
$\displaystyle+\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{t}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}(v_{t}-x_{t})^{\epsilon_{t}}\partial_{u}\partial_{v}\Delta_{(N,M)(t)}^{z{\bar{z}}}(u_{t},{\bar{u}}_{t};v_{t},{\bar{v}}_{t};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})]|_{v_{t}=u_{t}}\Big{]}$
(121)

Finally we can also consider

$\displaystyle\langle$
$\displaystyle\left(\partial^{n}X\sigma_{\epsilon_{t},f_{t}}\right)\left(\partial^{m}\bar{X}\sigma_{\epsilon_{\hat{t}},f_{\hat{t}}}\right)\prod_{{\bar{t}}\neq
t,{\hat{t}}}\sigma_{\epsilon_{\bar{t}},f_{\bar{t}}}\rangle=\frac{\partial^{2}}{\partial\bar{d}_{(t)nz}\partial
d_{({\hat{t}})mz}}V_{N}|_{d=\bar{d}=0}$
$\displaystyle=\langle\prod_{{\bar{t}}}\sigma_{\epsilon_{\bar{t}},f_{\bar{t}}}\rangle\lim_{u_{t,{\hat{t}}}\rightarrow
x_{t,{\hat{t}}}}\Big{[}\partial^{n-1}_{u_{t}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}\partial_{u}X_{cl}(u_{t})]\partial^{m-1}_{u_{\hat{t}}}[(u_{\hat{t}}-x_{\hat{t}})^{\bar{\epsilon}_{{\hat{t}}}}\partial_{u}X_{cl}(u_{\hat{t}})]$
$\displaystyle+\partial^{n-1}_{u_{t}}\partial^{m-1}_{u_{\hat{t}}}[(u_{t}-x_{t})^{\bar{\epsilon}_{t}}(u_{\hat{t}}-x_{\hat{t}})^{\epsilon_{{\hat{t}}}}\partial_{u_{t}}\partial_{u_{\hat{t}}}G_{(N,M)}^{z{\bar{z}}}(u_{t},{\bar{u}}_{t};u_{\hat{t}},{\bar{u}}_{\hat{t}};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})]\Big{]}$
(122)

This correlator is the correlator of eq. (4.83) of [5], i.e.
$\langle\tau_{\alpha}(x_{1})\tau_{1-\alpha}(x_{2})\sigma_{\beta}(x_{3})\sigma_{1-\beta}(x_{4})\rangle$
when we set $N=4$, $n=m=1$ and $\epsilon_{1}=\alpha$, $\epsilon_{2}=1-\alpha$,
$\epsilon_{3}=\beta$ and $\epsilon_{4}=1-\beta$.

###  4.5 Correlators of boundary operators and excited twists on
$\mathbb{R}^{2}$

Finally we can assemble the results from previous section to write the
generating function for correlators of boundary operators and excited twists
on $R^{2}$ to be

$\displaystyle V_{N+L}(K_{t},J_{i})$
$\displaystyle=\lim_{\\{u_{t}\\}\rightarrow\\{x_{t}\\}}\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle$
$\displaystyle\times$
$\displaystyle\prod_{t=1}^{N}\Bigg{\\{}e^{\sum_{n=1}^{\infty}d_{(t)nI}\partial^{n-1}_{u_{t}}[(u_{t}-x_{t})^{\epsilon_{tI}}\partial_{u}X^{I}_{cl}(u_{t},{\bar{u}}_{t})]}$
$\displaystyle\times$ $\displaystyle
e^{\frac{1}{2}\sum_{n,m=1}^{\infty}d_{(t)nI}d_{(t)mJ}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{t}}[(u_{t}-x_{t})^{\epsilon_{tI}}(v_{t}-x_{t})^{\epsilon_{tJ}}\partial_{u}\partial_{v}\Delta_{(N,M)(t)}^{IJ}(u_{t},{\bar{u}}_{t};v_{t},{\bar{v}}_{t};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})]|_{v_{t}=u_{t}}}\Bigg{\\}}$
$\displaystyle\times$
$\displaystyle\prod_{i=1}^{L}\Bigg{\\{}e^{\sum_{n=0}^{\infty}c_{(i)nI}\partial^{n}_{x_{i}}X^{I}_{cl}(x_{i},x_{i})}$
$\displaystyle\times$ $\displaystyle
e^{\frac{1}{2}\sum_{n=0}^{\infty}c_{(i)nI}\sum_{m=0}^{\infty}c_{(i)mJ}\partial^{n}_{x_{i}}\partial^{m}_{\hat{x}_{i}}\Delta^{IJ}_{(N,M),bou(i)}(x_{i},\hat{x}_{i};\\{x_{t},\epsilon_{t}\\})|_{\hat{x}_{i}=x_{i}}}\Bigg{\\}}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq t<{\hat{t}}\leq
N}e^{\sum_{n,m=1}^{\infty}d_{(t)nI}d_{({\hat{t}})mJ}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{\hat{t}}}[(u_{t}-x_{t})^{\epsilon_{tI}}(v_{\hat{t}}-x_{\hat{t}})^{\epsilon_{{\hat{t}}J}}\partial_{u}\partial_{v}G_{(N,M)}^{IJ}(u_{t},{\bar{u}}_{t};v_{\hat{t}},{\bar{v}}_{\hat{t}};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})]}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq i<j\leq
L}e^{\sum_{n=0}^{\infty}c_{(i)nI}\sum_{m=0}^{\infty}c_{(j)mJ}\partial^{n}_{x_{i}}\partial^{m}_{x_{j}}G^{IJ}_{(N,M),bou}(x_{i},x_{j};\\{x_{t},\epsilon_{t}\\})}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq t\leq N}\prod_{1\leq j\leq
L}e^{\sum_{n=1}^{\infty}d_{(t)nI}c_{(j)mJ}\partial^{n-1}_{u_{t}}\partial^{m}_{x_{j}}[(u_{t}-x_{t})^{\epsilon_{tI}}\partial_{u}G_{(N,M)}^{IJ}(u_{t},{\bar{u}}_{t};x_{j},x_{j};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})]}$
(123)

where the last line is the interaction between the twist fields and the
boundary operators. The generating function for correlators on $T^{2}$ can be
formally easily obtained as done in section 4.2 by summing over all possible
wrapping contributions as in eq. 107.

###  4.6 Correlators of bulk operators and excited twists on $\mathbb{R}^{2}$

We can now make an educated guess of the generating function of the
correlators of bulk operators and excited twists. As long as the bulk vertex
operators do not involve momenta there is no doubt on the result since the
bulk field can be written as the product of a chiral vertex times an
antichiral vertex therefore the generating function is nothing else but the
product of the generating function for the chiral part times the generating
function for the antichiral part times the obvious interaction of the chiral
current with the antichiral. What requires an educated guess is when the bulk
vertex operators involve momenta since in this case we know that the local
description requires to separate the right moving from the left moving part
and normal order them separately, i.e. to the abstract vertex
$e^{ik_{I}X^{I}(u,{\bar{u}})}$ corresponds the local version
$:e^{ik_{I}X^{I}_{(loc)L}({u_{loc}})}::e^{ik_{I}X^{I}_{(loc)R}({{\bar{u}}_{loc}})}:$
up to cocycles ([33], [26] but see also [34]) Hence we guess that in the
presence of momenta the twisted Green function must be split in its chiral-
chiral, chiral-antichiral and so on pieces, i.e.
$G=G_{(LL)}+G_{(LR)}+G_{(RL)}+G_{(RR)}$. This is obviously consistent with the
case where derivatives are present since applying a derivative like
$\partial_{u}$ of the Green function is actually projecting it on the chiral
piece. We therefore guess that the generating function of the correlators of
$L_{c}$ bulk operators and $N$ excited twists read up to phases due to
cocycles

