# 2D Fermion on the Strip with Boundary Defects as a CFT with Excited Spin
Fields

Riccardo Finotello E-mail<EMAIL_ADDRESS>Igor Pesando
Dipartimento di Fisica, Università di Torino
and I.N.F.N. - sezione di Torino
Via P. Giuria 1, I-10125 Torino, Italy
E-mail<EMAIL_ADDRESS>
###### Abstract

We consider a two-dimensional fermion on the strip in the presence of an
arbitrary number of zero-dimensional boundary changing defects. We show that
the theory is still conformal with time dependent stress-energy tensor and
that the allowed defects can be understood as excited spin fields. Finally we
compute correlation functions involving these excited spin fields without
using bosonization.

###### Contents

  1. 1 Introduction and Conclusion
  2. 2 Point-like Defect CFT: the Minkowskian Formulation
    1. 2.1 Action Principle and Boundary Conditions
  3. 3 Conserved Product and Charges
    1. 3.1 Definition of the Conserved Product
      1. 3.1.1 Flavour Vector Current
      2. 3.1.2 Stress-Energy Tensor
    2. 3.2 Basis of Solutions and Dual Modes
  4. 4 Point-like Defect CFT: the Euclidean Formulation
    1. 4.1 Fields on the Strip
    2. 4.2 Double Strip Formalism and Doubling Trick
    3. 4.3 Fields on the Upper Half Plane
    4. 4.4 Fields on the Complex Plane and the Doubling Trick
  5. 5 Algebra of Creation and Annihilation Operators
    1. 5.1 NS Complex Fermions
    2. 5.2 Twisted Complex Fermions: preliminaries
      1. 5.2.1 Usual Twisted Fermions
      2. 5.2.2 Generic Case With Defects
  6. 6 Representation of the Algebra: Definition of the In-Vacuum
    1. 6.1 NS Fermions
    2. 6.2 Twisted Fermion
      1. 6.2.1 Excited vacuum
      2. 6.2.2 Minimum Energy Vacuum
      3. 6.2.3 Relation Between Vacua
    3. 6.3 Generic Case with Defects
  7. 7 Asymptotic Fields and Relation Between Asymptotic Fields Vacua and the Vacuum
    1. 7.1 Asymptotic in-field and relation between its vacuum and generic case vacuum
    2. 7.2 Relation between generic case vacuum and asymptotic out-field vacuum
  8. 8 Contractions and Stress-Energy Tensor
    1. 8.1 NS Complex Fermion
    2. 8.2 Twisted Fermion
      1. 8.2.1 OPE and Stress-Energy Tensor
      2. 8.2.2 Virasoro Operators and Conformal Dimensions
    3. 8.3 Generic Case With Defects
  9. 9 Hermitian Conjugation
    1. 9.1 Usual Twisted Fermions
    2. 9.2 Generic Case With Defects
  10. 10 Definition of the Out-Vacuum
    1. 10.1 Usual Twisted Fermions
    2. 10.2 Generic Case With Defects
    3. 10.3 Asymptotic vacua
  11. 11 Spin Field Correlators
  12. A  Details on Reflection Condition on the Vacuum with an Arbitrary Number of Defects for Asymptotic Field

##  1 Introduction and Conclusion

The study of viable phenomenological models in the framework of String Theory
often involves the analysis of the properties of systems of D-branes. Clearly
the inclusion of the physical requirements needed for a consistent theory
deeply constrains the possible scenarios. In particular the chiral spectrum of
the Standard Model acts as a strong restriction on the possible D-brane setup.
Intersecting and magnetized branes represent relevant classes of such models
with interacting chiral matter. In particular most models involve a
compactification with factorized two-tori and magnetic backgrounds (see [1, 2]
for review) and only few attempts have been done to study more general cases.

The computation of interesting quantities such as Yukawa couplings involves
quite often correlators of excited spin and twist fields. The correlators of
(excited) spin fields has been a research subject for many years until the
formulation found in the seminal paper by Friedan, Martinec and Shenker [3]
based on bosonization. On the other side the correlators of excited twist
fields has not been algorithmic until recent [4, 5] (see for example [6, 7, 8,
9] for earlier work on excited twist fields and [10, 11, 12, 13, 14, 15, 16,
17, 18] for the basic correlators). Even for the spin field case the available
techniques allow to compute only correlators involving “Abelian”
configurations, i.e. configurations which can be factorized in sub-
configurations having $\mathrm{U}(1)$ symmetry and only few papers have
considered the non Abelian case [19, 20, 21, 22, 23, 24] which is
mathematically by far more complicated and related to the unresolved
connection problem of Fuchsian equations.

