We continue to develop the holographic interpretation of classical
conformal blocks in terms of particles propagating in an asymptotically
AdS
Article funded by SCOAP3
0$ for a conical defect, $\alpha^2<0$ for the BTZ black hole), while the light dimensions are identified with AdS masses via the standard formula for scalar fields, $m^2 = \Delta(\Delta-4)/R_{\rm AdS}$. Note that in this case the lightness parameter $\delta$ introduced in~\eqref{lightness} measures a backreaction of the background on a probe. Using symmetry arguments three-dimensional bulk analysis can be consistently reduced to a constant time slice identified with a two-dimensional disk. The corresponding bulk configuration of the worldlines is shown in figure~\bref{bulk}. To draw the bulk worldline graph in figure~\bref{bulk} associated to a given boundary diagram in figure~\bref{block} we have a simple mnemonic rule: the boundary diagram is to be pasted into the disc in such a way that outer ends of solid lines are attached to distinguished points on the boundary circle, while bold lines are collapsed into the origin of coordinates with one intermediate wavy line attached. \begin{figure}[t] \centering \vspace*{-25pt} \begin{tikzpicture}[line width=1pt] \draw (0,0) circle (3cm); \foreach \a in {1,2,...,40}{ \draw (\a*360/40: 3.5cm) coordinate(N\a){}; \draw (\a*360/40+5: 3.5cm) coordinate(D\a){}; \draw (\a*360/40: 4cm) coordinate(A\a){}; \draw (\a*360/40: 3cm) coordinate(K\a){}; \draw (\a*360/40: 2.5cm) coordinate(F\a){}; \draw (\a*360/40: 2cm) coordinate(L\a){}; \draw (\a*360/40: 1.5cm) coordinate(I\a){}; \draw (\a*360/40: 1.1cm) coordinate(J\a){}; \draw (\a*360/40: 1cm) coordinate(M\a){}; \draw (\a*360/40: 0.8cm) coordinate(C\a){}; ;} \draw plot [smooth, tension=1.0, line width=1pt] coordinates {(K35) (L34) (I30)}; \draw plot [smooth, tension=1.0, line width=1pt] coordinates {(K27) (L28) (I30)}; \draw plot [smooth, tension=1.0, line width=1pt] coordinates {(K23) (L23) (M25)}; \draw plot [smooth, tension=1.0, line width=1pt] coordinates {(K20) (L19) (M19)}; \draw plot [smooth, tension=1.0, line width=1pt] coordinates {(K16) (L14) (M12)}; \draw [smooth, tension=1.0, line width=1pt, decorate, decoration = {snake, segment length = 2mm, amplitude=0.4mm}] (M12) -- (0,0); \draw [smooth, tension=1.0, line width=1pt, decorate, decoration = {snake, segment length = 2mm, amplitude=0.4mm}] (I30) -- (J29) -- (M25); \draw [smooth, tension=1.0, line width=1pt, decorate, decoration = {snake, segment length = 2mm, amplitude=0.4mm}] (M12) -- (J16) -- (M19); \draw [smooth, tension=1.0, line width=1pt, decorate, decoration = {snake, segment length = 2mm, amplitude=0.4mm}] (J19) -- (M25); \fill (I30) circle (0.8mm); \fill (M25) circle (0.8mm); \fill (K35) circle (0.8mm); \fill (K27) circle (0.8mm); \fill (K23) circle (0.8mm); \fill (0,0) circle (0.8mm); \fill (M19) circle (0.8mm); \fill (M12) circle (0.8mm); \fill (K20) circle (0.8mm); \fill (K16) circle (0.8mm); \draw (N27) node {$w_2, \epsilon_2$}; \draw (N35) node {$w_1, \epsilon_1$}; \draw (A16) node {$w_{n-2}, \epsilon_{n-2}$}; \draw (D19) node {{\bf.}}; \draw (N19) node {{\bf.}}; \draw (D20) node {{\bf.}}; \draw (N20) node {{\bf.}}; \draw (D21) node {{\bf.}}; \draw (N21) node {{\bf.}}; \draw (D22) node {{\bf.}}; \draw (N22) node {{\bf.}}; \draw (D23) node {{\bf.}}; \draw (N23) node {{\bf.}}; \end{tikzpicture} \vspace*{-20pt} ]]>
