\subsection{Motivation}

As previously pointed out, the computation of quantities such as Yukawa couplings involves correlators of excited spin and twist fields.
After the analysis of the main contribution to amplitudes involving twist fields at the intersection of D-branes, we focus on the computation of correlators of (excited) spin fields.
This has been a research subject for many years until the formulation found in the seminal paper by Friedan, Martinec and Shenker~\cite{Friedan:1986:ConformalInvarianceSupersymmetry} based on bosonization.
In general the available techniques allow to compute only correlators involving Abelian configurations, that is configurations which can be factorized in sub-configurations having \U{1} symmetry.
Non Abelian cases have also been considered~\cite{Inoue:1987:NonAbelianOrbifolds,Inoue:1990:StringInteractionsNonAbelian,Gato:1990:VertexOperatorsNonAbelian,Pesando:2016:FullyStringyComputation}, though their mathematical formulation is by far more complicated.
  
Despite the existence of an efficient method based on bosonization~\cite{Friedan:1986:ConformalInvarianceSupersymmetry} and old methods based on the Reggeon vertex~\cite{Sciuto:1969:GeneralVertexFunction,DiVecchia:1990:VertexIncludingEmission,Nilsson:1990:GeneralNSRString,DiBartolomeo:1990:GeneralPropertiesVertices,Petersen:1989:CovariantSuperreggeonCalculus}, we take into examination the computation of spin field correlators and propose a new method to compute them.
We hope to be able to extend this approach to correlators involving twist fields and non Abelian spin and twist fields.
We would also like to investigate the reason of the non existence of an approach equivalent to bosonization for twist fields.
At the same time we are interested to explore what happens to a \cft in presence of defects.
It turns out that despite the defects it is still possible to define a radial time dependent stress-energy tensor which satisfies the canonical \ope
Moreover the boundary changing defects in the construction can be associated with excited spin fields enabling the computation of correlators involving excited spin fields without resorting to bosonization.



\subsection{Point-like Defect CFT: the Minkowskian Formulation}
\label{sec:Mink_theory}
  
Let $( \tau,\, \sigma ) \in \Sigma = ( -\infty,\, +\infty) \times \qty[ 0, \pi ]$ define a strip with Lorentzian metric and consider $N_f$ massless complex fermions $\psi^i$ such that $i = 1,\, 2,\, \dots,\, N_f$.
Their two-dimensional Minkowski action defined on the strip $\Sigma$ is:
\begin{equation}
  S
  =
  \frac{T}{2}
  \infinfint{\tau}
  \finiteint{\sigma}{0}{\pi}
  \qty(
    \frac{1}{2}\, \bpsi_i( \tau,\, \sigma )\,
    \qty( -i \gamma^{\alpha} \lripd{\alpha} )\,
    \psi^i( \tau,\, \sigma ) 
  ),
  \label{eq:cft-action_full}
\end{equation}
where the gamma matrices are
\begin{equation}
  \gamma^{\tau}
  =
  \mqty( & 1 \\ -1 & )
  =
  -\gamma_{\tau},
  \qquad
  \gamma^{\sigma}
  =
  \mqty( & 1 \\ 1 & )
  =
  \gamma_{\sigma},
\end{equation}
and the components of the massless fermions are
\begin{equation}
  \psi
  =
  \mqty( \psi_+ \\ \psi_- ),
  \qquad
  \bpsi
  =
  \psi^{\dagger}\, \gamma^{\tau}
  =
  \mqty( -\psi_-^* & \psi_+^* ).
\end{equation}
We then define the lightcone coordinates $\xi_{\pm} = \tau \pm \sigma$ such that $\ipd{\pm} = \frac{1}{2}\, \qty( \ipd{\tau} \pm \ipd{\sigma} )$.
In components the action reads:
\begin{equation}
  S
  =
  i \frac{T}{4}
  \infinfint{\xi_+}
  \infinfint{\xi_-}
  \qty(
    \psi^*_{-,\, i} \lripd{+} \psi^i_- + \psi^*_{+,\, i} \lripd{-} \psi^i_+
  ),
  \label{eq:cft-action}
\end{equation}
so the \eom are:
\begin{equation}
  \begin{split}
    \ipd{-} \psi_{+}^i( \xi_+, \xi_- )
    & =
    \ipd{+} \psi_{-}^i( \xi_+, \xi_- )
    =
    0,
    \\
    \ipd{-} \psi^*_{+,\, i}( \xi_+, \xi_- )
    & =
    \ipd{+} \psi^*_{-,\, i}( \xi_+, \xi_- )
    =
    0.
  \end{split}
  \label{eq:eom}
\end{equation}
Their solutions are the ``holomorphic'' functions $\psi_{+}^i(\xi_+)$ and $\psi_{-}^i(\xi_-)$ and their complex conjugates.\footnotemark{}
\footnotetext{%
  Notice that $\psi^*$ is indeed the complex conjugate of the field $\psi$, while it will no longer be the case in the Euclidean formalism.
}
\begin{figure}[tbp]
  \centering
  \import{tikz}{defects.pgf}
  \caption{Propagation of the string in the presence of the worldsheet defects.}
  \label{fig:point-like-defects}
\end{figure}
The boundary conditions are instead:
\begin{equation}
  \eval{
    \qty(
      \var{\psi}_{+,\, i}^* \psi_{+}^{ i} +
      \var{\psi}_{-,\, i}^* \psi_{-}^{ i} -
      \psi_{+,\, i}^* \var{\psi}_{+}^{ i} -
      \psi_{-,\, i}^* \var{\psi}_{-}^{ i}
    )
  }_{\sigma = 0}^{\sigma = \pi} = 0.
  \label{eq:boundary-conditions}
\end{equation}
We solve the constraint imposing the non trivial relations:
\begin{equation}
  \begin{cases}
    \psi_-^i( \tau, 0 )
    =
    \tensor{\qty( R_{(t)} )}{^i_j}
    \psi^j_+( \tau, 0 ),
    & \qquad
    \tau \in \qty( \htau_{(t)}, \htau_{(t-1)} ),
    \\
    \psi_-^i( \tau, \pi )
    =
    - \psi_+^i( \tau, \pi ),
    & \qquad
    \tau \in \R,
  \end{cases}
  \label{eq:boundary-conditions-solutions}
\end{equation}
where $t = 1, 2, \dots, N$.
This way we introduce $N$ zero-dimensional defects on the boundary, pictorially shown in~\Cref{fig:point-like-defects}.
They are located on the strip at $( \htau_{(t)}, 0 ) \in \Sigma$ such that $\htau_{(t)} < \htau_{(t-1)}$ with $\htau_{N+1} = -\infty$ and $\htau_0 = +\infty$.
Their characterisation is given by $N$ matrices $R_{(t)} \in \U{N_f}$.
    
In most of this paper we want the in- and out-vacua to be the usual NS vacuum.
We thus choose the boundary condition at $\sigma = \pi$ so that when there are no defects the system describes NS fermions.
We require also the cancellation of the action of the defects at $\htau = \pm\infty$, that is:
\begin{equation}
  R_{(N)} R_{(N-1)} \dots R_{(1)} = \1.
\end{equation}
More general cases where the asymptotic vacua are twisted can be worked out in similar fashion.
   
In order to connect to the Euclidean formulation we introduce $N_f$ ``double fields'' $\Psi^i$.\footnotemark{}
\footnotetext{%
  In this case they correspond to the fields $\psi^i_+$.
}
They can be obtained by gluing $\psi^i_+$ and $\psi^i_-$ along the $\sigma = \pi$ boundary and labeled by an index $i = 1,\, 2,\, \dots,\, N_f$:
\begin{equation}
  \Psi^i(\tau,\, \phi)
  =
  \begin{cases}
    \psi^i_+(\tau,\, \phi),
    & \qquad 0\le\phi\le \pi
    \\
    -\psi^i_-(\tau,\, 2\pi-\phi),
    & \qquad \pi \le \phi \le 2 \pi        
  \end{cases}
  \label{eq:double-field-Lorentzian}
\end{equation}
where $0 \le \phi \le 2 \pi$.
The boundary conditions become:
\begin{equation}
  \Psi^i(\tau, 2 \pi )
  =
  - \tensor{\qty( R_{(t)} )}{^i_j}
  \Psi^j(\tau,\, 0 ),
  \qquad
  \tau \in \qty( \htau_{(t)},\, \htau_{(t-1)} ).
\end{equation}
Using the equation of motion we get $\Psi^i(\tau,\, \phi) = \Psi^i(\tau + \phi)$ and the boundary conditions become the (pseudo-)periodicity conditions
\begin{equation}
  \Psi^i(\tau + 2 \pi )
  =
  - \tensor{\qty( R_{(t)} )}{^i_j}
  \Psi^j(\tau ),
  \qquad
  \tau \in \qty( \htau_{(t)}, \htau_{(t-1)} ).
\end{equation}
The main issue is now to expand $\Psi$ in a basis of modes and proceed to its quantization.
Even in the simplest case $N_f = 1$ the task of finding the Minkowskian modes turns out to be fairly complicated.
It is however possible to overcome the issue in the Euclidean formalism.


\subsection{Conserved Product and Charges}
\label{sec:product}

In order to promote the theory to its quantum formulation, we define a procedure to build a Fock space of states in the Heisenberg formalism.
Equal time anti-commutation relations must then be invariant in time.
We thus need a time independent internal product to extract the creation and annihilation operators and expand the fields on the basis of modes.


\subsubsection{Conserved Product and Current}

Start from a generic conserved current
\begin{equation}
  j( \tau,\, \sigma )
  =
  j_{\tau}( \tau,\, \sigma )\, \dd{\tau} + j_{\sigma}( \tau,\, \sigma )\, \dd{\sigma},
\end{equation}
and consider
\begin{equation}
  \star j
  =
  j_{\sigma} \dd{\tau} + j_{\tau} \dd{\sigma}
  \quad
  \Rightarrow
  \quad
  \dd{(\star j)}
  =
  \qty( \ipd{\tau} j_{\tau} - \ipd{\sigma} j_{\sigma} ) \dd{\tau} \dd{\sigma},
\end{equation}
where $\star$ is the Hodge dual operator.
Integration over the surface $\Sigma' = [ \tau_i, \tau_f ] \times [ 0, \pi ]$ yields:
\begin{equation}
  \int\limits_{\Sigma'} \dd{(\star j)}
  =
  \int\limits_{\pd \Sigma'}
  \star j
  =
  0
  \qquad
  \Leftrightarrow
  \qquad
  \finiteint{\sigma}{0}{\pi}
  \eval{j_{\tau}}_{\tau = \tau_f}^{\tau = \tau_i}
  =
  \finiteint{\tau}{\tau_i}{\tau_f}
  \eval{j_{\sigma}}_{\sigma = \pi}^{\sigma = 0}.
\end{equation}
The current $j_{\tau}( \tau,\, \sigma )$ is thus conserved in time if
\begin{equation}
  \finiteint{\tau}{\tau_i}{\tau_f}
  \qty( \eval{j_{\sigma}}_{\sigma = \pi} - \eval{j_{\sigma}}_{\sigma = 0} )
  =
  0.
  \label{eq:time-conservation}
\end{equation}
In this case we can define
\begin{equation}
  Q = \finiteint{\sigma}{0}{\pi} j_{\tau}( \tau,\, \sigma ) 
\end{equation}
as conserved quantity (that is $\ipd{\tau} Q = 0$).
We now consider explicitly the symmetries of the action~\eqref{eq:cft-action}.
We focus on diffeomorphism invariance and $\U{N_f}$ flavour symmetries of the bulk theory leading to the stress-energy tensor and a vector current.


\subsubsection{Flavour Vector Current}

Consider first the $\U{N_f}$ vector current of the flavour symmetry in~\eqref{eq:cft-action_full}.
We write it as
\begin{equation}
  j_{\alpha}^a ( \tau,\, \sigma )
  =
  \tensor{\qty( \rT^a )}{^i_j}\,
  \bpsi_i( \tau,\, \sigma )\, \gamma_{\alpha}\, \psi^j( \tau,\, \sigma ),
\end{equation}
where $\rT^a$ is a generator of $\U{N_f}$ such that $a = 1,\, 2,\, \dots,\, N_f^2$.\footnotemark{}
\footnotetext{%
  The results however are more general and hold for a generic matrix $M$ taking the place of any of the generators $\rT^a$.
  Spinors $\psi$ and $\bpsi$ can also be generalized to two different and arbitrary solutions of the \eom~\eqref{eq:eom} while keeping the current conserved.
}
In components we have:
\begin{eqnarray}
  j^a_{\tau}( \tau,\, \sigma )
  & = &
  \tensor{\qty( \rT^a )}{^i_j}\,
  \qty( \psi^*_{+,\, i} \psi^j_+ + \psi^*_{-,\, i} \psi^j_- )
  \\
  j^a_{\sigma}( \tau,\, \sigma )
  & = &
  \tensor{\qty( \rT^a )}{^i_j}\,
  \qty( \psi^*_{+,\, i} \psi^j_+ - \psi^*_{-,\, i} \psi^j_- ).
\end{eqnarray}

In order to define a conserved charge, we require:
\begin{equation}
  \finiteint{\tau}{\tau_i}{\tau_f}
  \qty(
    \eval{j_{\sigma}^a}_{\sigma = \pi} - \eval{j_{\sigma}^a}_{\sigma = 0}
  )
  =
  0,
\end{equation}
where
\begin{equation}
  \eval{j_{\sigma}^a( \tau,\, \sigma )}_{\sigma = \pi} \equiv 0
\end{equation}
using the boundary conditions~\eqref{eq:boundary-conditions}.
Moreover we have:
\begin{equation}
  \eval{j_{\sigma}^a( \tau,\, \sigma )}_{\sigma = 0}
  =
  \qty[
    \psi^*_+\,
    \qty( \rT^a - R_{(t)}^{\dagger} \rT^a R_{(t)} )\,
    \psi_+
  ]_{\sigma = 0},
  \qquad
  \tau \in \qty( \htau_{(t)}, \htau_{(t-1)} ).
\end{equation}
In general
\begin{equation}
  \eval{j_{\sigma}^a( \tau,\, \sigma )}_{\sigma = 0}
  =
  0
  \qquad
  \Leftrightarrow
  \quad
  \rT^a \propto \1
\end{equation}
so that $R_{(t)}^{\dagger} \rT^a = \rT^a R_{(t)}^{\dagger}$.
This shows that the presence of the point-like defects on the worldsheet breaks the $\U{N_f}$ symmetry down to a \U{1} phase.\footnotemark{}
\footnotetext{%
  The symmetry is $\SO{N_f} \times \SO{N_f}$ if we consider Majorana-Weyl fermions.
}
The \U{1} vector current then defines a conserved charge for a restricted class of functions.

Let $\alpha$ and $\beta$ be two arbitrary (bosonic) solutions to the \eom~\eqref{eq:eom}, we define a product
\begin{equation}
  \consprod{\alpha}{\beta}
  =
  \cN
  \finiteint{\sigma}{0}{\pi}
  \qty( \alpha_{+,\, i}^* \beta_+^i + \alpha_{-,\, i}^* \beta_-^i ),
  \label{eq:conserved-product}
\end{equation}
where $\cN \in \R$ is a normalisation constant and the integrand must not present non integrable singularities.
The product is such that $\consprod{\alpha}{\beta} = \consprod{\alpha}{\beta}^*$.
We can also rewrite the result to the double fields defined in~\eqref{eq:double-field-Lorentzian}.
Let in fact $A$ and $B$ be the ``double
fields'' corresponding to $\alpha$ and $\beta$ respectively:
\begin{equation}
  \consprod{\alpha}{\beta}
  =
  \cN
  \finiteint{\phi}{0}{2\pi}\,
  A_i^*( \tau + \phi )\, B^i( \tau + \phi ).
  \label{eq:conserved-product-double-field}
\end{equation}

\subsubsection{Stress-Energy Tensor}

Consider the stress-energy tensor of the bulk theory.
Using the usual Nöther's procedure we get the on-shell non vanishing components:
\begin{equation}
  \begin{split}
    \cT_{++}( \xi_+ )
    & =
    -i \frac{T}{4} \psi_{+,\, i}^*( \xi_+ ) \lripd{+} \psi_+^i( \xi_+ ),
    \\
    \cT_{--}( \xi_- )
    & =
    -i \frac{T}{4} \psi_{-,\, i}^*( \xi_- )\lripd{-} \psi_-^i( \xi_- ).
  \end{split}
  \label{eq:stress-energy-tensor-lightcone}
\end{equation}
As always the boundary of $\Sigma$ breaks the symmetry for translations in the $\sigma$ direction, while the defects break the time translations: the Hamiltonian is therefore time-dependent but it is constant between two consecutive point-like defects.
In fact, from the definition of the stress-energy tensor, we can in principle
build the hypothetical charges:
\begin{eqnarray}
  \rH( \tau )
  & = &
  \finiteint{\sigma}{0}{\pi}
  \cT_{\tau\tau}( \tau,\, \sigma )
  =
  \finiteint{\sigma}{0}{\pi}
  \qty( \cT_{++}( \tau + \sigma ) + \cT_{--}( \tau - \sigma ) ),
  \label{eq:hamiltonian}
  \\
  \rP( \tau )
  & = &
  \finiteint{\sigma}{0}{\pi}
  \cT_{\tau\sigma}( \tau,\, \sigma )
  =
  \finiteint{\sigma}{0}{\pi}
  \qty( \cT_{++}( \tau + \sigma ) - \cT_{--}( \tau - \sigma ) ),
  \label{eq:momentum}
\end{eqnarray}
which are conserved if~\eqref{eq:time-conservation} holds.