Despite the existence of an efficient method based on bosonization [3] for
computing correlators involving excited spin fields we re-examine the problem
and give a new method to compute such correlators which adds on the present
one and the very old one based on Reggeon vertex [25, 26, 27, 28, 29, 30, 31,
32].

One reason for such a research is that we hope to be able to extend this
approach to correlators involving twist fields and non Abelian spin and twist
fields. In particular we would also like to clarify the reason of the non
existence of an approach equivalent to bosonization for twist fields.

Another reason is that 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 with the right central charge. Moreover the
boundary changing defects in the construction can be associated with excited
spin fields and this allows to compute correlators involving excited spin
fields without resorting to bosonization.

The paper is organized as follows. In Section 2 we define the Minkowskian
formulation of the theory we are interested in and we introduce the notation.
Then in Section 3 we discuss the conserved quantities. In particular we
introduce a conserved product used to extract the coefficients of the
expansion of the fields in modes. In order to obtain these coefficients, which
we want to interpret as creation and annihilation operators, we are led to the
introduction of the space of dual modes.

In Section 4 we discuss the Euclidean formulation on the strip and the upper
plane not relying on the CFT properties since we have not yet shown that the
theory is a CFT.

In Section 5 we find the explicit expression of the modes which satisfy the
equations of motion and the boundary conditions. Then we compute the dual
modes and finally the algebra of the creators and annihilators. This step is
conceptually separated from the definition of the in-vacuum and the Fock space
where this algebra is represented. We take care of this in Section 6.

In Section 7 we relate the fermionic field with its asymptotic in- and out-
counterparts. This is useful in the last section in order to justify the new
way of computing the excited spin fields correlators.

With the definition of the vacuum we have an associated normal ordering. In
Section 8 we compute the contractions and OPEs of the operators and define the
stress-energy tensor which satisfies the canonical CFT algebra, thus showing
that the theory is a CFT. Then we argue that the defects are excited spin
fields.

In Section 9 we take care of the definition of the operation which we want to
interpret as Euclidean Hermitian conjugation when we define the bra of the
vacuum. This operation is almost the same as the $\star$ operator defined in
the algebraic approach to QFT. Using this definition, which we want to be the
Hermitian conjugation, in Section 10 we define the bra-vacuum as it is
necessary to compute the correlators.

Finally in Section 11 we compute correlators of excited spin fields using the
in- and out-vacua in presence of defects.

##  2 Point-like Defect CFT: the Minkowskian Formulation

In this section we introduce the theory we study by presenting its worldsheet
action and boundary conditions in the presence of $N$ zero-dimensional
defects. This theory is later shown to be a CFT despite the existence of
defects.

###  2.1 Action Principle and Boundary Conditions

Let
$\quantity(\uptau,\upsigma)\in\Upsigma=\quantity(-\infty,+\infty)\times\quantity[0,\uppi]$
define a strip with Lorentzian metric111 We consider the metric
$\differential{s}^{2}=-\differential{\uptau}^{2}+\differential{\upsigma}^{2},$
and the lightcone coordinates $\upxi_{\pm}=\uptau\pm\upsigma$ which allow to
define
$\partial_{\pm}=\frac{1}{2}\quantity(\partial_{\uptau}\pm\partial_{\upsigma})$
and
$\differential^{2}{\upxi}=\frac{1}{2}\differential{\upxi_{+}}\differential{\upxi_{-}}=\differential{\uptau}\differential{\upsigma}.$
The anti-symmetric tensor is
$\upepsilon_{\uptau\upsigma}=-\upepsilon^{\uptau\upsigma}=1$ and the gamma
matrices are  $\upgamma^{\uptau}=\matrixquantity(&1\\\
-1&)=-\upgamma_{\uptau},\qquad\upgamma^{\upsigma}=\matrixquantity(&1\\\
1&)=\upgamma_{\upsigma}.$ We also consider the two-dimensional spinor
$\uppsi=\matrixquantity(\uppsi_{+}\\\ \uppsi_{-}),$ whose conjugate is
$\overline{\uppsi}=\uppsi^{\dagger}\upgamma^{\uptau}=\matrixquantity(-\uppsi_{-}^{*}&\uppsi_{+}^{*})$.
and consider $\mathrm{N}_{f}$ massless complex fermions $\uppsi^{i}$ such that
$i=1,2,\dots,\mathrm{N}_{f}$. Their two-dimensional Minkowski action defined
on the strip $\Upsigma$ is:

$S_{\mathcal{M}}=\frac{T}{2}\int\limits_{-\infty}^{+\infty}\differential{\uptau}\int\limits_{0}^{+\uppi}\differential{\upsigma}\quantity(\frac{1}{2}\overline{\uppsi}_{i}(\uptau,\upsigma)\quantity(-i\upgamma^{\upalpha}\overset{\leftrightarrow}{\partial_{\upalpha}})\uppsi^{i}(\uptau,\upsigma)).$

In components the action reads:

$S_{\mathcal{M}}=i\frac{T}{2}\iint\differential^{2}{\upxi}\quantity(\uppsi^{*}_{-,\,i}\overset{\leftrightarrow}{\partial_{+}}\uppsi^{i}_{-}+\uppsi^{*}_{+,\,i}\overset{\leftrightarrow}{\partial_{-}}\uppsi^{i}_{+}),$
(2.1)

so the equations of motion are:

$\begin{split}\partial_{-}\uppsi_{+}^{i}(\upxi_{+},\upxi_{-})&=\partial_{+}\uppsi_{-}^{i}(\upxi_{+},\upxi_{-})=0,\\\
\partial_{-}\uppsi^{*}_{+,\,i}(\upxi_{+},\upxi_{-})&=\partial_{+}\uppsi^{*}_{-,\,i}(\upxi_{+},\upxi_{-})=0.\end{split}$
(2.2)

Their solutions are the usual “holomorphic” functions
$\uppsi_{+}^{i}(\upxi_{+})$ and $\uppsi_{-}^{i}(\upxi_{-})$, together with
their complex conjugates222 Notice that $\uppsi^{*}$ is indeed the complex
conjugate of the field $\uppsi$, while it will no longer be the case in the
Euclidean formalism. .

Figure 2.1: We describe the propagation of a string in the presence of point-
like defects in the time direction. Each point $\hat{\uptau}_{t}$ on the
boundary of the strip is in correspondence to a non trivial change in the
boundary conditions.

The boundary conditions are instead:

$\evaluated{\quantity(\variation{\uppsi}_{+,\,i}^{*}\uppsi_{+}^{i}+\variation{\uppsi}_{-,\,i}^{*}\uppsi_{-}^{i}-\uppsi_{+,\,i}^{*}\variation{\uppsi}_{+}^{i}-\uppsi_{-,\,i}^{*}\variation{\uppsi}_{-}^{i})}_{\upsigma=0}^{\upsigma=\uppi}=0.$
(2.3)

We solve the constraint imposing the non trivial relations:

$\begin{cases}\uppsi_{-}^{i}(\uptau,0)=\quantity(\mathrm{R}_{\quantity(t)})^{i}_{j}\uppsi^{j}_{+}(\uptau,0)&\mbox{\quad
for\quad}\uptau\in\quantity(\hat{\uptau}_{t},\hat{\uptau}_{t-1}),\\\
\uppsi_{-}^{i}(\uptau,\uppi)=-\uppsi_{+}^{i}(\uptau,\uppi)&\mbox{\quad
for\quad}\uptau\in\mathds{R},\end{cases}$ (2.4)

where $t=1,2,\dots,N$. This way we introduced $N$ zero-dimensional defects on
the boundary as in Figure 2.1. They are located on the strip at
$(\hat{\uptau}_{t},0)$ such that $\hat{\uptau}_{t}<\hat{\uptau}_{t-1}$ with
$\hat{\uptau}_{N+1}=-\infty$ and $\hat{\uptau}_{0}=+\infty$. We have also
introduced $N$ matrices
$\mathrm{R}_{\quantity(t)}\in\mathrm{U}(\mathrm{N}_{f})$ which characterize
the defects.

Since in most of this paper we want the in- and out-vacua to be the usual NS
vacuum, we have chosen the boundary condition at $\upsigma=\uppi$ so that when
there are no defects the system describes NS fermions. We require also the
cancellation of the action of the defects at $\hat{\uptau}=\pm\infty$, i.e.:

$\mathrm{R}_{\quantity(N)}\mathrm{R}_{\quantity(N-1)}\dots\mathrm{R}_{\quantity(1)}=\mathds{1}.$

More general cases where in- and/or out-vacua are twisted can be worked out
similarly to what we do.