$\displaystyle V_{N+L_{c}}(K_{t},J_{c})$
$\displaystyle=\lim_{\\{u_{t}\\}\rightarrow\\{x_{t}\\}}\langle\sigma_{\epsilon_{1},f_{1}}(x_{1})\dots\sigma_{\epsilon_{N},f_{N}}(x_{N})\rangle$
$\displaystyle\times$
$\displaystyle\prod_{t=1}^{N}\Bigg{\\{}e^{\sum_{n=1}^{\infty}d_{(t)nI}\partial^{n-1}_{u_{t}}[(u_{t}-x_{t})^{\epsilon_{tI}}\partial_{u}X^{I}_{cl}(u_{t},{\bar{u}}_{t})]}$
$\displaystyle\times$ $\displaystyle
e^{\frac{1}{2}\sum_{n,m=1}^{\infty}d_{(t)nI}d_{(t)mJ}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{t}}[(u_{t}-x_{t})^{\epsilon_{tI}}(v_{t}-x_{t})^{\epsilon_{tJ}}\partial_{u}\partial_{v}\Delta_{(N,M)(t)}^{IJ}(u_{t},{\bar{u}}_{t};v_{t},{\bar{v}}_{t};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})]|_{v_{t}=u_{t}}}\Bigg{\\}}$
$\displaystyle\times$
$\displaystyle\prod_{c=1}^{L_{c}}\Bigg{\\{}e^{\sum_{n=0}^{\infty}c_{(L,c)nI}\partial^{n}_{u_{c}}X^{I}_{(L)cl}(u_{c})}$
$\displaystyle\times$ $\displaystyle
e^{\frac{1}{2}\sum_{n=0}^{\infty}c_{(L,c)nI}\sum_{m=0}^{\infty}c_{(L,c)mJ}\partial^{n}_{u_{c}}\partial^{m}_{\hat{u}_{c}}\Delta^{IJ}_{(N,M),(LL,c)}(u_{c},\hat{u}_{c};\\{x_{t},\epsilon_{t}\\})|_{\hat{u}_{c}=u_{c}}}\Bigg{\\}}$
$\displaystyle\times$
$\displaystyle\prod_{c=1}^{L_{c}}\Bigg{\\{}e^{\sum_{n=0}^{\infty}c_{(R,c)nI}\partial^{n}_{{\bar{u}}_{c}}X^{I}_{(R)cl}({\bar{u}}_{c})}$
$\displaystyle\times$ $\displaystyle
e^{\frac{1}{2}\sum_{n=0}^{\infty}c_{(R,c)nI}\sum_{m=0}^{\infty}c_{(R,c)mJ}\partial^{n}_{{\bar{u}}_{c}}\partial^{m}_{\hat{\bar{u}}_{i}}\Delta^{IJ}_{(N,M),(RR,c)}(u_{c},\hat{{\bar{u}}}_{c};\\{x_{t},\epsilon_{t}\\})|_{\hat{\bar{u}}_{c}={\bar{u}}_{c}}}\Bigg{\\}}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq t<{\hat{t}}\leq
N}e^{\sum_{n,m=1}^{\infty}d_{(t)nI}d_{({\hat{t}})mJ}\partial^{n-1}_{u_{t}}\partial^{m-1}_{v_{\hat{t}}}[(u_{t}-x_{t})^{\epsilon_{tI}}(v_{\hat{t}}-x_{\hat{t}})^{\epsilon_{{\hat{t}}J}}\partial_{u}\partial_{v}G_{(N,M)}^{IJ}(u_{t},{\bar{u}}_{t};v_{\hat{t}},{\bar{v}}_{\hat{t}};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})]}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq c<d\leq
L_{c}}e^{\sum_{n=0}^{\infty}c_{(L,c)nI}\sum_{m=0}^{\infty}c_{(L,d)mJ}\partial^{n}_{u_{c}}\partial^{m}_{u_{d}}G^{IJ}_{(N,M)(LL)}(u_{c},u_{d};\\{x_{t},\epsilon_{t}\\})}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq c,d\leq
L_{c}}e^{\sum_{n=0}^{\infty}c_{(L,c)nI}\sum_{m=0}^{\infty}c_{(R,d)mJ}\partial^{n}_{u_{c}}\partial^{m}_{{\bar{u}}_{d}}\left[G^{IJ}_{(N,M)(LL)}(u_{c},{\bar{u}}_{d};\\{x_{t},\epsilon_{t}\\})+G^{JI}_{(N,M)(LL)}({\bar{u}}_{d},u_{c};\\{x_{t},\epsilon_{t}\\})\right]}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq c<d\leq
L_{c}}e^{\sum_{n=0}^{\infty}c_{(R,c)nI}\sum_{m=0}^{\infty}c_{(R,d)mJ}\partial^{n}_{{\bar{u}}_{c}}\partial^{m}_{{\bar{u}}_{d}}G^{IJ}_{(N,M)(RR)}({\bar{u}}_{c},{\bar{u}}_{d};\\{x_{t},\epsilon_{t}\\})}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq t\leq N}\prod_{1\leq c\leq
L_{c}}e^{\sum_{n=1}^{\infty}d_{(t)nI}c_{(L,c)mJ}\partial^{n-1}_{u_{t}}\partial^{m}_{u_{c}}[(u_{t}-x_{t})^{\epsilon_{tI}}\partial_{u}\left[G_{(N,M)(LL)}^{IJ}(u_{t};u_{c};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})+G_{(N,M)(RL)}^{IJ}({\bar{u}}_{t};u_{c};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})\right]]}$
$\displaystyle\times$ $\displaystyle\prod_{1\leq t\leq N}\prod_{1\leq c\leq
L_{c}}e^{\sum_{n=1}^{\infty}d_{(t)nI}c_{(R,c)mJ}\partial^{n-1}_{u_{t}}\partial^{m}_{{\bar{u}}_{c}}[(u_{t}-x_{t})^{\epsilon_{tI}}\partial_{u}\left[G_{(N,M)(LL)}^{IJ}(u_{t};{\bar{u}}_{c};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})+G_{(N,M)(RL)}^{IJ}({\bar{u}}_{t};{\bar{u}}_{c};\\{x_{\bar{t}},\epsilon_{\bar{t}}\\})\right]]}$
(124)

Using the previous generating function it would be interesting deriving the
boundary state with $N$ twist fields. This could be done as in [35] and would
be an interesting generalization of the boundary state with open string
interactions derived in [36]. This boundary state could be used as in [37] to
derive useful information about the long distance spacetime geometry generated
by branes at angles.

###  4.7 Rewriting the Reggeon vertex using auxiliary Fock spaces

In the previous section we have given the explicit form of the generating
function for correlators of boundary operators and excited twists.
Traditionally and for sewing a different expression is used where to any
operator insertion, i.e. external leg is associated an auxiliary Fock space.
If we associate to any twist operator an auxiliary Fock space with vacuum
$|T_{t}\rangle$ and we identify888 The normalization is chosen in the usual
way such that applying the map $d$ to auxiliary operators on eq. (81) we have
$\langle T_{aux}|{\cal
T}(d\rightarrow\bar{\alpha}_{aux},\bar{d}\rightarrow\alpha_{aux})~{}\lim_{u_{aux}\rightarrow
0}\partial_{u_{aux}}^{n}\left[{u_{aux}}^{\bar{\epsilon}}\partial_{u_{aux}}X_{loc}({u_{aux}},{\bar{u}_{aux}})\right]|T_{aux}\rangle=\partial_{u_{loc}}^{n}\left[{u_{loc}}^{\bar{\epsilon}}\partial_{u_{loc}}X_{loc}({u_{loc}},{{\bar{u}}_{loc}})\right]$
.

$(n-1)!\bar{d}_{(t)n}\leftrightarrow\frac{-2}{\alpha^{\prime}}{k_{\epsilon_{t}}}\frac{\alpha_{(t,aux)n-\bar{\epsilon}_{t}}}{n-\bar{\epsilon}_{t}},~{}~{}~{}~{}(n-1)!d_{(t)n}\leftrightarrow\frac{-2}{\alpha^{\prime}}{k_{\bar{\epsilon}_{t}}}\frac{\bar{\alpha}_{(t,aux)n-\epsilon_{t}}}{n-\epsilon_{t}},~{}~{}~{}~{}$
(125)

and we do the same for any boundary vertex operator to which we associate an
auxiliary Fock space with vacuum $|0_{a},p_{(j)}=0\rangle$ and we identify
($n\geq 0$)

$n!\bar{c}_{(j)n}\leftrightarrow\frac{-2}{\alpha^{\prime}}{k_{\epsilon_{t_{j}}}}\alpha_{(j,aux)n},~{}~{}~{}~{}n!c_{(j)n}\leftrightarrow\frac{-2}{\alpha^{\prime}}{k_{\bar{\epsilon}_{t_{j}}}}\bar{\alpha}_{(j,aux)n}$
(126)