We order the point-like defects as $\htau_{(t_0 - 1)} < \tau_i \le \htau_{(t_0)} < \htau_{(t_N)} \le \tau_f < \htau_{(t_N+1)}$.
For the linear momentum $\rP$ the condition of conservation
reads:\footnotemark{}
\footnotetext{%
  Notice that in the second term of the second line the differentiation with respect to $\tau$ is acting only on $R_{(t)}$ and $R_{(t)}^{\dagger}$.
}
\begin{equation}
  \begin{split}
    & \finiteint{\tau}{\tau_i}{\tau_f}
    \eval{\qty( \cT_{++}( \tau + \sigma ) + \cT_{--}( \tau - \sigma ) )}_{\sigma = 0}^{\sigma = \pi}
    \\
    & =
    - i \frac{T}{4}
    \int \Delta\tau
    \qty(
      2 \eval{\psi_{+,\, i}^*\, \lripd{\tau} \psi_+^i}^{\sigma = \pi}_{\sigma = 0}
      -
      \eval{\psi_{+,\, i}^* \tensor{\qty( R_{(t)}^{\dagger} \lripd{\tau} R_{(t)} )}{^i_j} \psi_+^j}_{\sigma = 0}
    )
    \neq
    0.
  \end{split}
\end{equation}
The corresponding condition for the Hamiltonian $\rH$ is:
\begin{equation}
  \begin{split}
    & \finiteint{\tau}{\tau_i}{\tau_f}
    \eval{\qty( \cT_{++}( \tau + \sigma ) - \cT_{--}( \tau - \sigma ) )}_{\sigma = 0}^{\sigma = \pi}
    \\
    & =
    - i \frac{T}{4}
    \int \Delta\tau
    \qty( \eval{\psi_{+,\, i}^* \tensor{\qty( R_{(t)}^{\dagger} \lripd{\tau} R_{(t)} )}{^i_j} \psi_+^j}_{\sigma = 0} )
    = 0
    \quad
    \Leftrightarrow
    \quad
    \qty( \tau_i, \tau_f ) \in \qty( \htau_{(t)}, \htau_{(t-1)} ).
  \end{split}
\end{equation}
In both cases we used the shorthand graphical notation
\begin{equation}
    \int \Delta \tau
    =
    \qty(
      \int\limits_{\tau_i}^{\htau_{t_0}}
      +
      \finitesum{t}{t_0}{t_N - 1}
      \int\limits_{\htau_{(t)}}^{\htau_{(t+1)}}
      +
      \int\limits_{\htau_N}^{\tau_f}
    )
    \dd{\tau}
\end{equation}
for simplicity.

These relations therefore prove that the generator of the $\sigma$-translations~\eqref{eq:momentum} is not conserved in time because of the boundary conditions, while the time evolution operator $\rH$ is only piecewise conserved and therefore globally time dependent.


\subsection{Basis of Solutions and Dual Modes}

Let $\qty{ \psi_{n,\, \pm}^i }_{n \in \Z}$ be a complete basis of modes such that:
\begin{equation}
  \begin{cases}
    \psi_{n,\, +}^i( \tau, 0 ) = \qty( R_{(t)} )^i_j \psi_{n,\, -}^j(
    \tau, 0 ) & \qfor \tau \in \qty( \htau_{(t)}, \htau_{(t-1)} )
    \\
    \psi_{n,\, +}^i( \tau, \pi ) = -\psi_{n,\, -}^i( \tau, \pi )
    & \qfor \tau \in \R
  \end{cases}.
\end{equation}
These fields are related to a complete basis of the modes of the ``double field'' $\Psi_n^i$ as in~\eqref{eq:double-field-Lorentzian}.
The modes $\psi_n$ (and their counterparts $\Psi_n$) are a basis of solutions of the \eom and the boundary conditions for $\tau \in \R \setminus \qty{ \htau_{(t)} }_{0 \le t \le N}$.
The fields $\psi^i$ (and the fields $\Psi^i$) are then a superposition of such modes:
\begin{equation}
  \psi^i_{\pm}( \xi_{\pm} )
  =
  \infinfsum{n} b_n\, \psi^i_{n,\, \pm}( \xi_{\pm} )
  \qquad
  \Rightarrow
  \qquad
  \Psi^i( \xi )
  =
  \infinfsum{n} b_n\, \Psi^i_n( \xi ).
\label{eq:usual-mode-expansion}
\end{equation}

In order to extract the ``coefficients'' $b_n$ we first introduce the dual basis $\dual{\psi}_{n,\, \pm}$ (and $\dual{\Psi}_n$) in an abstract sense such that:
\begin{itemize}
  \item the dual fields $\dual{\psi}_{n,\, \pm}$ (and $\dual{\Psi}_n$) must be solutions to the \eom,

  \item the dual fields $\dual{\psi}_{n,\, \pm}$ (and $\dual{\Psi}_n$) can differ from $\psi_{n,\, \pm}$ (and $\Psi_n$) in their behavior at the boundary,

  \item the functional form of $\dual{\psi}_{n,\, \pm}$ (and $\dual{\Psi}_n$) is fixed by the request of time invariance of the usual anti-commutation relations $\liebraket{b_n}{b_m^{\dagger}}_+$ (that is $b_n$ and $b_n^{\dagger}$ can evolve in time, but their anti-commutation relations must remain constant).
\end{itemize}
We then define the conserved product for the ``double fields''~\eqref{eq:conserved-product-double-field} in such a way that:
\begin{equation}
  \eval{\lconsprod{\dual{\Psi}_n}{\Psi_m}}_{\tau = \tau_0}
  =
  \cN
  \finiteint{\sigma}{0}{2\pi}
  \dual{\Psi}_{n,\, i}^{*}(\tau + \sigma)\,
  \Psi_m^i( \tau + \sigma )
  =
  \delta_{n,\, m}.
  \label{eq:conserved-product-dual-basis}
\end{equation}
In the previous expression we changed the notation of the product.
We are in fact dealing with the space of solutions whose basis is $\qty{ \Psi_n }$ and a dual space with basis $\qty{ \dual{\Psi}_n }$ which is not required to span entirely the original space but only to be a subset of it in order to be able to compute the anti-commutation relations among the annihilation and construction operators in a well defined way as in~\eqref{eq:Mink_can_anticomm_rel_ann_des}.
Given the previous product we can extract the operators as
\begin{eqnarray}
  \lconsprod{\dual{\Psi}_n}{\Psi} & = & b_n,
  \\
  \lconsprod{\dual{\Psi}_n^*}{\Psi^*} & = & b_n^{\dagger}.
\end{eqnarray}
As a consequence of the canonical anti-commutation relations
\begin{equation}
  \qty[
    \Psi^i\qty( \tau,\, \sigma ), \Psi^*_j\qty( \tau,\, \sigma' )
  ]_+
  =
  \frac{2}{T}\, \tensor{\delta}{^i_j}\, \delta( \sigma - \sigma' ),
\end{equation}
we have then:
\begin{equation}
  \eval{\liebraket{b_n}{b_m^{\dagger}}_+}_{\tau = \tau_0}
  =
  \frac{2}{T} \cN \eval{\lconsprod{\dual{\Psi}_n}{\dual{\Psi}_m}}_{\tau = \tau_0}.
  \label{eq:Mink_can_anticomm_rel_ann_des}
\end{equation}
By definition the product~\eqref{eq:conserved-product-dual-basis} is time independent as long as the integrand $\dual{\Psi}_n^* \Psi_m$ is free of singularities at $\tau = \htau_{(t)}$ for $t = 1, 2, \dots, N$.
Such request on the dual basis automatically fixes its functional form.
Clearly this does not exclude the possibility to have singularities in $\Psi_m$ or $\dual{\Psi}_n$ separately: they are instead deeply connected to the boundary changing primary operator hidden in the discontinuity of the boundary conditions, that is different singularities will be shown to be in correspondence to the excited spin fields.
Using the definition of the conserved product we therefore moved the focus from finding a consistent definition of the Fock space to the construction of the dual basis of modes.
This task is however easier to address in a Euclidean formulation.


\subsection{Point-like Defect CFT: the Euclidean Formulation}
\label{sec:eucl_formulation}

In the Euclidean reformulation the solution to the \eom might be easier to study than its Lorentzian worldsheet form.
This is specifically the case when $R_{(t)} \in \U{1}^{N_f} \subset \U{N_f}$.
The presence of a time dependent Hamiltonian is not standard and we can neither blindly apply the usual Wick rotation nor the usual \cft techniques.


\subsubsection{Fields on the Strip}

Performing the Wick rotation as $\tau_E = i \tau$ such that $e^{i S} = e^{-S_E}$, the Minkowskian action~\eqref{eq:cft-action} becomes:
\begin{equation}
  S_E
  =
  \frac{T}{2}
  \iint \dd{\xi} \dd{\bxi}\,
  \frac{1}{2}\,
  \qty(
    \hpsi_{E,\, +,\, i}^*\, \lripd{\bxi} \hpsi_{E,\, +}^i
    +
    \hpsi_{E,\, -,\, i}^*\, \lripd{\xi} \hpsi_{E,\, -}^i
  ),
  \label{eq:S_Eu_strip}
\end{equation}
where the Euclidean fermion on the strip is connected to the Minkowskian formulation through
\begin{equation}
  \hpsi_{E,\, \pm}^i( \xi,\, \bxi )
  =
  \psi_{\pm}^i( -i\xi,\, -i\bxi ).
\end{equation}
In the previous expressions we defined the coordinates $\xi = \tau_E + i \sigma$, $\bar \xi = \tau_E - i \sigma$ such that $\bxi = \xi^*$.
Moreover we get $\ipd{\xi} = \pdv{\xi} = \frac{1}{2} \qty( \ipd{\tau_E} - i\, \ipd{\sigma} )$, $\ipd{\bxi} = \pdv{\bxi} = \frac{1}{2} \qty( \ipd{\tau_E} + i\, \ipd{\sigma} )$.
As a consequence the Euclidean ``complex conjugation'' $\star$ (defined off-shell) acts as
\begin{equation}
  \qty[
    \hpsi_{E,\, \pm}^i(\xi,\, \bxi)
  ]^\star
  =
  \hpsi_{E,\, \pm i}^*(-\bxi,\, -\xi).
  \label{eq:off-shell-Hermitian-conjugate}
\end{equation}

The \eom are as usual
\begin{eqnarray}
  \ipd{\xi} \hpsi_{E,\, -}^i( \xi,\, \bxi )
  = &
  \ipd{\bxi} \hpsi_{E,\, +}^i( \xi,\, \bxi )
  = &
  0,
  \\
  \ipd{\xi} \hpsi_{E,\, -,\, i}^*( \xi,\, \bxi )
  = &
  \ipd{\bxi} \hpsi_{E,\, +,\, i}^*( \xi,\, \bxi )
  = &
  0,
\end{eqnarray}
whose solutions are the holomorphic functions $\hpsi_{E,\, +}( \xi )$ and $\hpsi_{E,\, -}( \bxi )$, together with $\hpsi_{E,\, +}^*( \xi )$ and $\hpsi_{E,\, -}^*( \bxi )$.
In these coordinates the boundary conditions~\eqref{eq:boundary-conditions-solutions} translate to:
\begin{equation}
  \begin{cases}
    \hpsi_{E,\, -}^i( \tau_E - i\, 0^+ )
    & =
    \tensor{\qty( R_{(t)} )}{^i_j}\,
    \hpsi_{E,\, +}^j(\tau_E + i\, 0^+ )
    \\
    \hpsi_{E,\, -,\, i}^{*}( \tau_E - i\, 0^+ )
    & =
    \tensor{\qty( R_{(t)}^* )}{_i^j}\,
    \hpsi_{E,\, +,\, j}^*(\tau_E + i\, 0^+ )
  \end{cases}
  \label{eq:bc_eu_strip}
\end{equation}
for $\tau_E \in \qty( \htau_{E,\, (t)}, \htau_{E,\, (t-1)} )$ and
\begin{equation}
  \begin{cases}
    \hpsi_{E,\, -}^i( \tau_E - i\, \pi )
    & =
    -\hpsi_{E,\, +}^i( \tau_E + i\, \pi )
    \\
    \hpsi_{E,\, -,\, i}^*( \tau_E - i\, \pi )
    & =
    -\hpsi_{E,\, +,\, i}^*( \tau_E + i\, \pi )
  \end{cases},
\end{equation}
where $t = 1,\, 2,\, \dots,\, N$ and $\htau_{E,\, (t)}$ are the Wick-rotated locations of the $N$ zero-dimensional defects, analytically continued to a real value.

The conserved product on the strip becomes:
\begin{equation}
  \consprod{\halpha^*_E}{\hbeta_E}
  =
  \cN
  \finiteint{\sigma}{0}{\pi}
  \qty(
    \halpha^*_{E,\, +,\, i} \hbeta_{E,\, +}^i
    +
    \halpha^*_{E,\, -,\, i}  \hbeta_{E,\, -}^i
  ),
  \label{eq:euclidean-conserved-product-strip}
\end{equation}
where $\halpha^*_E$ and $\hbeta_E$ are the Euclidean counterparts of the generic solutions in the original definition of the product in~\eqref{eq:conserved-product}.
In the Euclidean context we have to explicitly write $\halpha^*_E$ because it is no longer the ``complex conjugate''  of $\halpha_E$ in the traditional sense.
The product is conserved only when it couples two solutions which have different boundary conditions as in~\eqref{eq:bc_eu_strip}.

