text:
\section{Introduction}

Historically, the first instance of a string amplitude is the Veneziano amplitude, describing the scattering of four `would-be' pions but actually tachyons. Yet, thanks to the powerful constraints of conformal invariance and modular invariance in the perturbative regime, a systematic construction has been successfully achieved for closed-string theories  much earlier than for open-string theories \cite{Green:1987sp, Green:1987mn, Polchinski:1998rq, Polchinski:1998rr, Kiritsis:2007zza}. The systematic construction of theories with open unoriented strings was completed in the early 90's \cite{Bianchi:1990yu, Bianchi:1990tb}\footnote{For a review see {\it e.g.}~ \cite{Angelantonj:2002ct}.}. The advent of D-branes \cite{Angelantonj:2002ct} triggered an enormous interest in the subject that led to the construction of exactly solvable models with D-branes and $\Omega$-planes based on tori, orbifolds, free fermions, minimal models, group manifolds and Gepner models \cite{Bianchi:1988ux, Bianchi:1988fr, Bianchi:1989du, Bianchi:1990yu, Bianchi:1990tb, Bianchi:1991rd, Bianchi:1991eu, Angelantonj:1996mw, Angelantonj:1996uy,
Pradisi:1995qy, Recknagel:1997sb, Blumenhagen:2003su, Blumenhagen:2004cg, Dijkstra:2004ym, Anastasopoulos:2006da}. Most of the attention has been devoted to the spectrum {\it i.e.}~ to the one-loop partition function.

Much less work has been devoted to the computation of scattering amplitudes in non-trivial settings based on genuinely interacting CFT's \cite{Bianchi:1991rd, Angelantonj:1996mw, Pradisi:1995qy, Recknagel:1997sb, Blumenhagen:2003su, Blumenhagen:2004cg, Dijkstra:2004ym, Anastasopoulos:2006da}. The only prominent exceptions are twist field correlators on tori and orbifolds \cite{Gava:1997jt, David:2000um, David:2000yn, Cvetic:2003ch,Abel:2003vv,Abel:2003yx}. As reviewed in \cite{Light}, twist fields are essential ingredients in the vertex operators for open string states living at brane intersections. However one should keep in mind that while closed-string twists are quantized (rational numbers $k/n$), intersection angles are not, since they depend on both discrete (wrappings) and continuous (moduli) parameter.

Aim of the present note is to derive open-string twist-field correlators using algebraic techniques, devised in \cite{Bianchi:1990yu, Bianchi:1990tb} and applied to minimal models \cite{Bianchi:1991rd} and later on to WZW models \cite{Pradisi:1995qy}, and complementing the `stress tensor method' \cite{Gava:1997jt, David:2000um, David:2000yn, Cvetic:2003ch,Abel:2003vv,Abel:2003yx}. In this approach one has to carefully decompose the `parent' closed-string amplitude in conformal blocks and then take the `square root' in each sector, thus producing chiral blocks evaluated at a real argument that are to be combined with appropriate coefficients. The latter are tightly constrained by `planar duality' \cite{Bianchi:1991rd} and by correct factorization, {\it i.e.}~ correct normalization and positivity of the residues of the poles.

We consider correlators of four bosonic twist-fields with one or two independent angles, recalling the closed-string computation and then deriving the open string results. In order to take the relevant square-root, we need to rewrite the closed-string result as a sesqui-linear form in the conformal blocks. We comment on possible ambiguity and arbitrariness in the procedure. As mentioned before, open-string twist-field can be considered `descendants' of closed-string ones only for rational intersection angles. Since rational form a dense set, one can safely extend the result to irrational values (in units of $\pi$) for the angles.

We conclude with preliminary considerations on how to generalize our approach to more sophisticated cases (such as WZW or Gepner models) that are not related in such a simple way as orbifolds to free CFT's.

\section{General strategy \label{sec general strategy}}

In the works of \cite{Gava:1997jt, David:2000um, David:2000yn, Cvetic:2003ch,Abel:2003vv,Abel:2003yx} the authors compute open string correlators that contain multiple bosonic twist fields. In order to do so they extend the world-sheet, whose original domain is the upper half complex plane, via the ``doubling trick" to the full complex plane and employ conformal field theory techniques that are related to the study of bosonic twist fields in the context of closed string theory on orbifolds \cite{Dixon:1986qv,Burwick:1990tu}.

Here we follow a different approach based on the systematic construction of open-string theories from their parent closed-string  theories \cite{Bianchi:1990yu, Bianchi:1990tb}. This algebraic procedure has been mostly applied to the determination of the spectrum coded in the four contributions -- torus, Klein-bottle, annulus and M\"obius-strip -- to the one-loop partition function.
One notable extension of the systematic procedure is the computation of 4-point correlation functions of open strings at tree-level (disk) in minimal models that display the desired factorization properties and `planar duality \cite{Bianchi:1991rd}. Later on this computation has been generalized to $SU(2)$ WZW models \cite{Pradisi:1995qy}.

Quite remarkably both one-loop amplitudes and 4-point correlators at tree level depend on one `invariant', which is complex for oriented closed strings and `real' for open and unoriented strings. At one-loop one has the complex modular parameter $\tau$ for the torus, that becomes purely imaginary ($\tau = i \tau_2$) for Klein-bottle and Annulus and has a fixed real part ($\tau_1= 1/2$) for the M\"obius-strip \cite{Bianchi:1988ux, Bianchi:1988fr, Bianchi:1989du, Bianchi:1990yu, Bianchi:1990tb}. At tree level, one has the complex cross-ratio $z=z_{12}z_{34}/z_{13}z_{24}$ for the sphere and a real cross ratio $x=x_{12}x_{34}/x_{13}x_{24}$ for the upper half plane, topologically equivalent to a disk.

As we will momentarily see, the close similarity between tree-level 4-point correlators and one-loop partition functions is more than a mere analogy. Indeed, twist-field correlators on the sphere can be expressed as (fake) torus partition functions once an appropriate branched-cover of the sphere is considered that trivializes the monodromies.

Before entering the details of the twist-field correlators, let us describe our strategy
\cite{BianchiPHD, StanevMaster, Fuchs:1999fh, Fuchs:2000hn}. Generically a closed string correlator takes the form
\begin{align}\nonumber
{\cal A}_{closed} &= \langle \Phi_{h_1,\overline h_1}(z_1, \bar{z}_1) \Phi_{h_2,\overline h_2}(z_2, \bar{z}_2)\Phi_{h_3,\overline h_3}(z_3, \bar{z}_3)\Phi_{h_4,\overline h_4}(z_4, \bar{z}_4)\rangle \\ &= \prod_{i<j} |z_{ij}|^{-(h_i + \overline h_i +h_j +\overline h_j)+{\Delta +\overline \Delta \over 3}}\sum_{i,j} c_{ij} {\cal F}_i(z)   \overline {\cal F}_j (\overline z)
\end{align}
where $\Delta= h_1 + h_2 +h_3 +h_4$, $\overline \Delta= \overline h_1 + \overline h_2 + \overline h_3 +\overline h_4$ and  $ {\cal F}_i$ and $ \overline {\cal F}_j$ denote the holomorphic and anti-holomorphic conformal blocks, respectively. The index $i$ runs over the conformal families that appear in the s-channel {\it i.e.}~ in the OPE's of both $\Phi_{h_1}\Phi_{h_2}$ and $\Phi_{h_3}\Phi_{h_4}$ exposed in the limit $z\rightarrow 0$. Clearly one can express the same amplitude in the t- or u- channels that give rise to different OPE's and thus different bases of conformal blocks. A defining property of the conformal blocks ${\cal F}_i(z)$ is that, up to an overall factor $z^{h-h_1-h_2}$, they admit an expansion in integer powers of $z$ very much as one-loop characters $\chi_h(w)$ admit an expansion in integer powers of $w=\exp(2\pi i \tau)$, up to an overall factor $w^{h-{c\over 24}}$.

Closely following the systematic construction of one-loop amplitudes, open string 4-point correlators take the form
 \begin{align}
{\cal A}_{open} = \langle \phi_{h_1}(x_1) \phi_{h_2}(x_2) \phi_{h_3}(x_3) \phi_{h_4}(x_4)\rangle = \prod_{i<j} x_{ij}^{-(h_i  +h_j)+{\Delta  \over 3}}\sum_{i} a_i{\cal F}_i(x)
\end{align}
where $x$ is real. The coefficients $a_i$ are tightly constrained by planar duality \cite{Bianchi:1991rd, Pradisi:1995qy} and are adjusted in such a way that they give rise to the correct pole structures. For our purpose some specific limits correspond to the exchange of the identity sector or other untwisted states.

In the simple case of $Z_2$ twist-fields, the above procedure was essentially followed in \cite{Antoniadis:1993jp}. For generic twists, the closed string bosonic correlator with one or two independent angles takes the form
\begin{align}
{\cal A}_{closed} = |K(z)|^2 \, \sum_{\vec{k},\vec{v}} c_{\vec{k}\vec{v}} w(z)^{\frac{\alpha' p^2_L}{4}} \,\,\overline w(\overline z)^{\frac{\alpha' p^2_R}{4}}\,\,,
\label{eq closed string form}
\end{align}
where $w(z)$ is a holomorphic function and $p_L$ and $p_R$ denote the closed string momenta
\begin{align}
p^2_L =\left(\vec{k}+ \frac{\vec{v}}{\alpha'}\right)^2 \qquad \qquad p^2_R = \left(\vec{k}- \frac{\vec{v}}{\alpha'}\right)^2\,\,.
\end{align}
Note that $\vec{k}$ and $\vec{v}$ are two dimensional vectors in the dual lattice $\Lambda^*$ and the lattice $\Lambda$, respectively.
Then the corresponding open string result will be given by
\begin{align}
{\cal A}_{open} = K(x) \, \sum_{p,q} c_{p,q} w(x)^{\alpha' p^2_{open}}
\end{align}
with $p$ and $q$ being integers and $p^2_{open}$ denoting the open string mass. For a D1-brane wrapping a one-cycle on a two torus the mass is given
by  \cite{Kors:2001ku,Angelantonj:2005hs}
\begin{align}
p^2_{open}= \frac{1}{L^2} p^2 +\frac{1}{\alpha'^2}\frac{R^2_1 R^2_2}{L^2} q^2\,\,.
\end{align}
Here $R_1$ and $R_2$ are the radii of the two torus and $L$ denotes the length of the brane the open string is attached to. The first term corresponds to  the  Kaluza-Klein states along the brane while the latter to the winding excitations along the perpendicular direction.

Thus one of the tasks in our approach is to bring the closed string bosonic twist field correlator in the form of  \eqref{eq closed string form}. Such an expression allows one then to make an ansatz for the open string correlator as laid out above.
Given this ansatz one furthermore has to determine the coefficients $c_{p,q}$ by analyzing specific limits that correspond to the exchange of universal states like the gauge bosons (identity sector) or their excitations in the untwisted sector. Positivity of the residues of the poles in all channels puts severe constraints on the $c_{p,q}$ and guarantees `planar duality' \cite{Bianchi:1991rd, Pradisi:1995qy}.

\section{Bosonic twist field correlator with one independent angle}

Generically the closed string correlator is determined via the energy momentum tensor method and given in Lagrangian form. As already mentioned before in order to extract the open string correlator from the closed string one one has to manipulate the closed string result in such a way that it reveals the form \eqref{eq closed string form}. Specifically that implies a Poiss\`on resummation in one of the lattice variables.

\subsection{Closed string correlator}
Using the energy-momentum tensor method the  quantum part of the closed string bosonic twist field correlator with one independent angle has been shown to be \cite{Dixon:1986qv}
\begin{align}
|z_{\infty}|^{2 \theta (1-\theta)}\,\,\langle \sigma_{1-\theta}(0)\, \sigma_{\theta}(z,\overline z) \,
\sigma_{1-\theta}(1) \, \sigma_{\theta} (\infty)\rangle = {\cal C} \,\, \frac{|z(1-z)|^{-2\theta(1-\theta)}}{F(z)\overline F(1-\overline z) +\overline F( \overline z) F(1-z)}\,\,,
\end{align}
where $0< \theta < 1$ is a rational number encoding the twist ($k/n$) and $F(z)$ denotes the hypergeometric function
\begin{align}
F(z)={_2F}_1[\theta,1-\theta,1,z] \,\,.
\end{align}
The action for the classical solutions is given by (here  and in the following we set $\alpha'=2$ for the closed string computations)
\begin{align}
S_{cl}(v_1,v_2)=\frac{\pi}{4 \tau_2 \sin(\pi \theta)} \big[ v_2 \overline v_2 + \tau_1 \left( v_1 \overline v_2 \overline \beta + \overline v_1 v_2 \beta  \right) + |\tau|^2 v_1 \overline v_1\big]\,\,,
\end{align}
where $\beta=-i e^{-i\pi \theta}$ denotes a phase and $\tau(z)$ denotes the modulus of a ``fake torus'' introduced in \cite{Dixon:1986qv}
\begin{align}
\tau(z) =\tau_1+i\tau_2=i\frac{F(1-z)}{F(z)}  \qquad \qquad \qquad \overline \tau(\overline z) =\tau_1-i\tau_2=-i\frac{\overline F(1-\overline z)}{\overline F(\overline z)}\,\,.
\label{eq definition of tau}
\end{align}
Here $v_1$ and $v_2$ denote vectors belonging to some coset of the lattice $\Lambda$.