we can write the generating function as a usual Reggeon vertex as

$\displaystyle\langle V_{N+L}|=$ $\displaystyle\prod_{t=1}^{N}\langle T_{t}|$
$\displaystyle\prod_{i=1}^{L}\langle 0_{a},x_{(i)}=0|$
$\displaystyle\prod_{t=1}^{N}e^{\frac{-2}{\alpha^{\prime}}\oint_{z=x_{t}}\frac{dz}{2\pi
i}\left[\bar{\chi}_{(t,aux)}^{(+)}(z-x_{t})~{}\partial\chi_{cl}(z)+\chi_{(t,aux)}^{(+)}(z-x_{t})~{}\partial\bar{\chi}_{cl}(z)\right]}$
$\displaystyle\prod_{i=1}^{L}e^{\frac{-2}{\alpha^{\prime}}\oint_{z=x_{t}}\frac{dz}{2\pi
i}\left[\bar{\chi}_{(i,aux)}^{(+)}(z-x_{t})~{}\partial\chi_{cl}(z)+\chi_{(i,aux)}^{(+)}(z-x_{t})~{}\partial\bar{\chi}_{cl}(z)\right]}$
$\displaystyle\prod_{t,{\hat{t}}=1}^{N}\Bigg{[}$ $\displaystyle
e^{\frac{2}{\alpha^{\prime 2}}\oint_{z=x_{t}}\frac{dz}{2\pi
i}\oint_{w=x_{\hat{t}}}\frac{dw}{2\pi
i}\bar{\chi}_{(t,aux)}(z-x_{t})~{}\bar{\chi}_{({\hat{t}},aux)}(w-x_{\hat{t}})~{}\partial_{z}\partial_{w}G^{zz}_{(N,M)}(z,{\bar{z}};w,\bar{w})}$
$\displaystyle e^{\frac{2}{\alpha^{\prime 2}}\oint_{z=x_{t}}\frac{dz}{2\pi
i}\oint_{w=x_{\hat{t}}}\frac{dw}{2\pi
i}\chi_{(t,aux)}(z-x_{t})~{}\chi_{({\hat{t}},aux)}(w-x_{\hat{t}})~{}\partial_{z}\partial_{w}G^{{\bar{z}}{\bar{z}}}_{(N,M)}(z,{\bar{z}};w,\bar{w})}$
$\displaystyle e^{\frac{4}{\alpha^{\prime 2}}\oint_{z=x_{t}}\frac{dz}{2\pi
i}\oint_{w=x_{\hat{t}}}\frac{dw}{2\pi
i}\bar{\chi}_{(t,aux)}(z-x_{t})~{}\bar{\chi}_{({\hat{t}},aux)}(w-x_{\hat{t}})~{}\partial_{z}\partial_{w}G^{z{\bar{z}}}_{(N,M)}(z,{\bar{z}};w,\bar{w})}\Bigg{]}$
$\displaystyle\prod_{i,j=1}^{L}\Bigg{[}$ $\displaystyle
e^{\frac{1}{\alpha^{\prime 2}}\oint_{z=x_{i}}\frac{dz}{2\pi
i}\oint_{w=x_{j}}\frac{dw}{2\pi
i}\partial_{z}\bar{\chi}_{(i,aux)}(z-x_{t})~{}\partial_{w}\bar{\chi}_{(j,aux)}(w-x_{j})~{}G^{zz}_{(N,M)}(z,{\bar{z}};w,\bar{w})}$
$\displaystyle e^{\frac{1}{\alpha^{\prime 2}}\oint_{z=x_{i}}\frac{dz}{2\pi
i}\oint_{w=x_{j}}\frac{dw}{2\pi
i}\partial_{z}\chi_{(i,aux)}(z-x_{t})~{}\partial_{w}\chi_{(j,aux)}(w-x_{j})~{}G^{{\bar{z}}{\bar{z}}}_{(N,M)}(z,{\bar{z}};w,\bar{w})}$
$\displaystyle e^{\frac{2}{\alpha^{\prime 2}}\oint_{z=x_{i}}\frac{dz}{2\pi
i}\oint_{w=x_{j}}\frac{dw}{2\pi
i}\partial_{z}\bar{\chi}_{(i,aux)}(z-x_{t})~{}\partial_{w}\bar{\chi}_{(j,aux)}(w-x_{j})~{}G^{z{\bar{z}}}_{(N,M)}(z,{\bar{z}};w,\bar{w})}\Bigg{]}$
$\displaystyle\prod_{i=1}^{L}\prod_{t=1}^{N}\Bigg{[}$ $\displaystyle
e^{\frac{1}{\alpha^{\prime 2}}\oint_{z=x_{i}}\frac{dz}{2\pi
i}\oint_{w=x_{t}}\frac{dw}{2\pi
i}\partial_{z}\bar{\chi}_{(i,aux)}(z-x_{t})~{}\bar{\chi}_{(t,aux)}(w-x_{t})~{}\partial_{w}G^{zz}_{(N,M)}(z,{\bar{z}};w,\bar{w})}$
$\displaystyle e^{\frac{1}{\alpha^{\prime 2}}\oint_{z=x_{i}}\frac{dz}{2\pi
i}\oint_{w=x_{t}}\frac{dw}{2\pi
i}\partial_{z}\chi_{(i,aux)}(z-x_{t})~{}\chi_{(t,aux)}(w-x_{t})~{}\partial_{w}G^{{\bar{z}}{\bar{z}}}_{(N,M)}(z,{\bar{z}};w,\bar{w})}$
$\displaystyle e^{\frac{1}{\alpha^{\prime 2}}\oint_{z=x_{i}}\frac{dz}{2\pi
i}\oint_{w=x_{t}}\frac{dw}{2\pi
i}\partial_{z}\bar{\chi}_{(i,aux)}(z-x_{t})~{}\chi_{(t,aux)}(w-x_{t})~{}\partial_{w}G^{z{\bar{z}}}_{(N,M)}(z,{\bar{z}};w,\bar{w})}$
$\displaystyle e^{\frac{1}{\alpha^{\prime 2}}\oint_{z=x_{i}}\frac{dz}{2\pi
i}\oint_{w=x_{t}}\frac{dw}{2\pi
i}\partial_{z}\chi_{(i,aux)}(z-x_{t})~{}\bar{\chi}_{(t,aux)}(w-x_{t})~{}\partial_{w}G^{{\bar{z}}z}_{(N,M)}(z,{\bar{z}};w,\bar{w})}\Bigg{]}$
(127)

where we have used the doubled fields $\chi_{I}(z)$ defined as in eq. 16 for
the twisted case and in the usual way for the untwisted one. Notice that the
terms $t={\hat{t}}$ and $i=j$ must be regularized by properly computing the
two contour integrals as discussed in [24]

Acknowledgments

The author thanks R. Richter for useful comments on a preliminary draft.

This work is supported in part by the Compagnia di San Paolo contract ”Modern
Application in String Theory” (MAST) TO-Call3-2012-0088

##  Appendix A Details and useful formula for the untwisted string and $N=2$
case

We start considering the untwisted string associated to the $D_{t}$ brane,
i.e. the string with both ends on $D_{t}$. This has boundary conditions

$\displaystyle
Re(e^{-i\pi\alpha_{t}}{\partial_{y}}X_{loc}|_{y=0})=Im(e^{-i\pi\alpha_{t}}X_{loc}|_{y=0})-g_{t}=0$
(128)

in the upper half plane $H$ and has the expansion

$\displaystyle X_{loc}({u_{loc}},{{\bar{u}}_{loc}})$
$\displaystyle=e^{i\pi\alpha_{t}}\Bigg{[}x^{\hat{1}}+ig_{t}-i~{}2\alpha^{\prime}p^{\hat{1}}~{}\ln|{u_{loc}}|$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}\sum_{n=1}^{\infty}\left[\frac{\bar{\alpha}_{(t)n}}{n}{u_{loc}}^{-n}-\frac{\alpha_{(t)n}^{\dagger}}{n}{u_{loc}}^{n}\right]$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}\sum_{n=1}^{\infty}\left[-\frac{\bar{\alpha}_{(t)n}^{\dagger}}{n}{{\bar{u}}_{loc}}^{n}+\frac{\alpha_{(t)n}}{n}{{\bar{u}}_{loc}}^{-n}\right]\Bigg{]}$
$\displaystyle\bar{X}_{loc}({u_{loc}},{{\bar{u}}_{loc}})$
$\displaystyle=e^{-i\pi\alpha_{t}}\Bigg{[}x^{\hat{1}}-ig_{t}-2\alpha^{\prime}ip^{\hat{1}}~{}\ln|{u_{loc}}|$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}\sum_{n=1}^{\infty}\left[-\frac{\bar{\alpha}_{(t)n}^{\dagger}}{n}{u_{loc}}^{n}+\frac{\alpha_{(t)n}}{n}{u_{loc}}^{-n}\right]$
$\displaystyle+i\frac{1}{2}\sqrt{2\alpha^{\prime}}\sum_{n=1}^{\infty}\left[\frac{\bar{\alpha}_{(t)n}}{n}{{\bar{u}}_{loc}}^{-n)}-\frac{\alpha_{(t)n}^{\dagger}}{n}{{\bar{u}}_{loc}}^{n}\right]\Bigg{]}$
(129)

We have the non trivial commutators

$[\alpha_{(t)n},\alpha_{(t)m}^{\dagger}]=n\delta_{m,n},~{}~{}~{}~{}[\bar{\alpha}_{(t)n},\bar{\alpha}_{(t)m}^{\dagger}]=n\delta_{m,n},~{}~{}~{}~{}[x^{\hat{1}},p^{\hat{1}}]=i$
(130)

where $x^{\hat{1}},p^{\hat{1}}$ are the zero mode position and momentum of
string
$X^{\hat{1}}=\frac{1}{\sqrt{2}}(e^{-i\pi\alpha_{t}}X+e^{+i\pi\alpha_{t}}\bar{X})$
with $NN$ boundary condition. The vacuum is defined in the usual way by

$\alpha_{(t)n}|0_{t}\rangle=\bar{\alpha}_{(t)n}|0_{t}\rangle=p^{\hat{1}}|0_{t}\rangle=0~{}~{}~{}~{}n\geq
1$ (131)

even if care must be taken in order to deal with the $DD$ zero modes ***. Then
we can compute the untwisted Green functions