The definition of the stress-energy tensor in~\eqref{eq:stress-energy-tensor-lightcone} requires a change in the numerical pre-factor to use the usual \cft normalisation.
Introducing a spacetime variable central charge as well the components of the stress-energy tensor become:\footnotemark{}
\footnotetext{%
  The canonical coefficient in front of the \cft stress-energy tensor is such that the Euclidean Hamiltonian $\rL_{0}$ is normalized such that
  \begin{equation}
    \cT_{\zeta\zeta}( \zeta ) = \sum_n \rL_{n} e^{-n \zeta}
  \end{equation}
  (we are anticipating the double strip notation defined in the next subsection for simplicity).
  We thus get:
  \begin{equation}
     \rH_E
     =
     \rL_{0}
     =
     \int\limits_{0}^{2\pi} \frac{\dd{\phi}}{2 \pi}
     \cT_{\zeta \zeta}( \tau_E + i\, \phi )
  \end{equation}
  therefore $\cT_{\zeta\zeta}( \zeta ) = 2 \pi\, \cT^{(can)}_{\zeta\zeta}( \zeta )$.
}
\begin{equation}
  \begin{split}
    \cT_{\xi \xi}( \xi )
    & =
    - \frac{\pi T}{2}\,
    \hpsi_{E,\, +,\, i}^*( \xi )\, \lripd{\xi} \hpsi^i_{E,\, +}( \xi )
    +
    \widehat{\cC}( \xi ),
    \\
    \cT_{\bxi \bxi}( \bxi )
    & =
    - \frac{\pi T}{2}\,
    \hpsi_{E,\, -,\, i}^*( \bxi ) \lripd{\bxi}
    \hpsi^i_{E,\, -}( \bxi )
    +
    \widehat{\bcC}( \bxi ),
  \end{split}
\end{equation}
where $\widehat{\cC}$ and $\widehat{\bcC}$ are the leftover terms after the regularization of the singularities due to the normal ordering.
The canonical anti-commutation relations are then
\begin{equation}
  \eval{
    \qty[
      \hpsi_{E,\, \pm}^i( \xi_1,\, \bxi_1),
      \hpsi_{E,\, \pm,\, j}^*( \xi_2,\, \bxi_2 )
    ]_+
  }_{\Re\xi_1 = \Re\xi_2}
  =
  \frac{2}{T}\,
  \tensor{\delta}{^i_j}\,
  \delta\qty( \Im\xi_1 - \Im\xi_2 ).
\end{equation}

Given the Euclidean modes $\hpsi^i_{E,\, \pm,\, n}$ and $\hpsi^*_{E,\, \pm,\, n,\, i}$ where $n \in \Z$, we can then define the dual modes $\dual{\hpsi}^i_{E,\, n}$ and $\dual{\hpsi}^*_{E,\, n,\, i}$ such that the conserved product~\eqref{eq:euclidean-conserved-product-strip} between them gives:
\begin{equation}
  \lconsprod{\dual{\hpsi}^*_{E,\, n}}{\hpsi_{E,\, m}}
  =
  \lconsprod{\dual{\hpsi}_{E,\, n}}{\hpsi^*_{E,\, m}}
  =
  \delta_{n,m}.
\end{equation}
We can then expand the fields as
\begin{equation}
  \begin{cases}
    \hpsi^i_{E,\, +}(\xi)
    & =
    \infinfsum{n} b_n\, \hpsi^i_{E,\, +,\, n}(\xi)
    \\
    \hpsi^i_{E,\, -}(\bxi)
    & =
    \infinfsum{n} b_n\, \hpsi^i_{E,\, -,\, n}(\bxi)
  \end{cases}
\end{equation}
and
\begin{equation}
  \begin{cases}
    \hpsi^*_{E,\, +,\, i}(\xi)
    & =
    \infinfsum{n} b^*_n\, \hpsi^*_{E,\, +,\, n,\, i}(\xi)
    \\
    \hpsi^*_{E,\, -,\, i}(\bxi)
    & =
    \infinfsum{n} b^*_n\, \hpsi^*_{E,\, -,\, n,\, i}(\bxi)
  \end{cases}
\end{equation}
in order to extract the operators through the conserved product
\begin{equation}
  b_n
  =
  \lconsprod{\dual{\hpsi}^*_{E,\, n}}{\hpsi_{E}},
  \qquad
  b^*_n
  =
  \lconsprod{\dual{\hpsi}_{E,\, n}}{\hpsi^*_{E}},
\end{equation}
and get the anti-commutation relations at fixed Euclidean time as
\begin{equation}
  \eval{ \liebraket{b_n}{b^*_m}_+ }_{\tau_E = \tau_{E,\, (0)}}
  =
  \frac{2 \cN}{T}
  \lconsprod{\dual{\hpsi}^*_{E,\, n}}{\dual{\hpsi}_{E,\, m}}.
\end{equation}


\subsubsection{Double Strip Formalism and Doubling Trick}

It is natural to use the doubling trick on the strip to simplify the previous expressions by gluing the holomorphic and anti-holomorphic fields along the $\sigma = \pi$ boundary.
Define the coordinate $\zeta = \tau_E + i\, \phi$ with $0 \le \phi \le 2\pi$.
We then have
\begin{equation}
  \hPsi( \zeta )
  =
  \begin{cases}
    \hpsi_{E,\, +}(\zeta)
    &
    \qfor
    \phi = \sigma \in \qty[ 0, \pi ]
    \\
    -\hpsi_{E,\, -}(\zeta - 2 \pi i)
    &
    \qfor
    \phi = 2\pi - \sigma \in \qty[ \pi, 2 \pi ]
  \end{cases}
\end{equation}
on-shell (and similarly for $\hPsi^*( \zeta )$ with the substitution $\hpsi_{E,\, \pm} \to \hpsi_{E,\, \pm}^*$).
The ``complex conjugation'' $\star$ acts on the off-shell double fields as
\begin{equation}
  \qty[ \hPsi^i(\zeta,\, \bzeta) ]^\star = \hPsi_i^*(-\bzeta,\, -\zeta),
\end{equation}
while the boundary conditions are translated into
\begin{equation}
  \begin{cases}
    \hPsi^i( \tau_E + 2 \pi i^- )
    =
    -\tensor{\qty( R_{(t)} )}{^i_j}\,
    \hPsi^j( \tau_E + i\, 0^+ )
    \\
    \hPsi^{* i}( \tau_E + 2 \pi i^- )
    =
    -\tensor{\qty( R_{(t)}^* )}{^i_j}\,
    \hPsi^{* j}( \tau_E + i\, 0^+ )
  \end{cases}
\end{equation}
for $\tau_E \in \qty( \htau_{E,\, (t)}, \htau_{E,\, (t-1)} )$.
The conserved product can then be defined as
\begin{equation}
  \consprod{\widehat{A}^*}{\widehat{B}}
  =
  \cN
  \finiteint{\phi}{0}{2\pi}\,
  \widehat{A}^*_i(\tau_E + i\, \phi )\,
  \widehat{B}^i( \tau_E + i\, \phi ),
\end{equation}
where $\widehat{A}^*$ and $\widehat{B}$ are the double fields connected to $\halpha^*_E$ and $\hbeta_E$ in the previous definition on the strip.
The holomorphic stress-energy tensor is then
\begin{equation}
  \cT_{\zeta \zeta}( \zeta )
  =
  - \frac{\pi T}{2}\,
  \hPsi_{i}^*( \zeta )\, \lripd{\zeta} \hPsi^i( \zeta ) 
  +
  \widehat{\cC}(\zeta)
\end{equation}
and the canonical anti-commutation relations are now
\begin{equation}
  \eval{
    \qty[
      \hPsi^i( \zeta_1 )
      ,
      \hPsi_{j}^*( \zeta_2 )
    ]_+
  }_{\Re\zeta_1 = \Re\zeta_2}
  =
  \frac{2}{T}\, \tensor{\delta}{^i_j} \delta\qty( \Im\zeta_1 - \Im\zeta_2 ).
\end{equation}


The double field formulation shows that we need only one coefficient $b_n$ (or $b_n^*$) for both $\psi_{E,\, +}$ and $\psi_{E,\, -}$ (or for both $\psi^*_{E,\, +}$ and $\psi^*_{E,\, -}$).
In fact, given the Euclidean modes $\hPsi^i_{n}$ and $\hPsi^*_{n,\, i}$ where $n \in \Z$, we define the dual modes $\dual{\hPsi}^i_{n}$ and $\dual{\hPsi}^*_{n,\, i}$ such that:
\begin{equation}
  \lconsprod{\dual{\hPsi}^*_{n}}{\hPsi_{m}}
  =
  \lconsprod{\dual{\hPsi}_{n}}{\hPsi^*_{m}}
  =
  \delta_{n,m}.
\end{equation}
We expand the double fields as
\begin{equation}
  \hPsi^i(\zeta)
  =
  \infinfsum{n} b_n \hPsi^i_{n}(\zeta),
  \qquad
  \hPsi^*_{i}(\zeta)
  =
  \infinfsum{n} b^*_n \hPsi^*_{n}(\zeta)
\end{equation}
Operators are then extracted as
\begin{equation}
  b_n
  =
  \lconsprod{\dual{\hPsi}^*_{n}}{\hPsi},
  \qquad
  b^*_n
  =
  \lconsprod{\dual{\hPsi}_{n}}{\hPsi^*}.
  \label{eq:upper-half-extraction}
\end{equation}
Finally we get the anti-commutation relations as
\begin{equation}
  \eval{ \liebraket{b_n}{b^*_m}_+ }_{\tau_E = \tau_{E,\, 0}}
  =
  \frac{2 \cN}{T} \lconsprod{\dual{\hPsi}^*_{n}}{\dual{\hPsi}_m}.
\end{equation}


\subsection{Fields on the Upper Half Plane}

\begin{figure}[tbp]
  \centering
  \import{tikz}{complex_plane_defects.pgf}
  \caption{%
    Fields are glued on the $x < 0$ semi-axis with non trivial discontinuities for $x_{(t)} < x < x_{(t-1)}$ for $t = 1,\, 2,\, \dots,\, N$ and where $x_{(t)} = \exp( \htau_{E,\, (t)} )$.
  }
  \label{fig:complex-plane}
\end{figure}

We consider a set of coordinates on the upper half \ccH of the complex plane:
\begin{equation}
  u = e^{\xi} \in \ccH,
\end{equation}
where $\xi = \tau_E + i \sigma$ and $\sigma \in \qty[ 0, \pi ]$ define the usual strip, and $\ccH = \qty{ w \in \C \mid \Im w \ge 0 }$.
These coordinates can then be extended to the entire complex plane by considering
\begin{equation}
  z = e^{\zeta} \in \C,
\end{equation}
where $\zeta = \tau_E + i \phi$ and $\phi \in \qty[0, 2\pi ]$ define the double strip.
Under this change of coordinates the Euclidean action~\eqref{eq:S_Eu_strip} becomes
\begin{equation}
  \begin{split}
    S_E
    & =
    \frac{T}{2}
    \iint \dd{u}\dd{\baru}\,
    \frac{1}{2}\,
    \qty(
      \frac{1}{u}\, \hpsi_{E,\, +,\, i}^* \lripd{\baru} \hpsi_{E,\, +}^i
      +
      \frac{1}{\baru}\, \hpsi_{E,\, -,\, i}^* \lripd{u} \hpsi_{E,\, -}^i
    )
    \\
    & =
    \frac{T}{2}
    \iint \dd{u}\dd{\baru}\,
    \frac{1}{2}\,
    \qty(
      \psi_{E,\, +,\, i}^* \lripd{\baru} \psi_{E,\, +}^i
      +
      \psi_{E,\, -,\, i}^* \lripd{u} \psi_{E,\, -}^i
    ),
  \end{split}
\end{equation}
where we introduce the off-shell field redefinitions:
\begin{equation}
  \psi_{E,\, +}^i(u,\, \baru)
  =
  \frac{1}{\sqrt{u}}\, \hpsi_{E,\, +}^i( \xi,\, \bxi ),
  \qquad
  \psi_{E,\, -}^i(u,\, \baru)
  =
  \frac{1}{\sqrt{\baru}}\, \hpsi_{E,\, -}^i( \xi,\, \bxi ).
  \label{eq:euclidean-off-shell-redefinitions}
\end{equation}
Fields with the hat sign on top thus represent strip and double strip definitions, while fields without the hat sign are defined on $\ccH$ or $\C$.\footnotemark{}
\footnotetext{%
  We could have anticipated these redefinitions from a \cft argument where
    \begin{equation}
      \psi( u ) = \eval{\qty( \dv{u}{\xi} )^{-\frac{1}{2}}
        {\hpsi}(\xi)}_{\xi = \ln( u )},
    \end{equation}
  but we cannot and do not rely on \cft properties since we have not shown that the theory is a \cft yet.
}
Notice that this is the result one would expect from the engineering dimension: in this case it works since the theory is essentially free.
Using the redefinitions~\eqref{eq:euclidean-off-shell-redefinitions}, the off-shell ``complex conjugation'' $\star$ then becomes
\begin{equation}
  \qty[ \psi_{E,\, +,\, i}( u,\, \baru ) ]^\star
  =
  \frac{1}{\baru}\, \psi_{E,\, +,\, i}^*\qty( \frac{1}{\baru},\, \frac{1}{u} ),
  \qquad
  \qty[ \psi_{E,\, -,\, i}( u,\, \baru ) ]^\star
  =
  \frac{1}{u}\, \psi_{E,\, -,\, i}^*\qty( \frac{1}{\baru},\, \frac{1}{u} ).
\end{equation}

We choose to insert the cut of the square root on the real negative axis.
The boundary conditions are then translated into
\begin{equation}
  \begin{cases}
    \psi_{E,\, -}^i( x - i\, 0^+ )
    =
    \tensor{\qty( R_{(t)} )}{^i_j} \psi_{E,\, +}^j( x + i\, 0^+ )
    \\
    \psi^{*}_{E,\, -,\, i}( x - i\, 0^+ ) =
    \tensor{\qty( R_{(t)}^* )}{_i^j} \psi^{*}_{E,\, +,\, j}( x + i\, 0^+ )
  \end{cases}
\end{equation}
for $x \in \qty( x_{(t)}, x_{(t-1)} )$, where $x_{(t)} = \exp( \htau_{E,\, (t)} ) > 0$, and
\begin{equation}
  \psi_{E,\, -}^i( x - i\, 0^+ )
  =
  \psi_{E,\, +}^i( x + i\, 0^+ ),
  \qquad
  \psi^{*}_{E,\, -,\, i}( x - i\, 0^+ )
  =
  \psi^{*}_{E,\, +,\, i}( x + i\, 0^+ )
\end{equation}
for $x<0$.

The product~\eqref{eq:euclidean-conserved-product-strip} is then
\begin{equation}
  \consprod{\alpha^*}{\beta}
  =
  -i \cN\,
  \qty[
    \int\limits_{\widehat{\Sigma}} \dd{u}
    \alpha^*_{+,\, i}(u) \beta_+^i(u)
    -
    \int\limits_{\widehat{\overline{\Sigma}}} \dd{\baru}
    \alpha^*_{-,\, i}(\baru) \beta_-^i(\baru)
  ],
  \label{eq:prod_H}
\end{equation}
where integrations are computed over two semi-circles $\widehat{\Sigma} = \qty{ u \in \C \mid \abs{u} = \exp( \htau_E ),\, 0 \le \Im u \le \pi }$ and $\widehat{\overline{\Sigma}} = \qty{ u \in \C \mid \abs{u} = \exp( \htau_E ),\, -\pi \le \Im u \le 0}$.
The stress-energy tensor becomes:\footnotemark{}
\footnotetext{%
  While rewriting the operator part of the stress-energy tensor from the strip
  formulation into the coordinates in $\ccH$ we actually get
  \begin{equation}
    \cT_{\xi \xi}( \xi(u) ) = u^2\, \cT_{u u}( u ).
  \end{equation}
  The reason of the presence of $u^2$ factor can be explained in two ways.
  Using GR we know that $\cT_{\xi \xi}( \xi ) \dss[2]{\xi} = \cT_{u u}( u ) \dss[2]{u}$.
  Another way is to notice that a translation in $\xi$ is a dilatation of $u$: the infinitesimal generator of $\xi$ translation must be the infinitesimal
  generator of $u$ dilatation, that is:
  \begin{equation}
    P_{\xi}
    \sim
    \int \dd{\sigma}\, \cT_{\xi \xi}
    \sim
    D_u
    \sim 
    \int \dd{u}\, u\, \cT_{u u}.
  \end{equation}
}
\begin{equation}
  \begin{split}
    \cT_{u u}( u )
    & =
    - \frac{\pi T}{2}\,
    \psi_{E,\, +,\, i}^*( u )\, \lripd{u} \psi^i_{E,\, +}( u )
    +
    \widehat{\cC}(u ),
    \\
    \cT_{\baru \baru}( \baru )
    & =
    - \frac{\pi T}{2}\,
    \psi_{E,\, -,\, i}^*( \baru )\, \lripd{\baru} \psi^i_{E,\, -}( \baru )
    +
    \widehat{\bcC}( \baru ).
  \end{split}
\end{equation}
Finally the anti-commutation relations are
\begin{equation}
  \begin{cases}
    \eval{
      \qty[
        \psi_{E,\, +}^i( u_1,\, \baru_1 )
        ,
        \psi_{E,\, +,\, j}^*( u_2,\, \baru_2 )
      ]_+
    }_{\abs{u_1} = \abs{u_2}}
    & =
    \frac{2}{T u_1}\, \tensor{\delta}{^i_j}\, \delta( \arg(u_1) - \arg(u_2) )
    \\
    \eval{
      \qty[
        \psi_{E,\, -}^i( u_1,\, \baru_1 )
        ,
        \psi_{E,\, -,\, j}^*( u_2,\, \baru_2 )
      ]_+
    }_{\abs{u_1} = \abs{u_2}}
    & =
    \frac{2}{T \baru_1}\, \tensor{\delta}{^i_j}\, \delta( \arg(u_1) - \arg(u_2) ),
  \end{cases}
\end{equation}
which despite the strange look of the expression are perfectly compatible with the definition~\eqref{eq:upper-half-extraction} leading to:
\begin{equation}
  \liebraket{b_n}{b^*_m}_+
  =
  \frac{2 \cN}{T}\,
  \lconsprod{\dual{\hpsi}^*_{E,\, n}}{\dual{\hpsi}_{E,\, m}}
  =
  \frac{2 \cN}{T}\,
  \lconsprod{\dual{\psi}^*_{E,\, n}}{\dual{\psi}_{E,\, m}}
\end{equation}
when the product $\lconsprod{\cdot}{\cdot}$ is defined according to~\eqref{eq:prod_H}.
We expand the fields in modes as:
\begin{equation}
  \begin{cases}
    \psi^i_{E,\, +}(u)
    =
    \infinfsum{n} b_n\, \psi^i_{E,\, +,\, n}(u)
    \\
    \psi^i_{E,\, -}(\baru)
    =
    \infinfsum{n} b_n\, \psi^i_{E,\, -,\, n}(\baru)
  \end{cases}
\end{equation}
and
\begin{equation}
  \begin{cases}
    \psi^*_{E,\, +,\, i}(u)
    =
    \infinfsum{n} b^*_n\, \psi^*_{E,\, +,\, n,\, i}(u)
    \\
    \psi^*_{E,\, -,\, i}(\baru)
    =
    \infinfsum{n} b^*_n\, \psi^*_{E,\, -,\, n,\, i}(\baru)
  \end{cases}
\end{equation}
and $\dual{\psi}_{E,\, n}$ and $\dual{\psi}^*_{E,\, n}$ are the corresponding dual modes on the upper half plane.   