Combining both results gives
\begin{align}
|z_{\infty}|^{2 \theta (1-\theta)}\,\, \langle \sigma_{1-\theta}(0)\, \sigma_{\theta}(z,\overline z) \,
\sigma_{1-\theta}(1) \, \sigma_{\theta} (\infty)\rangle = {\cal C} \,\, \frac{|z(1-z)|^{-2\theta(1-\theta)}}{\tau_2(z,\overline z) |F(z)|^2} \,\, \sum_{v_1,v_2} e^{-S_{cl}(v_1,v_2)}\,\,,
\end{align}
where we used \eqref{eq definition of tau} to simplify the quantum part. This is the Lagrangian form of the correlator. In the following we will perform a Poiss\`on resummation over the variable $v_2$ to obtain the Hamiltonian form which is much more convenient to extract the corresponding open string result.

After Poiss\`on resumming one ends up with a sum over momenta in the dual lattice $\Lambda^*$, however since the sum over $v_2$ is only over a subset of the lattice $\Lambda^*$ we will substitute $v_2=-2\,\beta^{-1}\,\sin(\pi \theta) \left( f_{23} + q \right)$, where the summation over $q$ is now over the entire lattice $\Lambda$.  Poiss\`on resummation over $q$ yields
\begin{align}
\label{eq hamiltonian form closed 1 angle}
 |z_{\infty}|^{2 \theta (1-\theta)}\,\, \langle \sigma_{1-\theta}(0)\, \sigma_{\theta}(z,\overline z) \,
\sigma_{1-\theta}(1) \, \sigma_{\theta}& (\infty)\rangle = \frac{{\cal C}}{V_{\Lambda}} \,\, \frac{|z(1-z)|^{-2\theta(1-\theta)}}{ |F(z)|^2}\\ \nonumber
\times &\sum_{k \in \Lambda^*, v_1 \in \Lambda_c} \exp\left[-2 \pi i f_{23} \cdot k \right] \,\, w(z)^{1/2(k+\frac{v}{2})^2}\,  \overline w(\overline z)^{1/2(k-\frac{v}{2})^2},
\end{align}
where $\Lambda_c= (1-\theta)(f_{21}+\Lambda)$ denotes the coset over which $v$ runs, $V_{\Lambda}$ is the volume of the unit cell of the lattice $\Lambda$, and  $w(z)$ is given by
\begin{align}
w(z)=\exp\left[ \frac{i\pi \tau(z)}{\sin(\pi \theta)}\right]\,\,.
\end{align}
Note that the result \eqref{eq hamiltonian form closed 1 angle} takes the form \eqref{eq closed string form} that allows us to extract the open string result.

\subsection{Open string correlator}

Before we extract the open string correlator from the closed string one, let us briefly discuss such a correlator in the context of intersecting D-branes on a compactified six-torus $T^6=T^2 \times T^2 \times T^2$. In this work we assume that all branes go through the origin of the respective two-tori, thus all Wilson lines vanish. Moreover, we further simplify the setup by assuming that both branes $a$ and $b$ wrap such one-cycles on the respective two-tori that they intersect exactly once. Our results can be easily generalized to the more generic case of arbitrary number of intersections and non-vanishing Wilson-lines. We refer the interested reader to \cite{Cremades:2003qj,Cvetic:2003ch,Abel:2003yx, Abel:2003vv,Cremades:2004wa} for more details.

Figure \ref{fig twistoneangle}  depicts the intersection of two D-branes labeled by $a$ and $b$ on one of the three two-tori. At the intersection there exists a massless fermion going from brane $b$ to brane $a$, whose vertex operator contains the bosonic twist field $\sigma_{\theta}$, where $\theta$ denotes the intersection angle in one two torus.
At the cost of being pedantic, contrary to the twist in the `parent' closed string, $\theta$ is not quantized in the open-string case.

At the very same intersection there will be its antiparticle, a string going from brane $a$ to brane $b$, whose vertex operator contains a anti-twist field $\sigma_{1-\theta}$ \footnote{For a detailed discussion on vertex operators of massless states for arbitrary intersection angles, see \cite{Cvetic:2006iz,Bertolini:2005qh}, for a generalization to massive states see \cite{Light} and for a discussion on instantonic states at the intersection of  D-instanton and D-brane at arbitrary angles, see \cite{Cvetic:2009mt}. For an analysis of vertex operators for massive states in heterotic string theories, see \cite{Bianchi:2010es}.}.
\begin{figure}[h]
\begin{center}
\includegraphics[height=5cm]{4sigmas1angle.eps}
\end{center}
\caption{Intersection of two D-brane stacks, $a$ and $b$, in one complex dimension.}
\label{fig twistoneangle}
\end{figure}

Even though in this work we only consider the twist field correlator and not the amplitude of four matter fields that would contain additional conformal fields, it is worthwile to notice that the non fixed twist field operator position $x$ should remain in the interval $[0,1]$ since otherwise the Chan-Paton factors make the whole amplitude vanishing.

Finally, before we  perform the steps laid out in section \ref{sec general strategy} to derive the open string correlator let us discuss our expectations for the two limits $x \rightarrow 0$ and  $x \rightarrow 1$. In both limits we expect the exchange of untwisted string states, just as one observes in the closed string correlator. We will use gauge-boson exchange to normalize the correlator.

The result \eqref{eq hamiltonian form closed 1 angle} is of the type that allows us to extract the open string correlator. Since the open string result should only depend on real parameters we  expect $w(x)$ to take the form
\begin{align}
w(x)=\exp\left[- \frac{\pi t(x)}{\sin(\pi \theta)}\right]
\label{eq def w(x) one angle}
\end{align}
with $t(x)$ given by  \eqref{eq definition of tau}
\begin{align}
t(x) = \frac{1}{2 i} \left( \tau(x) - \overline \tau(x )\right)= \frac{F(1-x)}{F(x)}\,\,.
\end{align}
In the last step we used the fact that $\overline F(x) = F(x)$ for $0\leq x\leq 1 $.
Concerning the quantum part one expects only the holomorphic part to survive. Finally the mass of the open strings in the untwisted sector is given by \cite{Kors:2001ku,Angelantonj:2005hs}
\begin{align}
M^2_{open}(p,q)=  \frac{1}{L^2_a} p^2+ \frac{1}{\alpha'^2}\frac{R^2_1 \, R^2_2}{L^2_a} q^2\,\,,
\label{eq mass KK and winding}
\end{align}
where $R_1, R_2$ are the radii of the two-torus and $L_a$ is the length of the D-brane
\begin{align}
L_a = \sqrt{n^2_a R^2_1+ m^2_a R^2_2 }
\end{align}
with $n_a$, $m_a$ being the wrapping numbers of the brane $a$ on the two torus.

Thus the open string result takes the form
\begin{align}
x^{ \theta (1-\theta)}_{\infty} \langle \sigma_{1-\theta}(0)\, \sigma_{\theta}(x) \,
\sigma_{1-\theta}(1) \, \sigma_{\theta} (\infty)\rangle = {\cal C}_{open} \frac{[x(1-x)]^{-\theta(1-\theta)}}{F(x)} \sum_{p,q} w(x)^{ \left(  \frac{\alpha'}{L^2_a}  p^2 + \frac{1}{\alpha'}\frac{R^2_1 \, R^2_2}{L^2_a} q^2 \right)}
\label{eq suggested result open one angle}
\end{align}
with $w(x)$ given in \eqref{eq def w(x) one angle}.

This ansatz has to satisfy various non-trivial constraints, that generalize `planar duality', namely in the limit $x\rightarrow 0$ as well as for $x \rightarrow 1$ we expect an exchange of untwisted states. This condition also allows us to fix the constant ${\cal C}_{open}$.

Let us start with the limit $x \rightarrow 0$. In that limit $t(x)$ behaves
\begin{align}
\lim_{x\rightarrow 0} t(x)  =\frac{1}{\pi} \sin(\pi \theta)\, \left( -\ln(x) +\ln(\delta)\right)
\end{align}
with
\begin{align}
\ln(\delta) = 2 \psi(1) -\psi(\theta)- \psi(1-\theta)\,\,.
\end{align}
Plugging in that limit one obtains
\begin{align}
 {\cal C}_{open} x^{-\theta(1-\theta)} \sum_{p,q} \left(\frac{x}{\delta}\right)^{ \left(  \frac{\alpha'}{L^2_a}  p^2 + \frac{1}{\alpha'}\frac{R^2_1 \, R^2_2}{L^2_a} q^2 \right)}
 \label{eq result one angle a}
\end{align}

Let us turn to the $x \rightarrow 1$ limit which also exhibits the exchange of untwisted states, corresponding to strings starting and ending on the brane $b$ rather than on brane $a$. In order to determine this limit we perform a Poiss\`on resummation in the variables $p$ and $q$ and obtain
\begin{align}
{\cal C}_{open} \frac{[x(1-x)]^{-\theta(1-\theta)}}{F(x)} \, \frac{\sin(\pi \theta)}{t(x)} \frac{L^2_a}{R_1 \, R_2} \, \sum_{\tilde{p} \in \Lambda_{p}, \tilde{q} \in  {\Lambda}^*_{q}} \exp\left[-\frac{\pi \sin(\pi \theta)}{t(x)} \left( \frac{L^2_a}{\alpha'} \tilde{p}^2 +\frac{ \alpha'\, L^2_a}{R^2_1\, R^2_2} \tilde{q\,}^2 \right) \right]
\end{align}
With the identification (recall that brane $a$ and brane $b$  intersect exactly once on the torus)
\begin{align}
\sin(\pi \theta) = \frac{R_1 R_2}{L_a L_b}
\end{align}
this further simplifies to
\begin{align}
{\cal C}_{open}  \frac{[x(1-x)]^{-\theta(1-\theta)}}{F(1-x)} \, \frac{L_a}{L_b} \,\, \sum_{\tilde{p} \in \Lambda_{p}, \tilde{q} \in  \Lambda^*_{q}} \exp\left[-\frac{\pi }{\sin(\pi \theta) t(x)} \left( \frac{R^2_1 R^2_2}{ \alpha' \, L^2_b} \tilde{p}^2 +\frac{\alpha'}{L^2_b} \tilde{q}^2 \right) \right]\,\,.
\end{align}
Now performing the limit $x \rightarrow 1$ yields
\begin{align}
 {\cal C}_{open} \frac{L_a}{L_b} (1-x)^{-\theta(1-\theta)} \sum_{\tilde{p},\tilde{q}} \left(\frac{1-x}{\delta}\right)^{ \left(  \frac{\alpha'}{L^2_b} \,  \tilde{q}^2+ \frac{R^2_1 \, R^2_2}{\alpha' L^2_b} \, \tilde{p}^2\right)}\,\,.
\label{eq result one angle b}
\end{align}
Comparing the limits \eqref{eq result one angle a} and \eqref{eq result one angle b} it is natural to assume that the normalization constant is given by  ${\cal C}_{open}= \frac{\sqrt{\alpha'}}{L_a}$.

Let us now compare the derived open string expression to the ones in \cite{Gava:1997jt, David:2000um, David:2000yn, Cvetic:2003ch,Abel:2003vv,Abel:2003yx} derived via the energy momentum tensor method after extending the world-sheet to the whole complex plane. The first thing to notice is that \eqref{eq suggested result open one angle} is given in the Hamiltonian form while the correlator in \cite{Gava:1997jt, David:2000um, David:2000yn, Cvetic:2003ch,Abel:2003vv,Abel:2003yx} is of Lagrangian type. Thus we have to Poiss\`on resum \eqref{eq suggested result open one angle} over the variable $p$ which gives
\begin{align}
& \sqrt{\sin(\pi \theta)}\frac{[x(1-x)]^{-\theta(1-\theta)}}{\sqrt{F(x) F(1-x)}} \\ & \hspace{1cm}\times \sum_{\tilde{p} \in \Lambda_{p}, \tilde{q} \in  {\Lambda}^*_{q}} \exp\left[-\frac{\pi}{\alpha'} \sin(\pi \theta) F(x) F(1-x) \left(\left( \frac{\tilde{p} L_a}{F(1-x)}\right)^2+
\left( \frac{q L_b}{F(x)}\right)^2\right) \right] \,\,.\nonumber
\end{align}
This is indeed the result obtained via employing conformal field theory techniques. We conclude that in this case there is no ambiguity in the choice of the coefficients of the linear combination of conformal blocks.