0$). In section~\bref{sec:5line} we explicitly formulate the system of equations underlying the respective five-line graph in the bulk. Section~\bref{sec:deformation} discusses the perturbation method that treats the $5$-pt case as a deformation of the $4$-pt case. In section~\bref{sec:pert} we find an exact formula for the corresponding worldline action and compare it with the boundary results. In the last section~\bref{sec:multi} we propose a multi-line generalization of the approach supported in the 5-line case. Section~\bref{sec:conclusion} contains our conclusions and outlooks. ]]>
0$ will be associated with the background metric generated by the heavy fields, in the first order in $\delta$ we find the expansion \par\vskip-\baselineskip {\small\begin{align}\label{5ptdec1} f_{\delta}(q_1,q_2|t) &= \bigg[\!-\frac{\epsilon_3 q_1}{2} \!-\!\frac{\epsilon_3 q_2}{2}\bigg] t \!+\!\bigg[\!-\frac{\epsilon_3 q_1^2}{8} \!-\!\frac{\epsilon_3^2 q_1^2}{16 \tilde{\epsilon}_1} \!+\!\frac{\epsilon_3 q_1 q_2}{4} \!-\!\frac{\tilde{\epsilon}_1 q_1 q_2}{2} \!-\!\frac{7 \epsilon_3 q_2^2}{24} \!+\!\frac{\epsilon_3 q_2^2 \alpha^2}{6} \!-\!\frac{\epsilon_3^2 q_2^2 \alpha^2}{16 \tilde{\epsilon}_1}\bigg] t^2 \notag\\ &\quad +\!\bigg[\!-\frac{\epsilon_3 q_1^3}{24} \!-\!\frac{\epsilon_3^2 q_1^3}{16 \tilde{\epsilon}_1} \!-\!\frac{\epsilon_3 q_1^2 q_2}{8} \!+\!\frac{\epsilon_3^2 q_1^2 q_2}{16 \tilde{\epsilon}_1} \!+\!\frac{\epsilon_3 q_1 q_2^2}{24} \!-\!\frac{5 \epsilon_3 q_2^3}{24} \!-\!\frac{\epsilon_3 q_1 q_2^2 \alpha^2}{6} \!+\!\frac{\epsilon_3^2 q_1 q_2^2 \alpha^2}{16 \tilde{\epsilon}_1} \notag\\ &\quad \hphantom{+\bigg[} \!+\!\frac{\epsilon_3 q_2^3 \alpha^2}{6} \!-\!\frac{\epsilon_3^2 q_2^3 \alpha^2}{16 \tilde{\epsilon}_1}\bigg] t^3 +\cO(t^4) \,. \end{align}} The bulk computation procedure discussed in the next sections allows to reconstruct the classical conformal block using a series expansion around some exact seed solution to the bulk equations. In section~\bref{sec:deformation} we will explain this procedure taking as a seed solution the five-point classical block with the unity operator insertion $\Phi_3 = \mathbb{I}$. This means that the deformation parameter is identified with the conformal dimension $\epsilon_3$ and in the limit $\epsilon_3 = 0$ the 5-pt block goes to the $4$-pt one. Hence, to compare with the results of the bulk computation we will need an expansion of the 5-pt classical conformal block with respect to the parameter $\epsilon_3$, i.e., \be\label{pertu} f_{\delta}(q_1,q_2|t) = f^{(0)}_{\delta}(q_1,q_2|t) +\epsilon_3 f^{(1)}_{\delta}(q_1,q_2|t) +\epsilon_3^2 f^{(2)}_{\delta}(q_1,q_2|t) +\ldots \,. \ee Here, the leading term $f^{(0)}_{\delta}(q_1,q_2|1)$ is identified with the 4-pt classical conformal block, while the sub-leading terms perturbatively reconstruct the 5-pt classical conformal block. The explicit form of the first two terms read off from~\eqref{5ptdec1} is \begin{align} f^{(0)}_{\delta}(q_1,q_2|t) &= -\frac{1}{2} \tilde{\epsilon}_1 q_1 q_2 t^2 +\frac{1}{48} \big(-4 \epsilon_1 q_1^2 q_2^2 -10 \tilde{\epsilon}_1 q_1^2 q_2^2 +4 \epsilon_1 q_1^2 q_2^2 \alpha^2 +\tilde{\epsilon}_1 q_1^2 q_2^2 \alpha^2\big) t^4 \notag\\ &\quad +\frac{1}{48} \big(-4 \epsilon_1 q_1^3 q_2^3 -6 \tilde{\epsilon}_1 q_1^3 q_2^3 +4 \epsilon_1 q_1^3 