\subsection{Fields on the Complex Plane and the Doubling Trick}
  
We use again the doubling trick to define the fields on the subset $\C \setminus \qty[ x_{(N)},\, x_{(1)} ]$:
\begin{equation}
  \Psi( z )
  =
  \begin{cases}
    \psi_{E,\, +}(u)
    & \qfor
    z = u \in \ccH \setminus \qty[ x_{(N)}, x_{(1)} ]
    \\
    \psi_{E,\, -}(\baru)
    & \qfor
    z = \baru \in \overline{\ccH} \setminus \qty[ x_{(N)}, x_{(1)} ]
  \end{cases}
\end{equation}
where $z = \exp( \tau_E + i\, \phi ) = x + i y$ and $\overline{\ccH} = \qty{ w \in \C \mid \Im w \le 0 }$.
The same procedure applies also to $\Psi^*$ with the exchange $\psi_{E,\, \pm} \leftrightarrow \psi_{E,\, \pm}^*$.

In this case the ``complex conjugation'' $\star$ acts off-shell as
\begin{equation}
  \qty[ \Psi^i( z, \barz ) ]^\star
  =
  \frac{1}{\barz}\, \Psi_i^*\qty(\frac{1}{\barz}, \frac{1}{z}).
  \label{eq:complex-plane-conjugate}
\end{equation}
The boundary conditions then become:
\begin{equation}
  \begin{cases}
    \Psi^i( x - i\, 0^+ )
    & =
    \tensor{\qty( R_{(t)} )}{^i_j} \Psi^j( x + i\, 0^+ ),
    \\
    \Psi^{*\, i}( x - i 0^+ )
    & =
    \tensor{\qty( R_{(t)}^* )}{^i_j} \Psi^{*\, j}( x + i\, 0^+ ),
  \end{cases}
  \label{eq:boundary-condition-euclidean}
\end{equation}
for $x \in \qty( x_{(t)}, x_{(t-1)} )$, where $x_{(t)} = \exp( \htau_{E,\, (t)} ) > 0$ for $t \in \qty{ 1,\, 2,\, \dots,\, N }$.
When $x < 0$ we get
\begin{equation}
  \begin{cases}
    \Psi( x - i\, 0^+ ) & = \Psi( x + i\, 0^+ ),
    \\
    \Psi^*( x - i\, 0^+ ) & = \Psi^*( x + i\, 0^+ )
  \end{cases}.
  \label{eq:gluing-conditions-euclidean}
\end{equation}

Given the relations $\dd{z} = i\, z\, \dd{\phi}$, we can write the conserved product \eqref{eq:prod_H} as:
\begin{equation}
  \consprod{A^*}{B}
  =
  2\pi \cN
  \oint\limits_{\abs{z} = \exp( \tau_E )}
  \frac{\dd{z}}{2 \pi i}\, A^*_i( z )\, B^i( z ),
  \label{eq:conserved-product-complex-plane}
\end{equation}
where we explicitly stressed that the integral has to be performed at a fixed Euclidean time $\tau_E$: in the new coordinate on the plane the conserved product becomes a contour integral at a fixed radius from the origin.

In the same way we can recast the stress-energy tensor components~\eqref{eq:stress-energy-tensor-lightcone} in the new coordinates:
\begin{equation}
  \cT( z )
  =
  - \frac{\pi T}{2}\,
  \Psi^*_i( z )\, \lripd{z} \Psi^i( z )
  +
  \cC( z ),
\end{equation}
where $\cT = \cT_{zz}$ for simplicity.

Finally the canonical anti-commutation relations between the fields are:
\begin{equation}
  \eval{
    \qty[
      \Psi^i( z_1 ),\,
      \Psi_{j}^*( z_2 )
    ]_+
  }_{\abs{z_1} = \abs{z_2}}
  =
  \frac{2}{T z_1}\, \tensor{\delta}{^i_j}\, \delta( \arg(z_1) - \arg(z_2) ).
\end{equation}
The fields expansion in modes thus reads
\begin{equation}
  \Psi^i(z)
  =
  \infinfsum{n} b_n\, \Psi^i_{n}(z),
  \qquad
  \Psi^*_{i}(z)
  =
  \infinfsum{n} b^*_n\, \Psi^*_{n. i}(z).
  \label{eq:complex-plane-mode-expansion}
\end{equation}
The anti-commutation relations among the operators are
\begin{equation}
  \liebraket{b_n}{b^*_m}_+
  =
  \frac{2 \cN}{T}\, \lconsprod{\dual{\Psi}^*_{n}}{\dual{\Psi}_{m}},
\end{equation}
when we introduce the dual modes
$\dual{\Psi}_{n}(z)$ and $\dual{\Psi}^*_{n}(z)$ whose normalisation is
\begin{equation}
  \lconsprod{\dual{\Psi}^*_{n}}{{\Psi}_{m}}
  =
  \lconsprod{\dual{\Psi}_{n}}{{\Psi}^*_{m}}
  =
  \delta_{m,n}.
\end{equation}


\subsection{Algebra of Creation and Annihilation Operators}
\label{sec:modes_and_algebra}
  
In this section we find the explicit expression of the modes which satisfy the \eom and the boundary conditions.
We then compute the dual fields and finally the algebra of the creators and annihilators.


\subsubsection{NS Complex Fermions}
\label{sec:ns-complex-fermions}

We start from NS complex fermions to show that the formalism can recover known results.
Consider the usual definition:
\begin{equation}
  \begin{cases}
    \psi_-^i( \tau, 0 )
    & =
    \psi_+^i( \tau, 0 ),
    \\
    \psi_-^i( \tau, \pi )
    & =
    -\psi_+^i( \tau, \pi )
  \end{cases}
\end{equation}
for $\tau \in \R$, which can be recovered from~$\eqref{eq:boundary-conditions-solutions}$ setting $R_{(t)} \equiv \1$.
In the Euclidean formulation we use~\eqref{eq:boundary-condition-euclidean} and~\eqref{eq:gluing-conditions-euclidean} to get:
\begin{equation}
  \begin{cases}
    \Psi( x - i\, 0^+ ) & =  \Psi( x + i\, 0^+ )
    \\
    \Psi^*( x - i\, 0^+ ) & =  \Psi^*( x + i\, 0^+ )
  \end{cases}
\end{equation}
for $x \in \R$.

We define:
\begin{eqnarray}
  \Psi^i_{( n,\, i_0 )}( z )
  & = &
  \cN_{\Psi}\, \delta^i_{i_0}\, z^{-n},
  \\
  \dual{\Psi}_{( m,\, j_0 ),\, j}( z )
  & = &
  \qty( 2 \pi \cN\, \cN_{\Psi} )^{-1}\, \delta_{j, j_0}\, z^{m-1}
\end{eqnarray}
to recover the definition of the dual modes~\eqref{eq:conserved-product-dual-basis} using the Euclidean conserved product \eqref{eq:conserved-product-complex-plane}.
We then proceed similarly for $\Psi^*$ in such a way that
\begin{equation}
  \lconsprod{\dual{\Psi}_{( n,\, i_0 )}^*}{\Psi_{( m,\, j_0 )}}
  =
  \lconsprod{\dual{\Psi}_{( m,\, j_0 )}}{\Psi^*_{( n,\, i_0 )}}
  =
  \delta_{n, m}\, \delta_{i_0, j_0}.
\end{equation}
As a consequence we find
\begin{equation}
  \lconsprod{\dual{\Psi}_{( n,\, i_0 )}^*}{\dual{\Psi}_{( m,\, i_1 )}}
  =
  \frac{1}{2 \pi \cN\, \cN^2_{\Psi}}\, \delta_{i_0, i_1}\, \delta_{n + m, 1}.
\end{equation}

Consider the  NS expansion in modes of the double fields:
\begin{eqnarray}
  \Psi^i( z )
  & = &
  \infinfsum{n}\,
  \sum\limits_{i_0}\,
  b_{(n,\, i_0)}\, \Psi^i_{( n,\, i_0 )}( z ),
  \\
  \Psi^{*}_i( z )
  & = &
  \infinfsum{n}\,
  \sum\limits_{i_0}\,
  b^*_{(n,\, i_0)}\, \Psi^{*}_{( n,\, i_0 ),\, i}( z ),
\end{eqnarray}
then
\begin{eqnarray}
  b_{( n,\, i_0 )}
  & = &
  \lconsprod{\dual{\Psi}_{( n,\, i_0 )}^*}{\Psi},
  \\
  b^*_{( n,\, i_0 )}
  & = &
  \lconsprod{\dual{\Psi}_{( n,\, i_0)}}{\Psi^*},
\end{eqnarray}
and
\begin{equation}
  \liebraket{b_{( n, i_0 )}}{b^*_{( m, j_0 )}}_+
  =
  \frac{1}{\pi T \cN_{\Psi}^2}\, \delta_{i_0, j_0}\, \delta_{n + m, 1}.
  \label{eq:ns-algebra}
\end{equation}


\subsubsection{Twisted Complex Fermions: Preliminaries}

We can then generalise the discussion about $N_f = 1$ complex fermions in the presence of $N$ point-like defects which we will show to be primary boundary changing operators (i.e. plain and excited spin fields).
Let
\begin{equation}
  \begin{cases}
    R_{(t)} & = e^{i \pi \alpha_{( t )}} \in \U{ 1 }
    \\
    R_{(t)}^* & = e^{-i \pi \alpha_{( t )}} \in \U{ 1 }
  \end{cases}
\end{equation}
such that $0 < \alpha_{( t )} < 2$. The boundary conditions are:
\begin{equation}
  \begin{cases}
    \Psi( x - i\, 0^+ ) & =  e^{i \pi \alpha_{( t )}} \Psi( x + i\, 0^+)
    \\
    \Psi^*( x - i\, 0^+ ) & = e^{-i \pi \alpha_{( t )}} \Psi^*( x + i\, 0^+ )
  \end{cases},
\end{equation}
for $x \in ( x_{(t)}, x_{(t-1)} )$, and
\begin{equation}
  \begin{cases}
    \Psi( x - i\, 0^+ ) & =  \Psi( x + i\, 0^+)
    \\
    \Psi^*( x - i\, 0^+ ) & = \Psi^*( x + i\, 0^+ )
  \end{cases},
\end{equation}
for $x < 0$.
These boundary conditions can be recast in the form of monodromy factors.
With a loop around $x_{(t)}$ we find
\begin{equation}
  \Psi\qty( x_{(t)} + \delta e^{i 0^+} )
  =
  e^{i \pi \qty( \alpha_{( t )} - \alpha_{( t + 1 )} )}\,
  \Psi( x_{(t)} + \delta e^{2 \pi i} ),
\end{equation}
where $\delta \in \R^+$ is small enough and the $\pm$ in the phase represents the position relative to the real axis ($+$ is in the upper half plane \ccH, while $-$ in the lower half plane \bccH).\footnotemark{}
\footnotetext{%
  More precisely $0 < \delta < \min\qty( \abs{x_{(t-1)} - x_{(t)}}, \abs{x_{(t)} - x_{t+1}} )$.
}
We then define the convenient combination:
\begin{equation}
  \epsilon_{(t)}
  =
  \alpha_{( t+1 )} - \alpha_{( t )}
  +
  \theta( \alpha_{( t )} -\alpha_{( t+1 )} - 1 )
  -
  \theta( \alpha_{( t+1 )} - \alpha_{( t )} - 1 )
\end{equation}
such that:\footnotemark{}
\footnotetext{%
  \label{foot:other_range}
  Notice that the choice of the range for $\epsilon_{(t)}$ is not unique.
  We can choose $0 < \alpha_{( t )} < 2$ leading to $\epsilon_{(t)} = \alpha_{( t+1 )} - \alpha_{( t )} + 2 \theta( \alpha_{( t )} - \alpha_{( t+1 )} )$.
  Then in this case $\epsilon_{(t)} = 2 - \bepsilon_{(t)}$ and $\epsilon_{(t)}, \, \bepsilon_{(t)} \in \qty( 0, 2 )$.
  We will however stick to the first definition in the following sections since it allows us to consider the NS case as special.
}
\begin{equation}
  -1 < \epsilon_{(t)} < 1, \qquad \forall t = 1, 2, \dots, N.
\end{equation}
The previous loop around $x_{(t)}$ induces a monodromy 
\begin{equation}
  \begin{cases}
    \Psi( x_{(t)} + \delta e^{i 0^+} )
    & =
    e^{-i \pi \epsilon_{(t)}} \Psi( x_{(t)} + \delta e^{2 i \pi^+} )
    \\
    \Psi^*( x_{(t)} + \delta e^{i 0^+} )
    & =
    e^{-i \pi \bepsilon_{(t)}} \Psi^*( x_{(t)} + \delta e^{2 i \pi^+} ),
  \end{cases}
  \label{eq:monodromy-factors}
\end{equation}
where $\bepsilon_{(t)} =  - \epsilon_{(t)} \Rightarrow -1 < \bepsilon_{(t)} < 1$, thus showing a symmetry under the exchange of:
\begin{equation}
  \Psi \longleftrightarrow \Psi^*
  \qquad \Rightarrow \qquad
  \epsilon_{(t)} \longleftrightarrow \bepsilon_{(t)}.
\end{equation}


\subsubsection{Usual Twisted Fermions}
\label{sec:usual-twisted-fermion}

As a reference for future discussion, we consider the case of one complex fermion in the presence of one twisted boundary condition with the defects located at zero and infinity.
We take $N = 2$ and $x_{(1)} = \infty$ and $x_{(2)} = 0$.
For simplicity we denote with $\epsilon$ the argument of the monodromy factor arising from the presence of the cut on the interval $( 0, +\infty)$.