Before we turn to the bosonic twist correlator with two independent angles let us determine the sub-dominant poles for the correlator \eqref{eq suggested result open one angle} in the limit $x \rightarrow 0$. This requires the knowledge of the behaviour of $F(x)$ for small $x$, which is (see appendix \ref{app hypergeometric})
\begin{align}
\lim_{x\rightarrow 0} F(x) = \frac{1}{\Gamma(\theta) \, \Gamma(1-\nu)} \sum^{\infty}_{n=0} \frac{\Gamma(\theta+n) \, \Gamma(1-\nu+n)}{\Gamma(n)}  \frac{x^n}{n !}\,\,.
\end{align}
Given that the correlator \eqref{eq suggested result open one angle} behaves in the limit $x \rightarrow 0$ as
\begin{align}
\frac{\sqrt{\alpha'}}{L_a} x^{-\theta(1-\theta)} \left( 1 - \theta(1-\theta) x -\frac{1}{2} \theta (1-\theta)  ((\theta^2 -  \theta + 2) x^2+ ... \right)
 \sum_{p,q} \left(\frac{x}{\delta}\right)^{ \left(  \frac{\alpha'}{L^2_a}  p^2 + \frac{1}{\alpha'}\frac{R^2_1 \, R^2_2}{L^2_a} q^2 \right)}
\end{align}
suggests the following OPE
\begin{align}
\sigma_{\theta}(z) \, \sigma_{1-\theta} (w) & \sim \sqrt{\frac{\sqrt{\alpha'}}{L_a}} (z-w)^{-\theta(1-\theta)} \,\, \mathbf{1} \\ \nonumber
& \hspace{15mm} + \sqrt{\frac{\sqrt{\alpha'}}{L_a}} \sqrt{\theta(1-\theta)} (z-w)^{-\theta(1-\theta)+1} \left( e^{i\alpha} \partial X +   e^{-i\alpha} \partial \overline X \right) +...
\end{align}
Here the phase $\alpha$ indicates the arbitrariness in the definition of the conformal fields $\sigma$ as well as $\partial X$ and $\overline \partial X$, and can be eliminated by an appropriate redefinition of the latter. One can easily extend this OPE to higher order poles which involve then conformal fields with larger conformal dimension.

\section{Bosonic twist field correlator with two independent angles}

Now we turn to the bosonic twist correlator with two independent angles. We follow the same procedure as above for the correlator with just one independent angle.  We will manipulate the closed string result to obtain an expression of type \eqref{eq closed string form} and extract from that the open string result.

\subsection{Closed string correlator}

Let us display the complete result, containing the quantum as well as the classical part, for the four twist field correlator with two independent angles determined in
\cite{Burwick:1990tu}\footnote{See also \cite{Stieberger:1992bj,Erler:1992gt}.} (again we set $\alpha'=2$)
\begin{align}
\label{eq correlator 2 independent angles}
& |z_{\infty}|^{2 \nu (1-\nu)}\,\,\langle \sigma_{1-\theta}(0)\, \sigma_{\theta}(z,\overline z) \,
\sigma_{1-\nu}(1) \, \sigma_{\nu} (\infty)\rangle \\ & \hspace{2cm} =|z|^{-2\theta(1-\theta)}
\left(1-z\right)^{-\nu(1-\theta)}\, \left(1-\overline z\right)^{-\theta(1-\nu)}
I^{-1}( z, \bar{z}) \,\,\sum_{v_1, v_2}e^{-S_{cl}} \nonumber
\end{align}
with $I(z, \bar{z})$ given by
\begin{align*}
I(z, \bar{z})= \frac{1}{2\pi} \big[B_1(\theta,\nu) \,G_2( z)
\overline H_1(1-\overline z)+ B_2(\theta,\nu) \, \overline G_1(\overline z) {H_2}(1- z)\big]\,\,,
\end{align*}
where
\begin{align*}
\begin{gathered}
B_1(\theta,\nu)=\frac{\Gamma(\theta)\,\Gamma(1-\nu)}{\Gamma(1+\theta-\nu)}\qquad
B_2(\theta,\nu)=\frac{\Gamma(\nu)\,\Gamma(1-\theta)}{\Gamma(1+\nu-\theta)}\\
\\
G_1(z)= {_2F}_1[\theta,1-\nu,1;z]\qquad
G_2(z)= {_2F}_1[1-\theta,\nu,1;z]\\
\\
H_1(z)= {_2F}_1[\theta,1-\nu,1+\theta-\nu;z]\qquad
 H_2(z)={_2F}_1[1-\theta,\nu,1-\theta+\nu;z]\,\,.
 \end{gathered}
\end{align*}
and $S_{cl}$ is
\begin{align}
S_{cl}=V_{11} v_1 \overline v_1 +V_{12} v_1 \overline v_2 +V^*_{21} v_2 \overline v_1 +V_{22} v_2 \overline v_2 \,\,.
\end{align}
Here the $V_{ij}$ are given by
\begin{align} \nonumber
V_{11} &= \frac{1}{4} \left(\frac{\sin(\pi \theta)}{\pi}\right)^2 |I(z,\overline z)|^{-2} \\ & \nonumber
 \hspace{1cm}\left( \,\, B_2 \big|H_2(1-z)\big|^2\Big[ B_1 G_1(z) \overline H_1(1-\overline z)+  B_1 \overline G_1(\overline z)  H_1 (1- z) \right.
\\& \left.\nonumber
  \hspace{5cm}
  + \pi \left( \cot(\pi \nu) - \cot(\pi \theta) \right)\left| G_1(z)\right|^2\Big]
\right. \\ & \left.   \nonumber
 \hspace{1cm} +B_1 \big|H_1(1-z)\big|^2\Big[ B_2 G_2(z) \overline H_2(1-\overline z)+  B_2\overline G_2(\overline z)  H_2 (1- z)
 \right. \\ &\left.
   \hspace{5cm}
  + \pi \left( \cot(\pi \theta) - \cot(\pi \nu) \right)\left| G_2(z)\right|^2\Big] \right)
\end{align}
\begin{align} \nonumber
V_{12} &= \frac{e^{i\pi \theta}}{4} \frac{\sin(\pi \theta)}{\pi} |I(z,\overline z)|^{-2} \\ & \nonumber
 \hspace{1cm}\left( \,\, B_2 G_2 (z) \overline H_2(1-\overline z)\Big[ B_1 G_1(z) \overline H_1(1-\overline z)+  B_1 \overline G_1(\overline z)  H_1 (1- z) \right.
\\& \left.\nonumber
  \hspace{5cm}
  + \pi \left( \cot(\pi \nu) - \cot(\pi \theta) \right)\left| G_1(z)\right|^2\Big]
\right. \\ & \left.   \nonumber
 \hspace{1cm} -B_1 \overline G_1(\overline z ) \overline H_1(1-z)\big|^2\Big[ B_2 G_2(z) \overline H_2(1-\overline z)+  B_2\overline G_2(\overline z)  H_2 (1- z)
 \right. \\ &\left.
   \hspace{5cm}
  + \pi \left( \cot(\pi \theta) - \cot(\pi \nu) \right)\left| G_2(z)\right|^2\Big] \right)
\end{align}
\begin{align} \nonumber
V_{22} = \frac{1}{4} |I(z,\overline z)|^{-2} \,\, &\left( \,\, G_1(z) G_2 (z) \Big[ B_1 \overline G_2(\overline z) \overline H_1(1-\overline z)+  B_2 \overline G_1(\overline z)  H_2 (1-\overline z) \right.
\\& \left.
 \hspace{0.1cm} + \overline G_1( \overline z) \overline G_2 (\overline z) \Big[ B_1 G_2( z) H_1(1- z)+  B_2  G_1(z)  H_2 (1-z)\right)\,\,.
\end{align}
In order to extract the open-string correlator from the closed-string one let us perform a Poiss\`on resummation, just as we did for the four bosonic twist correlator with just one independent angle. The substitution $v_2 = 2 i e^{-i\pi \theta} \sin(\pi \theta) \left( f_{23} + q \right)$ yields to a summation over the lattice $\Lambda$
\begin{align}
\sum_{q, v_1 \in \Lambda} e^{-S_{cl}}   = \sum_{q, v_1 \in \Lambda} &\exp\Big[ -V_{11} |v_1|^2 + 2 i\, \sin(\pi \theta) e^{i\pi \theta}  V_{12}\,\, v_1  \left( \overline f_{23} + \overline q \right)  \\ \nonumber
&   -2 i\, \sin(\pi \theta) e^{-i\pi \theta}  V^*_{12}\,\, \overline v_1  \left(  f_{23} + q \right)  - 4 \sin^2(\pi \theta) V_22 \,\, \left(  f_{23} + q \right) \left( \overline f_{23} + \overline q \right) \Big]\,\,,
\end{align}
which allows us to perform Poiss\`on resummation. One obtains (note that here we omit the quantum part)
\begin{align*}
\frac{\pi}{4 \sin^2(\pi \theta) V_{22}}
\,\,\sum_{k \in \Lambda^*, v \in \Lambda}  & \exp\Big[ -\frac{\pi^2}{4 \sin^2(\pi \theta) V_{22} }
|\vec{k}|^2  - 2 \pi i \vec{f}_{23} \cdot \vec{k} + \pi \frac{\left( e^{-i \pi \theta} V_{12} -e^{i \pi \theta}  {V_{12}}^*\right)}{2 \sin( \pi \theta)\,\, V_{22} } \, \vec{k} \cdot \vec{v}\\
& + i \pi \frac{\left( e^{-i \pi \theta} V_{12} +e^{i \pi \theta} {V_{12}}^*\right)}{2 \sin( \pi \theta)
\,\, V_{22} }  \vec{p}^{\,T}
\left(\begin{array}{cc}
0&1\\
-1& 0
\end{array}
\right) \vec{v} + \left(\frac{V_{12}  {V_{12}}^*}{V_{22}} -V_{11} \right) |\vec{v}|^2
 \Big]\,\,.
\end{align*}

Some statements are in order. First of all note that $v_1 \rightarrow v$. Moreover, in the last line the $\vec{k}$ and $\vec{v}$ are two-dimensional vectors rather than complex numbers. In addition one can verify the following, very useful relations
\begin{align}
\left(\frac{V_{12}  {V_{12}}^*}{V_{22}} -V_{11} \right)  = -\frac{\pi^2}{16 \sin^2(\pi \theta) V_{22} }
\end{align}
\begin{align}
 \frac{\left( e^{-i \pi \theta} V_{12} +e^{i \pi \theta} {V_{12}}^*\right)}{2 \sin( \pi \theta)
\,\, V_{22} } = \frac{1}{4} \big( \cot(\pi \nu) - \cot(\pi \theta) \big)\,\,.
\end{align}
Finally, it is convenient to define the following quantities
\begin{align}
\tau_1(z,\overline z) = i \, \frac{1}{2 \, V_{22}} \left({V_{12}}^* - V_{12} \right)
\end{align}
and
\begin{align}
\tau_{2} (z, \overline z) =\frac{\pi}{4 \, \sin(\pi \theta)} \,\, \frac{1}{V_{22}} \,\,.
\end{align}
They combine to the holomorphic ({\it a priori} far from obvious) quantity
\begin{align}
\tau(z) =\tau_1(z,\overline z) + i \tau_{2} (z, \overline z)  = i \frac{\sin(\pi \theta)}{2\, \pi} \left( \frac{B_1 H_1(1-z)}{G_1(z)} + \frac{B_2 H_2(1-z)}{G_2(z)}\right) \,\,,
\end{align}
which for the special case of equal angles, $\theta =\nu$, takes the form \eqref{eq definition of tau}. With these identifications we can write the Poiss\`on resummed expression as (here we still omit the quantum part)
\begin{align}
\frac{\pi}{4 \sin^2(\pi \theta) V_{22}}
\,\,\sum_{k \in \Lambda^*, v \in \Lambda}  e^{i\pi \vec{k}^T B \vec{v} }  e^{-2 \pi i \vec{f}_{23} \cdot \vec{k}}\,\, w(z)^{1/2(k+v/2)^2} \,\, \overline w(\overline z)^{1/2(k-v/2)^2}
\label{eq hamiltonian without quantum}
\end{align}
with
\begin{align}
B= \frac{1}{4} \big( \cot(\pi \nu) - \cot(\pi \theta) \big) \,\, \left(\begin{array}{cc}
0&1\\
-1& 0
\end{array} \right)
\end{align}
and
\begin{align}
w(z)=\exp\left[{i\frac{\pi \tau(z)}{\sin(\pi \theta)}} \right]\,\,.
\end{align}
Adding the quantum part (see eq. \eqref{eq correlator 2 independent angles}) to \eqref{eq hamiltonian without quantum} one obtains
\begin{align}
|z_{\infty}|^{2 \nu (1-\nu)}\,\,\langle \sigma_{1-\theta}(0)\, \sigma_{\theta}(z,\overline z) \,
\sigma_{1-\nu}(1) \, \sigma_{\nu} (\infty)\rangle  =&
|z|^{-2\theta(1-\theta)}
|1-z|^{-\theta(1-\nu)}\, \frac{1}{|G_1(z)|^2}  \\ & \hspace{-2cm} \times
\,\,\sum_{k \in \Lambda^*, v \in \Lambda}  e^{i\pi \vec{k}^T B \vec{v} }  e^{-2 \pi i \vec{f}_{23} \cdot \vec{k}}\,\, w(z)^{\frac{(k+v/2)^2}{2}} \,\, \overline w(\overline z)^{\frac{(k-v/2)^2}{2}}\,\,. \nonumber
\label{eq poisson resum closed result}
\end{align}
This is exactly  the same as the complicated expression \eqref{eq correlator 2 independent angles}. Apart from the obviously much simpler form it is also of the type \eqref{eq closed string form} that allows us to extract the open string result. Before we turn to this derivation we analyze the two interesting limits $z \rightarrow 0$ and $z \rightarrow 1$.