q_2^3 \alpha^2 +\tilde{\epsilon}_1 q_1^3 q_2^3 \alpha^2\big) t^6 +\cO(t^8) \,, ~~ \end{align} and \begin{align} f^{(1)}_{\delta}(q_1,q_2|t) &= -\frac{1}{2}(q_1+ q_2) t +\frac{1}{24} \big(-3 q_1^2 +6 q_1 q_2 -7 q_2^2 +4 q_2^2 \alpha^2\big) t^2 \notag\\ &\quad +\frac{1}{24} \big(-q_1^3 -3 q_1^2 q_2 +q_1 q_2^2 -5 q_2^3 -4 q_1 q_2^2 \alpha^2 +4 q_2^3 \alpha^2\big) t^3 +\cO(t^4) \,. ~~ \end{align} As expected, function $f^{(0)}_{\delta}(q_1,q_2|t)$ depends on the combination $q_1q_2$ only which is identified with the coordinate $z_2$, cf.~\eqref{zq5}. Setting $t=1$ one can recognize in the above series expression the following functions \be\label{finblock1} f^{(0)}_{\delta}(q_1,q_2) = 2 \epsilon_1 \ln\bigg[\!-\frac{2 \sinh\big[\frac{\alpha \ln[1 -q_1 q_2]}{2}\big]}{\alpha q_1 q_2}\bigg] -\tilde{\epsilon}_1 \ln \bigg[\!-\frac{4 \tanh\big[\frac{\alpha \ln[1-q_1 q_2]}{4}\big]}{\alpha q_1 q_2}\bigg] +\epsilon_1 \ln[1-q_1 q_2] \,, \ee and \be\label{finblock2} f^{(1)}_{\delta}(q_1,q_2) = \ln \bigg[\frac{\sinh\big[\frac{\alpha (\ln[1-q_1 q_2] -2 \ln[1-q_2])}{2}\big]}{\alpha q_2}\bigg] +\ln[1-q_2] \,. \ee After a particular conformal transformation of the coordinates these are the expansion coefficients that are seen on the bulk side in section~\bref{sec:pert}. A few comments are in order. Firstly, the choice of conformal dimensions~\eqref{PPPP} is not assumed to be clear and requires some explanation. As we already discussed, the bulk computations corresponding to our boundary configuration (discussed in the next sections) rely on the specially developed perturbation procedure around known exactly seed solution. So, in order to find explicitly general 5-point conformal blocks we have to fix a number of perturbation parameters. In our case we consider one possible choice corresponding to the value $\epsilon_3=0$~\eqref{pertu}. On the general physical grounds (namely, taking into account fusion rules) in this case we forced to fix $\tilde{\epsilon}_1=\tilde{\epsilon}_2$. This explains our choice~\eqref{PPPP}. More general conformal block within the framework of this perturbation procedure can be reconstructed order by order with respect to each perturbation parameter. Thus, to evaluate the conformal block with $\tilde{\epsilon}_1\neq\tilde{\epsilon}_2$ we have to introduce additional small parameter $\delta'=\tilde{\epsilon}_1-\tilde{\epsilon}_2$, and develop corresponding perturbation theory which is based on the same idea, and so represents not conceptual but technical difference. Secondly, we note that the AGT method used here allows to get only series expansion of the conformal block functions. So that, essentially what we perform is the check that the series expansion of the resulting expression obtained in the bulk computation gives exactly the coefficients of the classical conformal block calculated up to rather high order $\cO(q_1^mq_2^n)$. Thus, the bulk computation of section~\bref{sec:pert} allows to derive the exact result, while the boundary computation allows only to conjecture this expression and check the lower level coefficients comparing with the bulk. It would be interesting to actually derive this exact result from the boundary point of view. ]]>