$\displaystyle
G^{zz}_{U(t)}({u_{loc}},{{\bar{u}}_{loc}};{v_{loc}},{{\bar{v}}_{loc}})$
$\displaystyle=[X_{loc}^{(+)}({u_{loc}},{{\bar{u}}_{loc}}),X_{loc}^{(-)}({v_{loc}},{{\bar{v}}_{loc}})]={k_{\epsilon}}^{2}\ln|{u_{loc}}-{{\bar{v}}_{loc}}|^{2}$
$\displaystyle
G^{{\bar{z}}{\bar{z}}}_{U(t)}({u_{loc}},{{\bar{u}}_{loc}};{v_{loc}},{{\bar{v}}_{loc}})$
$\displaystyle=[\bar{X}_{loc}^{(+)}({u_{loc}},{\bar{u}}),\bar{X}_{loc}^{(-)}({v_{loc}},{{\bar{v}}_{loc}})]=k_{\bar{\epsilon}}^{2}\ln|{u_{loc}}-{{\bar{v}}_{loc}}|^{2}$
$\displaystyle
G^{z{\bar{z}}}_{U(t)}({u_{loc}},{{\bar{u}}_{loc}};{v_{loc}},{{\bar{v}}_{loc}})$
$\displaystyle=[X_{loc}^{(+)}({u_{loc}},{{\bar{u}}_{loc}}),\bar{X}_{loc}^{(-)}({v_{loc}},{{\bar{v}}_{loc}})]={k_{\epsilon}}k_{\bar{\epsilon}}\ln|{u_{loc}}-{v_{loc}}|^{2}$
(132)

where
${k_{\epsilon_{t}}}=-i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{i\pi\alpha_{t}}$
and
${k_{\bar{\epsilon}_{t}}}=-i\frac{1}{2}\sqrt{2\alpha^{\prime}}e^{-i\pi\alpha_{t}}$
as in the main text. Notice that $G^{z{\bar{z}}}_{U(t)}$ does not feel that
the brane is rotated while both $G^{zz}_{U(t)}$ and
$G^{{\bar{z}}{\bar{z}}}_{U(t)}$ do because of the phase in
${k_{\epsilon_{t}}}^{2}$ and ${k_{\bar{\epsilon}_{t}}}^{2}$.

In a similar way we can compute the $N=2$ twisted Green functions

$\displaystyle
G^{zz}_{N=2}(u,{\bar{u}};v,{\bar{v}};\\{0,\epsilon;\infty,\bar{\epsilon}\\})$
$\displaystyle=[X_{loc}^{(+)}({u_{loc}},{{\bar{u}}_{loc}}),X_{loc}^{(-)}({v_{loc}},{{\bar{v}}_{loc}})]$
$\displaystyle=-{k_{\epsilon_{t}}}^{2}\left[\frac{1}{\epsilon}\left(\frac{{v_{loc}}}{{{\bar{u}}_{loc}}}\right)^{\epsilon}{}_{2}F_{1}(1,\epsilon;1+\epsilon;\frac{{v_{loc}}}{{{\bar{u}}_{loc}}})+\frac{1}{{\bar{\epsilon}}}\left(\frac{{{\bar{v}}_{loc}}}{{u_{loc}}}\right)^{\bar{\epsilon}}{}_{2}F_{1}(1,{\bar{\epsilon}};1+{\bar{\epsilon}};\frac{{{\bar{v}}_{loc}}}{{u_{loc}}})\right]$
$\displaystyle
G^{{\bar{z}}{\bar{z}}}_{N=2}(u,{\bar{u}};v,{\bar{v}};\\{0,\epsilon;\infty,\bar{\epsilon}\\})$
$\displaystyle=[\bar{X}_{loc}^{(+)}({u_{loc}},{{\bar{u}}_{loc}}),\bar{X}_{loc}^{(-)}({v_{loc}},{{\bar{v}}_{loc}})]$
$\displaystyle=-{k_{\bar{\epsilon}_{t}}}^{2}\left[\frac{1}{\epsilon}\left(\frac{{{\bar{v}}_{loc}}}{{u_{loc}}}\right)^{\epsilon}{}_{2}F_{1}(1,\epsilon;1+\epsilon;\frac{{{\bar{v}}_{loc}}}{{u_{loc}}})+\frac{1}{{\bar{\epsilon}}}\left(\frac{{v_{loc}}}{{{\bar{u}}_{loc}}}\right)^{\bar{\epsilon}}{}_{2}F_{1}(1,{\bar{\epsilon}};1+{\bar{\epsilon}};\frac{{v_{loc}}}{{{\bar{u}}_{loc}}})\right]$
$\displaystyle
G^{z{\bar{z}}}_{N=2}(u,{\bar{u}};v,{\bar{v}};\\{0,\epsilon;\infty,\bar{\epsilon}\\})$
$\displaystyle=[X_{loc}^{(+)}({u_{loc}},{{\bar{u}}_{loc}}),\bar{X}_{loc}^{(-)}({v_{loc}},{{\bar{v}}_{loc}})]$
$\displaystyle=G^{{\bar{z}}z}_{N=2}(v,{\bar{v}};u,{\bar{u}};\\{0,\epsilon;\infty,\bar{\epsilon}\\})=G^{{\bar{z}}z}_{N=2}(u,{\bar{u}};v,{\bar{v}};\\{0,\epsilon;\infty,\bar{\epsilon}\\})$
$\displaystyle=-{k_{\epsilon_{t}}}{k_{\bar{\epsilon}_{t}}}\left[\frac{1}{\epsilon}\left(\frac{{{\bar{v}}_{loc}}}{{{\bar{u}}_{loc}}}\right)^{\epsilon}{}_{2}F_{1}(1,\epsilon;1+\epsilon;\frac{{{\bar{v}}_{loc}}}{{{\bar{u}}_{loc}}})+\frac{1}{{\bar{\epsilon}}}\left(\frac{{v_{loc}}}{{u_{loc}}}\right)^{\bar{\epsilon}}{}_{2}F_{1}(1,{\bar{\epsilon}};1+{\bar{\epsilon}};\frac{{v_{loc}}}{{u_{loc}}})\right]$
(133)

where we have used

${}_{2}F_{1}(1,\epsilon;1+\epsilon;x)=\sum_{n=0}^{\infty}\frac{\epsilon}{n+\epsilon}x^{n}~{}~{}~{}~{}|x|<1$
(134)

as follows from the general expression for the hypergeometric function
${}_{2}F_{1}(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{n!(c)_{n}}x^{n}$
with $(a)_{n}=\Gamma(a+n)/\Gamma(a)$ the Pochhammer symbol. They have the
following symmetry properties

$\displaystyle
G^{IJ}_{N=2}(u,{\bar{u}};v,{\bar{v}};\\{0,\epsilon;\infty,\bar{\epsilon}\\})=G^{JI}_{N=2}(v,{\bar{v}};u,{\bar{u}};\\{0,\epsilon;\infty,\bar{\epsilon}\\})=G^{IJ}_{N=2}(v,{\bar{v}};u,{\bar{u}};\\{0,\bar{\epsilon};\infty,\epsilon\\})$
(135)

which follow from the hypergeometric transformation properties in particular
${}_{2}F_{1}(1,\epsilon;1+\epsilon;x)=\frac{\epsilon}{\bar{\epsilon}x}{}_{2}F_{1}(1,\bar{\epsilon};1+\bar{\epsilon};1/x)$.

For future use f.x. in eq. (175) we notice that

$\displaystyle g(u,v;\epsilon)$
$\displaystyle=({k_{\epsilon_{t}}}{k_{\bar{\epsilon}_{t}}})^{-1}\partial_{u}\partial_{v}G^{z{\bar{z}}}=z^{-\bar{\epsilon}}w^{-\epsilon}\frac{\bar{\epsilon}z+\epsilon
w}{(u-v)^{2}}$ $\displaystyle l(u,v;\epsilon)$
$\displaystyle=\partial_{u}\partial_{v}G^{zz}=0$ $\displaystyle
h(u,v;\epsilon)$
$\displaystyle=\partial_{u}\partial_{v}G^{{\bar{z}}{\bar{z}}}=0$ (136)

In order to write $\Delta^{IJ}$, the regularized Green function, in a more
compact and transparent way we introduce the quantity

$D(x;\epsilon)=\sum_{n=0}^{\infty}\frac{x^{n+\epsilon}}{n+\epsilon}-\sum_{n=1}^{\infty}\frac{x^{n}}{n}=\frac{x^{\epsilon}}{\epsilon}{}_{2}F_{1}(1,\epsilon;1+\epsilon;x)+\log(1-x)~{}~{}|arg(x)|<\pi$
(137)

which can be expanded around $x=1$ as

$D(x;\epsilon)=\psi(1)-\psi(\epsilon)-\sum_{n=1}^{\infty}\left(\begin{array}[]{c}\epsilon-1\\\
n\end{array}\right)\frac{(x-1)^{n}}{n}$ (138)

where $\psi(x)=d\log\Gamma(x)/dx$ is the digamma function. All of this
expansion but the constant term can be easily obtained by computing
$D^{\prime}(x;\epsilon)$ and the integrating on $x$. To get the constant term
is necessary to use the