In the limit $z \rightarrow 0$ we expect the exchange  of untwisted states. In that limit $w(z)$ behaves as
\begin{align}
\lim_{z \rightarrow 0} w(z) = \frac{z}{\delta}
\end{align}
with $\ln \delta= 2\psi(1) - \psi(\theta) -\psi(1-\theta) -\psi(\nu) - \psi(1-\nu) $. Here we used various relations and limits displayed in appendix \ref{app hypergeometric}.

This gives the expected result which looks similar to the limits $z \rightarrow 0$ and $z \rightarrow 1$ for the closed string correlator with just one independent twist/angle \cite{Dixon:1986qv}
\begin{align}
|z|^{-2\theta(1-\theta)}
\,\,\sum_{k \in \Lambda^*, v \in \Lambda}  e^{i\pi \vec{k}^T B \vec{v} }  \frac{e^{-2 \pi i \vec{f}_{23} \cdot \vec{k}}}{\delta^{h+\overline h}}\,\,z^h\,\overline z^{\overline h}
\end{align}
with the conformal weights $h$ and $\overline h$ given by
\begin{align}
h=\frac{1}{2}\left(k+\frac{v}{2}\right)^2\qquad \qquad  \qquad \overline h=\frac{1}{2}\left(k-\frac{v}{2}\right)^2\,\,.
\end{align}

In contrast in the limit $z \rightarrow 1$ we expect the exchange of twisted states. Thus the closed string twist field correlator OPE's can be derived from taking the limit
$z \rightarrow 1$ of \eqref{eq correlator 2 independent angles} where we ignore the classical part of the correlator. For $z \rightarrow 1$ the quantum part of \eqref{eq correlator 2 independent angles} behaves as
\begin{align}
\label{eq expansion nu>theta}
2 \pi \,|1-z|^{-2 \theta(1-\nu)} &\frac{\Gamma(1-\theta) \, \Gamma( \nu) \, \Gamma(\nu-\theta)}{\Gamma(\theta) \, \Gamma( 1-\nu) \, \Gamma(1+\theta-\nu)}\\  & \times  \left[ 1-\frac{\Gamma^2(1-\theta) \, \Gamma^2(\nu) \, \Gamma(1+\theta-\nu) \, \Gamma(\theta -\nu) }{\Gamma^2(\theta) \, \Gamma^2(1-\nu) \, \Gamma(1-\theta +\nu) \, \Gamma(\nu-\theta) } |1-z|^{2\nu-2\theta} + ...\right] \nonumber
\end{align}
for $\nu >\theta$ and\footnote{The angle $\alpha$ appearing in the bosonic twist field is in the open interval $(0,1)$. Thus for $\nu >\theta$ the resulting angle on the right-hand side is not the naively expected angle $1-\theta+\nu$, but rather $\nu-\theta$.}
\begin{align}
\label{eq expansion nu<theta}
2 \pi \,|1-z|^{-2 \nu(1-\theta)} &\frac{\Gamma(\theta) \, \Gamma( 1-\nu) \, \Gamma(1-\theta+\nu)}{\Gamma(1-\theta) \, \Gamma( \nu) \, \Gamma(\theta-\nu)}\\  & \times  \left[ 1-\frac{\Gamma^2(\theta) \, \Gamma^2(1-\nu) \, \Gamma(\nu- \theta) \, \Gamma(1-\theta +\nu) }{\Gamma^2(1-\theta) \, \Gamma^2(\nu) \, \Gamma(\theta -\nu) \, \Gamma(1+\theta-\nu) } |1-z|^{2\theta-2 \nu} + ...\right] \nonumber
\end{align}
for $\theta > \nu$. Using  the properties displayed in appendix \ref{app hypergeometric} it is straightforward to show that hypergeometric functions behave as
\begin{align} \nonumber
&\lim_{z\rightarrow 1} G_1(z) = \frac{\Gamma(\nu-\theta)}{\Gamma(1-\theta) \, \Gamma(\nu)} +(1-z)^{\nu-\theta} \frac{\Gamma(\theta-\nu)}{\Gamma(\theta) \, \Gamma(1-\nu)}\\
&\lim_{z\rightarrow 1} G_2(z) = \frac{\Gamma(\theta-\nu)}{\Gamma(\theta) \, \Gamma(1-\nu)} +(1-z)^{\theta-\nu} \frac{\Gamma(\nu-\theta)}{\Gamma(1-\theta) \, \Gamma(\nu)}\\
&\lim_{z\rightarrow 1} H_1(1-z)  = \lim_{z\rightarrow 1} H_2(1-z) =1\nonumber\,\,.
\end{align}
which eventually gives rise to the limits \eqref{eq expansion nu>theta} and \eqref{eq expansion nu<theta}.

Thus the OPE of two bosonic twist fields is given by
\begin{align}
\sigma_{\theta} (z, \overline z) \, \sigma_{1-\nu} (w, \overline w) \sim\,\, & C^-_{\sigma}\,  \,|z-w|^{-2 \theta(1-\nu)}\,\sigma_{1+\theta-\nu}\,  (w, \overline w) \\
& \hspace{10mm}+  C^-_{\widetilde{\tau}} \,\, |z-w|^{-2 \theta(2-\nu)+2 \nu} \widetilde{\tau}_{1+\theta-\nu}\, (w , \overline w)\,  \nonumber
\end{align}
for $\nu > \theta$, and takes the form
\begin{align}
\sigma_{\theta} (z, \overline z) \, \sigma_{1-\nu} (w, \overline w) \sim\,\, & C^+_{\sigma}\,  \,|z-w|^{-2 \nu(1-\theta)}\,\sigma_{\theta-\nu}\,  (w, \overline w) \\
& \hspace{10mm}+  C^+_{{\tau}} \,\, |z-w|^{-2 \nu(2-\theta)+2 \theta} {\tau}_{\theta-\nu}\, (w , \overline w)\,  \nonumber
\end{align}
for $\theta > \nu$. Here the coefficients are
\begin{align}
\label{eq OPE closed yukawa1}
C^-_{\sigma} &= \sqrt{2 \pi \,\,\frac{\Gamma(1-\theta)\,\Gamma(\nu)\, \Gamma(1+\theta-\nu)}{\Gamma(\theta)\,\Gamma(1-\nu)\, \Gamma(\nu-\theta)}}\\
C^-_{\tau} &=\sqrt{ 2 \pi \,\,\frac{\Gamma^3(1-\theta)\,\Gamma^3(\nu)\, \Gamma^2(1+\theta-\nu)\, \Gamma(\theta-\nu)}{\Gamma^3(\theta)\,\Gamma^3(1-\nu)\, \Gamma^2(\nu-\theta)\, \, \Gamma(1-\theta+\nu)}}\\
\label{eq OPE closed yukawa2}
C^+_{\sigma} &= \sqrt{2 \pi\,\, \frac{\Gamma(\theta)\,\Gamma(1-\nu)\, \Gamma(1-\theta+\nu)}{\Gamma(1-\theta)\,\Gamma(\nu)\, \Gamma(\theta-\nu)}}\\
C^+_{\widetilde{\tau}} &=\sqrt{ 2 \pi \,\,\frac{\Gamma^3(\theta)\,\Gamma^3(1-\nu)\, \Gamma^2(1-\theta+\nu)\, \Gamma(\nu-\theta)}{\Gamma^3(1-\theta)\,\Gamma^3(\nu)\, \Gamma^2(\theta-\nu)\, \, \Gamma(1+\theta-\nu)}}\,\,,
\end{align}
where \eqref{eq OPE closed yukawa1} and \eqref{eq OPE closed yukawa2} are related to the quantum part of the physical Yukawa couplings determined in
\cite{Burwick:1990tu,Stieberger:1992bj}. The  weights of the respective conformal fields are given by
\begin{align}
h_{\sigma_{\alpha}} &= \frac{1}{2} \alpha \left(1-\alpha \right) \qquad \qquad \qquad \hspace{11mm}\overline h_{\sigma_{\alpha}} = \frac{1}{2} \alpha \left(1-\alpha \right)\\
h_{\widetilde{\tau}_{\alpha}} &= \frac{1}{2} \left(1-\alpha \right)\left(2+\alpha \right) \qquad \qquad \qquad \overline h_{\widetilde{\tau}_{\alpha}} = \frac{1}{2} \left(1-\alpha \right)\left(2+\alpha \right)\\
h_{\tau_{\alpha}} &= \frac{1}{2} \alpha \left(3-\alpha \right) \qquad \qquad \qquad \hspace{11mm}\overline h_{\tau_{\alpha}} = \frac{1}{2} \alpha \left(3-\alpha \right)\,\,.
\end{align}
Let us make a few remarks regarding this result. So far we have not distinguished between the bosonic twist field $\sigma^+_{\alpha}$ and the bosonic anti-twist field $\sigma^{-}_{\alpha}$. As shown in the appendix \ref{app twists} the anti-twist field $\sigma^{-}_{\alpha}$ can be identified with  $\sigma^+_{1-\alpha}$. Thus for the case at hand, namely the OPE $\sigma_{1-\theta}(z, \overline z) \, \sigma_{\nu}(w, \overline w) $, it is not surprising to obtain two different scenarios, depending on which angle is larger. One should interpret that OPE as an OPE between a twist field and an anti-twist field. Depending on the choice angles these two fields couple to a bosonic twist field or an bosonic anti-twist field. Similar arguments apply to the the sub-dominant couplings to the excited twist fields $\tau_{\alpha}(z, \overline z)$ and $\widetilde{\tau}_{\alpha}(z, \overline z )$. That suggests that $\tau_{\alpha}(z, \overline z)$
should be interpreted as $\tau^+_{\alpha}(z, \overline z)$, while $\widetilde{\tau}_{\alpha}(z, \overline z)$
should be interpreted as excited anti-twist field $\tau^-_{\beta}(z, \overline z)$, with $\beta= 1-\alpha$. This is also exposes the analogy between the two fields, since they have now the same conformal dimension $h_{\tau^+_{\alpha}}=\overline h_{\tau^+_{\alpha}}=h_{\tau^-_{\alpha}}=\overline h_{\tau^-_{\alpha}}=\frac{1}{2} \alpha(3-\alpha)$.

One can further extend the OPE's \eqref{eq OPE closed yukawa1} and \eqref{eq OPE closed yukawa2} by looking at higher order terms in the expansions  \eqref{eq expansion nu>theta} and \eqref{eq expansion nu<theta}, respectively. One finds that the two bosonic twist fields couple to  doubly excited twist fields as well. This can be generalized to the statement that they couple to N-times excited twist fields for any $N$. As we will see momentarily this is in contrast to the open string, where one observes an even grading, namely the bosonic twist fields couple only to  N-times excited twist fields where $N$ is even.

\subsection{Open string correlator \label{sec open two angles}}

In contrast to the previously discussed case the setup here consists of three D-branes denoted by $a$, $b$ and $c$. Again we assume a simplified setup in which all D-branes go through the origin and all three D-branes intersect each other once. Figure \ref{fig twisttwoangles} depicts the above described setup for one of the three two-tori. The intersection angle between $a$ and $b$ is given by $\theta$ while between  $a$ and $c$  it is $\nu$. As for the correlator with just one independent angle only for the interval $0 \leq x \leq 1$ the result is non-vanishing\footnote{This statement is slightly modified in un-oriented theories in which one of the D-branes is the orientifold image of one of the other two. Such a situation has been discussed in \cite{Cvetic:2006iz}.}.
\begin{figure}[h]
\begin{center}
\includegraphics[height=5cm]{4sigmas2angles.eps}
\end{center}
\caption{Intersection of three D-brane stacks, $a$, $b$ and $c$, in one complex dimension.}\label{fig twisttwoangles}
\end{figure}

In contrast to the correlator with just one independent angle here only in the limit $x \rightarrow 0$ we expect the exchanges of untwisted states. This limit allows us to normalize the correlator. On the other hand in the limit $x \rightarrow 1$ one observes the exchange of twisted states. As for the closed string correlator this limit allows us then to determine the proper operator product expansion of two bosonic twist fields even beyond the dominant pole.