0$ parameterizes an angle deficit. The physical singularity is placed at $\rho = 0$, where the Riemann tensor component $R_{\rho\phi\rho\phi}$ blows up. The conformal boundary corresponds to points $\rho = \pi/2$. To approach the boundary we use the regularization $\cos \rho = \Lambda^{-1}$ at $\Lambda \rightarrow \infty$. Particle propagation between initial and final positions is described by the worldline action \be\label{OPA} S = m \int_{\lambda^{'}}^{\lambda^{''}} d\lambda \,\sqrt{g_{\mu\nu}(x) \dot{x}^\mu\dot{x}^\nu} \equiv m \int_{\lambda^{'}}^{\lambda^{''}} d\lambda \,\sqrt{g_{tt} \dot{t}^2+g_{\phi\phi} \dot{\phi}^2+g_{\rho\rho} \dot{\rho}^2} \,, \ee where $m$ is the mass of a particle, $\lambda$ is the evolution parameter, and $\dot{x}^\mu = d x^{\mu}/d \lambda$. In general, ending points are parameters of the theory and the variation of the on-shell action reads \be\label{variation_bound} \delta S = p_\mu^{''}\delta x^{'' \mu} -p_\mu^{'}\delta x^{' \mu} , \ee where $p^{'}_\mu$ and $p^{''}_\mu$ are momenta in the initial and final positions, $\delta x^{' \mu}$ and $\delta x^{'' \mu}$ are respective coordinate variations. The corresponding Euler-Lagrange equations of motion are the geodesic equation provided that $\lambda$ is identified with a length of the path. In this case one arrives at the normalization condition \be\label{properpar} |\dot{x}| \equiv \sqrt{g_{\mu\nu}(x) \dot{x}^\mu\dot{x}^\nu} = 1 \,. \ee The bulk dynamics can be reduced to a constant time disk with polar coordinates $\rho$ and $\phi$. Indeed, $t$ and $\phi$ are cyclic coordinates resulting in the conservation laws, $\dot{p}_t = 0$ and $\dot{p}_\phi = 0$, where $p_t = g_{tt}\dot{t}$ and $p_\phi = g_{\phi\phi} \dot{\phi}$ are the corresponding momenta. Choosing a particular value $p_t = 0$ one arrives at $\dot{t} = 0$ $\rightarrow$ $t(\lambda) = \text{const}$. From now on we choose a constant time slice $t=0$. Taking into account the normalization condition~\eqref{properpar} integration constants can be identified with initial and final radial positions and the value of the conserved angular momentum $p_\phi$. Solving equations of motion explicitly is superfluous as we need just an action value evaluated on a given path. Using~\eqref{properpar} one finds that the action is a length of the path \be\label{actlambda} S = \int_{\lambda^{'}}^{\lambda^{''}} d \lambda = \lambda^{''} -\lambda^{'} . \ee A useful trick is that the normalization condition~\eqref{properpar} is sufficient to express a proper parameter $\lambda$ as a function of radius and angular momentum values~\cite{Hijano-ml-2015rla}. For $\dot{t} = 0$ the condition~\eqref{properpar} can be cast into the form \be\label{geod} \frac{1}{\cos^2\rho} \dot{\rho}^2 +\frac{p_{\phi}^2}{\alpha^2}\cot^2\rho = 1 \,, \ee from which it follows that the radial velocity is expressed as \be\label{rhorad} \dot \rho = \pm \cos\rho\, \sqrt{1-\frac{p_\phi^2}{\alpha^2}\cot^2 \rho} \,, \ee where the overall sign depends on the direction of the $\lambda$ flow. The minimal radial distance between the particle path and the singularity is therefore given by $\tan^2 \rho_{\rm min} = \big(\frac{p_\phi}{\alpha}\big)^2$. Obviously, the maximal radial distance corresponds to the point $\rho_{\rm max} = \pi/2$ located on the boundary. Changing variables as $y = \cot^2 \rho$ at $\dot \rho \geq 0$, and introducing notation \be\label{s} s = \frac{|p_\phi|}{\alpha} \,, \ee equation~\eqref{geod} can be directly integrated