$\displaystyle{}_{2}F_{1}(a,b;a+b;x)=$
$\displaystyle\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(n!)^{2}}\left[2\psi(n+1)-\psi(a+n)-\psi(b+n)-\log(1-x)\right](1-x)^{n}$
$\displaystyle~{}~{}~{}~{}|arg(1-x)|<\pi,~{}~{}|1-x|<1$ (139)

We can now write the boundary $\Delta_{bou}^{IJ}$ defined as

$\Delta_{bou}^{IJ}({x_{1}};{x_{2}};\epsilon)=\left\\{\begin{array}[]{c
l}\scriptstyle
G^{IJ}_{N=2}({x_{1}}+i0^{+},{x_{1}}-i0^{+};{x_{2}}+i0^{+},{x_{2}}-i0^{+};\\{0,\epsilon;\infty,\bar{\epsilon}\\})&\scriptstyle-G^{IJ}_{U(t)}({x_{1}}+i0^{+},{x_{1}}-i0^{+};{x_{2}}+i0^{+},{x_{2}}-i0^{+})\\\
&{x_{1}},{x_{2}}>0\\\ \scriptstyle
G^{IJ}_{N=2}({x_{1}}+i0^{+},{x_{1}}-i0^{+};{x_{2}}+i0^{+},{x_{2}}-i0^{+};\\{0,\epsilon;\infty,\bar{\epsilon}\\})&\scriptstyle-G^{IJ}_{U(t+1)}({x_{1}}+i0^{+},{x_{1}}-i0^{+};{x_{2}}+i0^{+},{x_{2}}-i0^{+})\\\
&{x_{1}},{x_{2}}<0\end{array}\right.$ (140)

as

$\displaystyle\Delta_{bou}^{zz}({x_{1}};{x_{2}};\epsilon)$
$\displaystyle=\left\\{\begin{array}[]{c
c}e^{i2\pi\alpha_{t}}~{}\Delta_{bou}(x_{1},x_{2})&{x_{1}},{x_{2}}>0\\\
e^{i2\pi\alpha_{t+1}}~{}\Delta_{bou}(x_{1},x_{2})&{x_{1}},{x_{2}}<0\end{array}\right.$
(143)
$\displaystyle\Delta_{bou}^{{\bar{z}}{\bar{z}}}({x_{1}};{x_{2}};\epsilon)$
$\displaystyle=\left\\{\begin{array}[]{c
c}e^{-i2\pi\alpha_{t}}~{}\Delta_{bou}(x_{1},x_{2})&{x_{1}},{x_{2}}>0\\\
e^{-i2\pi\alpha_{t+1}}~{}\Delta_{bou}(x_{1},x_{2})&{x_{1}},{x_{2}}<0\end{array}\right.$
(146) $\displaystyle\Delta_{bou}^{z{\bar{z}}}({x_{1}};{x_{2}};\epsilon)$
$\displaystyle=~{}\Delta_{bou}(x_{1},x_{2})~{}~{}~{}$ (147)

where we have defined the common factor

$\frac{-2}{\alpha^{\prime}}\Delta_{bou}(x_{1},x_{2})=D\left(\frac{{x_{1}}}{{x_{2}}};\epsilon\right)+D\left(\frac{{x_{1}}}{{x_{2}}};\bar{\epsilon}\right)+\log({x_{1}}^{2})\sim\log({x_{1}}^{2})+R^{2}(\epsilon_{t})$
(148)

We have to consider the cases ${x_{1}},{x_{2}}>0$ and ${x_{1}},{x_{2}}<0$
because, for example,
$\frac{{v_{loc}}}{{{\bar{u}}_{loc}}}=\frac{{x_{2}}}{{x_{1}}}$ when
${x_{1}},{x_{2}}>0$ and
$\frac{{v_{loc}}}{{{\bar{u}}_{loc}}}=\frac{|{x_{2}}|}{|{x_{1}}|}e^{i2\pi}$
when ${x_{1}},{x_{2}}<0$ and this gives raise to different phases. This is
issue is not present for $\Delta_{bou}^{z{\bar{z}}}$ because
$\frac{{v_{loc}}}{{u_{loc}}}=\frac{{x_{2}}}{{x_{1}}}$ independently on
${x_{1}},{x_{2}}>0$ or ${x_{1}},{x_{2}}<0$. Notice that these phases are
fundamental for projecting an arbitrary momentum $(k,\bar{k})$ in the
direction parallel to the $D_{t}$ brane as shown explicitly in section 4.1.

In a similar way we define the regularized Green function with the twist
$\sigma_{\epsilon,f}$ at ${x_{loc}}=0$ and the anti-twist
$\sigma_{\bar{\epsilon},f}$ in ${x_{loc}}=\infty$ used in chiral operators
correlators. In particular because of the fact that there are at least
$\partial_{u}\partial_{v}$ the only piece which contributes is

$\displaystyle\Delta_{c}^{z{\bar{z}}}({u_{loc}},{{\bar{u}}_{loc}};{v_{loc}},{{\bar{v}}_{loc}};\epsilon)$
$\displaystyle=G^{z{\bar{z}}}_{N=2}({u_{loc}},{{\bar{u}}_{loc}};{v_{loc}},{{\bar{v}}_{loc}};\\{0,\epsilon;\infty,\bar{\epsilon}\\})-G^{z{\bar{z}}}_{U(t)}(({u_{loc}},{{\bar{u}}_{loc}};{v_{loc}},{{\bar{v}}_{loc}})$
$\displaystyle=\frac{-\alpha^{\prime}}{2}\left[D\left(\frac{{{\bar{v}}_{loc}}}{{{\bar{u}}_{loc}}};\epsilon\right)+D\left(\frac{{v_{loc}}}{{u_{loc}}};\bar{\epsilon}\right)+\log|{u_{loc}}|^{2}\right]$
(149)

which is again independent on the phase of ${k_{\epsilon}}_{t}$.

##  Appendix B Classical solutions

In this appendix we would like to summarize the results of the previous work
[22] (see also [15] and [14]). Defined the anharmonic ratio for a complex
variable $z\in\mathbb{C}$ to be

$\omega_{z}=\frac{z-x_{2}}{z-x_{N}}\frac{x_{1}-x_{N}}{x_{1}-x_{2}}$ (150)

so that $\omega_{1}=1$, $\omega_{2}=0$ and $\omega_{N}=-\infty$, a basis of
the derivatives of zero modes of the two dimensional laplacian satisfying the
boundary conditions (22) is

$\displaystyle\partial_{\omega}{\cal X}^{(n)}(\omega_{z})$
$\displaystyle=~{}\prod_{t=1}^{N-1}(\omega_{z}-\omega_{t})^{-(1-\epsilon_{t})}~{}\omega_{z}^{n},~{}~{}~{}~{}0\leq
n\leq N-M-2$ $\displaystyle\partial_{\omega}{\bar{\cal X}}^{(r)}(\omega_{z})$
$\displaystyle=~{}\prod_{t=1}^{N-1}(\omega_{z}-\omega_{t})^{-\epsilon_{t}}~{}\omega_{z}^{r},~{}~{}~{}~{}0\leq
r\leq M-2$ (151)

so that we can write the classical solution

$\displaystyle X_{cl}(u,{\bar{u}};\\{x_{t},\epsilon_{t},f_{t}\\})=f_{N}$
$\displaystyle+\sum_{n=0}^{N-M-2}a_{n}(\omega_{t})\int_{-\infty;\omega\in
H^{+}}^{\omega_{u}}d\omega~{}\partial_{\omega}{\cal X}^{(n)}(\omega)$
$\displaystyle+\sum_{r=0}^{M-2}b_{r}(\omega_{t})\left[\int_{-\infty;\omega\in
H^{+}}^{\omega_{u}}d\omega~{}\partial_{\omega}{\bar{\cal
X}}^{(r)}(\omega)\right]^{*}$ (152)

which satisfies also the global constraints (23). The real coefficients
$e^{-i\pi\alpha_{1}}a_{n}(\omega_{t})$ and
$e^{-i\pi\alpha_{1}}b_{r}(\omega_{t})$ are fixed by the constraints

$X_{cl}(x_{t+1},\bar{x}_{t+1})-X_{cl}(x_{t},\bar{x}_{t})=f_{t+1}-f_{t}~{}~{}~{}~{}t=2,\dots
N-1$ (153)

The “reality” of the coefficients can be easily seen once we introduce the
following functions which are real on the real axis

$\displaystyle\partial_{\omega}{\mathfrak{X}}^{(n)}(\omega)$
$\displaystyle=~{}\prod_{t=1}^{N-1}|\omega-\omega_{t}|^{-\bar{\epsilon}_{t}}~{}\omega^{n},~{}~{}~{}~{}0\leq
n\leq N-M-2$ $\displaystyle\partial_{\omega}{\mathfrak{X}}^{(r)}(\omega)$
$\displaystyle=~{}\prod_{t=1}^{N-1}|\omega-\omega_{t}|^{-\epsilon_{t}}~{}\omega^{r},~{}~{}~{}~{}0\leq
r\leq M-2$ (154)

and their integrals (which can be expressed using the type D Lauricella
functions)