Following the same steps as for the four bosonic twist field correlator with just one independent angle we obtain for the correlator
 \begin{align} \nonumber
 & x^{\nu (1-\nu)}_{\infty}\,\,  \langle \sigma_{1-\theta}(0)\, \sigma_{\theta}(x) \,
\sigma_{1-\nu}(1) \, \sigma_{\nu} (\infty)\rangle = \\ & \hspace{2cm} {\cal C}_{open} \frac{x^{-\theta(1-\theta)} (1-x)^{-\theta(1-\nu)}}{G_1(x)} \sum_{p,q} w(x)^{ \left(  \frac{\alpha'}{L^2_a}  p^2 + \frac{1}{\alpha'}\frac{R^2_1 \, R^2_2}{L^2_a} q^2 \right)}
\label{eq suggested result open two angles}
 \end{align}
with $w(x)$ given by
\begin{align}
w(x)=\exp\left[- \frac{\pi t(x)}{\sin(\pi \theta)}\right]
\end{align}
where $t(x)$ is
\begin{align}
t(x) = \frac{\sin(\pi \theta)}{2 \pi} \left( \frac{B_1 H_1(1-x)}{G_1(x)} + \frac{B_2 H_2(1-x)}{G_2(x) }
 \right) \,\,.
 \end{align}
Here we use the fact that  $G_i (x) = \overline G_i (x) $ and $H_i (x) = \overline H_i (x)$ for $0\leq x\leq 1$. Note that, as expected, \eqref{eq suggested result open two angles} reduces to \eqref{eq suggested result open one angle} for $\theta= \nu$ .

Let us determine the constant ${\cal C}_{open}$ which we of course expect to be $\sqrt{\alpha'}/L_a$. Recall that we expect the following OPE
\begin{align}
\sigma_{1-\theta}(z) \, \sigma_{\theta} (w) \sim (z-w)^{-\theta(1-\theta)} \,\,\frac{\sqrt{\alpha'}}{L_a} \,\,\mathbf{1}\,\,,
\label{eq expected result}
\end{align}
where the normalization factor $\sqrt{\alpha'}/L_a$ is crucial for consistency of the four bosonic twist field correlator with just one independent angle. In order to determine the  constant ${\cal C}_{open}$ we analyze the limit $x \rightarrow 0 $, which corresponds to the exchange of untwisted states. In that limit $t(x)$ behaves like
\begin{align}
t(x) \approx \frac{\sin (\pi\theta)}{ \pi}\left( -\ln(x) + \ln(\delta)\right)
\end{align}
with $\ln (\delta)$ given by
\begin{align}
\ln(\delta) = 2 \psi(1) -\frac{1}{2} \left(\psi(\theta) + \psi(1-\theta) + \psi(\nu) + \psi(1-\nu) \right)\,\,.
\end{align}
That leads to the following expression for \eqref{eq suggested result open two angles} in the limit $x \rightarrow 0$
\begin{align}
{\cal C}_{open}\, x^{-\theta(1-\theta)} \sum_{p,q} \left(\frac{x}{\delta}\right)^{\left(  p^2 \frac{\alpha'}{L^2_a} + q^2 \frac{R^2_1 \, R^2_2}{\alpha'\, L^2_a}\right)}
\label{eq lim x -> 0 two angles}
\end{align}
Thus comparing \eqref{eq lim x -> 0 two angles} for $p=q=0$ with \eqref{eq expected result}  we indeed get for the normalization constant
${\cal C}_{open} =\sqrt{\alpha'}/L_a$.

Eventually we are interested in the limit $x \rightarrow 1$ that corresponds to the exchange of twisted states. In order to take the limit it is convenient to perform a Poiss\`on resummation over the variable $p$, such that both variables we sum over live in the lattice $\Lambda$. We obtain (taking already into account the normalization constant)
\begin{align}
\frac{x^{-\theta(1-\theta)} (1-x)^{-\theta(1-\nu)}}{G_1(x)}  \sqrt{\frac{\sin(\pi \theta)}{t(x)}}
\sum_{\widetilde{p},q}
\exp\left[-\pi \frac{\sin(\pi \theta)} {t(x)} \, \frac{L^2_a}{\alpha'} \, \widetilde{p}^2- \pi \frac{t(x) }{\sin(\pi \theta)}  \frac{ R^2_1\, R^2_2}{\alpha' \,  L^2_a}  \, q^2\right]\,\,.
\label{eq Poisson resummed result}
\end{align}
A comparison with the results obtained in \cite{Cvetic:2003ch,Abel:2003vv,Abel:2003yx} shows that both methods, the one using the doubling trick to extend the world-sheet to the complex plane and then using conformal field theory techniques and the other one in which one extracts the open string correlator directly from the closed string correlator, give the exactly same result. Once again we conclude that there are no ambiguity in the choice of the constants $c_{p,q}$ in this case either.

For the limit $x \rightarrow 1$ one has to distinguish two different cases
\begin{align} \label{eq limits t(x) theta>nu}
\lim_{x\rightarrow 1}  t (x) &= \frac{\sin(\pi(\theta-\nu) )}{2 \sin(\pi \nu)} \qquad  \qquad \text{for} \qquad \theta > \nu\\
\label{eq limits t(x) theta<nu}
\lim_{x\rightarrow 1} t (x) &= \frac{\sin(\pi(\nu-\theta) )}{2 \sin(\pi \nu)} \qquad  \qquad \text{for} \qquad \theta < \nu\,\,.
\end{align}
This can be easily derived from
\begin{align} \nonumber
&\lim_{x\rightarrow 1} G_1(x) = \frac{\Gamma(\nu-\theta)}{\Gamma(1-\theta) \, \Gamma(\nu)} +(1-x)^{\nu-\theta} \frac{\Gamma(\theta-\nu)}{\Gamma(\theta) \, \Gamma(1-\nu)}\\ \label{eq limits of hypergeometric}
&\lim_{x\rightarrow 1} G_2(x) = \frac{\Gamma(\theta-\nu)}{\Gamma(\theta) \, \Gamma(1-\nu)} +(1-x)^{\theta-\nu} \frac{\Gamma(\nu-\theta)}{\Gamma(1-\theta) \, \Gamma(\nu)}\\
&\lim_{x\rightarrow 1} H_1(1-x)  = \lim_{x\rightarrow 1} H_2(1-x) =1\nonumber\,\,.
\end{align}
Thus for $\nu>\theta$ we get \eqref{eq suggested result open two angles}
\begin{align}
(1-x)^{-\theta(1-\nu)} &\frac{\Gamma(1-\theta)}{\Gamma(\nu)}{\Gamma(\nu -\theta)} \sqrt{\frac{2 \sin(\pi \theta) \, \sin(\pi \nu)}{\sin(\pi (\nu-\theta))}} \\
&\times \sum_{\widetilde{p},q}  \exp\left[-2\pi \frac{\sin(\pi \theta) \, \sin(\pi \nu) }{\sin(\pi(\theta-\nu)} \, \frac{ L^2_a}{\alpha'}\, \widetilde{p}^2- \frac{\pi}{2} \frac{\sin(\pi (\theta-\nu))}{\sin(\pi \theta)\,\sin(\pi \nu) }  \, \,\frac{ R^2_1\, R^2_2}{ \alpha' \,L^2_a} q^2\right]\,\,. \nonumber
\end{align}

The angles can be expressed in the following matter (recall that all the branes intersect exactly once)
\begin{align}
\sin(\pi \theta) =  \frac{R_1 R_2}{ L_a L_b} \hspace{1cm} \sin(\pi \nu) =  \frac{R_1 R_2}{ L_c L_a} \hspace{1cm} \sin(\pi (\nu-\theta)) =  \frac{R_1 R_2}{ L_b L_c}\,\,.
\label{eq sinus definitions}
\end{align}
Given \eqref{eq sinus definitions} one obtains
\begin{align}
\sqrt{2 \pi}     (1-x)^{-\theta(1-\nu)} & \sqrt{       \frac{\Gamma(1-\theta)\Gamma(\nu)\Gamma(1+\theta-\nu)}
{\Gamma(\theta)\Gamma(1-\nu)\Gamma(\nu-\theta)} } \sum_{\widetilde{p},q}  \exp\left[-\frac{2\pi}{\alpha'} R_1 R_2 \tilde{p}^2- \frac{\pi}{2 \alpha'} R_1 R_2 q^2\right]
\end{align}
which, with the redefinition of the summation variables (note that with this definition  $r_1$ and $r_2$ are integers)
\begin{align}
r_1 = 2 \tilde{p} +q \qquad \qquad r_2 = 2 \tilde{p}- q
\end{align}
leads to
 \begin{align}
\sqrt{2 \pi}     (1-x)^{-\theta(1-\nu)} & \sqrt{       \frac{\Gamma(1-\theta)\Gamma(\nu)\Gamma(1+\theta-\nu)}
{\Gamma(\theta)\Gamma(1-\nu)\Gamma(\nu-\theta)} } \sum_{r_1, r_2}  \exp\left[-\frac{\pi}{4 \alpha'} R_1 R_2 \left( r^2_1 +r^2_2 \right)\right]\,\,.
\label{eq Yukawa final theta<nu}
\end{align}
Analogously one obtains for $\theta> \nu$
 \begin{align}
\sqrt{2 \pi}     (1-x)^{-\nu(1-\theta)} & \sqrt{       \frac{\Gamma(\theta)\Gamma(1-\nu)\Gamma(1-\theta+\nu)}
{\Gamma(1-\theta)\Gamma(\nu)\Gamma(\theta-\nu)} } \sum_{r_1, r_2}  \exp\left[-\frac{\pi}{4 \alpha'} R_1 R_2 \left( r^2_1 +r^2_2 \right)\right]\,\,.
\label{eq Yukawa final theta>nu}
\end{align}
Again $r_1$ and $r_2$ are positive integers. Note that the classical part is exactly twice the world-sheet instanton contribution to the Yukawa coupling arising from a two-torus determined in \cite{Cremades:2003qj, Abel:2003vv}. Moreover, \eqref{eq Yukawa final theta<nu} and \eqref{eq Yukawa final theta>nu}, quantum and classical part, agree with the results of \cite{Cremades:2003qj,Cvetic:2003ch, Abel:2003vv} where the authors investigated Yukawa couplings in intersecting brane worlds.

As we saw for the closed string correlator, to extract the OPE between two bosonic twist fields it is sufficient to look at the quantum part of the correlator. For large volumes, namely in the limit $R_1, R_2 \rightarrow \infty$ the only contribution comes from $(\widetilde{p}, q) =(0,0)$ and the whole correlator reduces to the quantum part given by\footnote{Here we used also the identities displayed in eq. \ref{eq sinus definitions}.}
\begin{align}
\frac{x^{-\theta(1-\theta)} (1-x)^{-\theta(1-\nu)}}{G_1(x)}  \sqrt{\frac{\sin(\pi \theta)}{t(x)}}
\end{align}
which is exactly the quantum part computed in \cite{Cvetic:2003ch,Abel:2003yx} using the same conformal field theory techniques as for the closed string after extending the world-sheet to the whole complex plane via the doubling trick.

We already showed above that the dominant pole gives the expected Yukawa couplings. Now we would like to take a look at the sub-dominant terms. Again we will start with $\nu>\theta$. Using \eqref{eq limits t(x) theta<nu} and \eqref{eq limits of hypergeometric} one gets
\begin{align}
\label{eq expansion theta<nu}
(1-x)^{-\theta(1-\nu)} & \sqrt{2 \pi} \sqrt{       \frac{\Gamma(1-\theta)\Gamma(\nu)\Gamma(1+\theta-\nu)}
{\Gamma(\theta)\Gamma(1-\nu)\Gamma(\nu-\theta)} } \\ & \left( 1-\frac{\Gamma^2(1-\theta) \Gamma^2(\nu) \Gamma(\theta-\nu) \Gamma(1+\theta-\nu)}{\Gamma^2(\theta) \Gamma^2(1-\nu) \Gamma(\nu-\theta) \Gamma(1-\theta+\nu)} (1-x)^{2(\nu-\theta)} + ... \right)\,\,.\nonumber
\end{align}
This implies that in the OPE between the two bosonic twist fields there is no coupling to the excited twist field $\tau$ but the first sub-dominant pole rather indicates a coupling to the doubly excited twist field $\rho$. Let us display the OPE explicitly (keep in mind one has to take the square root out of the different coefficients in front of the poles)
\begin{align}
\sigma_{\theta}(w) \sigma_{1-\nu} (z) \sim & \left( 2\pi \frac{\Gamma(1-\theta)\Gamma(\nu)\Gamma(1+\theta-\nu)}
{\Gamma(\theta)\Gamma(1-\nu)\Gamma(\nu-\theta)} \right)^{\frac{1}{4}} (z-w)^{-\theta(1-\nu)}  \sigma_{1+\theta-\nu} (z)   \\
& \hspace{-2cm} + \left( 2\pi \frac{\Gamma^5(1-\theta) \Gamma^5(\nu) \Gamma^2(\theta-\nu) \Gamma^3(1+\theta-\nu)}{\Gamma^5(\theta) \Gamma^5(1-\nu) \Gamma^3(\nu-\theta) \Gamma^2(1-\theta+\nu)} \right)^{\frac{1}{4}} (z-w)^{-\theta(3-\nu)+2 \nu}\widetilde{\rho}_{1+\theta-\nu}(z) + ...\nonumber
\end{align}
Analogously one obtains for $\theta>\nu$ in the limit $x \rightarrow 1$
\begin{align}
\label{eq expansion theta>nu}
   (1-x)^{-\nu(1-\theta)} & \sqrt{2 \pi}   \sqrt{       \frac{\Gamma(\theta)\Gamma(1-\nu)\Gamma(1-\theta+\nu)}
{\Gamma(1-\theta)\Gamma(\nu)\Gamma(\theta-\nu)} }\\ & \left( 1-\frac{\Gamma^2(\theta) \Gamma^2(1-\nu) \Gamma(\nu-\theta) \Gamma(1-\theta+\nu)}{\Gamma^2(1-\theta) \Gamma^2(\nu) \Gamma(\theta-\nu) \Gamma(1+\theta-\nu)} (1-x)^{2(\nu-\theta)} + ... \right)\nonumber
\end{align}
which leads to the OPE
\begin{align}
\sigma_{\theta}(w) \sigma_{1-\nu} (z) \sim & \left( 2\pi \frac{\Gamma(\theta)\Gamma(1-\nu)\Gamma(1-\theta+\nu)}
{\Gamma(1-\theta)\Gamma(\nu)\Gamma(\theta-\nu)} \right)^{\frac{1}{4}} (z-w)^{-\nu(1-\theta)}  \sigma_{\theta-\nu} (z)   \\
& \hspace{-2.3cm} + \left( 2\pi \frac{\Gamma^5(\theta) \Gamma^5(1-\nu) \Gamma^2(\nu-\theta) \Gamma^3(1-\theta+\nu)}{\Gamma^5(1-\theta) \Gamma^5(\nu) \Gamma^3(\theta-\nu) \Gamma^2(1+\theta-\nu)} \right)^{\frac{1}{4}} (z-w)^{-\nu(3-\theta)+2 \theta} \,\,\rho_{\theta-\nu}(z) + ...\nonumber\,\,.
\end{align}
Note that in contrast to the closed string, two bosonic open twist fields $\sigma_{\alpha}(w)$ and  $\sigma_{\beta}(z)$ do not couple to the excited twist field $\tau_{\alpha+\beta}$. Their first sub-dominant pole in the OPE indicates the coupling to the doubly excited twist field $\widetilde{\rho}$ and $\rho$ respectively. This can be generalized by looking at higher order terms in the expansions \eqref{eq expansion theta<nu} and \eqref{eq expansion theta>nu}
to the statement that the two bosonic twist fields do only couple to $N$-times excited twist fields, where $N$ is an even number.

