Progress on fermions with defects

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>

sec/app/parameters.tex

@@ -76,7 +76,7 @@ This would then imply
 We can finally show in details the computation of the parameters of the basis of hypergeometric functions used in the main text.
 The relation between these and the \SU{2} matrices can be computed requiring that the monodromies induced by the choice of the parameters equal the monodromies produced by the rotations of the D-branes.
 
-The monodromy in $\omega_{\bt-1} = 0$ is simpler to compute given that we choose $\cL(\vb{n}_{\vb{0}})$ and $\cR(\widetilde{\vb{m}}_{\vb{0}})$ to be diagonal.
+The monodromy in $\omega_{\bart-1} = 0$ is simpler to compute given that we choose $\cL(\vb{n}_{\vb{0}})$ and $\cR(\widetilde{\vb{m}}_{\vb{0}})$ to be diagonal.
 We impose:
 \begin{eqnarray}
   \mqty( \dmat{1, e^{-2\pi i c^{(L)}}} )
@@ -163,7 +163,7 @@ We therefore have:
   k_{a b}\in \Z.
   \label{eq:aL-bL}
 \end{equation}
-The allowed values for $k_{\delta^{(L)}_{\vb{\infty}}}$ follow a construction similar to the monodromy around $\omega_{\bt-1} = 0$.
+The allowed values for $k_{\delta^{(L)}_{\vb{\infty}}}$ follow a construction similar to the monodromy around $\omega_{\bart-1} = 0$.
 The main difference is given by the fact that $\frac{1}{2}(a^{(L)} + b^{(L)})$  may a priori take values in an interval of width $1$.
 As in the previous case we have $\alpha \le \delta_{\vb{\infty}}^{(L)} \le \alpha + \frac{1}{2}$ with $\alpha$ technically arbitrary.
 We cannot thus choose a vanishing $k_{\delta^{(L)}_{\vb{\infty}}}$ but we have to consider $k_{\delta^{(L)}_{\infty}} = 0,\, 1$.
@@ -205,7 +205,7 @@ where
   \frac{n_{\vb{\infty}}^3}{n_{\vb{\infty}}}.
 \label{eq:cos_n1}
 \end{equation}
-This expression is connected with rotation parameter in the third interaction point $\omega_{\bt+1} = 1$.
+This expression is connected with rotation parameter in the third interaction point $\omega_{\bart+1} = 1$.
 In fact $\cos(2\pi \cA^{(L)}) = \cos(2\pi {n}_{\vb{1}})$.
 We then write
 \begin{equation}

sec/app/reflection.tex

@@ -0,0 +1,95 @@
+We provide details on how~\eqref{eq:reflection condition_out_field_generic_vacuum} can be computed.
+First we introduce the projector of positive frequency and negative frequency modes for the NS fermion as
+\begin{eqnarray}
+  P^{(+,\, 0)}(z,w)
+  & = &
+  \frac{+1}{z-w},
+  \qquad
+  \abs{z} > \abs{w}
+  \\
+  P^{(-,\, 0)}(z,w)
+  & = &
+  \frac{-1}{z-w},
+  \qquad
+  \abs{z} < \abs{w},
+\end{eqnarray}
+such that
+\begin{equation}
+  \oint\limits_{\abs{z} > \abs{w}} \ddw
+  P^{(+,\, 0)}(z,w)\,
+  \Psi^{(0)}( 0 )
+  =
+  \Psi^{(0,\, +)}( z ),
+\end{equation}
+and similarly for the negative frequency modes.
+Likewise we introduce the projectors for the field with defects as
+\begin{eqnarray}
+  P^{(+)}(z,\, w)
+  & = &
+  \frac{P\qty(z;\, \qty{x_{(t)},\, \rE_{(t)}} )\,
+        P\qty(w;\, \qty{x_{(t)},\, -\rE_{(t)}} )
+       }{z-w},
+  \qquad
+  \abs{z} > \abs{w}
+  \\
+  P^{(-)}(z,\, w)
+  & = &
+  -
+  \frac{P\qty(z;\, \qty{x_{(t)},\, \rE_{(t)}} )\,
+        P\qty(w;\, \qty{x_{(t)}, -\rE_{(t)}} )
+       }{z-w},
+  \qquad
+  \abs{z} < \abs{w},
+\end{eqnarray}
+with $P\qty(z;\, \qty{x_{(t)},\, \rE_{(t)}} ) = \finiteprod{t}{1}{N} \qty( 1- \frac{z}{x_{(t)}} )^{\rE_{(t)}}$ as in the main text.
+    
+We then compute
+\begin{equation}
+  \begin{split}
+    \qty(P^{(+)}\, P^{(+,\,0)})(z,\, w)
+    & =
+    \oint\limits_{\abs{z} > \abs{\zeta} > \abs{w}} \ddz
+    P^{(+)}(z,\, \zeta)\,
+    P^{(+,\, 0)}(\zeta,\, w)
+    =
+    P^{(+,\, 0)}(z,\, w)
+    \\
+    \qty(P^{(+)}\, P^{(-,\, 0)})(z,\, w)
+    & =
+    \frac{P\qty(z;\, \qty{x_{(t)},\, \rE_{(t)}} )\,
+          P\qty(w;\, \qty{x_{(t)},\, -\rE_{(t)}} ) -1
+         }{z-w}.
+  \end{split}
+\end{equation}
+The last equation is valid when $\rM=\finitesum{t}{1}{N} \rE_{(t)} \le 0$ and for $\abs{z}$ and $\abs{w}$ arbitrary.
+Specializing the previous expressions to $\Psi^{(out)}( z )$, we need to constrain $\abs{z} > x_{(1)}$ and $\abs{w} > x_{(1)}$.
+
+Finally the vacuum in presence of defects can be  described by
+\begin{equation}
+  \begin{split}
+    \Psi^{(+)}( z ) \Gexcvacket
+    & =
+    \qty(P^{(+)}\, \Psi)( z ) \Gexcvacket
+    \\
+    & =
+    \qty(P^{(+)}\, \Psi^{(out)})( z ) \Gexcvacket
+    \\
+    & =
+    \left\lbrace
+      \qty(P^{(+)}\, P^{(+,\, 0)}\, \Psi^{(out)})( z )
+    \right.
+    \\
+    & +
+    \left.
+      \qty(P^{(+)}\, P^{(-,\, 0)}\, \Psi^{(out)})( z )
+    \right\rbrace
+    \Gexcvacket
+    \\
+    & =
+    0,
+  \end{split}
+\end{equation}
+where we assumed $\abs{z} > x_{(1)}$.
+The expression finally becomes~\eqref{eq:reflection condition_out_field_generic_vacuum}.
+
+% vim: ft=tex