$\displaystyle I^{(N)}_{t,n}(\bar{\epsilon})$
$\displaystyle=\int_{\omega_{t+1}}^{\omega_{t}}d\omega~{}\partial_{\omega}{\mathfrak{X}}^{(n)}(\omega)$
$\displaystyle I^{(N)}_{t,r}(\epsilon)$
$\displaystyle=\int_{\omega_{t+1}}^{\omega_{t}}d\omega~{}\partial_{\omega}{\mathfrak{X}}^{(r)}(\omega)$
(155)

so that we can write the constraints as

$\displaystyle(-1)^{t-1}\sum_{n=0}^{N-M-2}a_{n}I^{(N)}_{t,n}(\bar{\epsilon})+\sum_{r=0}^{M-2}b_{r}I^{(N)}_{t,r}(\epsilon)=e^{-i\pi\alpha_{1}}\left[e^{-i\pi\alpha_{t}}(f_{t}-f_{t+1})\right]~{}~{}~{}~{}$
(156)

where the quantity between square brackets on the right hand side is real.
Finally the classical action can be written as

$\displaystyle S_{cl}=$
$\displaystyle\frac{1}{8\pi\alpha^{\prime}}\Bigg{[}\sum_{n,m=0}^{N-M-2}(e^{-i\pi\alpha_{1}}a_{n})~{}(e^{-i\pi\alpha_{1}}a_{m})\sum_{t=1}^{N-2}\sum_{{\hat{t}}=t+1}^{N-1}\sin\left(\pi\sum_{u=t+1}^{\hat{t}}\bar{\epsilon}_{u}\right)I^{(N)}_{t,n}(\bar{\epsilon})I^{(N)}_{{\hat{t}},m}(\bar{\epsilon})$
$\displaystyle+$
$\displaystyle\sum_{r,s=0}^{M-2}(e^{-i\pi\alpha_{1}}b_{r})(e^{-i\pi\alpha_{1}}b_{s})\sum_{t=1}^{N-2}\sum_{{\hat{t}}=t+1}^{N-1}\sin\left(\pi\sum_{u=t+1}^{\hat{t}}\epsilon_{u}\right)I^{(N)}_{t,r}(\epsilon)I^{(N)}_{{\hat{t}},s}(\epsilon)\Bigg{]}$
(157)

##  Appendix C Green function

In ([2]) following previous works on the subjects we defined the derivatives
of the Green function on the whole complex plane $\mathbb{C}$ using the
doubling trick as

$\displaystyle g_{(N,M)}(z,w;\\{x_{t}\\})$
$\displaystyle=\frac{-2}{\alpha^{\prime}}\frac{\langle\partial{\cal
X}_{q}(z)\partial{\bar{\cal
X}}_{q}(w)\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}{\langle\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}$
$\displaystyle h_{(N,M)}(z,w;\\{x_{t}\\})$
$\displaystyle=\frac{-2}{\alpha^{\prime}}\frac{\langle\partial{\bar{\cal
X}}_{q}(z)\partial{\bar{\cal
X}}_{q}(w)\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}{\langle\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}$
$\displaystyle l_{(N,M)}(z,w;\\{x_{t}\\})$
$\displaystyle=\frac{-2}{\alpha^{\prime}}\frac{\langle\partial{\cal
X}_{q}(z)\partial{\cal
X}_{q}(w)\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}{\langle\sigma_{\epsilon_{1},f}(x_{1})\dots\sigma_{\epsilon_{N},f}(x_{N})\rangle}$
(158)

with expansions

$\displaystyle g_{(N,M)}(z,w;\\{x_{t}\\})$
$\displaystyle=\frac{\partial\omega_{z}}{\partial
z}\frac{\partial\omega_{w}}{\partial
w}\frac{1}{(\omega_{z}-\omega_{w})^{2}}\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{\hat{n}\hat{s}}(\omega_{t\neq
1,2,N})~{}\partial_{\omega}{\cal
X}^{(\hat{n})}(\omega_{z})~{}\partial_{\omega}{\bar{\cal
X}}^{(\hat{s})}(\omega_{w})$
$\displaystyle=\frac{1}{(z-w)^{2}}\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{\hat{n}\hat{s}}(\omega_{t\neq
1,2,N})~{}\partial_{\omega}{\cal
X}^{(\hat{n})}(\omega_{z})~{}\partial_{\omega}{\bar{\cal
X}}^{(\hat{s})}(\omega_{w})$ $\displaystyle h_{(N,M)}(z,w;\\{x_{t}\\})$
$\displaystyle=e^{-i2\pi\alpha_{1}}\frac{\partial\omega_{z}}{\partial
z}\frac{\partial\omega_{w}}{\partial w}\sum_{r,s=0}^{M-2}b_{rs}(\omega_{t\neq
1,2,N})~{}\partial_{\omega}{\bar{\cal
X}}^{(r)}(\omega_{z})~{}\partial_{\omega}{\bar{\cal X}}^{(s)}(\omega_{w})$
$\displaystyle l_{(N,M)}(z,w;\\{x_{t}\\})$
$\displaystyle=e^{i2\pi\alpha_{1}}\frac{\partial\omega_{z}}{\partial
z}\frac{\partial\omega_{w}}{\partial
w}\sum_{n,m=0}^{N-M-2}c_{nm}(\omega_{t\neq 1,2,N})~{}\partial_{\omega}{\cal
X}^{(n)}(\omega_{z})~{}\partial_{\omega}{\cal X}^{(m)}(\omega_{w})$ (159)

where we have used the anharmonic ratio as defined in eq. (150) and we have
extended the range of definition from $N-M-2$ to $N-M$ for ${\cal X}^{(n)}$
and from $M-2$ to $M$ for ${\bar{\cal X}}^{(s)}$ in order to write in a more
compact way $g_{(N,M)}$ and have therefore used hatted indexes. The previous
quantities are subject to the constraints

$\displaystyle\left[\prod_{t=1}^{N-1}\frac{(\omega_{z}-\omega_{t})^{\epsilon_{t}-1}}{(\omega_{w}-\omega_{t})^{\epsilon_{t}}}\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{\hat{n}\hat{s}}(\omega_{t\neq
1,2,N})\omega_{z}^{\hat{n}}\omega_{w}^{\hat{s}}\right]\Big{|}_{\omega_{w}=\omega_{z}}=1$
$\displaystyle\frac{\partial}{\partial\omega_{w}}\Big{|}_{\omega_{w}=\omega_{z}}\left[\prod_{t=1}^{N-1}\frac{(\omega_{z}-\omega_{t})^{\epsilon_{t}-1}}{(\omega_{w}-\omega_{t})^{\epsilon_{t}}}\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{\hat{n}\hat{s}}(\omega_{t\neq
1,2,N})\omega_{z}^{\hat{n}}\omega_{w}^{\hat{s}}\right]=0$ (160)

because $g$ must have only a double pole with coefficient $1$ and

$\displaystyle\int_{x_{t+1}}^{x_{t}}dx~{}g_{(N,M)}(x+i0^{+},w)+e^{i2\pi\alpha_{1}}h_{(N,M)}(x-i0^{+},w)=0$
$\displaystyle\int_{x_{t+1}}^{x_{t}}dx~{}l_{(N,M)}(z,x-i0^{+})+e^{i2\pi\alpha_{1}}g_{(N,M)}(z,x+i0^{+})=0$
(161)

due to the boundary conditions. This set of equations is an overdetermined but
consistent one as discussed in [22]. Using the previous quantities we wrote
that the Green function in presence of $N$ twists is given by