In order to fulfill the requests \eqref{eq:monodromy-factors} we write the
modes as:
\begin{equation}
  \begin{split}
    \Psi_n^{( \rE )} & = \cN_{\Psi}\, z^{-n + \rE},
    \\
    \Psi_n^{*\, ( \brE )} & = \cN_{\Psi}\, z^{-n + \brE},
  \end{split}
  \label{eq:usual-twisted-modes}
\end{equation}
such that
\begin{equation}
  \begin{split}
    \rE = n_{\rE} + \frac{\epsilon}{2},
    & \quad
    n_{\rE} \in \Z,
    \\
    \brE = n_{\brE} + \frac{\bepsilon}{2},
    & \quad
    n_{\brE} \in \Z.
  \end{split}
\end{equation}
Together with the integer factor $n_{\rE}$ and $n_{\brE}$ we also define a third integer for later convenience:\footnotemark
\footnotetext{%
  The choice discussed in \Cref{foot:other_range} implies $\rL = n_{\rE} + n_{\brE} +1$.   
  We can swap the definitions by exchangin $\bepsilon_{(t)} \leftrightarrow \bepsilon_{(t)} + 2$ and $n_{\brE} \leftrightarrow n_{\brE} - 1$.
}
\begin{equation}
  \rL = \rE + \brE = n_{\rE} + n_{\brE} \in \Z.
  \label{eq:usual_integer_factor}
\end{equation}
To extract creation and annihilation operators with the conserved product~\eqref{eq:conserved-product-complex-plane}, we define the dual basis as:
\begin{eqnarray}
  \dual{\Psi}_n^{( \brE )}( z )
  & = &
  \frac{1}{2 \pi \cN\, \cN_{\Psi}} z^{n - 1 - \brE},
  \\
  \dual{\Psi}_n^{*\, ( \rE )}( z )
  & = &
  \frac{1}{2 \pi \cN\, \cN_{\Psi}} z^{n - 1 - \rE}.
\end{eqnarray}
This way we compute the usual anti-commutation relations as
\begin{equation}
  \lconsprod{\dual{\Psi}_n^{*\, ( \rE )}}{\dual{\Psi}_m^{( \brE )}}
  =
  \frac{\delta_{n + m, 1 + \rL}}{2 \pi \cN\, \cN_{\Psi}^2}
  \quad \Rightarrow \quad
  \liebraket{b_n}{b_m^*}_+
  =
  \frac{1}{\pi T \cN_{\Psi}^2} \delta_{n + m, 1 + \rL},
  \label{eq:twisted-fermion-algebra}
\end{equation}
which are constant in time independently from $\rE$ or $\brE$ since the only possible singularities are located at $z = 0$ and $z = \infty$.
We can then expand the fields $\Psi(z)$ and $\Psi^*( z )$ using this or a more conventional basis:
\begin{eqnarray}
  \Psi( z )
  & = &
  \infinfsum{n} b^{(\rE)}_n \Psi_n^{( \rE )}( z )
  =
  \infinfsum{n} b_{n + n_{\rE}} \Psi_n^{( \frac{\epsilon}{2} )}( z ),
  \label{eq:usual-twisted-mode-expansion}
  \\
  \Psi^*( z )
  & = &
  \infinfsum{n} b^{*\, (\brE)}_n\, \Psi_n^{*\, (\brE)}(z)
  =
  \infinfsum{n} b^{*}_{n+n_{\brE}}\, \Psi_n^{*\, (-\frac{\epsilon}{2})}(z),
  \label{eq:usual-twisted-mode-expansion-conjugate}
\end{eqnarray}
where we used the notation $b = b^{( \frac{\epsilon}{2} )}$ and $b^* = b^{*\,( \frac{\epsilon}{2} )}$.


\subsubsection{Generic Case With Defects}
\label{sec:generic-twisted}

We consider one complex fermion in the presence of $N$ defects such that the modes satisfy:
\begin{equation}
  \Psi_n( x_{(t)} + \delta e^{2 \pi i^+} )
  =
  e^{i \pi \epsilon_{(t)}}\, \Psi_n( x_{(t)} + \delta e^{i 0^+} )
\end{equation}
for $t = 1, 2, \dots, N$ and $\delta > 0$.
We define the basis of solutions as:
\begin{eqnarray}
  \Psi_n( z;\, \qty{x_{(t)},\, \rE_{(t)}} )
  & = &
  \cN_{\Psi}\,
  z^{-n}\,
  \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{\rE_{(t)}},
  \label{eq:generic-case-basis}
  \\
  \Psi^*_n( z;\, \qty{x_{(t)},\, \brE_{(t)}} )
  & = &
  \cN_{\Psi}\,
  z^{-n}\,
  \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{\brE_{(t)}},
  \label{eq:generic-case-basis-conjugate}
\end{eqnarray}
where we generalise the definition of
\begin{eqnarray}
  \rE_{(t)}
  & = &
  n_{\rE_{(t)}} + \frac{\epsilon_{(t)}}{2},
  \quad
  n_{\rE_{(t)}} \in \Z,
  \\
  \brE_{(t)}
  & = &
  n_{\brE_{(t)}} + \frac{\bepsilon_{(t)}}{2}
  \quad
  n_{\brE_{(t)}} \in \Z
\end{eqnarray}
and we define $N$ integer factors analogous to~\eqref{eq:usual_integer_factor}:
\begin{equation}
  \rL_{(t)}
  =
  \rE_{(t)} + \brE_{(t)}
  =
  n_{\rE_{(t)}} + n_{\brE_{(t)}}  \in \Z,
  \qquad
  \forall t \in \qty{ 1,\, 2,\, \dots,\, N }.
\end{equation}
From the definition of the conserved product~\eqref{eq:conserved-product-complex-plane}, we compute the dual basis:
\begin{eqnarray}
  \dual{\Psi}_n( z )
  & = &
  \frac{1}{2 \pi \cN\, \cN_{\Psi}}\,
  z^{n-1}\,
  \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{-\brE_{(t)}},
  \\
  \dual{\Psi}^*_n( z )
  & = &
  \frac{1}{2 \pi \cN\, \cN_{\Psi}}\,
  z^{n-1}\,
  \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{-\rE_{(t)}},
\end{eqnarray}
and the conserved products between dual modes:
\begin{equation}
  \lconsprod{\dual{\Psi}_n^*}{\dual{\Psi}_m}
  =
  \frac{1}{2 \pi \cN\, \cN_{\Psi}^2}\,
  \oint \ddz\,
  z^{n+m-2}\,
  \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{-\rL_{(t)}}.
\end{equation}
Notice that the products are radially invariant only if
\begin{equation}
  \rL_{(t)} \le 0,
  \qquad
  \forall t \in \qty{ 1,\, 2,\, \dots, N },
  \label{eq:generic-case-negativity-condition}
\end{equation}
since the integrand must not present time dependent singularities on the integration path, thus
\begin{equation}
  \begin{split}
    \lconsprod{\dual{\Psi}_n^*}{\dual{\Psi}_m} 
    & =
    \frac{1}{2 \pi \cN\, \cN_{\Psi}^2}\,
    \oint \ddz\,
    \finiteprod{t}{1}{N}\,
    \finitesum{k_t}{0}{\abs{\rL_{(t)}}}\,
    \binom{\abs{\rL_{(t)}}}{k_t}
    \qty( - \frac{1}{x_{(t)}} )^{k_t}\,
    z^{k_t + n + m - 2}
    \\
    & =
    \frac{1}{2 \pi \cN\, \cN_{\Psi}^2}\, p_{1 - n - m},
  \end{split}
\end{equation}
where we defined
\begin{equation}
  p_k
  =
  \finiteprod{t}{1}{N}\,
  \finitesum{k_t}{0}{\abs{\rL_{(t)}}}\,
  \binom{\abs{\rL_{(t)}}}{k_t}
  \qty( - \frac{1}{x_{(t)}} )^{k_t}\,
  \delta_{\finitesum{t}{1}{N} k_t,\, k}
  \label{eq:generic-conserved-product-factor}
\end{equation}
such that
\begin{eqnarray}
  p_{0 \le k \le \finitesum{t}{1}{N} \abs{\rL_{(t)}}}
  & \neq &
  0,
  \\
  p_{k \le -1}
  =
  p_{k \ge \finitesum{t}{1}{N} \abs{\rL_{(t)}} + 1}
  & = &
  0.
\end{eqnarray}
We can finally write
\begin{equation}
  \liebraket{b_n}{b^*_m}_+
  =
  \frac{1}{\pi T \cN_{\Psi}^2}\, p_{1 - n - m},
  \qquad
  1 - \finitesum{t}{1}{N} \abs{\rL_{(t)}} \le n + m \le 1.
  \label{eq:generic-case-anti-commutation}
\end{equation}


\subsection{Representation of the Algebra: Definition of the In-Vacuum}
\label{sec:invacuum}
  
In the previous section we computed the algebra of the operators for different theories.
We now define in-vacua where they are represented to be able to compute the relevant amplitudes.
We show how to recover the NS vacuum and the usual twisted vacuum.
Finally we discuss the vacuum in the presence of a generic number of defects.
  

\subsubsection{NS Fermions}

The in-vacuum can be correctly obtained either requiring $\Psi( z )$ and $\Psi^*( z )$ to be non singular as $z \to 0$ when applied on the vacuum.
The same request can also be made on $\hPsi( \xi )$ and $\hPsi^*( \xi )$.
In both cases we get the same vacuum which  turns out to be \SL{2}{\R} invariant:
\begin{equation}
  b_{( n,\, i_0 )} \regvacuum
  =
  b^*_{( n,\, i_0 )} \regvacuum
  =
  0,
  \qquad
  n \ge 1.
\label{eq:NS_SL2_vacuum}
\end{equation}
The spectrum of the theory is constructed acting with operators $b_{( n,\, i_0 )}$ and $b^*_{( n,\, i_0 )}$ with $n \le 0$.
    

\subsubsection{Twisted Fermion}
\label{sec:twisted-fermion}

Consider the case of the usual twisted fermion in \Cref{sec:usual-twisted-fermion}.
Define the excited vacuum $\excvacket$ as:
\begin{equation}
  b^{(\rE)}_n \excvacket
  =
  b^{*\, ( \brE )}_n \excvacket
  =
  0,
  \qquad n \ge 1.
  \label{eq:usual-excited-vacuum}
\end{equation}
The introduction of $\rE$ and $\brE$ is necessary to define the vacuum as in previous cases, that is with a range of $n$ independent from them and without singularities as $z \to 0$.
Explicitly we have:
\begin{equation}
  \Psi(z) \excvacket
  \sim
  z^{\rE}\, (\dots),
  \qquad
  \Psi^*(z) \excvacket
  \sim
  z^{\brE}\, (\dots).
  \label{eq:asymp_beha_Psi_on_exc_vac}
\end{equation}
Comparing with~\eqref{eq:generic-case-basis} and~\eqref{eq:generic-case-basis-conjugate}, the behavior suggests the existence of a hidden operator in $x_{(t)}$ creating $\excvacket$ with $\rE = \rE_{(t)}$ and $\brE = \brE_{(t)}$.

These relations are subject to consistency conditions since
\begin{equation}
  \excvacket
  =
  \pi T \cN_{\Psi}^2 \qty[b^{(E)}_n,\, b^{*\, ( \brE )}_{\rL+1-n}]_+ \excvacket,
\end{equation}
that is we cannot have two in-annihilators (namely both $b^{(\rE)}_n$ and $b^{*\, ( \brE )}_{\rL+1-n}$) with non vanishing anti-commutation relations.
Specifically we have:
\begin{equation}
  1 \le n \le \rL
  \qquad \Rightarrow \qquad
  b^{(\rE)}_n \excvacket = 0,
  \quad
  b^{*\, ( \brE )}_{\rL + 1 - n} \excvacket = 0,
\end{equation}
that is
\begin{equation}
  \excvacket
  =
  \pi T \cN_{\Psi}^2 \liebraket{b^{(\rE)}_n}{b^{*\, ( \brE )}_{\rL + 1 - n}}_+ \excvacket
  =
  0,
\end{equation}
which is not consistent (see~\Cref{fig:inconsistent-theories} for a graphical reference): the theory does not exist.
We shall therefore consider only cases such that
\begin{equation}
  \rL \le 0,
\end{equation}
analogously to~\eqref{eq:generic-case-negativity-condition}.

Moreover notice that for $\rL \le -1$ both $b^{(\rE)}_{\rL \le n \le 0}$ and $b^{*\, ( \brE )}_{\rL \le n \le 0}$ are in- and out-creation operators.

\begin{figure}[tbp]
  \centering
  \import{tikz}{inconsistent_theories.pgf}
  \caption{As a consistency condition, we have to exclude the values of
  $\rL$ for which both $b^{(
E)}_n$ and $b^{*\, ( \brE )}_{\rL + 1 - n}$ are in-annihilators
  with a non vanishing anti-commutation relation.}
  \label{fig:inconsistent-theories}
\end{figure}

The vacuum $\excvacket$ is not however associated to the lowest energy.
In fact the usual way to build the vacuum would be to require $\Psi( z )$ and $\Psi^*( z )$ to be non singular as $z \to 0$ for the in-vacuum so that $b^{(\rE)}_n \twsvacket = 0$ for $n > \rE$, and $b^{*\, ( \brE )}_n \twsvacket = 0$ for $n > \brE$.
However this procedure almost always fails to give a good  definition of the vacuum.
In fact it works only for NS fermions.
For example when $\epsilon > 0$ we have:
\begin{equation}
  0
  =
  \pi T \cN_{\Psi}^2\,
  \liebraket{b^{(\rE)}_{1 + n_{\rE} }}{b^{*\, ( \brE )}_{n_{\brE} }}_+
  \twsvacket
  =
  \twsvacket,
\end{equation}
which is not consistent since both  $b^{(\rE)}_{1 + n_{\rE}}$ and $b^{*\, ( \brE )}_{n_{\brE}}$ are annihilators as $1 + n_{\rE} > E$ and $n_{\brE} > \brE$.

The minimum energy vacuum is instead defined in a proper way on the strip.
Requiring that the action of $\hPsi( \xi )$ and $\hPsi^*( \xi )$ for $\xi \to -\infty$ on the vacuum is well defined we get
\begin{eqnarray}
  b^{(\rE)}_n \twsvacket
  & = &
  0,
  \qquad
  n > \rE + \frac{1}{2},
  \\
  b^{*\, ( \brE )}_m \twsvacket
  & = &
  0,
  \qquad
  m > \brE + \frac{1}{2}.
\end{eqnarray}
This is a good definition of the vacuum as $-\frac{1}{2} < \frac{\epsilon}{2} = -\frac{\bepsilon}{2} < \frac{1}{2}$ implies that $b^{(\rE)}_n$ and $b^{*\, ( \brE )}_m$ are annihilation operators for $n \ge n_{\rE}+1 > \rE + \frac{1}{2}$ and $m \ge n_{\brE}+1 > \brE + \frac{1}{2}$ so that
\begin{equation}
  0
  =
  \pi T \cN_{\Psi}^2\,
  \liebraket{b^{(\rE)}_{n}}{b^{*\, ( \brE )}_{m}}_+
  \twsvacket
  =
  \delta_{n + m,\, \rE+\brE+1 }
  \twsvacket
  =
  0.
\end{equation}
This way we get a consistent definition of the twisted vacuum.
This is however not generally \SL{2}{\R} invariant as we will show later when defining stress-energy tensor.\footnotemark{}
\footnotetext{%
  Notice that the second choice of $\epsilon$ interval discussed in~\Cref{foot:other_range} needs to distinguish between two cases: $0 < \frac{\epsilon}{2} < \frac{1}{2}$ (and $\frac{1}{2} < \frac{\bepsilon}{2} < 1$) and $\frac{1}{2} < \frac{\epsilon}{2} < 1$ (and $0 < \frac{\bepsilon}{2} < \frac{1}{2}$).
}