In order to connect to the Euclidean formulation we introduce $\mathrm{N}_{f}$
“double fields”333In this case they correspond to the fields $\uppsi^{i}_{+}$.
$\Uppsi^{i}$ obtained by gluing $\uppsi^{i}_{+}$ and $\uppsi^{i}_{-}$ along
the $\upsigma=\uppi$ boundary and labeled by an index
$i=1,2,\dots,\mathrm{N}_{f}$:

$\Uppsi^{i}(\uptau,\upphi)=\begin{cases}\uppsi^{i}_{+}(\uptau,\upphi)&\mbox{\quad
for\quad}0\leq\upphi\leq\uppi\\\
-\uppsi^{i}_{-}(\uptau,2\uppi-\upphi)&\mbox{\quad for\quad}\uppi\leq\upphi\leq
2\uppi\end{cases}$ (2.5)

where $0\leq\upphi\leq 2\uppi$. The boundary conditions become:

$\Uppsi^{i}(\uptau,2\uppi)=-\quantity(\mathrm{R}_{\quantity(t)})^{i}_{j}\Uppsi^{j}(\uptau,0),\quad\uptau\in\quantity(\hat{\uptau}_{t},\hat{\uptau}_{t-1}).$

Using the equation of motion we get
$\Uppsi^{i}(\uptau,\upphi)=\Uppsi^{i}(\uptau+\upphi)$ and the boundary
conditions become the (pseudo)periodicity conditions

$\Uppsi^{i}(\uptau+2\uppi)=-\quantity(\mathrm{R}_{\quantity(t)})^{i}_{j}\Uppsi^{j}(\uptau),\quad\uptau\in\quantity(\hat{\uptau}_{t},\hat{\uptau}_{t-1}).$

We will use them to write some expressions similar to the Euclidean ones.

The main issue is now to expand $\Uppsi$ in a basis of modes and proceed to
its quantization. Even in the simplest case $\mathrm{N}_{f}=1$ 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.

##  3 Conserved Product and Charges

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

###  3.1 Definition of the Conserved Product

Start from a generic conserved current

$j(\uptau,\upsigma)=j_{\uptau}(\uptau,\upsigma)\differential{\uptau}+j_{\upsigma}(\uptau,\upsigma)\differential{\upsigma},$

and consider

$\star
j=j_{\upsigma}\differential{\uptau}+j_{\uptau}\differential{\upsigma}\quad\Rightarrow\quad\differential{(\star
j)}=\quantity(\partial_{\uptau}j_{\uptau}-\partial_{\upsigma}j_{\upsigma})\differential{\uptau}\differential{\upsigma},$

where $\star$ is the Hodge dual operator. Integrating the 2-form over a
surface
$\Upsigma^{\prime}=\quantity[\uptau_{i},\uptau_{f}]\times\quantity[0,\uppi]$
yields:

$\begin{split}\int\limits_{\Upsigma^{\prime}}\differential{(\star
j)}=\int\limits_{\partial\Upsigma^{\prime}}\star j&=0\Leftrightarrow\\\
\Leftrightarrow\int\limits_{0}^{\uppi}\differential{\upsigma}\quantity(\evaluated{j_{\uptau}}_{\uptau=\uptau_{f}}-\evaluated{j_{\uptau}}_{\uptau=\uptau_{i}})&=\int\limits_{\uptau_{i}}^{\uptau_{f}}\differential{\uptau}\quantity(\evaluated{j_{\upsigma}}_{\upsigma=\uppi}-\evaluated{j_{\upsigma}}_{\upsigma=0}).\end{split}$

The current $j_{\uptau}(\uptau,\upsigma)$ is thus conserved in time if

$\int\limits_{\uptau_{i}}^{\uptau_{f}}\differential{\uptau}\quantity(\evaluated{j_{\upsigma}}_{\upsigma=\uppi}-\evaluated{j_{\upsigma}}_{\upsigma=0})=0.$
(3.1)

If this is the case, then we can say that

$Q=\int\limits_{0}^{\uppi}\differential{\upsigma}j_{\uptau}(\uptau,\upsigma)$

is conserved (that is $\partial_{\uptau}Q=0$).

