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