The vacua $\excvacket$ and $\twsvacket$ are related.
Consider for example the case $n_{\rE} \ge 1$ and the definition:
\begin{eqnarray}
  b^{(\rE)}_n \excvacket & = & 0, \qquad n \ge 1,
  \\
  b^{(\rE)}_n \twsvacket & = & 0, \qquad n \ge 1 + n_{\rE}.
\end{eqnarray}
Then for $1 \le n \le n_{\rE}$ the modes $b^{(\rE)}_n$ act as a annihilation operator on $\excvacket$ and as a creation operator on $\twsvacket$:
\begin{equation}
  \excvacket
  \propto
  b^{(\rE)}_{n_{\rE}}\, b^{(\rE)}_{n_{\rE} - 1}\, \dots\, b^{(\rE)}_1
  \twsvacket.
  \label{eq:usual-twisted-vacuum-relation}
\end{equation}
Moreover, since $\rL = n_{\rE} + n_{\brE} \le 0 \Rightarrow n_{\brE} \le -1$, we have:
\begin{eqnarray}
  b^{*\, ( \brE )}_m \excvacket & = & 0, \qquad m \ge 1,
  \\
  b^{*\, ( \brE )}_n \twsvacket & = & 0, \qquad m \ge 1 - \abs{n_{\brE}},
\end{eqnarray}
which leads to:
\begin{equation}
  \twsvacket
  \propto
  b^{*\, ( \brE )}_0\, b^{*\, ( \brE )}_1\, \dots\, b^{*\, ( \brE )}_{1 - \abs{n_{\brE}}}
  \excvacket.
  \label{eq:usual-twisted-fermion-conformal-twisted}
\end{equation}
The consistency of the definition can be checked requiring
\begin{equation}
  \excvacket
  =
  \qty( \pi T \cN_{\Psi}^2 )^{n_{\rE}}\,
  b^{(\rE)}_{n_{\rE}}\,
  b^{(\rE)}_{n_{\rE}-1}\,
  \dots\,
  b^{(\rE)}_1\,
  b^{*\, ( \brE )}_0\,
  b^{*\, ( \brE )}_1\,
  \dots\,
  b^{*\, ( \brE )}_{1 - \abs{n_{\brE}}}
  \excvacket,
\end{equation}
where the number of $b$ operators has to match the number of $b^*$
 operators:
\begin{equation}
  n_{\rE} + n_{\brE}
  =
  \rE + \brE
  =
  \rL
  =
  0.
  \label{eq:twisted-fermion-consistency}
\end{equation}
The same procedure applies also in the case $n_{\rE} \le 0$, leading to the same result.
As a consequence of~\eqref{eq:twisted-fermion-consistency}, we express the twisted vacuum as:
\begin{eqnarray}
  b^{(\rE)}_n \twsvacket & = & 0, \qquad n \ge 1 + n_{\rE},
  \\
  b^{*\, ( \brE )}_m \twsvacket & = & 0, \qquad m \ge 1 - n_{\rE}.
\end{eqnarray}
      

\subsubsection{Generic Case with Defects}
    
Since the fields in presence of defects behave as NS fields in the limit $z \to 0$, we define the vacuum in the usual fashion by requiring a finite limit $\lim\limits_{z \to 0} \Psi( z ) \Gexcvacket$.
As in the NS we get the representation:
\begin{equation}
  b_{n} \Gexcvacket
  =
  b^*_{n} \Gexcvacket
  =
  0,
  \qquad n \ge 1.
  \label{eq:generic_vacuum}
\end{equation}


\subsection{Asymptotic Fields and Vacua}
\label{sect:asymp_fields}
    
In this section we define the asymptotic in-field and out-field and discuss how their vacua are related to the theory with defects.
The relation is ``radial time dependent'' explicitly showing that an interaction is hidden in the defects.
In particular the vacuum for the theory with defects can be identified with \SL{2}{\R} in-field vacuum while it is connected by a Bogoliubov transformation to the \SL{2}{\R} out-field vacuum.
    
In the following we use the expansion of
\begin{equation}
  P\qty(z;\, \qty{ x_{(t)},\, \rE_{(t)} } )
  =
  \finiteprod{t}{1}{N}
  \qty( 1- \frac{z}{x_{(t)}} )^{\rE_{(t)}},
\end{equation}
around the origin and infinity with coefficients   
\begin{eqnarray}
  \cmode{k}{0}{\rE_{(t)}}{x_{(t)}}
  & = &
  \sum_{ \qty{k_t} \in \N^N} 
  \finiteprod{t}{1}{N}
  \qty[ \binom{\rE_{(t)}}{k_t} \qty( - \frac{1}{x_{(t)}} )^{k_t} ]
  \delta_{\finitesum{t}{1}{N} k_t, k}
  \\
  \cmode{k}{\infty}{\rE_{(t)}}{x_{(t)}}
  & = &
  \sum_{ \qty{k_t} \in \N^N}
  \finiteprod{t}{1}{N}
  \qty[ \binom{\rE_{(t)}}{k_t} \qty( - x_{(t)} )^{k_t - \rE_{(t)}} ]
  \delta_{\finitesum{t}{1}{N} k_t, k}, 
\end{eqnarray}
so that we can write
\begin{equation}
  \begin{split}
    P\qty(z;\, \qty{x_{(t)}, \rE_{(t)}})
    =
    \begin{cases}
      \zeroinfsum{k}
      \cmode{k}{0}{\rE_{(t)}}{x_{(t)}}\, z^k,
      & \qfor
      \abs{z} < x_{(N)}
      \\
      \zeroinfsum{k}
      \cmode{k}{\infty}{\rE_{(t)}}{x_{(t)}}\, z^{-k+\finitesum{t}{1}{N} \rE_{(t)}},
      & \qfor
      \abs{z} > x_{(1)}
    \end{cases}
  \end{split}
\end{equation}
We do not discuss intermediate fields, that is expansions for $x_{(t)} < \abs{z} < x_{(t-1)}$, as it is not possible to disentangle the effects  of defects before and after this range.
The vacuum in the presence of defects is in fact related to the radial ordering of the operators associated with the defects as we argue later on.
    

\subsubsection{Asymptotic In-field and Its Vacuum}

Consider the definitions of the basis of solutions~\eqref{eq:generic-case-basis} and expand around $z = 0$.\footnotemark{}
\footnotetext{%
  Similarly we could have considered~\eqref{eq:generic-case-basis-conjugate} to begin with.
  Analogous relations can in fact be written for $b_n^{*\, ( 0 )}$ with the substitutions of $\rE_{(t)} $ with $\brE_{(t)}$.
}
We get for $0 \le \abs{z} < x_{(N)}$:
\begin{equation}
  \Psi_n( z )
  =
  \zeroinfsum{k}
  \cmode{k}{0}{\rE_{(t)}}{x_{(t)}}\,
  \Psi^{( 0 )}_{n-k}( z ),
  \label{eq:expansion-around-0}
\end{equation}
where $\Psi^{( 0 )}_n( z ) = \cN_{\Psi} z^{-n}$ as in~\eqref{eq:usual-twisted-modes} with $\rE = 0$ which are the modes of a untwisted fermion, i.e.\ a plain NS fermion.
The previous expansion connects the asymptotic behavior of the modes of the fermion with defects with the modes of a NS fermion close to the origin.
We can now connect the operators of the system with defects with those of the asymptotic in-field.
To this purpose we substitute the expansion~\eqref{eq:expansion-around-0} with the usual expression of the modes~\eqref{eq:complex-plane-mode-expansion}:
\begin{equation}
  \Psi( z )
  =
  \infinfsum{n}
  b_n\, \Psi_n( z )
  \underset{\abs{z} < x_{(N)}}{=}
  \Psi^{(\text{in})}( z )
  =
  \infinfsum{n}
  b_n^{( 0 )}\, \Psi_n^{( 0 )}( z )
\end{equation}
thus leading to
\begin{equation}
    b_n^{( 0 )}
    =
    \zeroinfsum{k}
    b_{n + k}\, \cmode{k}{0}{\rE_{(t)}}{x_{(t)}}.
\end{equation}
These expressions can be inverted writing $\Psi^{( 0 )}_{n}( z ) = \Psi_{n}( z )\, P\qty(z;\, \qty{x_{(t)},\, -\rE_{(t)}} )$.
We then get:
\begin{equation}
  \begin{split}
    b_n
    & =
    \zeroinfsum{k}
    \cmode{k}{0}{-\rE_{(t)}}{x_{(t)}}\, b^{(0)}_{n + k},
  \end{split}
\end{equation}
Annihilation operators of the asymptotic theory, i.e.\ operators with positive index, are therefore expressed only using annihilation operators of the theory with defects.
We can thus set
\begin{equation}
  \Gexcvacket = \regvacuumin.
\end{equation}

  
\subsubsection{Asymptotic Vacua and Bogoliubov Transformations}

Revisiting the previous section, we can also explicitly compute the expansion for $\abs{z} > x_{(1)}$.
Define for simplicity $\rM = \finitesum{t}{1}{N} \rE_{(t)}$).
We then get:
\begin{equation}
  \Psi_n( z )
  =
  \zeroinfsum{k}
  \cmode{k}{\infty}{\rE_{(t)}}{x_{(t)}}\,
  \Psi^{( 0 )}_{n+k-\rM}( z ).
\end{equation}
The formula connects the asymptotic behavior of the modes of the fermion with defects to modes of a NS fermion which can be seen close to the infinity.

Out-operators can thus be connected to the theory with defect through:\footnotemark{}
\footnotetext{%
  To avoid a redundant notation we do not introduce an object $\Psi^{(\infty)}_n( z )$.
  Even though it would have been in principle correct, we would have also found that $\Psi^{(\infty)}_n( z ) = \Psi^{(0)}_n( z )$.
}
\begin{equation}
  \Psi(z)
  =
  \infinfsum{n}
  b_n\, \Psi_n( z )
  \underset{|z|>x_1}{=}
  \Psi^{(\text{out})}( z )
  =
  \infinfsum{n}
  b_n^{( \infty )}\, \Psi_n^{( 0 )}( z ).
\end{equation}
We get:
\begin{equation}
  b_n^{( \infty )}
  =
  \zeroinfsum{k}
  b_{n + \rM - k}\,
  \cmode{k}{\infty}{\rE_{(t)}}{x_{(t)}}.
  \label{eq:b_inf-b}
\end{equation}
The inverse of the expression is:
\begin{equation}
  \begin{split}
    b_n
    & =
    \zeroinfsum{k}
    \cmode{k}{\infty}{-\rE_{(t)}}{x_{(t)}}\,
    b^{( \infty )}_{n + \rM - k}.
  \end{split}
\end{equation}
As we will show later however $\rM = 0$.
Annihilation operators of the asymptotic theory, i.e.\ operators with positive index, are thus expressed using both annihilation and creation operators of the theory with defects while creators, i.e.\ operators with non negative index, are expressed by means of creation operators only.
It follows from the vacuum definition that:
\begin{equation}
  \begin{split}
    \qty(%
      \tffC_0\, b_1^{(\infty)}
      +
      \text{creation op.}
    ) \Gexcvacket
    & =
    0,
    \\
    \qty(%
      \tffC_0\, b_2^{(\infty)}
      +
      \tffC_1\, b_1^{(\infty)}
      +
      \text{creation op.}
    ) \Gexcvacket
    & =
    0,
    \\
    & \vdots
  \end{split}
\end{equation}
where $\tffC_n = \cmode{n}{\infty}{-\rE_{(t)}}{x_{(t)}}$ for brevity.
This means that the vacuum for the asymptotic out-field is non trivially connected to the vacuum of the theory with defects.
More explicitly we have:
\begin{equation}
  \begin{split}
    \Gexcvacket
    & =
    \cN_{(\text{out})}(\qty{x_{(t)},\, \rE_{(t)},\, \brE_{(t)}})
    \\
    & \times
    e^{%
      \infzerosum{m} \infzerosum{n}
      \cM_{m n}^{(\text{out})}\qty(\qty{x_{(t)},\, \rE_{(t)},\, \brE_{(t)}})\,
      b^{*\, ( \infty )}_{m}\, b^{( \infty )}_{n}
    }
    \regvacuumout,
  \end{split}
\end{equation}
so that the two \SL{2}{\R} vacua are connected by a Bogoliubov transformation.
More precisely we get (see appendix \ref{sec:details_reflection} for details)
\begin{equation}
  \qty(%
    \Psi^{(\text{out},\, +)}( z )
    +
    \ccL_{(1)}\qty[\Psi^{(\text{out},\, -)}]( z )
  )
  \Gexcvacket
  =
  0,
  \qquad
  \abs{z} > x_{(1)},
  \label{eq:reflection condition_out_field_generic_vacuum}
\end{equation}
where
\begin{equation}
  \ccL_{(t)}\qty[\Psi]( z )
  =
  \oint\limits_{\abs{w} > x_{(t)}} \ddw
  \frac{P\qty( z;\, \qty{x_{(t)},\, \rE_{(t)}} )\, P\qty( w;\, \qty{x_{(t)},\, -\rE_{(t)}} ) - 1}{z - w}\,
  \Psi( w ).
\end{equation}
The corresponding equation for $\Psi^{*\, (\text{out})}( z )$ can be found with the substitution $\rE  \rightarrow \brE$.
Notice that the kernel of the integral is nothing else (up to a multiplicative constant) but the regularised propagator, that is the propagator in the presence of the defects to which the NS propagator has been subtracted. 
The previous equation is solved explicitly by:
\begin{equation}
  \begin{split}
    \Gexcvacket
    & =
    \cN\qty(\qty{x_{(t)},\, \rE_{(t)},\, \brE_{(t)}})~
    \\
    & \times
    e^{%
      \oint_{\abs{z} > x_{(1)}} \ddz
      \Psi^{*\, (\text{out},\, -)}( z )\,
      \ccL\qty[ \Psi^{(\text{out},\, -)} ]( z )
    }
    \regvacuumout.
  \end{split}
\end{equation}
In the previous equation there is no need to specify the relation between $\abs{z}$ and $\abs{w}$ since $\Psi^{*\, (\text{out}, -)}( z )$ and $\Psi^{(\text{out}, -)}( w )$ anti-commute.
From the same expression for $\Psi^{*\, (\text{out}, -)}( z )$ we deduce that $\brE_{(t)} = -\rE_{(t)}$.


\subsection{Contractions and Stress-energy Tensor}
\label{sec:contraction_and_T}

Given the definition of the algebra of the operators and its representation, we can finally define the normal ordering operation and proceed to compute the contractions and \ope of the operators.
The procedure ultimately leads to the definition of the stress-energy tensor. This is enough to show that the theory is a time dependent \cft since the stress-energy tensor satisfies the canonical \ope


\subsubsection{NS Complex Fermion}

First of all we deal with the simple case of NS fermions.
Using the algebra~\eqref{eq:ns-algebra} we compute the \ope of fermion fields as:
\begin{equation}
  \Psi^i( z )\, \Psi^*_j( w )
  =
  \no{\Psi^i( z )\, \Psi^*_j( z )}
  +
  \frac{1}{\pi T} \frac{\tensor{\delta}{^i_j}}{ z - w },
  \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}.
We then get the expression of the stress-energy tensor:
\begin{equation}
  \begin{split}
    \cT( z )
    & =
    \lim\limits_{w \to z}
    \qty[%
      -\frac{ \pi T }{2}
      \qty( \Psi^*_i( z )\, \ipd{w} \Psi^i( w ) - \ipd{z} \Psi^*_i( z )\, \Psi^i( w ) )
      +
      \frac{N_f}{ ( z - w )^2 }
    ]
    \\
    & =
    -\frac{\pi T}{2} \no{ \Psi^*_i( z ) \lripd{z} \Psi^i( z ) }.
  \end{split}
\end{equation}
We are then able to derive the necessary minimal subtraction:
\begin{equation}
  \ffh( z - w ) = \frac{N_f}{( z - w )^2}.
\end{equation}


\subsubsection{Twisted Fermion}

We now focus on $N_f = 1$ theories.
First of all we consider the simplest case of the usual twisted fermion with the mode expansion~\eqref{eq:usual-twisted-mode-expansion} and~\eqref{eq:usual-twisted-mode-expansion-conjugate}.
Using the anti-commutation relations~\eqref{eq:twisted-fermion-algebra} we can compute the \ope:
\begin{equation}
  \Psi( z )\, \Psi^*( w )
  =
  \noE{ \Psi( z )\, \Psi^*( w ) }
  +
  \frac{1}{\pi T} \qty( \frac{z}{w} )^{\rE} \frac{1}{z - w},
  \qquad
  \abs{w} < \abs{z},
\end{equation}
and
\begin{equation}
  \Psi^*( w )\, \Psi( z )
  =
  \noE{ \Psi^*( w )\, \Psi( z ) }
  +
  \frac{1}{\pi T} \qty( \frac{w}{z} )^{\brE} \frac{1}{w - z},
  \qquad
  \abs{w} > \abs{z}.
\end{equation}
If we require that the previous results can be assembled in a well defined continuous radial ordering $\rR\qty( \Psi( z ) \Psi^*( w ) )$ we need to set $\rE = -\brE$ so we can write
\begin{equation}
  \rR\qty( \Psi( z ) \Psi^*( w ) )
  =
  \noE{ \Psi( z )\, \Psi^*( w ) }
  +
  \frac{1}{\pi T} \qty( \frac{z}{w} )^{\rE} \frac{1}{z - w}.
\end{equation}

The same result can be reached by computing the stress-energy tensor starting from the previous expressions.
We have two ways to construct it depending on the ordering of the classical expression:
\begin{equation}
  \begin{split}
    \cT( z )
    & =
    \lim\limits_{\substack{w \to z \\ \abs{w} < \abs{z}}}
    \qty[%
      -\frac{\pi T}{2}
      \qty(%
        \Psi^*( z ) \ipd{w} \Psi( w )
        -
        \ipd{z} \Psi^*( z ) \Psi( w )
      )
      +
      \frac{1}{( z- w )^2}
    ]
    \\
    & =
    -\frac{\pi T}{2}
    \no{ \Psi^*( z ) \lripd{z} \Psi( z ) }
    +
    \frac{\rE^2}{2 z^2},
  \end{split}
  \label{eq:T_excited_vacuum}
\end{equation}
or
\begin{equation}      
  \begin{split}
    \cT( z )
    & =
    \lim\limits_{\substack{w \to z \\ \abs{w} < \abs{z}}}
    \qty[%
      -\frac{\pi T}{2}
      \qty(%
        -
        \ipd{z} \Psi( z )\, \Psi^*( w )
        +
        \Psi( z )\, \ipd{w} \Psi^*( w )
      )
    +
    \frac{1}{( z -  w )^2}
    ]
    \\
    & =
    -\frac{\pi T}{2}
    \no{ \Psi^*( z ) \lripd{z} \Psi( z ) }
    +
    \frac{\brE^2}{2 z^2},
  \end{split}
\end{equation}
which however must coincide for consistency.
Since
\begin{equation}
  \no{ \Psi( z ) \lripd{z} \Psi^*( z ) }
  =
  \no{ \Psi^*( z ) \lripd{z} \Psi( z ) }
\end{equation}
then we must then require $\rE^2 = \brE^2$.
We can  get a stronger constraint by computing the \ope $\cT( z )\, \cT( w )$.
In fact the cancellation of the cubic divergence requires $\rE + \brE = 0$.
From now on we will therefore use the notation \eexcvacket instead of \excvacket.