We now consider explicitly the symmetries of the action (2.1). In particular
we focus on the diffeomorphism invariance and $\mathrm{U}(\mathrm{N}_{f})$
flavour symmetries of the bulk theory leading to the stress-energy tensor and
a vector current. We apply the aforementioned procedure to these objects to
study their properties and (non-)conservation.

####  3.1.1 Flavour Vector Current

Consider first the $\mathrm{U}(\mathrm{N}_{f})$ vector current generated by
the flavour symmetry of the action (2.1). In general we can write it as

$j_{\upalpha}^{a}(\uptau,\upsigma)=\quantity(\mathrm{T}^{a})^{i}_{j}\overline{\uppsi}_{i}(\uptau,\upsigma)\upgamma_{\upalpha}\uppsi^{j}(\uptau,\upsigma),$

where $\mathrm{T}^{a}$ is in principle a generator of
$\mathrm{U}(\mathrm{N}_{f})$ ($a=1,2,\dots,\mathrm{N}_{f}^{2}$), but the
result holds for a generic matrix. The spinors $\uppsi$ and
$\overline{\uppsi}$ can also be generalized to two different and arbitrary
solutions to the equations of motion (2.2) while keeping the current
conserved. In components we have:

$\displaystyle j^{a}_{\uptau}(\uptau,\upsigma)$ $\displaystyle=$
$\displaystyle\quantity(\mathrm{T}^{a})^{i}_{j}\quantity(\uppsi^{*}_{+,\,i}\uppsi^{j}_{+}+\uppsi^{*}_{-,\,i}\uppsi^{j}_{-})$
$\displaystyle j^{a}_{\upsigma}(\uptau,\upsigma)$ $\displaystyle=$
$\displaystyle\quantity(\mathrm{T}^{a})^{i}_{j}\quantity(\uppsi^{*}_{+,\,i}\uppsi^{j}_{+}-\uppsi^{*}_{-,\,i}\uppsi^{j}_{-}).$

In order to define a conserved charge, we require:

$\int\limits_{\uptau_{i}}^{\uptau_{f}}\differential{\uptau}\quantity(\evaluated{j_{\upsigma}^{a}}_{\upsigma=\uppi}-\evaluated{j_{\upsigma}^{a}}_{\upsigma=0})=0,$

where

$\evaluated{j_{\upsigma}^{a}(\uptau,\upsigma)}_{\upsigma=\uppi}\equiv 0$

using the boundary conditions (2.3), and

$\evaluated{j_{\upsigma}^{a}(\uptau,\upsigma)}_{\upsigma=0}=\quantity[\uppsi^{*}_{+}\quantity(\mathrm{T}^{a}-\mathrm{R}_{\quantity(t)}^{\dagger}\mathrm{T}^{a}\mathrm{R}_{\quantity(t)})\uppsi_{+}]_{\upsigma=0},\quad\uptau\in\quantity(\hat{\uptau}_{t},\hat{\uptau}_{t-1}).$

In general

$\evaluated{j_{\upsigma}^{a}(\uptau,\upsigma)}_{\upsigma=0}=0\quad\Leftrightarrow\quad\mathrm{T}^{a}\propto\mathds{1}$

so that
$\mathrm{R}_{\quantity(t)}^{\dagger}\mathrm{T}^{a}=\mathrm{T}^{a}\mathrm{R}_{\quantity(t)}^{\dagger}$.
This shows that the presence of the point-like defects on the worldsheet
generally breaks the $\mathrm{U}(\mathrm{N}_{f})$ symmetry
($\mathrm{SO}(\mathrm{N}_{f})\times\mathrm{SO}(\mathrm{N}_{f})$ if we consider
Majorana-Weyl fermions) down to a $\mathrm{U}(1)$ phase because of the
boundary conditions (2.3). The $\mathrm{U}(1)$ vector current then defines a
conserved charge for a restricted class of functions.