The conformal dimensions of the fields $\sigma_{\alpha}$ and $\widetilde{\rho}_{\alpha}$  and ${\rho}_{\alpha}$ are
\begin{align}
h_{\sigma_{\alpha}}= \frac{1}{2} \alpha\left(1-\alpha \right) \qquad  \hspace{4mm} h_{\widetilde{\rho}_{\alpha}}= \frac{1}{2} \left(1-\alpha\right)\left(4+\alpha \right) \qquad   \hspace{4mm} h_{\rho_{\alpha}}= \frac{1}{2} \alpha\left(5-\alpha \right) \,\,.
\end{align}
Analogously to the closed string result $\widetilde{\rho}_{\alpha}$ can be interpret as a doubly-excited anti-twist field $\rho^-_{\beta}$ with $\beta=1-\alpha$. On the other hand $\rho_{\alpha}$ is the doubly-excited twist field $\rho^-_{\alpha}$. As for the closed string this exhibits the analogy of these two conformal fields due to their same conformal dimension $h_{\rho^+_{\alpha}}=h_{\rho^-_{\alpha}}=\frac{1}{2} \alpha \left(5-\alpha \right)$ \footnote{For a more detailed discussion on this issue see \cite{Light} and appendix \ref{app twists}.}.

\section{Summary}

In this note we revisited the correlator of four open string bosonic twist-fields with one and two independent angles, respectively. In contrast to previous works we extracted the open string correlator directly from the closed string result. This way we provide a non-trivial check for the method advocated in \cite{Gava:1997jt,David:2000um,Cvetic:2003ch,Abel:2003vv} where the authors extend  the world-sheet, whose original domain is the upper half complex plane, via the ``doubling trick'' to the full complex plane and employ conformal field theory techniques.

Given the four-point correlators, we investigated various limits that allowed us to extract the OPE's between two bosonic twist fields beyond the dominant terms. Interestingly one finds, in contrast to the closed string, that two bosonic twist fields do not couple to an excited twist field, but rather to a doubly excited twist field. This can be generalized to
the statement that two open string bosonic twist fields couple only to $N$-times excited twist fields with $N$ being an even positive integer.

Finally, we found an interesting identification between bosonic twist field and anti-twist field even for higher excited twist fields. This allows to compute
the twist field correlator for just one combination of ``twist'' and ``anti-twist'' fields. Any other combination can be determined by the appropriate identifications.

Before concluding we would like to sketch how to generalize our discussion to genuinely interacting albeit rational CFT's. While in the case of twist-fields the number of block was actually infinite, in a RCFT the number of blocks is finite. Conformal blocks can be determined by means of null vectors and the resulting Ward identities. Explicit expressions are available for WZW models and for various minimal models. For RCFT the number of D-branes is at most equal to the number of characters. Given a parent closed-string model, specified by the choice of the one-loop modular invariant combination of characters, there may be many open string `descendants'. By the same token, given a closed-string 4-point correlator we expect different open-string correlators, at least as many as open string descendants. In some simple cases, as minimal models ({\it e.g.}~ Ising model), only the simplest open-string descendant with real Chan-Paton charges was considered \cite{Bianchi:1991rd} to give 4-point amplitudes compatible with planar duality. Later on, based on the analysis of $SU(2)$ WZW models \cite{Pradisi:1995qy} it was noticed that complex Chan-Paton factors were also allowed.
It would be interesting to explore the problem of computing open-string correlator based on the knowledge of their parent closed-string correlators in WZW or even better in Gepner models. In the latter case only limited knowledge of chiral blocks is available at present, however. Given their relation to solvable compactifications on CY manifolds this is a rather sad state of affairs.

\section*{Acknowledgements}

We acknowledge M. Cveti{\v c}, F. Fucito, E. Kiritsis, G. Leontaris, J. F. Morales Morera, I. Pesando, G. Pradisi, B. Schellekens, O. Schlotterer, M. Schmidt-Sommerfeld, Ya. Stanev, P. Teresi and T. Weigand for interesting discussions and correspondence. P.~A. is supported by FWF P22000.  P.~A. and R.~R. are grateful to the organizers of the school and workshop of ITN ``UNILHC" PITN-GA-2009-237920 for hospitality during parts of this work. The work of R.~R. was partly supported by the German Science Foundation (DFG) under the Collaborative Research Center (SFB) 676 "Particles, Strings and the Early Universe".
The work of M.~B. was partially supported by the ERC Advanced Grant n.226455 Superfields, by the Italian MIUR-PRIN contract 20075ATT78, by the NATO grant PST.CLG.978785. M.~B. and P.~A. would like to thank NORDITA, Stockholm for hospitality during the completion of this project.

\newpage

redpajama_set_name: RedPajamaArXiv


WHAT IS DIFFERENT THIS TIME THAT WILL MAKE YOU SUCCESSFUL: I have learned how to motivate myself and I have learned coping skills to help me deal with the things that may happen down the road.
ANYTHING ELSE YOU WOULD LIKE PEOPLE TO KNOW ABOUT YOU? I'm opiniated. Just thought you should know that.
HOBBIES/INTERESTS: Reading, listening to music, writing, being with family and friends, the outdoors, and watching sports.
WHAT IS YOUR BIGGEST PROBLEM IN WEIGHT LOSS: Trying to exercise and if I don't see results right away I get frustrated.
WHAT IS DIFFERENT THIS TIME THAT WILL MAKE YOU SUCCESSFUL: Wanting to be healthy for myself and my daughter. Finally taking care of everything in my life.
ANYTHING ELSE YOU WOULD LIKE PEOPLE TO KNOW ABOUT YOU? I have been diagnosed with PCOS, Depression/Anxiety/traits of Personality Disorder. I just had bloodwork show my cholesterol and sugars are getting up there. I don't want anymore health problems. I want to lose the weight and maybe take care of everything.
CHILDREN/AGES: 2 sons steve 36 and robert 34, both married and have 3 girls between them.
WHAT IS YOUR BIGGEST PROBLEM IN WEIGHT LOSS: depression, feelings, binging, etc.
WHAT IS DIFFERENT THIS TIME THAT WILL MAKE YOU SUCCESSFUL: not really sure just want to get weight off.
ANYTHING ELSE YOU WOULD LIKE PEOPLE TO KNOW ABOUT YOU? i have 3 granddaughters, jordan is 14, jenna is 10. they are my youngest sons girls.
hannah is 16 months, she is our oldest sons daughter. they are the apples of my eye, my heart swells.
ANYTHING ELSE YOU WOULD LIKE PEOPLE TO KNOW ABOUT YOU? i am a very busy mother of triplets + 1 and I work full time, go to school part time. I love food and I have severe issues with depression. I have been diagnosed Bipolar.
CHILDREN/AGES: Son-Robbie/24, his fiance Teri/24, daughter-Heather/21, her husband-Luis/25, granddaughter-Faith Anne/3, granddaughter-Nevaeh Hope/18months.
HOBBIES/INTERESTS: I love to read and listen to music and travel.
WHAT IS YOUR BIGGEST PROBLEM IN WEIGHT LOSS: Emotional eating and not knowing when to stop.
WHAT IS DIFFERENT THIS TIME THAT WILL MAKE YOU SUCCESSFUL: I am sick of being fat, my feet hurt all the time.
ANYTHING ELSE YOU WOULD LIKE PEOPLE TO KNOW ABOUT YOU? I have a real hard time sticking to things.
WHAT IS DIFFERENT THIS TIME THAT WILL MAKE YOU SUCCESSFUL: i am determined this time. i've gotten way too big.
PETS: My kid if you will, she's a miniature Doberman/sheltie mix and tons of attitude! I've had her since she was 8 weeks old.
WHAT IS YOUR BIGGEST PROBLEM IN WEIGHT LOSS: Falling off the wagon and eating all sorts of unhealthy things – pasta, chocolate, chips, cookies, you know. . .
WHAT IS DIFFERENT THIS TIME THAT WILL MAKE YOU SUCCESSFUL: Organic and natural foods are more available/affordable and they taste good.
I just started back on Prozac yesterday and will be going back to counseling. I've done both before (I was in counseling from age 11 – 20. Prozac for about a year or so when I turned 19) and I'm wondering if I really have Dysthemia (chronic depression) because I seem to have a major episode every year or so.
I like this idea, it makes things easier; something like this would've been easier in the main introduction forum too.
CHILDREN/AGES: None yet and not until I get my Master's Degree and tenure, LOL!
GOAL WEIGHT: 163 to start; 135 when it's all said and done.
WHAT IS DIFFERENT THIS TIME THAT WILL MAKE YOU SUCCESSFUL: If I don't lose weight for my health I won't be around much longer, let alone even be able to think about having a child in the future.
ANYTHING ELSE YOU WOULD LIKE PEOPLE TO KNOW ABOUT YOU? I recovered from an eating disorder my senior year of high school only to go the opposite way into compulsive overeating. I suffer from anxiety/panic disorder and major depressive disorder which ultimately affects my actions in trying to lose weight.
HOBBIES/INTERESTS: All things Italian, bass guitar, singing, creative writing, producing my music (just started so I'm not very good at this yet), learning Italian.
WHAT IS YOUR BIGGEST PROBLEM IN WEIGHT LOSS: Weight loss.
WHAT IS DIFFERENT THIS TIME THAT WILL MAKE YOU SUCCESSFUL: This is really the first time I've ever been overweight. I have always been a skinny person until the past couple of years.
WHAT IS YOUR BIGGEST PROBLEM IN WEIGHT LOSS: I food.
WHAT IS YOUR BIGGEST PROBLEM IN WEIGHT LOSS: I don't have enough time to exercise, plan healthy meals. I'm also addicted to sugar.
ANYTHING ELSE YOU WOULD LIKE PEOPLE TO KNOW ABOUT YOU? I'm a workaholic.
Last edited by sct; 10-22-2008 at 03:04 PM.
HEIGHT: 5' 7 3/4" ~ 5' 8"
ANYTHING ELSE YOU WOULD LIKE PEOPLE TO KNOW ABOUT YOU? I have a very dry sense of humor, very sarcastic. Be forewarned.
HOBBIES/INTERESTS:Crocheting, Gym, Reading, Cooking, Music & Movies.
GOAL WEIGHT: 175LBS, but eventually maybe lower like 150-155lbs.
- Eating whatever I want because it tastes good, and taking too much.
- I NEED to do it for my health, no one else is going to take care of me but me.
- I don't really know, can ask if you want, lol.


text:
Q: what is the best way to find longest path between set of nodes(certain nodes) in neo4j? I have set of nodes which will be declared by user(I mean user enters some words and those which exist among my nodes will be included in my set of nodes),and I want to find the longest path which exists between just these nodes(does not have to include all of the nodes in set ,but should not include a node which is not exist in my set of nodes)in neo4j embedded java database.(I do not know what is the start node,all of them can be the start node.)
First I thought maybe I have to make a subgraph of these nodes and relations between them,and then find the longest path in that subgraph,but I don't know what is the right and best thing to do?
Then i thought,I can directly find longest path between the set of nodes(can that be done?).
Or if this can't be done,how about finding the relations of the certain nodes(set of nodes) and make a set of relations,and then find the longest path between set of relations?
I want the optimum and fastest way in performance,and I don't want cypher query,because I am using java core api(it is faster),so please tell me what is the best way?
Consider it as a large scale database,with so many nodes.
Thanks in advance.