From the usual definition of the stress-energy tensor in terms of the Virasoro generators $\cT( z ) = \infinfsum{k} L_k z^{-k-2}$, we extract the operators $L_k$ from any of the previous definitions:
\begin{equation}
  \begin{split}
    L_{(\rE) k}
    & =
    -\frac{\pi T}{2}\, \cN_{\Psi}^2\,
    \infinfsum{k}
    \noE{b^{*\, ( \brE )}_n\, b^{(\rE)}_{k + 1 - n}}
    ( 2n - k + 2 \rE - 1 ) 
    +
    \frac{\rE^2}{2} \delta_{k,0}
    \\
    & =
    \frac{\pi T}{2}\, \cN_{\Psi}^2\,
    \finitesum{n}{1}{+\infty}
    \left[
      ( 2n - k + 2 \rE - 1 ) 
      \noE{b^{(\rE)}_{k + 1 - n}\, b^{*\, ( \brE )}_n}
    \right.
      \\
    & +
    \left.
      ( 2n - k - 2 \rE - 1 ) 
      \noE{b^{*\, ( \brE )}_{k + 1 - n}\, b^{(\rE)}_n} 
    \right]
    +
    \frac{\rE^2}{2} \delta_{k,0}.
  \end{split}
\end{equation}

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}
    L_{(\rE)\, k} \eexcvacket
    & =
    0,
    \qquad
    k > 0,
    \\
    L_{(\rE)\, 0} \eexcvacket
    & =
    \frac{\rE^2}{2} \eexcvacket,
  \end{split}
\end{equation}
with non trivial conformal dimensions $\Delta\qty( \eexcvacket ) =  \frac{\rE^2}{2}$.
This operator is an excited spin field $\rS_{\rE_{(t)}}\qty( x )$ inserted at $x = 0$.
Its equivalent expression using bosonization is:
\begin{equation}
  \rS_{\rE}\qty( x ) = e^{i \rE \phi( x )},
\end{equation}
where $\phi$ is such that
\begin{equation}
  \left\langle \phi( z ) \phi( w ) \right\rangle
  =
  -\frac{1}{( z - w)^2}.
\end{equation}
The minimal conformal dimension is achieved for $n_{\rE}=n_{\brE}=0$, i.e.\ $\Delta\qty( \twsvacket ) =  \frac{\epsilon^2}{8}$, and identifies a plain spin field.
We can further check this idea  by showing that the conformal dimensions are consistent.
Using~\eqref{eq:usual-twisted-fermion-conformal-twisted} we get:
\begin{equation}
  \begin{split}
    L_{(\rE)\, 0} \twsvacket
    & =
    L_0\,
    \qty(%
      b^{*\, ( \brE )}_0\,
      b^{*\, ( \brE )}_{-1}\,
      \dots\,
      b^{*\, ( \brE )}_{2-n_{\rE}}
      \eexcvacket
    )
    \\
    & =
    \left[%
      \finitesum{n}{1}{n_{\rE}}
      \qty( n - \frac{\rE +  1}{2} )
      +
      \frac{\rE^2}{2}
    \right]
    \twsvacket
    =
    \frac{\epsilon^2}{8} \twsvacket.
  \end{split}
\end{equation}


\subsubsection{Generic Case With Defects}

We then consider the generic case of one complex fermion in the presence of an arbitrary number of spin fields with respect to the vacuum we introduced in~\eqref{eq:generic_vacuum}.
We consider the mode expansion~\eqref{eq:generic-case-basis} and~\eqref{eq:generic-case-basis-conjugate} as well as the anti-commutation relations~\eqref{eq:generic-case-anti-commutation}.

As in the usual twisted case we first consider the contraction of the field $\Psi$ and $\Psi^*$ and then move to the stress-energy tensor.
Using the anti-commutation relations and
\begin{equation}
  \infinfsum{k} p_k z^k = \finiteprod{t}{1}{N} \qty( 1 - \frac{z}{x_{(t)}} )^{-\rL_{(t)}}
\end{equation}
where $p_k$ is defined in~\eqref{eq:generic-conserved-product-factor}.
We have:
\begin{equation}
  \Psi( z )\, \Psi^*( w )
  =
  \no{ \Psi( z )\, \Psi^*( w ) }
  +
  \frac{1}{\pi T}\,
  \frac{1}{z - w}\,
  \finiteprod{t}{1}{N}
  \qty( 1 - \frac{z}{x_{(t)}} )^{\rE_{(t)}}
  \qty( 1 - \frac{w}{x_{(t)}} )^{-\rE_{(t)}},
\end{equation}
as well as
\begin{equation}
  \Psi^*( z )\, \Psi( w )
  =
  \no{ \Psi^*( z )\, \Psi( w ) }
  +
  \frac{1}{\pi T}\,
  \frac{1}{z - w}\,
  \finiteprod{t}{1}{N}
  \qty( 1 - \frac{z}{x_{(t)}} )^{\brE_{(t)}}
  \qty( 1 - \frac{w}{x_{(t)}} )^{-\brE_{(t)}},
\end{equation}
both for $\abs{w} < \abs{z}$.
To assemble the expressions in a well defined continuous radial ordering $\rR\qty[ \Psi( z )\, \Psi^*( w ) ]$ we need to set $\rE_{(t)} = -\brE_{(t)}$ such that we can write
\begin{equation}
  \rR\qty[ \Psi( z )\, \Psi^*( w ) ]
  =
  \no{ \Psi( z )\, \Psi^*( w ) }
  +
  \frac{1}{\pi T}\,
  \frac{1}{z - w}\,
  \finiteprod{t}{1}{N}
  \qty( 1 - \frac{z}{x_{(t)}})^{\rE_{(t)}}
  \qty( 1 - \frac{w}{x_{(t)}} )^{-\rE_{(t)}}.
  \label{eq:gen_Radial_order}
\end{equation}
We can then expand the results around $z$:
\begin{equation}
  \begin{split}
    \rR\qty[\Psi( z )\, \Psi^*( w )]
    & =
    \no{ \qty(\Psi \Psi^*)( z ) }
    +
    \no{ \qty(\Psi\, \pd \Psi^*)( z ) }\, (w-z)
    \\
    & +
    \frac{1}{\pi T}
    \left[%
      \frac{-1}{w- z}
      +
      \finitesum{t}{1}{N} \frac{\rE_{(t)}}{z - x_{(t)}}
    \right.
    \\
    & \left.
      -
      \frac{1}{2}
      \qty(%
        \finitesum{t}{1}{N}
        \sum\limits_{u \neq t}
        \frac{\rE_{(t)}\, \rE_{(u)}}{( z - x_{(t)}) ( z - x_{(u)} )}
        +
        \finitesum{t}{1}{N}
        \frac{\rE_{(t)}\, \qty( \rE_{(t)} - 1 )}{( z - x_{(t)} )^2}
      )
      \qty( w - z )
    \right]
    \\
    & +
    \order{( w - z )^2},
  \end{split}
\end{equation}
and around $w$
\begin{equation}
  \begin{split}
    \rR\qty[\Psi( z )\, \Psi^*( w )]
    & =
    \no{ \qty(\Psi \Psi^*)( w ) }
    +
    \no{ \qty(\pd \Psi\, \Psi^*)( w ) } (z-w)
    \\
    & +
    \frac{1}{\pi T}
    \left[%
      \frac{1}{z- w}
      +
      \finitesum{t}{1}{N} \frac{\rE_{(t)}}{w - x_{(t)}}
    \right.
    \\
    & \left.
      + \frac{1}{2}
      \qty(%
        \finitesum{t}{1}{N}
        \sum\limits_{u \neq t}
        \frac{\rE_{(t)}\, \rE_{(u)}}{(w - x_{(t)}) (w - x_{(u)})}
        +
        \finitesum{t}{1}{N}
        \frac{\rE_{(t)}\, \qty( \rE_{(t)} - 1 )}{( w - x_{(t)} )^2}
      )
      \qty( z - w )
    \right]
    \\
    & +
    \order{( z - w )^2},
  \end{split}
\end{equation}      
so that the stress-energy tensor becomes:
\begin{equation}
  \begin{split}
    \cT( z )
    & =
    -\frac{\pi T}{2}
    \no{ \Psi( z ) \lripd{z} \Psi^*( z ) } 
    +
    \frac{1}{2}
    \qty( \finitesum{t}{1}{N}  \frac{\rE_{(t)}}{z - x_{(t)}} )^2
    \\
    & =
    \frac{\pi T }{2}\, \cN_{\Psi}^2
    \infinfsum{n,\, m}
    \no{b_n\, b^*_m}
    z^{-n-m}
    \qty[%
      \frac{m-n}{z}
      +
      2\, \finitesum{t}{1}{N} \frac{\rE_{(t)}}{z-x_{(t)}}
    ]
    +
    \frac{1}{2} \qty( \finitesum{t}{1}{N} \frac{\rE_{(t)}}{z - x_{(t)}} )^2.
  \end{split}
\end{equation}
The last expression shows that the energy momentum tensor $\cT( z )$ is radial time dependent but it satisfies the usual \ope

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.
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
\begin{equation}
  \begin{split}
    \GGexcvacket
    & =
    \cN(\qty{x_{(t)},\, \rE_{(t)}})~
    \rS_{\rE_{(1)}}\qty( x_{(1)} )\,
    \rS_{\rE_{(2)}}\qty( x_{(2)} )
    \dots
    \rS_{\rE_{(N)}}\qty( x_{(N)} )
    \regvacuum
    \\
    & =
    \cN(\qty{x_{(t)},\, \rE_{(t)} })~
    \rR\qty[%
      \finiteprod{t}{1}{N} \rS_{\rE_{(t)}}\qty( x_{(t) } )
    ]
    \regvacuum,
    \label{eq:vacuum_R_prod_spin_fields}
  \end{split}
\end{equation}
then we get
\begin{equation}
  \cT( z )
  \GGexcvacket
  =
  \cN(\qty{x_{(t)},\, \rE_{(t)} })~
  \rR\qty[%
    \cT( z )
    \finiteprod{t}{1}{N}
    \rS_{\rE_{(t)}}\qty( x_{(t)} )
  ]
  \regvacuum.
\end{equation}
The fact that $\cT( z )$ enters the radial ordering may seem strange but the left hand side is well defined for all $z$ and the only well defined expression for the right hand side is with the radial ordering.
In fact an operator expression like $\cT( z ) \rR\qty[ \ipd{x_{(1)}} \phi(x_{(1)})\, \ipd{x_{(2)}} \phi(x_{(2)})] \regvacuum$ is only defined for $\abs{z} > x_{(1),\, (2)}$.
It then follows that
\begin{equation}
  \cT( z )
  \GGexcvacket
  =
  \finitesum{t}{1}{N}
  \qty(%
    \frac{\rE_{(t)}^2 / 2}{(z - x_{(t)})^2}
    +
    \frac{\ipd{x_{(t)}} - \ipd{x_{(t)}} \log\cN}{z - x_{(t)}}
  )
  \GGexcvacket
  +
  \order{1},
\end{equation}
which allows us to write
\begin{equation}
  \begin{split}
    &
    \cN(\qty{x_{(t)}, \rE_{(t)} })~
    \rR\qty[%
      \ipd{x_{(t)}} \rS_{\rE_{(t)}}\qty( x_{(t)} )
      \prod_{u \ne t} \rS_{\rE_{(u)}}\qty( x_{(u)} )
    ]
    \regvacuum
    \\
    = &
    \rE_{(t)}\,
    \qty[%
      \pi T \cN_{\Psi}^2\,
      \zeroinfsum{n,\, m}
      \frac{b_n\, b^*_m}{x_{(t)}^{n + m}}
      +
      \sum_{u \ne t}
      \frac{\rE_{(u)}}{x_{(t)} - x_{(u)}}
    ]
    \GGexcvacket.
  \end{split}
\end{equation}        
This result shows the way non primary operators are represented in this formalism.


\subsection{Hermitian Conjugation}
\label{sec:hermitian_and_outvacuum}

Before we can define the amplitudes involving spin and matter fields, we still need to introduce some of the necessary tools.
In this section we focus on the operation of ``Hermitian conjugation'' in a broad sense: the usual Hermitian conjugation requires the existence of an inner product which is not yet available since we have not defined the out-vacuum.
The operation we define is similar to the  $\star$ operator of $C^\star$ algebras even though the $\star$ operator sends an element of an algebra to another element of the same algebra.
This is not what happens in the generic case since the $\star$ is essentially associated with the inversion $z \rightarrow \barz^{-1}$, i.e.\ in evolving from $\tau = +\infty$ to $\tau = -\infty$ so that the order of boundary singularities is reversed.
  