$\displaystyle\frac{-2}{\alpha^{\prime}}G^{z{\bar{z}}}_{(N,M)}(u,{\bar{u}};v,{\bar{v}};\\{x_{t}\\})$
$\displaystyle=\int_{x_{t_{1}};u^{\prime}\in
H}^{u}du^{\prime}\int_{x_{t_{2}};v^{\prime}\in
H}^{v}dv^{\prime}~{}g_{(N,M)}(u^{\prime},v^{\prime};\\{x_{t}\\}))$
$\displaystyle+e^{-i2\pi\alpha_{1}}\int_{x_{t_{1}};u\in
H}^{u}du^{\prime}\int_{x_{t_{2}};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}l_{(N,M)}(u^{\prime},{\bar{v}}^{\prime};\\{x_{t}\\}))$
$\displaystyle+e^{i2\pi\alpha_{1}}\int_{x_{t_{1}};{\bar{u}}^{\prime}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{t_{2}};v^{\prime}\in
H}^{v}dv^{\prime}~{}h_{(N,M)}({\bar{u}}^{\prime},v^{\prime};\\{x_{t}\\}))$
$\displaystyle+\int_{x_{t_{1}};{\bar{u}}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{t_{2}};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}g_{(N,M)}({\bar{v}}^{\prime},{\bar{u}}^{\prime};\\{x_{t}\\}))$
(162)

and

$\displaystyle\frac{-2}{\alpha^{\prime}}G^{zz}_{(N,M)}(u,{\bar{u}};v,{\bar{v}};\\{x_{t}\\})$
$\displaystyle=\int_{x_{t_{1}};u^{\prime}\in
H}^{u}du^{\prime}\int_{x_{t_{2}};v^{\prime}\in
H}^{v}dv^{\prime}~{}l_{(N,M)}(u^{\prime},v^{\prime};\\{x_{t}\\}))$
$\displaystyle+e^{i2\pi\alpha_{1}}\int_{x_{t_{1}};u\in
H}^{u}du^{\prime}\int_{x_{t_{2}};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}g_{(N,M)}(u^{\prime},{\bar{v}}^{\prime};\\{x_{t}\\}))$
$\displaystyle+e^{i2\pi\alpha_{1}}\int_{x_{t_{1}};{\bar{u}}^{\prime}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{t_{2}};v^{\prime}\in
H}^{v}dv^{\prime}~{}g_{(N,M)}(v^{\prime},{\bar{u}}^{\prime};\\{x_{t}\\}))$
$\displaystyle+e^{i4\pi\alpha_{1}}\int_{x_{t_{1}};{\bar{u}}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{t_{2}};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}h_{(N,M)}({\bar{u}}^{\prime},{\bar{v}}^{\prime};\\{x_{t}\\}))$
(163)

and

$\displaystyle\frac{-2}{\alpha^{\prime}}G^{{\bar{z}}{\bar{z}}}_{(N,M)}(u,{\bar{u}};v,{\bar{v}};\\{x_{t}\\})$
$\displaystyle=\int_{x_{t_{1}};u^{\prime}\in
H}^{u}du^{\prime}\int_{x_{t_{2}};v^{\prime}\in
H}^{v}dv^{\prime}~{}h_{(N,M)}(u^{\prime},v^{\prime};\\{x_{t}\\}))$
$\displaystyle+e^{-i2\pi\alpha_{1}}\int_{x_{t_{1}};u\in
H}^{u}du^{\prime}\int_{x_{t_{2}};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}g_{(N,M)}({\bar{v}}^{\prime},u^{\prime};\\{x_{t}\\}))$
$\displaystyle+e^{-i2\pi\alpha_{1}}\int_{x_{t_{1}};{\bar{u}}^{\prime}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{t_{2}};v^{\prime}\in
H}^{v}dv^{\prime}~{}g_{(N,M)}({\bar{u}}^{\prime},v^{\prime};\\{x_{t}\\}))$
$\displaystyle+e^{-i4\pi\alpha_{1}}\int_{x_{t_{1}};{\bar{u}}\in
H^{-}}^{\bar{u}}d{\bar{u}}^{\prime}\int_{x_{t_{2}};{\bar{v}}^{\prime}\in
H^{-}}^{\bar{v}}d{\bar{v}}^{\prime}~{}l_{(N,M)}({\bar{u}}^{\prime},{\bar{v}}^{\prime};\\{x_{t}\\}))$
(164)

where the normalization is needed in order to match the singularity of the
untwisted Green function (132) 999 This is obvious for
$G^{z{\bar{z}}}_{(N,M)}$ but for $G^{zz}_{(N,M)}$ there would seem to be a
mismatch of phases since
$\partial_{u}\partial_{\bar{v}}G^{zz}_{U(t)}=e^{i2\pi\alpha_{t}}\frac{-2}{\alpha^{\prime}}\frac{1}{(u-{\bar{v}})^{2}}$
has a phase which depends on the brane while naively
$\partial_{u}\partial_{\bar{v}}G^{zz}_{(N,M)}=e^{i2\pi\alpha_{1}}\frac{-2}{\alpha^{\prime}}g_{(N,M)}(u,{\bar{v}})\sim
e^{i2\pi\alpha_{1}}\frac{-2}{\alpha^{\prime}}\frac{1}{(u-{\bar{v}})^{2}}$. The
issue is solved by noticing that the singularity is only there when
$u,{\bar{v}}\rightarrow x\in\mathbb{R}$ and therefore
$g_{(N,M)}(x+i\delta_{1},x-i\delta_{2})$ ($\delta_{1,2}>0$) has the
singularity with the required phase, explicitly for $x_{t}<x<x_{t-1}$
$g_{(N,M)}(x+i\delta_{1},x-i\delta_{2})=e^{i2\pi(\alpha_{t}-\alpha_{1})}g_{(N,M)}(x+i\delta_{1},x+i\delta_{2})$
since
$\prod_{u=1}^{N-1}(\omega_{x-i\delta}-\omega_{u})^{-\epsilon_{u}}=e^{i2\pi(\alpha_{t}-\alpha_{1})}\prod_{u=1}^{N-1}(\omega_{x+i\delta}-\omega_{u})^{-\epsilon_{u}}$.
. The arbitrariness of the lower integration limit is due to the constraints
(161) which allow to change $x_{t_{1}}\rightarrow x_{t_{3}}$ for whichever
${t_{3}}$ ans similarly for $x_{t_{2}}$. We would now justify this result
since it is important in computing correlators involving momenta. The reason
why we fixed the lower integration limit to one of the twist location is
because we want

$G^{IJ}(x_{t_{1}},x_{t_{1}};v,{\bar{v}};\\{x_{t}\\})=G^{IJ}(u,{\bar{u}};x_{t_{1}},x_{t_{1}};\\{x_{t}\\})=0$
(165)

as follows from the boundary condition (23) in the case of the quantum
fluctuation where $f_{t}\rightarrow 0$.

##  Appendix D Boundary Green functions and their regularized version
$\Delta_{(N,M),bou}$

For the computation of the boundary correlators it is interesting and useful
to notice that all the components of the Green function are proportional,
analogously to eq.s (147) we have

$\displaystyle G^{zz}_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle=e^{i\pi(\alpha_{t_{1}}+\alpha_{t_{2}})}G_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle G^{{\bar{z}}{\bar{z}}}_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle=e^{-i\pi(\alpha_{t_{1}}+\alpha_{t_{2}})}G_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle G^{z{\bar{z}}}_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle=e^{i\pi(\alpha_{t_{1}}-\alpha_{t_{2}})}G_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle G^{{\bar{z}}z}_{(N,M),bou}(x_{1},x_{2})$
$\displaystyle=e^{i\pi(-\alpha_{t_{1}}+\alpha_{t_{2}})}G_{(N,M),bou}(x_{1},x_{2})$
(166)

where the point $x_{1}$ is on the brane $D_{t_{1}}$, i.e.
$x_{t_{1}}<x_{1}<x_{t_{1}-1}$ and similarly for $x_{2}$ and we have defined
the common real symmetric function
$G_{(N,M),bou}(x_{1},x_{2})=G_{(N,M),bou}(x_{2},x_{1})$ 101010The easiest way
to verify that it is symmetric is to notice that
$G^{zz}_{(N,M),bou}(x_{1},x_{2})$ is symmetric.  to be

$\displaystyle\frac{-2}{\alpha^{\prime}}$ $\displaystyle
G_{(N,M),bou}(x_{1},x_{2})=$ $\displaystyle
e^{i\pi(\bar{N}_{t_{1}}+\bar{N}_{t_{2}})}\sum_{n,m=0}^{N-M-2}c_{nm}(\omega_{t\neq
1,2,N})~{}\int_{\omega_{t_{1}}}^{\omega_{x_{1}}}d\omega~{}\partial_{\omega}{\mathfrak{X}}^{(n)}(\omega)~{}\int_{\omega_{t_{2}}}^{\omega_{x_{2}}}d\omega~{}\partial_{\omega}{\mathfrak{X}}^{(m)}(\omega)$
$\displaystyle+$ $\displaystyle
e^{i\pi(N_{t_{1}}+N_{t_{2}})}\sum_{r,s=0}^{M-2}b_{rs}(\omega_{t\neq
1,2,N})~{}\int_{\omega_{t_{1}}}^{\omega_{x_{1}}}d\omega~{}\partial_{\omega}{\mathfrak{X}}^{(r)}(\omega)~{}\int_{\omega_{t_{2}}}^{\omega_{x_{2}}}d\omega~{}\partial_{\omega}{\mathfrak{X}}^{(s)}(\omega)$
$\displaystyle+$ $\displaystyle
e^{i\pi(\bar{N}_{t_{1}}+N_{t_{2}})}\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{\hat{n}\hat{s}}(\omega_{t\neq
1,2,N})~{}\int_{\omega_{t_{1}}}^{\omega_{x_{1}}}d\omega_{z}~{}~{}\int_{\omega_{t_{2}}}^{\omega_{x_{2}}}d\omega_{w}\frac{\partial_{\omega}{\mathfrak{X}}^{(\hat{n})}(\omega_{z})\partial_{\omega}{\mathfrak{X}}^{(\hat{s})}(\omega_{w})}{(\omega_{z}-\omega_{w})^{2}}$
$\displaystyle+$ $\displaystyle
e^{i\pi(N_{t_{1}}+\bar{N}_{t_{2}})}\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{ns}(\omega_{t\neq
1,2,N})~{}\int_{\omega_{t_{1}}}^{\omega_{x_{1}}}d\omega_{z}~{}~{}\int_{\omega_{t_{2}}}^{\omega_{x_{2}}}d\omega_{w}\frac{\partial_{\omega}{\mathfrak{X}}^{(\hat{n})}(\omega_{w})\partial_{\omega}{\mathfrak{X}}^{(\hat{s})}(\omega_{z})}{(\omega_{z}-\omega_{w})^{2}}$
(167)

where $\partial_{\omega}{\mathfrak{X}}^{(n)}(\omega)$ and
$\partial_{\omega}{\mathfrak{X}}^{(s)}(\omega)$ are the functions (154) which
are real when $\omega\in\mathbb{R}$ and we have introduced the integers

$\displaystyle N_{t}$
$\displaystyle=\sum_{u=1}^{t-1}\theta(\alpha_{u}-\alpha_{u+1})$
$\displaystyle\bar{N}_{t}$ $\displaystyle=N_{t}+(t-1)$ (168)

which enter the game because of the way $\epsilon$ is defined in eq. (8).