A: i'm afraid this cannot be done by any standard algorithm except traversing the whole graph. thus, there is no optimum way in performance. neo4j has no build-in option for it.
this is an cypher query (sorry i don't know other input languages in neo4j):
start n1=node(*), n2=node(user)
match n1--n2    //filter only the just declared user nodes, or any other matching condition
with n1, n2
match p=n1-[r:..*]-m, m--n2    //all paths between declared user nodes
return p
order by length(r) desc
limit 1

redpajama_set_name: RedPajamaStackExchange


Fashion week is back in Rome, from June 27th to July 1st the eternal city has become the main spotlight for young talents as well as famous Italian and international designers.
As you know I love attending Altaroma and as usual I captured the best moments from Rome fashion week in order to share them with you and bring you along with me into the magic world of fashion.
So, I created a visual diary featuring my favourite shows, as well as my looks, the most interesting events and some behind the scenes moments.
Let me know in the comments down below what do you think about my "ALTAROMA Fashion Week June 2018 | A Visual Diary" and also, share with me your favourite shows from this edition.
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Wow, what an incredible experience! Love your white dress!
Very gorgeous white dress! Those gowns from fashion week are very amazing.
Che evento super e... mi piace molto il tuo look per l'0occasione!
OMG, this is amazing :3 loved all combos!
Thanks for sharing these lovely pics.
im sure it has to be an amazing experience!
I really love the bñack dress of your first day, it's so pretty!! amazing fashion week.
Thanks for this inspirational diary!
Que chulada!! Tu vestido me encanta, es monisimo!
It's alwaays so amazing to see other fashion weeks (apart from the one in your own country).
Congrats for this amazing experience and sharing the looks with us. So lovely.
Hello Camila, thanks a lot!
Ohh I love this post... seeing your fashionable looks and the stunning runway designs. So fun!
grazie mille Giulia, un bacione!
amazing photos! i really like!
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text:
María José Marenco (born September 27, 1976) is a Salvadoran former swimmer. She represented El Salvador at the age of 15 at the 1992 Summer Olympics in Barcelona in the 200 metre freestyle, 400 metre freestyle and 800 metre freestyle competing in the heats. She also acted as the flag bearer for the El Salvador team of 4 athletes during the opening ceremony.

Marenco represented El Salvador at the Central American Games and the Central American and Caribbean Swimming Championships on a number of occasions. At the 1994 Central American Games hosted in her home country, she was considered such a good medal hope in the freestyle event the then President of El Salvador Alfredo Cristiani attended her competition.

Record at the 1992 Olympics

References

External links

Living people
1976 births
Salvadoran female swimmers
Swimmers at the 1992 Summer Olympics
Olympic swimmers of El Salvador
20th-century Salvadoran women

redpajama_set_name: RedPajamaWikipedia


White chocolate Raspberry mousse with fresh raspberry in the center..
moist Vanilla cake, White chocolate mousse and raspberry, ….
Filling: White chocolate mousse and red current.
Chocolate raspberry cake covered with a Red white chocolate glaze.
Short Bread cookie in heart shape dipped in fondant.


text:
Prime Minister Scott Morrison to travel to the US next week for Quad leaders meeting
By Emily Bennett
Prime Minister Scott Morrison will travel to the US next week to meet with President Joe Biden.
Mr Morrison will travel to Washington, DC from September 21 to 24, in his first visit to the US since Mr Biden took office.
"I look forward to reaffirming our 70-year alliance, and to discussing ways to further strengthen our security and economic partnership," Mr Morrison said today.
READ MORE: Flight attendant's 'jet lag' turned out to be deadly disease
Prime Minister Scott Morrison will travel to the US this month to meet with President Joe Biden. (Jacky Ghossein/SMH)
Mr Morrison will also participate in the first face-to-face meeting of Quad leaders, with Mr Biden, Prime Minister Narendra Modi of India and Prime Minister Yoshihide Suga of Japan.
"Reconvening this important group reinforces our commitment to the Indo-Pacific COVID-19 recovery, and our efforts towards peace, prosperity and stability in our region," Mr Morrison said.
READ MORE: Beach party chaos as revellers flee croc-infested waters
"The Quad represents four great democracies working in partnership for an Indo-Pacific region that is open, inclusive, resilient, and anchored by shared principles.
"On my return to Australia on September 26, I will be complying with the health advice and quarantine requirements, as will accompanying staff, officials and media."

redpajama_set_name: RedPajamaCommonCrawl


The day will be clear with very few clouds in the sky in Shelland.
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Religion and the family, God and home, marriage and mother love, all are being wiped out.
Freedom of the press, freedom of speech, freedom of assemblage, the right to organize and strike — all the great institutions of democracy have disappeared.
Russian Information Bureau in the U. S.


Recruitments of mercenaries began after Russia's annexation of Crimea in March 2014 and amid the conflict in eastern Ukraine between Ukrainian and pro-Russian separatist forces.
The Ukrainian Security Service, SBU on Monday published a list of 206 individuals who it claimed have been fighting for private military company Wagner in Syria and eastern Ukraine, among them three Moldovan citizens.
The new name revealed among the Moldovan fighters on the list was 47-year-old Serghey Prida from Soroca in north-east Moldova, Balkan Insight reported.
Prida was listed alongside two Moldovans who have been named previously - Veaceslav Apostol, who has already been killed in Syria, and Radu Donciu, an active fighter with the Wagner unit. The men's names could not be independently verified.
"Wagner's mercenaries are instruments of Russia's hybrid aggression against the rest of the world," the SBU said.
Moldova's Security and Intelligence Service, SIS, told BIRN in April 2018 that it has identified 62 people from the country as taking part in mercenary activities in Ukraine.
"Evidence about these persons was handed over to the Prosecutor's Office of Moldova. It is worth mentioning that 17 mercenaries have been convicted by the courts of Moldova," the SIS said in April 2018.
Wagner, an Argentine-registered private military company, employs mercenaries who are believed to have played an important role in battles fought by pro-Russian separatists in eastern Ukraine and, from 2016, also in Syria.
Russian officials have long admitted the presence of volunteer fighters in the Donbass region of eastern Ukraine, and in Syria, where Moscow supports the Damascus government but denies any links to the private company Wagner.
An investigation by RISE Moldova has also shown how Moldovan citizens have opted to fight both for the separatists in Donbass but also for Ukrainian battalions, some for money and others for ideological reasons.
Russia has recruited fighters from all over Moldova, but many come from the breakaway region of Transnistria and some from the Gagauzia autonomous region inside Moldova, according to RISE Moldova.
Ukraine also expressed irritation in 2017 about Belgrade's failure to stop Serbians from fighting for pro-Russian separatists in eastern Ukraine.
Ukrainian Foreign Minister Pavlo Klimkin called on Serbia in November 2017 to respect his country's sovereignty and territorial integrity by taking action to stop its citizens joining separatist forces.
In December 2017, Ukraine's security service issued arrest warrants for six alleged Serbian mercenaries. The Kyiv authorities maintain that almost 300 Serbs are still fighting in separatist-controlled areas.
The Wagner company has also a so-called Serbian platoon allegedly led by a Bosnian Serb, Davor Dragolobovic Savicic, 42, nicknamed 'The Wolf' by his men for his courage and strength.


I'm new to this but have a problem. I was using Vista with Windows Live mail and when my hard drive started to get full I decided it was time to upgrade to Windows 8. I used the Windows Easy Transfer program and downloaded it onto my old PC and saved all my data to an external drive. So far so good. I then booted up my new PC and used Windows Easy Transfer to complete the operation. Everything worked fine until I looked at my email - nothing there. I have looked in the folder Users/Pete/Appdata/Local/Microsoft/Windows Mail/local folders and there are all the emails however all the names of each email are changed to 00f22E01-00001194 etc. and none will show in the inbox. I tried using EML viewer and it shows each one but it would be a nightmare having to open each one just to see which one it was. Are these all now corrupted or is there a way to recover them?
Try extracting the content of MIG-file with migrecover (if MIG was created in Vista, you need version 3). If e-mail names are fine there, see if replacing the local folder content will do the trick.
I tried restoring an easy transfer backup from Windows 7 to Windows 8 and it failed twice. I don't know if it is compatible but Vadi's solution sounds good.
1. Can I see a screenshot of the mail folder content after extraction from MIG?
Thanks for the quick response. I did a clean install - or rather I ditched my old PC and bought a new Dell XPS as I needed more of everything - disk space, memory etc. so there is no windows.old. I'll try and get a screen shot but the mail came into a folder called "Inbox replied" and I didn't create that.
the mail came into a folder called "Inbox replied" and I didn't create that.
It's also possible that the Live Mail versions are different. Did you have the latest version in Vista? If not, you may want to find the old version (google old apps), install it and try to reimport with Easy Transfer. Then upgrade the mail version.


This page is a place for me to rant and rave about pointless things that you could probably care less about. Keep reading!
A/V Tronics, a cool transcription firm where I spent several years in an IT Manager position.
If you don't like Guinness you don't know beer.
Coffee isn't Irish Coffee without Bushmills Irish Whiskey.
The first computer I ever owned. Viva La C= 64.
My 486DX4 100 mhz, HP Scanjet 4C and lots of Yuban to help along workflow decisions.
1993 Chrysler LeBaron Convertible - such fun.
Meet "Harmony", the laptop that powered the design of this very website you are reading now.


Tap Tap Revenge 2 is the addicting sequel to the most popular game in the App Store. You can play against another player, someone online, or even play a career. To start, choose a level of difficulty, then choose a song to tap to. And then just tap the targets when the tappers reach them. Tap tap tap away, and you'll be hitting the leaderboards in no time!


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Plants from the Chelsea Flower Show help bring colour to Ebbsfleet Garden City
PUBLISHED: 09:00 10 June 2018
Olivia Gantzer
Cherry Orchard Primary School teacher Sabina Borthwick with pupils. Picture: The Imageworks
The Imageworks
Schools and businesses have been given 250 plants from the Chelsea Flower Show to help bring more colour to Ebbsfleet Garden City.
Ebbsfleet Academy teacher Dr Rebecca Handley with pupils, from left, Emmanuel Adewunmi William Blackmoor Grace Barrett Lewis Clarke Harini Gopi. Picture: The Imageworks
The plants have been distributed around seven locations, some of which are in the Edible Ebbsfleet scheme, which allows members of the public to pick fruit and veg for free to encourage healthy eating.
The plants were donated by the show's re-use scheme and the Woodland Trust also donated fruit trees to the Garden City project.
They were taken to Cherry Orchard and Knockhall primary schools, Ebbsfleet Academy, Ebbsfleet International, Eastgate community centre and the two edible gardens set up in the Edible Ebbsfleet scheme at Swanscombe community garden and No Walls Gardens in Northfleet.
Beetroot, cabbage, peas and strawberries will now be grown at the edible garden sites for the public to enjoy free of charge while looking at the array of blooms which include allium, delphinium and lupin.
Kevin McGeough, director of Ebbsfleet Garden City's Healthy New Towns project, said: "The community couldn't wait to get involved in this exciting project, and the donation from the Chelsea Flower Show means many more members of the public will enjoy free, healthy fruit and veg in the weeks and months to come."
Ebbsfleet is the largest of ten designated NHS England healthy new towns and is a joint project with Dartford, Gravesham and Swanley Clinical Commissioning Group and Ebbsfleet Development Corporation.
A number of innovative new ideas are already up and running in addition to the edible gardens, including an app where people can be rewarded with shopping vouchers for being active and a Fitbit scheme where members of the public have their fitness levels monitored.
This will be the third year that Wayward is the Official Reuse Partner of the Chelsea Flower Show through a partnership with Barking Riverside London and the landscape contractors DF Clark.
The initiative is part of the House of Wayward Plants, a long-term project run by Wayward that facilitates large-scale community plant exchanges and adoption events for unwanted plants.
To find out more about Ebbsfleet Garden City Healthy New Towns visit ebbsfleetdc.org.uk/healthynewtowns/.
Commissioning Group
Ebbsfleet Development Corporation


