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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-11-10 17:32:50 +01:00
parent 334b7613c5
commit 1ffe028306
3 changed files with 11 additions and 11 deletions
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10
sec/part1/fermions.tex
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@@ -2039,7 +2039,7 @@ Using the algebra~\eqref{eq:ns-algebra} we compute the \ope of fermion fields as
\qquad
\abs{w} < \abs{z},
\end{equation}
where the operation $\no{\cdot}$ is the normal ordering with respect to the \SL{2}{R} vacuum defined in~\eqref{eq:NS_SL2_vacuum}.
where the operation $\no{\cdot}$ is the normal ordering with respect to the \SL{2}{\R} vacuum defined in~\eqref{eq:NS_SL2_vacuum}.
We then get the expression of the stress-energy tensor:
\begin{equation}
\begin{split}
@@ -2189,7 +2189,7 @@ From the usual definition of the stress-energy tensor in terms of the Virasoro g
\end{split}
\end{equation}
We already hinted to the fact that the vacua state involved are not in general \SL{2}{R} invariant.
We already hinted to the fact that the vacua state involved are not in general \SL{2}{\R} invariant.
In particular we can see that that the excited vacua \eexcvacket is a primary field
\begin{equation}
\begin{split}
@@ -2394,7 +2394,7 @@ The last expression shows that the energy momentum tensor $\cT( z )$ is radial t
First of all we notice that the vacuum $\Gexcvacket$ is actually $\GGexcvacket$, i.e.\ it depends only on $x_{(t)}$ and $\rE_{(t)}$.
We can try to interpret the previous result in the light of the usual \cft approach.
In particular we can refine the idea we discussed after~\eqref{eq:asymp_beha_Psi_on_exc_vac} that the singularity in the modes~\eqref{eq:generic-case-basis} and~\eqref{eq:generic-case-basis-conjugate} at the point $x_{(t)}$ is associated with a primary conformal operator which creates \eexcvacket with $\rE = \rE_{(t)}$.
By comparison with the stress energy tensor of an excited vacuum~\eqref{eq:T_excited_vacuum}, from the second order singularity we learn that at the points $x_{(t)}$ there is an operator which creates the excited vacuum \GGexcvacket from the \SL{2}{R} vacuum \regvacuum.
By comparison with the stress energy tensor of an excited vacuum~\eqref{eq:T_excited_vacuum}, from the second order singularity we learn that at the points $x_{(t)}$ there is an operator which creates the excited vacuum \GGexcvacket from the \SL{2}{\R} vacuum \regvacuum.
Given the discussion in the previous section this is an excited spin field $\rS_{\rE_{(t)}}\qty( x_{(t)}) = e^{i \rE_{(t)} \phi( x_{(t)} )}$.
The first order singularities in $x_{(u)} - x_{(t)}$ are then the result of the interaction between two of the previous excited spin fields.
Using the \cft operator approach we postulate that the following identification holds
@@ -2818,7 +2818,7 @@ Therefore we have
\end{equation}
which can be solved by
\begin{equation}
\ln \left\langle
\left\langle
\rR\qty[%
\rS_{\rE_{(t)}}\qty(x_{(t)})
\prod\limits_{\substack{u = 1 \\ u \neq t}}^N
@@ -2828,7 +2828,7 @@ which can be solved by
=
\cN_0
\qty( \qty{\rE_{(t)}} )
\prod\limits_{\substack{t = 1}{t > u}}^N
\prod\limits_{\substack{t = 1 \\ t > u}}^N
\qty( x_{(u)} - x_{(t)} )^{\rE_{(u)} \rE_{(t)}}.
\end{equation}
The constant $\cN_0\qty( \qty{\rE_{(t)}} )$ which depends on the $\rE_{(t)}$ only can be fixed by using the \ope