Another interesting point is to study the behavior of
$\Delta_{(N,M),bou}(x_{1},x_{2})$ with $x_{1},x_{2}\in(x_{t},x_{t-1})$, i.e.
they are on $D_{t}$ when $x_{1},x_{2}\rightarrow x_{t}$. This is an important
check since we should recover both the singularity and the form factor
$R^{2}(\epsilon_{t})$ of the function $\Delta_{bou}$ given in eq. (148). In
the limit $x_{1},x_{2}\rightarrow x_{t}$ only the last two lines of eq. (167)
contributes, if we change variables as
$\omega_{z}=\omega_{t}+(\omega_{x_{1}}-\omega_{t})y_{1}$ and
$\omega_{w}=\omega_{t}+(\omega_{x_{1}}-\omega_{t})y_{2}$ in the third line and
in a similar way in the forth one we get

$\displaystyle\frac{-2}{\alpha^{\prime}}$ $\displaystyle
G_{(N,M),bou}(x_{1},x_{2})\sim_{x_{1},x_{2}\rightarrow
x_{t}}\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{\hat{n}\hat{s}}(\omega_{t\neq
1,2,N})\Big{[}$ $\displaystyle(-)^{t-1}$
$\displaystyle\int_{0}^{1}dy_{1}\int_{0}^{\hat{y}}dy_{2}\frac{\partial_{\omega}{\mathfrak{X}}^{(\hat{n})}(\omega_{t}+(\omega_{x_{1}}-\omega_{t})y_{1})\partial_{\omega}{\mathfrak{X}}^{(\hat{s})}(\omega_{t}+(\omega_{x_{2}}-\omega_{t})y_{2})+(y_{1}\leftrightarrow
y_{2})}{(y_{1}-y_{2})^{2}}\Big{]}$
$\displaystyle=\int_{0}^{1}dy_{1}\int_{0}^{\hat{y}}dy_{2}\frac{y_{1}^{-\bar{\epsilon}_{t}}y_{2}^{-\epsilon_{t}}\left(\bar{\epsilon}_{t}y_{1}+\epsilon_{t}y_{2}+O(y^{2})\right)+(y_{1}\leftrightarrow
y_{2})}{(y_{1}-y_{2})^{2}}$ (169)

where we used eq.s (160) which imply
$\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{\hat{n}\hat{s}}(\omega_{u\neq
1,2,N})\omega_{t}^{\hat{n}+\hat{s}}=0$,
$\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{\hat{n}\hat{s}}(\omega_{u\neq
1,2,N})\hat{n}\omega_{t}^{\hat{n}+\hat{s}-1}=\bar{\epsilon}_{t}\prod_{u\neq
t,N}(\omega_{t}-\omega_{u})$ and the analogous equation
$\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{\hat{n}\hat{s}}(\omega_{u\neq
1,2,N})s\omega_{t}^{\hat{n}+\hat{s}-1}=\epsilon_{t}\prod_{u\neq
t,N}(\omega_{t}-\omega_{u})$. In the previous equation we have introduced also
$\hat{y}=\frac{\omega_{2}-\omega_{t}}{\omega_{1}-\omega_{t}}=\frac{x_{2}-x_{t}}{x_{2}-x_{N}}/\frac{x_{1}-x_{t}}{x_{1}-x_{N}}$
where $x_{1},x_{2}$ are not the location of the twist fields but the points
where the Green function is evaluated. This expression has to be compared with
the analogous for $N=2$ which can be written as

$\displaystyle\frac{-2}{\alpha^{\prime}}G_{N=2}(\bar{x}_{1},\bar{x}_{2})$
$\displaystyle=\int_{0}^{1}dy_{1}\int_{0}^{\bar{x}_{2}/\bar{x}_{1}}dy_{2}\frac{y_{1}^{-\bar{\epsilon}_{t}}y_{2}^{-\epsilon_{t}}\left(\bar{\epsilon}_{t}y_{1}+\epsilon_{t}y_{2}\right)+(y_{1}\leftrightarrow
y_{2})}{(y_{1}-y_{2})^{2}}$ (170)

then we can write

$G_{(N,M),bou}(x_{1},x_{2})\sim
G_{N=2}(1,\hat{y})=G_{N=2}(\frac{x_{1}-x_{t}}{x_{1}-x_{N}},\frac{x_{2}-x_{t}}{x_{2}-x_{N}})$
(171)

In the limit $x_{1},x_{2}\rightarrow x_{t}$ we notice that
$\hat{y}=\frac{x_{1}-x_{t}}{x_{2}-x_{t}}+O(x_{1}-x_{2})$ then we can use the
previous result (148) to write

$\displaystyle\Delta_{(N,M),bou}(x_{1},x_{2})\sim\log({x_{1}}^{2})+R^{2}(\epsilon_{t})+O(x_{1}-x_{2})$
(172)

##  Appendix E Functions entering excited twist fields correlators

If we look at eq. (119) and the more general eq. (123) we see immediately that
the key quantities are

$\displaystyle(u_{t}-x_{t})^{\epsilon_{tI}}\partial_{u}X^{I}_{cl}(u_{t})$
$\displaystyle(u_{t}-x_{t})^{\epsilon_{tI}}(v_{t}-x_{t})^{\epsilon_{tJ}}\partial_{u}\partial_{v}\Delta_{(N,M)(t)}^{IJ}(u_{t},v_{t};\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})$
$\displaystyle(u_{t}-x_{t})^{\epsilon_{tI}}(v_{\hat{t}}-x_{\hat{t}})^{\epsilon_{{\hat{t}}J}}\partial_{u}\partial_{v}G_{(N,M)}^{IJ}(u_{t},{\bar{u}}_{t};v_{\hat{t}},{\bar{v}}_{\hat{t}};\\{x_{t},\epsilon_{t}\\})$
$\displaystyle(u_{t}-x_{t})^{\epsilon_{tI}}\partial_{u}G_{(N,M)}^{IJ}(u_{t},{\bar{u}}_{t};x_{j},x_{j};\\{x_{t},\epsilon_{t}\\})$
(173)

We would now like to give a more explicit expression for these quantities
while a completely explicit expression is given in the next section for few
cases. Actually the previous expressions except the cases with only one
derivative can be written as

$\displaystyle(u-x_{t})^{\bar{\epsilon}_{t}}\partial_{u}X_{cl}(u)=\sum_{n=0}^{N-M-2}a_{n}(\omega_{\hat{t}})~{}(u-x_{t})^{\bar{\epsilon}_{t}}\partial_{u}{\cal
X}^{(n)}(\omega_{u})$
$\displaystyle(u-x_{t})^{\epsilon_{t}}\partial_{u}\bar{X}_{cl}(u)=\sum_{r=0}^{M-2}b_{r}(\omega_{\hat{t}})~{}(u-x_{t})^{\epsilon_{t}}\partial_{u}{\cal
X}^{(r)}(\omega_{u})$ (174)

and

$\displaystyle\frac{-2}{\alpha^{\prime}}(u-x_{t})^{\bar{\epsilon}_{t}}$
$\displaystyle(v-x_{t})^{\epsilon_{t}}\partial_{u}\partial_{v}\Delta_{(N,M)(t)}^{z{\bar{z}}}(u,v;\\{x_{\hat{t}},\epsilon_{\hat{t}}\\})=$
$\displaystyle\frac{1}{(u-v)^{2}}\Bigg{[}$
$\displaystyle\sum_{\hat{n}=0}^{N-M}\sum_{\hat{s}=0}^{M}a_{\hat{n}\hat{s}}(\omega_{{\hat{t}}\neq
1,2,N})~{}~{}(u-x_{t})^{\bar{\epsilon}_{t}}\partial_{\omega}{\cal
X}^{(\hat{n})}(\omega_{u})~{}~{}(v-x_{t})^{\epsilon_{t}}\partial_{\omega}{\bar{\cal
X}}^{(\hat{s})}(\omega_{v})$