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Girl Scouts of Gulfcoast Florida announces scholarship recipients
Community Submitted
May 30, 2019 at 10:21 AM Jun 12, 2019 at 12:56 PM
Girl Scouts of Gulfcoast Florida has announced their 2019 college scholarship recipients, including eight Girl Scouts from Sarasota County. A total of 18 girls each received a one-time scholarship, ranging from $500-$1,000. Local recipients include Julia Poncelet of Riverview High School; Hallie Hart and Krizelle Garcia from North Port High School; and Bianca Carey, Katrina Dedovets, Jade Goyette, Megan Hartnett, and Catherine Alvaro from Pine View School.
The Dr. Elinor Crawford Girl Scout College Scholarship Fund was established to recognize and celebrate graduating high school seniors who demonstrate the characteristics of the Girl Scout Mission, Promise, and Law; leadership to self and the community; and outstanding academic success. It is named for the late Dr. Elinor Crawford, a former resident of Sarasota. As a professor of physical education at the University of Northern Iowa for 30 years, Dr. Crawford served on the committee that contributed to the writing and passage of Title IX, the most significant piece of federal legislation impacting girls and women in education.
Catherine Alvaro also received the Lyndi Cyphers Memorial College Scholarship, named in honor of the late Lyndi Cyphers. Cyphers was a lifetime member of Girl Scouts, as well as a troop leader and service unit manager in Sarasota.
The scholarship recipients were honored at the 2019 Young Women of Distinction Ceremony in Punta Gorda, Florida on May 19. The annual ceremony celebrates the accomplishments of hundreds of Girl Scouts from across the area for service projects and programmatic achievements.
For more information about Girl Scouts of Gulfcoast Florida, visit www.gsgcf.org.
— Submitted by Lori Tomlinson
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A Space Program
ByDesign 2016
(Van Neistat & Tom Sachs, United States, 2015, DCP, 72 min)
Thursday, Apr 14 at 07:00PM
Playfully experimental filmmaker Van Neistat has given artist Tom Sachss 2012 performance piece of the same name the film treatment. Sachss A Space Program takes DIY to another level with the meticulous simulation of a NASA mission to Mars. Using found objects and available materials like plywood and Tyvek, the astronauts construct everything from their space suits to the mission control console to the ship itself. Even the film credits are hand lettered. Yet alongside the projects earnestness is a charmingly oddball sense of humor. The missions stated objective is to find love on Mars, but if not, to leave some love there. Supplies include whiskey, scented candles, and Al Green tapes.
For all A Space Programs whimsy, Sachss passionate attention to detail creates a deeply immersive atmosphere, fueled by a childlike sense of wonder and imagination.
...as a tribute to NASA, A Space Program is rich in the core elements that have always propelled humanitys flights of fancy: imagination and the right tools. -The New York Times


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SPCA Israel
Pet Boarding Services
All her life before her
At the age of nine, having been abandoned twice, Simba is still optimistic about finding a warm and loving home
Many dogs are not privileged to spend their lives with one family, are abandoned and, against their will, have to adapt to new realities again and again. For dogs abandoned more than once, life is extremely hard.
Simba is such a dog – a mixed golden retriever, nine years of age, who reached the Society for the first time at the age of one when the family who first raised her, abandoned her. Simba found a new home and lived there for eight years, until three months ago when her owners asked to put her in the Dog's Home for a short time. We are sorry to say, Simba's owners never came back for her and didn't respond to our calls or letters.
Simba is now looking for a new home once again, seemingly the fourth in her life. Simba is an easy-natured, friendly dog who gets along well with other dogs, people and, in particular, children. She is healthy, active, loves to play and go out for walks, and will bring lots of love to any family who want to make her part of their lives. The Society's team, who fell in love with her very quickly, are giving her tons of attention and spoiling her and we all hope that she will soon get what she truly deserves – a stable home and responsible family who will raise her with love.
Update: Four days after Simba's story was published, she was adopted by the Cohen family from Zichron Ya'akov.
Simba in her new home
It's Cold Outside
But inside it's cozy and warm. Help us equip our kennel with infra-red heating lamps before winter arrives in full force
DIY – Art Projects for Dogs and Cats
Do it Yourself: A video series of designer items for dogs and cats. A special project of SPCA Israel in collaboration with "Crafty Studio"
Neutering and castration at a young age
Neutering and castrating are the first operations that humans performed on domestic animals
Come and choose on Election Day
The Society will be open on Election Day between the hours of 8 am until 4 pm. Come and adopt
The Big Stars of 24/7: The Society's eight adorable puppies
Society for Prevention of Cruelty to Animals in Israel,
159 Herzl Street, Tel Aviv. Tel: *4553
Accessability policy Jobs
for the contribution from all of us!


Wind turbines in Scotland provided enough power to meet the average needs of 87 per cent of Scottish households last month, new data suggests.
The figures from WeatherEnergy show turbines produced 792,717MWh of electricity to the National Grid in October, up more than a quarter on last year.
However when viewed against total electricity consumption, including homes, business and industry, wind power generated 38 per cent of the 2,080,065MWh of electricity consumed in October.
Karen Robinson of WeatherEnergy said: "According to the Met Office, Scotland had the sunniest and one of the driest Octobers since records began.
"However, the month also witnessed some powerful winds leading to a significant increase in wind power output when compared to last year.
WWF Scotland director Lang Banks said: "Thanks to a combination of increased capacity and stronger winds, output from turbines surged by more than a quarter compared to the same period last year - supplying power equivalent to the electrical needs of over two million homes.


The Kemsley Courier keeps parents and carers up-to-date events that will happen in school and also shares and celebrates experiences and achievements the children have had. A list of dates and events are included in each letter.
Class newsletters share with parents and carers information about their children's up and coming curriculum content and explain what the children will be learning.
Other notes and letters inform parents and carers about trips, opportunities and events, and invite parents into school to share learning experiences with their children or to come along to groups and meetings.
We use our texting facility to remind parents and carers about events and to leave a quick message for a range of reasons. In the event that arrangements in school have to be altered or if the school has to close, the text message service will be used to inform parents of this. This service allows us to keep in regular contact with parents and carers. Please do let the school office know if your mobile phone number changes so we can ensure that we can keep in touch.
All class teachers have access to 'Marvellous Me'. This is an app that allows teachers to send you positive messages about what your child/ren has been up to. The messages are sent to your Marvellous Me account and you can enable notifications to your phone or other device. Our aim is to have 100% of children having at least one adult signed up to the MM app. If you have not yet joined the app, please do contact your child's class teacher who can let you have the unique PIN code that you need to join the app and link into your child's account.
Each half term, we hold a Parent Pow-wow which all parents and carers are warmly invited to attend. This is an open meeting with at least two members of the Senior Leadership Team. Each meeting will have a focus that the school sets with regard to something that we would really like to get parental views upon. In addition, there will be half an hour set aside for parents to raise questions and queries with the Senior Leadership Team. Minutes of the Pow-wows are published here (see download section).


Faculty who embrace Wi-Fi take advantage of the technology inside and outside of the classroom, says Matt Holmes of Johnson County Community College in Kansas.
While college IT administrators have made Wi-Fi ubiquitous on their campuses, the question remains: Do faculty members even want their students accessing their mobile devices during lectures? College IT staffers say they leave the decision up to the faculty member.
Discussions with faculty have gone on for years about the benefits of wireless in classrooms and whether to let students access the network during lectures, says Andrew Heiser, executive director of information services at Morningside College in Sioux City, Iowa.
In general, faculty members at Morningside are strong supporters of a reliable wireless network. With wireless, faculty are empowered to make their own choice and decide whether they want their students to have notebooks open during class or not, he says.
"The controversy should be curricular, not technological," Heiser says.
Similarly, at Bucknell University in Lewisburg, Pa., faculty have their own classroom policies in place on whether to allow students to access Wi-Fi devices in class. Some allow it, while others don't, says Param Bedi, Bucknell's vice president of library and information technology.
Faculty who embrace Wi-Fi take advantage of the technology inside and outside the classrooms, says Matt Holmes, director of data center operations at Johnson County Community College in Overland Park, Kan.
Automotive students use wireless diagnostic equipment, while dental hygiene students run more mobile devices in their classes. Many instructors also use new technologies, such as lecture capture, and make educational videos for students to watch outside of class, he says.
"As instructors use more technology tools to get information out, students want to get to that information, so they're bringing their own devices on campus to get access," Holmes explains.
At Cleveland State Community College in Cleveland, Tenn., Wi-Fi is widely accepted on campus, and faculty do not have concerns about students bringing their mobile devices into classrooms, says Andy Semak, manager of network and support services.
But the testing center on campus — which lets students take makeup tests and placement tests — requires students to check in their notebook computers and other electronic gadgets to ensure no cheating occurs during the test-taking, he says.


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This image, taken by NASA's Dawn spacecraft, shows the rim of Ikapati Crater on Ceres. The rough features around the rim of this impact scar give way to smoother, lightly cratered terrain in the lower half of the image.
This 3-D image shows the topography of craters and grooves of asteroid Vesta's south polar region obtained by the framing camera instrument aboard NASA's Dawn spacecraft on Aug. 23 and 28, 2011. Yo...
Anaglyph of Craters in the South Polar Region
Saturn's moon Tethys appears to float between two sets of rings in this view from NASA's Cassini spacecraft, but it's just a trick of geometry.
Ices and Shadows
From just below the plane of Saturn's rings, the Cassini spacecraft looks at the rings edge-on and sees the planet's second largest moon beyond. Although Rhea may appear to be in the foreground of...
Rhea and Edge-On Rings
The Cassini spacecraft awaits placement of its payload fairing at Launch Pad 40 at Cape Canaveral Air Station (CCAS) to protect Cassini during launch.
Weeks Before Launch
This close-up view of the inner A ring shows intriguing variations in brightness along the direction of ring motion -- from top to bottom. Close examination reveals dark regions that appear to wide...
Azimuthal Variability
Janus is spotted over Saturn's north pole in this image while Mimas' shadow glides across Saturn. Janus is the faint dot that appears just above Saturn's north pole. Mimas' shadow can be seen i...
Saturn Looms
Each Voyager spacecraft carries a copy of the Golden Record, which has been featured in several works of science fiction.
What's that bright point of light in the outer A ring? It's a star, bright enough to be visible through the ring!
Beyond the expanse of the rings sits Saturn's innermost large moon, Mimas. The rim of the large crater Herschel is visible as a flattening of the moon's leading side, at left. The view looks towar...
Herschel on the Edge
The terminator between light and dark throws Rhea's cratered surface into stark relief while the southern hemisphere is scored by bright icy cliffs. North on Rhea is up and rotated 42 degrees to t...
Crater Terminator
NASA's Galileo spacecraft acquired its highest resolution images of Jupiter's moon Io on 3 July 1999 during its closest pass to Io since orbit insertion in late 1995.
Global Image of Io (True Color)
The moon Enceladus passes behind the larger moon Tethys, as seen in this pair of Cassini spacecraft images. The image on the left was taken a little more than a minute before the image on the righ...
Enceladus Behind Tethys
The Cassini spacecraft, using its radar system, has discovered very strong evidence for hydrocarbon lakes on Titan. Dark patches, which resemble terrestrial lakes, seem to be sprinkled all over th...
Lakes on Titan
The exquisitely formed shadow of the moon Mimas graces Saturn's A ring in this Cassini portrait. Saturn's moons cast shadows onto the rings as the planet approaches its August 2009 equinox. To lea...
Moon Shadow Adornment
Voyager 2 obtained this high-resolution picture of Saturn's rings on August 22, 1981 when the spacecraft was 4 million kilometers (2.5 million miles) away. Evident here are the numerous "spoke" fea...
Ring Spokes
Revealing Titan's Surface October 26, 2004 Full-Res: PIA06125 These three pictures were created from a sequence of images acquired by...
Revealing Titan's Surface
+ View Full Res This raw, unprocessed image of Saturn's moon Enceladus was taken on Oct. 19, 2011. As the spacecraft passed Enceladus, its infrared instruments, cameras and other instruments moni...
Enceladus Flyby E-15 (Raw Image #4)
Sometimes at Saturn you can see things almost as if from every angle at once, the way a Cubist might imagine things
Cubist Saturn
Deep craters riddle the pulverized, icy surface of Saturn's moon Mimas. This view looks toward southern latitudes on Mimas from a vantage point 47 degrees below the moon's equator. North is toward...
Blasted by Impacts
This image from NASA's Dawn mission shows the varied minerals on the surface of the giant asteroid Vesta in false color.
Colorized Infrared View of Vesta
Saturn's small moon Pan casts a long shadow across the A ring in this image captured by the Cassini spacecraft a few days after the planet's August 2009 equinox. Pan (28 kilometers, or 17 miles ac...
Pan's Shadow
NASA's Cassini imaging scientists processed this view of Saturn's moon Hyperion, taken during a close flyby on May 31, 2015.
Spongy Surface
This image, taken by NASA's Dawn spacecraft, shows dwarf planet Ceres from an altitude of 2,700 miles (4,400 kilometers). The image, with a resolution of 1,400 feet (410 meters) per pixel, was take...
Dawn Survey Orbit Image 27
NASA's Dawn spacecraft views Abellio Crater (20 miles, 32 kilometers wide) on Ceres in this image. Dawn took this image on June 2, 2016, from its low-altitude mapping orbit, at a distance of about ...
Portrait of the prime crew of the Apollo 12 lunar landing mission.
Apollo 12 Crew