\subsubsection{Usual Twisted Fermions}

In general for a chiral primary conformal operator of dimension $\Delta$ in $z$ coordinates the Euclidean Hermitian conjugation is
\begin{equation}
  \qty[ O(z) ]^\dagger = \eval{ \qty( w^{2\Delta} O(w) ) }_{w = \barz^{-1}}.
\end{equation}
As a matter of fact we cannot use the words ``Euclidean Hermitian conjugation'' since we do not have an inner product.
We define the operation $\star$ which mimics its behavior.
Therefore we define
\begin{equation}
  \qty[ \Psi\qty( z; \rE ) ]^{\star}
  =
  \eval{\qty[ w\, \tPsi^*\qty( w; -\trE ) ]}_{w = \barz^{-1}},
  \qquad
  \qty[ \Psi^*\qty( z; \rE ) ]^{\star}
  = \eval{\qty( w\, \tPsi\qty( w; \trE ) )}_{w = \barz^{-1}}.
  \label{eq:star_on_Psi}
\end{equation}
In the last expression we did not assume that the action of $\star$ is an automorphism and we wrote $\Psi\qty( z; \rE )$ to explicitly show the dependence on the parameter $\rE$ which enters in the modes.
The previous action agrees with~\eqref{eq:complex-plane-conjugate}.
In terms of the basis~\eqref{eq:usual-twisted-modes} we write:\footnotemark{}
\footnotetext{%
  The second possibility $\qty[ \Psi_n^{( \rE )}(z) ]^{\star} = \eval{\qty[ w\, \Psi_{-n}^{*\, ( -\rE-1 )}\qty( w ) ]}_{w = \barz^{-1}}$ is inconsistent with the anti-commutation relations.
}
\begin{equation}
  \qty[ \Psi_n^{( \rE )}(z) ]^{\star}
  =
  \eval{\qty[ w\, \Psi_{1-n}^{*\, ( -\rE )}\qty( w ) ]}_{w = \barz^{-1}},
  \qquad
  \qty[ \Psi_n^{( -\rE )}(z) ]^{\star}
  =
  \eval{\qty[ w\, \Psi_{1-n}^{*\, ( \rE )}\qty( w ) ]}_{w = \barz^{-1}},
\end{equation}
which shows that in this case the image of the $\star$ operator is the same as the support.
Using the mode expansion of \eqref{eq:star_on_Psi} it follows that
\begin{equation}
  \qty[ b^{(\rE)}_n ]^{\star} = b^{*\, ( \brE )}_{1-n},
  \qquad
  \qty[ b^{*\, ( \brE )}_n ]^{\star} = b^{(\rE)}_{1-n}.
  \label{eq:star_usual_twisted}
\end{equation}
The $\star$ action is compatible with the anti-commutation relations as we can show by explicitly computing them:
\begin{equation}
  \qty(%
    \liebraket{b^{(\rE)}_n}{b^{*\, ( \brE )}_m}_+
  )^{\star}
  =
  \liebraket{b^{*\, ( \brE )}_{1-n}}{b^{(\rE)}_{1-m}}_+
  =
  \frac{1}{\pi T \cN_{\Psi}^2} \delta_{n+m, 1}.
\end{equation}
Furthermore $\star$ is involutive since:
\begin{equation}
  \qty[ \Psi_n^{( \rE )}( z ) ]^{\star \star} 
  =
  \Psi_n^{( \rE )}( z )
  \quad
  \Rightarrow
  \quad
  \qty[ b^{(\rE)}_n ]^{\star \star}
  =
  b^{(\rE)}_n.
\end{equation}


\subsubsection{Generic Case With Defects}

Consider the modes~\eqref{eq:generic-case-basis}.
We define the action of the $\star$ operator on them as:
\begin{equation}
  \begin{split}
    \qty[ \Psi_{n}\qty( z;\, \qty{x_{(t)},\, \rE_{(t)}} ) ]^{\star}
    & =
    \cN_{\Psi}\,
    \barz^{-n}\,
    \finiteprod{t}{1}{N} \qty( 1 - \frac{\barz}{x_{(t)}} )^{\rE_{(t)}}
    \\
    & =
    \eval{ \qty(
    w \, \finiteprod{t}{1}{N} \qty( - \frac{1}{x_{(t)}} )^{\rE_{(t)}} \,
    \widetilde{\Psi}_{M + 1 - n}^*
    \qty(w; \qty{\tildex_{(t)}, \overline{\trE}_{(t)}})
    ) }_{w = \barz^{-1}}
  \end{split}
\end{equation}
where we used $\rM = \finitesum{t}{1}{N} \rE_{(t)}$ and $\tPsi_l( w;\, \qty{ y,\, \rF } ) = \cN_{\Psi}\, w^{-l}\, \finiteprod{t}{1}{N} \qty( 1 - \frac{w}{y} )^{-\rF}$.
In this case the image of the $\star$ operator is a different space where the defects are located in $\tildex_{(t)}$ and the critical exponents are $\trE_{(t)}$ and $\overline{\trE}_{(t)}$ with
\begin{equation}
  \tildex_{(t)} = \frac{1}{x_{(t)}},
  \qquad
  \trE_{(t)} = -\rE_{(t)} = \brE_{(t)},
  \qquad
  \overline{\trE}_{(t)} = \rE_{(t)} = - \brE_{(t)},
\end{equation}
where we used $\rE_{(t)} + \brE_{(t)} = 0$.
We can then compute the action of the $\star$ operator on the creation and annihilation operators:
\begin{equation}
  b_n^{\star}
  =
  \finiteprod{t}{1}{N}
  \qty( - \frac{1}{x_{(t)}} )^{-\rE_{(t)}}~
  \tildeb^*_{M + 1- n},
  \qquad
  \qty(b_n^*)^{\star}
  =
  \finiteprod{t}{1}{N}
  \qty( - \frac{1}{x_{(t)}} )^{\rE_{(t)}}~
  \tildeb_{-M + 1- n}.
\end{equation}
As in the previous situation the anti-commutation relations are preserved by the $\star$ operator.
Explicitly we have:
\begin{equation}
  \qty( \liebraket{b_n}{b_m^*}_+ )^{\star}
  =
  \liebraket{\tildeb_{-M + 1 - m}}{\tildeb_{M + 1 - n}^*}_+
  =
  \frac{1}{\pi T \cN_{\Psi}^2}\,
  \delta_{n + m, 1}.
\end{equation}
Finally the $\star$ operator is involutive.


\subsection{Definition of the Out-Vacuum}
\label{sec:out-vacuum}

With the definition of the $\star$ operator we can now proceed to define the out-vacuum to as the Hermitian conjugation in the usual cases.
It is a conceptually separated step from the definitions of the algebra of operators and their representation on the in-vacuum.
We first consider the usual twisted theory from which we learn how to define the out-vacuum and then move to the generic case in the presence of multiple defects.


\subsubsection{Usual Twisted Fermions}

Consider the definition of the in-vacuum~\eqref{eq:usual-excited-vacuum} for the fields image of the $\star$ operator, i.e.\ $\tPsi\qty( w;\, \trE )$ and $\tPsi^*\qty( w;\, \overline{\trE} )$.
It is defined as
\begin{equation}
  \tildeb^{( \trE )}_n \ket{\tildeT_{\trE ,\overline{\trE}}}
  = 
  \tildeb^{*\, ( \overline{\trE} )}_n \ket{\tildeT_{\trE ,\overline{\trE}}}
  = 0,
  \qquad
  n \ge 1.
\end{equation}
The usual Hermitian conjugation gives
\begin{equation}
  \bra{\tildeT_{\trE ,\overline{\trE}}}
  \qty( \tildeb^{( \trE )}_n)^\dagger 
  = 
  \bra{\tildeT_{\trE ,\overline{\trE}}}
  \qty( \tildeb^{*\, ( \overline{\trE} )}_n)^\dagger 
  = 0,
  \qquad
  n \ge 1.
\end{equation}
Given the action of the $\star$ operator~\eqref{eq:star_usual_twisted}, the identification with the Hermitian conjugate is possible if
\begin{equation}
  \eexcvacbra b^{(\rE)}_n = \eexcvacbra b^{*\, ( \brE )}_n = 0,
  \qquad
  n \le 0.
\end{equation}


\subsubsection{Generic Case With Defects}

We can now analyse the case of an arbitrary number of defects.
Following the steps of the previous section we define the in-vacuum for the tilded theory as
\begin{equation}
  \tildeb_n \GGexcvacket
  = 
  \tildeb_n \GGexcvacket
  = 0,
  \qquad
  n \ge 1,
\end{equation}
and interpret it as the out-vacuum of the initial theory.     
The definition of the out-vacuum is therefore:
\begin{eqnarray}
  \GGexcvacbra b_n & = & 0,
  \qquad n \le \rM,
  \\
  \GGexcvacbra b^*_n & = & 0,
  \qquad n \le -\rM.
\end{eqnarray}
Since the action of the $\star$ operator is compatible with the anti-commutation relations, the definition of the out-states is consistent.
If we assume that
\begin{equation}
  \braket{\GGexcvac}{\GGexcvac} \neq 0,
\end{equation}
using the anti-commutation relations we get
\begin{equation}
  \begin{split}
    \frac{1}{\pi T \cN_{\Psi}^2} \braket{\GGexcvac}{\GGexcvac}
    & =
    \GGexcvacbra
    \liebraket{b_{\rM}}{b_{-\rM+1}^*}_+
    \GGexcvacket
    \\
    & =
    \GGexcvacbra
    b_{-\rM+1}^*\,  b_{\rM}
    \GGexcvacket
    \neq 0,
  \end{split}
\end{equation}
which requires $b_{\rM} \GGexcvacket \neq 0$.
A similar condition exists for $b^*_{-\rM}$, thus we must require $\rM \le 0$ and $-\rM \le 0$:
\begin{equation}
 \rM =\finitesum{t}{1}{N} \rE_{(t)} = 0.
\end{equation}
The situation is therefore analogous to the case depicted in \Cref{fig:inconsistent-theories} where $\rM$ and $\brM$ have the same role of $\rL$ for the twisted fermion.


\subsubsection{Asymptotic vacua}

The discussion is essentially the same as in~\Cref{sect:asymp_fields} with the role of asymptotic in- and out-fields swapped.
In particular we get
\begin{equation}
  \GGexcvacbra = \regvacuumoutconj,
\end{equation}
and
\begin{equation}
  \GGexcvacbra
  =
  \cN_{(in)}\qty(\qty{x_{(t)},\, \rE_{(t)}}) \regvacuuminconj
  e^{%
    \finitesum{m,\, n}{1}{+\infty}
    \cM_{m n}\qty(\qty{x_{(t)},\, \rE_{(t)}})\,
    b^{*\, (0)}_m b^{(0)}_n
  }.
\end{equation}


\subsection{Correlators in the Presence of Spin Fields}
\label{sec:spin_correlators}
      
The definitions of the in- and out-vacua and the stress-energy tensor are critical to compute any correlation function of operators in the presence of the point-like defects.
In fact we need to know both the algebra of the operators and their representation, usually defined on the in-vacuum (the ket vector), as well as their Hermitian conjugation in order to build the action of the operators on the out-vacuum (the bra vector).

Starting from~\eqref{eq:vacuum_R_prod_spin_fields} we can compute the spin field correlators
\begin{equation}
  \braket{\GGexcvac}
  =
  \cN(\qty{x_{(t)},\, \rE_{(t)} })
  \left\langle
    \rR\qty[%
      \finiteprod{t}{1}{N} \rS_{\rE_{(t)}}\qty(x_{(t)})
    ]
  \right\rangle.
\end{equation}
At first sight both \GGexcvacket and \GGexcvacbra might seem to contain $\rR\qty[ \finiteprod{t}{1}{N} \rS_{\rE_{(t)}}\qty(x_{(t)}) ]$.
However this it is not the case and it can be seen in different ways.
The simplest is to realise that such a square would be divergent while the product seems to be perfectly finite.
A more rigorous way is to consider what the previous product is from the point of view of asymptotic out field: we have $\GGexcvacket= \cN_{(out)}\, \rR\qty[ \finiteprod{t}{1}{N} \rS_{\rE_{(t)}}\qty(x_{(t)}) ] \regvacuumout$ and $\GGexcvacbra= \regvacuumoutconj$ so that $\cN_{(out)} = \cN$.
Moreover $\cT\qty(z) \underset{\abs{z} > x_{(1)}}{=} \cT_{(out)}\qty(z)$ when the two energy momentum tensors are normal ordered with respect to their different sets of operators which are related in~\eqref{eq:b_inf-b}.
Hence all the expressions are surely valid for $\abs{z} > x_{(1)}$ and can be analytically extended to the whole plane. 
The same result can be obtained from the point of view of asymptotic in-fields. 
   
Unfortunately the normalisation factor cannot be uniquely fixed.
The result depends on the normalisation chosen for the single spin field and effectively shows only when we relate the $N$ points to $N-1$ points correlators, recursively down to two points correlators.
Therefore we need to consider quantities where the normalisation is not present.
In particular we consider
\begin{equation}
  \begin{split}
    & \pdv{x_{(t)}}
    \ln\left\langle
      \rR\qty[%
        \rS_{\rE_{(t)}}\qty(x_{(t)})
        \prod\limits_{\substack{u = 1 \\ u \neq t}}^N \rS_{\rE_{(u)}}\qty(x_{(u)})
      ]
    \right\rangle
    \\
    = &
    \oint\limits_{\abs{z} = x_{(t)}} \ddz
    \frac{
      \left\langle
        \rR\qty[%
          \cT( z )
          \finiteprod{t}{1}{N} \rS_{\rE_{(t)}}\qty(x_{(t)})
        ]
      \right\rangle
    }{
      \left\langle
        \rR\qty[%
          \finiteprod{t}{1}{N} \rS_{\rE_{(t)}}\qty(x_{(t)})
        ]
      \right\rangle
    }
    \\
    = &
    \qty(%
      \oint\limits_{\abs{z} > x_{(t)}} \ddz
      -
      \oint\limits_{\abs{z} < x_{(t)}} \ddz
    )
    \frac{\GGexcvacbra \cT( z ) \GGexcvacket}{\braket{\GGexcvac}}
    \\
    =  &
    \frac{%
      \GGexcvacbra
      \qty(L_{-1}^{x_{(t)}^+} - L_{-1}^{x_{(t)}^-})
      \GGexcvacket
    }{\braket{\GGexcvac}}
  \end{split},
\end{equation}
since $\liebraket{L_{-1}}{\cO_h( z )} = \ipd{z} \cO_h( z )$ for a quasi-primary operator $\cO_h$.
From the definition of $\cT( z )$ it follows that:
\begin{equation}
  L_{-1}^{x_{(t)}^+} - L_{-1}^{x_{(t)}^-}
  =
  \oint\limits_{\cC_{x_{(t)}}} \ddz \cT( z )
  =
  \uppi T\, \cN_{\Psi}^2\, \rE_{(t)}\,
  \infinfsum{n,\, m}
  \no{b_n\, b^*_m}\,
  x_{(t)}^{-m-n}
  +
  \sum\limits_{\substack{u = 1 \\ u \neq t}}^N
  \frac{\rE_{(u)} \rE_{(t)}}{x_{(t)} - x_{(u)}},
\end{equation}
where $\cC_{x_{(t)}}$ is a small loop around $x_{(t)}$.
Therefore we have
\begin{equation}
  \pdv{x_{(t)}}
  \ln \left\langle
    \rR\qty[%
      \rS_{\rE_{(t)}}\qty(x_{(t)})
      \prod\limits_{\substack{u = 1 \\ u \neq t}}^N
      \rS_{\rE_{(u)}}\qty(x_{(u)})
    ]
  \right\rangle
  =
  \sum\limits_{u \neq t} \frac{\rE_{(u)} \rE_{(t)}}{x_{(t)} - x_{(u)}},
\end{equation}
which can be solved by
\begin{equation}
  \left\langle
    \rR\qty[%
      \rS_{\rE_{(t)}}\qty(x_{(t)})
      \prod\limits_{\substack{u = 1 \\ u \neq t}}^N
      \rS_{\rE_{(u)}}\qty(x_{(u)})
    ]
  \right\rangle
  =
  \cN_0
  \qty( \qty{\rE_{(t)}} )
  \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
The last equation reproduces  the usual bosonization procedure.

In similar way we can compute all correlators
\begin{equation}
  \begin{split}
    &\frac{%
      \GGexcvacbra
      \rR\qty[ \prod\limits_i \Psi(x_i)\, \prod\limits_j \Psi^*(x_j) ]
      \GGexcvacket
    }{\braket{\GGexcvac}}
    \\
    = &
    \frac{%
      \left\langle
        \rR\qty[
          \prod\limits_i \Psi(x_i)\,
          \prod\limits_j \Psi^*(x_j)\,
          \finiteprod{t}{1}{N} \rS_{\rE_{(t)}}\qty(x_{(t)})
        ]
      \right\rangle
    }{
      \left\langle
        \rR\qty[
          \finiteprod{t}{1}{N} \rS_{\rE_{(t)}}\qty(x_{(t)})
        ]
      \right\rangle
    },
  \end{split}
\end{equation}
using Wick's theorem since the algebra and the action of creation and annihilation operators is the usual.
In particular taking one $\Psi(z)$ and one $\Psi^*(w)$ we get the Green function which is nothing else but the contraction in equation~\eqref{eq:gen_Radial_order}.


% vim: ft=tex