to yield the on-shell action \be\label{lambda} S = \ln \frac{\sqrt{\eta}}{\sqrt{1+\eta} +\sqrt{1-s^2 \eta}}\,\bigg|_{\eta^{'}}^{\eta^{''}} , \ee where $\eta^{'} = \cot^2 \rho^{'}$ and $\eta^{''} = \cot^2 \rho^{''}$ are initial/final radial positions. Parameter $s$ is an integration constant that defines a particular form of the geodesic segment. The simplest case of a geodesic segment is the radial line starting (or ending, depending on the $\lambda$ flow direction) at the singularity point $\rho_2 = 0$, see figure~\bref{line}. \begin{figure}[t] \centering \vspace*{-10pt} \begin{tikzpicture}[line width=1pt] \draw (0,0) circle (2.0cm); \foreach \a in {1,2,...,40}{ \draw (\a*360/40: 2.0cm) coordinate(K\a){}; \draw (\a*360/40: 1.4cm) coordinate(L\a){}; \draw (\a*360/40: 2.5cm) coordinate(M\a){}; \draw (\a*360/40: 1cm) coordinate(I\a){};} \draw plot [smooth, tension=1.0, line width=1pt] coordinates {(K34) (L33) (I30)}; \draw plot [smooth, tension=1.0, line width=1pt] coordinates {(K26) (L27) (I30)}; \draw [smooth, tension=1.0, line width=1pt, decorate, decoration = {snake, segment length = 2mm, amplitude=0.4mm}] (I30) -- (0,0); \fill (I30) circle (0.8mm); \fill (K34) circle (0.8mm); \fill (K26) circle (0.8mm); \fill (0,0) circle (0.8mm); \draw (M34) node {$0$}; \draw (M26) node {$w$}; \draw (0,0.4) node {$\rho_2$}; \draw (0,-1.4) node {$\rho_1$}; \end{tikzpicture} \vspace*{-8pt} ]]>
0$, and $\dot{\rho}_{(1)} > 0$ since the proper parameter $\lambda$ increases away from the vertex. \paragraph{Independent integration constants.} The conserved angular momenta play the role of integration constants. In our case, there are five angular momenta subjected to three independent constraints. Namely, as the radial line $\b$ has vanishing angular velocity and taking~\eqref{lin1} and~\eqref{lin2} into account one finds \be \tilde p_\phi^2 = 0 \,, \qquad \epsilon_3 p_\phi^3 +\tilde\epsilon_1 p_\phi^a = 0 \,, \qquad \tilde \epsilon_1 \tilde p_\phi^1 +\epsilon_1 p_\phi^1 +\epsilon_2 p_\phi^2 = 0 \,. \ee Using~\eqref{s} the above relations can be represented as follows \be\label{rels} \tilde s_2 = 0 \,, \qquad \epsilon_3 s_3 -\tilde \epsilon_1 \tilde s_1 = 0 \,, \qquad \epsilon_1 s_1 -\epsilon_2 s_2 -\tilde \epsilon_1 \tilde s_1 = 0 \,. \ee Note that quantities $s_I$ are non-negative and therefore the relative signs are fixed according to the slopes of worldlines on figure~\bref{5bulk}. On the other hand, equations~\eqref{vertcoord1} and~\eqref{vertcoord2} can be used to find $\tan\rho_1$ and $\tan \rho_2$ as functions of two independent angular momenta (integration constants). \paragraph{\boldmath Vertex $\a-\b -3$ radial position.} The consideration here is similar to that one for the $4$-pt block. From~\eqref{vertcoord1} we have \be\label{vertcoord11} \epsilon_3 \sqrt{1-s_3^2 \eta_2} +\tilde \epsilon_1 \sqrt{1-\tilde s_1^2 \eta_2} = \tilde \epsilon_2 \,, \ee where angular parameters $s_{3}$ and $\tilde s_1$ are defined according to~\eqref{s}, and \be\label{etabar} \eta_2 = \cot^2{\rho_2} \,. \ee Solving equation~\eqref{vertcoord11} for $\eta_2$ one finds \be\label{1tan} \eta_2 = -\frac{(\epsilon_3 -\tilde \epsilon_1 -\tilde \epsilon_2) (\epsilon_3 +\tilde \epsilon_1 -\tilde \epsilon_1) (\epsilon_3 -\tilde\epsilon_1 +\tilde\epsilon_2) (\epsilon_3 +\tilde\epsilon_1 +\tilde\epsilon_2)}{4\, \epsilon_3^2\, \tilde\epsilon_2^2\, s_3^2} \equiv \frac{\tau^2}{s_3^2} \,, \ee so that $\tau^2$ is a function of the conformal dimensions only \be\label{tau} \tau^2 = -\frac{(\epsilon_3 -\tilde\epsilon_1 -\tilde\epsilon_2) (\epsilon_3 +\tilde\epsilon_1 -\tilde\epsilon_2) (\epsilon_3 -\tilde\epsilon_1 +\tilde\epsilon_2) (\epsilon_3 +\tilde\epsilon_1 +\tilde\epsilon_2)}{4\, \epsilon_3^2\,\tilde\epsilon_2^2} \,. \ee Function $\tau^2$ can be represented as $\tau^2 = \mu^2/\epsilon_3^2$, where $\mu^2$ already appeared in the 4-pt case~\cite{Hijano-ml-2015rla}, \be\label{mu} \mu^2 = \frac{\epsilon_{3}^2 +\tilde\epsilon_1^2 -\tilde\epsilon_2^2/2}{2} -\frac{(\epsilon_3^2 -\tilde\epsilon_1^2)^2}{4\tilde\epsilon_2^2} \,. \ee \paragraph{\boldmath Vertex $\a -1-2$ radial position.} Equation~\eqref{vertcoord2} can be cast into the form \be\label{vertcoord21} \epsilon_1 \sqrt{1-s_1^2 \eta_1} +\epsilon_2 \sqrt{1-s_2^2 \eta_1} = \tilde\epsilon_1 \sqrt{1-\tilde s_1^2 \eta_1} \,, \ee where momenta $s_{1,2}$ and $\tilde s_1$ are defined according to~\eqref{s}, and $\eta_1$ is given by \be\label{etatilde} \eta_1 = \cot^2{\rho_1} \,. \ee The general solution to~\eqref{vertcoord21} reads \be\label{solution} \eta_1 = \frac{1-\sigma^2}{s_1^2+s_2^2 -2 s_1s_2 \sigma} \,, \qquad \sigma = \frac{\epsilon_1^2 +\epsilon_2^2 -\tilde\epsilon_1^2}{2\epsilon_1\epsilon_2} \,. \ee Equivalently, \be\label{cottilde} \eta_1 = \frac{(\epsilon_1 -\epsilon_2 -\tilde\epsilon_1) (\epsilon_1 +\epsilon_2 -\tilde\epsilon_1) (\epsilon_1 -\epsilon_2 +\tilde\epsilon_1) (\epsilon_1 +\epsilon_2 +\tilde\epsilon_1)}{4\epsilon_1 \epsilon_2 \big(s_1 s_2 (\epsilon_1^2 +\epsilon_2^2 -\tilde\epsilon_1^2) -\epsilon_1 \epsilon_2 (s_1^2 +s_2^2)\big)} \,, \ee cf.~\eqref{1tan}. \pagebreak It is worth noting that parameter functions $\tau = \tau(\epsilon, \tilde\epsilon)$~\eqref{tau} and $\sigma = \sigma(\epsilon, \tilde\epsilon)$~\eqref{solution} are homogeneous functions of the conformal dimensions. We note also that both $\eta_1$ and $\eta_2$ contain a classical ``fusion polynomial'' factor \be\label{fusion} \Pi(\epsilon_I, \epsilon_J, \epsilon_K) = (\epsilon_I -\epsilon_J -\epsilon_K) (\epsilon_I +\epsilon_J -\epsilon_K) (\epsilon_I -\epsilon_J +\epsilon_K) (\epsilon_I +\epsilon_J +\epsilon_K) \,, \ee where $I,J,K = 1,2,3,\a,\b$. There are two useful propositions. \begin{itemize} \item From $\Pi(\epsilon_I, \epsilon_J, \epsilon_K) \leq 0$ it follows that \be\label{lem1} \epsilon_I \leq \epsilon_J +\epsilon_K \,, \qquad I \neq J \neq K \,. \ee The proof is straightforward. In particular, it implies that $\eta_2 \geq 0$ and $\eta_1 \geq 0$. The first inequality is obvious from the definition~\eqref{1tan}, while to show the second one we recall that $\epsilon_1 s_1 \geq \epsilon_2 s_2$~\eqref{rels}. We note that if~\eqref{lem1} is not satisfied then the bulk vertex disappears. In the boundary description it exactly corresponds to the case where the fusion rules in the corresponding vertex of the pant decomposition are violated. It explains the notion of the classical fusion polynomial introduced above. \item In the limit $\tilde s_1 = 0$ the radial vertex coordinates are related as \be\label{lem2} \eta_1 = \eta_2 [\a \rightarrow 2,\b \rightarrow \a, 3 \rightarrow 1] \,. \ee Using the third relation in~\eqref{rels} the proof is straightforward. \end{itemize} ]]>