Let $\upalpha$ and $\upbeta$ be two arbitrary (bosonic) solutions to the
equations of motion (2.2), we can in fact define a product

$\left\langle\upalpha,\upbeta\right\rangle=\mathcal{N}\int\limits_{0}^{\uppi}\differential{\upsigma}\quantity(\upalpha_{+,\,i}^{*}\upbeta_{+}^{i}+\upalpha_{-,\,i}^{*}\upbeta_{-}^{i}),$
(3.2)

where $\mathcal{N}\in\mathds{R}$ is a normalization constant and the integrand
must be free of non integrable singularities. The product is such that

$\left\langle\upalpha,\upbeta\right\rangle=\left\langle\upalpha,\upbeta\right\rangle^{*}.$

We can also rewrite the result to the double fields defined in (2.5). Let $A$
and $B$ be the “double fields” corresponding to $\upalpha$ and $\upbeta$
respectively, then we have:

$\left\langle\upalpha,\upbeta\right\rangle=\mathcal{N}\int\limits_{0}^{2\uppi}\differential{\upphi}A_{i}^{*}(\uptau+\upphi)B^{i}(\uptau+\upphi).$
(3.3)

####  3.1.2 Stress-Energy Tensor

We now consider the stress-energy tensor of the bulk theory. The Noether
procedure gives the off-shell tensor components

$\begin{split}\mathcal{T}_{\pm\pm}=&-i\frac{T}{4}\uppsi_{\pm,\,i}^{*}\overset{\leftrightarrow}{\partial_{\pm}}\uppsi_{+}^{i},\\\
\mathcal{T}_{\pm\mp}=&+i\frac{T}{4}\uppsi_{\mp,\,i}^{*}\overset{\leftrightarrow}{\partial_{\pm}}\uppsi_{\mp}^{i},\end{split}$

which become

$\begin{split}\mathcal{T}_{++}(\upxi_{+})=&-i\frac{T}{4}\uppsi_{+,\,i}^{*}(\upxi_{+})\overset{\leftrightarrow}{\partial_{+}}\uppsi_{+}^{i}(\upxi_{+}),\\\
\mathcal{T}_{--}(\upxi_{-})=&-i\frac{T}{4}\uppsi_{-,\,i}^{*}(\upxi_{-})\overset{\leftrightarrow}{\partial_{-}}\uppsi_{-}^{i}(\upxi_{-})\end{split}$
(3.4)

when on-shell. The boundary breaks the symmetry for translations in the
$\upsigma$ direction, while the defects break the time translations: the
Hamiltonian is therefore time-dependent but it is constant between two
consecutive point-like defects.

From the definition of the stress-energy tensor we can in principle build the
hypothetical charges:

$\displaystyle\mathrm{H}(\uptau)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{\uppi}\differential{\upsigma}\mathcal{T}_{\uptau\uptau}(\uptau,\upsigma)=\int\limits_{0}^{\uppi}\differential{\upsigma}\quantity(\mathcal{T}_{++}(\uptau+\upsigma)+\mathcal{T}_{--}(\uptau-\upsigma)),$
(3.5) $\displaystyle\mathrm{P}(\uptau)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{\uppi}\differential{\upsigma}\mathcal{T}_{\uptau\upsigma}(\uptau,\upsigma)=\int\limits_{0}^{\uppi}\differential{\upsigma}\quantity(\mathcal{T}_{++}(\uptau+\upsigma)-\mathcal{T}_{--}(\uptau-\upsigma)),$
(3.6)

which are conserved if (3.1) holds. Let the point-like defects be ordered as
$\hat{\uptau}_{t_{0}-1}<\uptau_{i}\leq\hat{\uptau}_{t_{0}}<\hat{\uptau}_{t_{N}}\leq\uptau_{f}<\hat{\uptau}_{t_{N}+1}$,
then for the linear momentum $\mathrm{P}$ the condition of conservation
reads444 Notice that in the second term of the second line, the
differentiation with respect to $\uptau$ is acting only on
$\mathrm{R}_{\quantity(t)}$ and $\mathrm{R}_{\quantity(t)}^{\dagger}$. :

$\begin{split}&\int\limits_{\uptau_{i}}^{\uptau_{f}}\differential{\uptau}\evaluated{\quantity(\mathcal{T}_{++}(\uptau+\upsigma)+\mathcal{T}_{--}(\uptau-\upsigma))}_{\upsigma=0}^{\upsigma=\uppi}\\\
&=-i\frac{T}{4}\int\Updelta\uptau\quantity(2\evaluated{\uppsi_{+,\,i}^{*}\overset{\leftrightarrow}{\partial_{\uptau}}\uppsi_{+}^{i}}_{\upsigma=0}^{\upsigma=\uppi}-\evaluated{\uppsi_{+,\,i}^{*}\quantity(\mathrm{R}_{\quantity(t)}^{\dagger}\overset{\leftrightarrow}{\partial_{\uptau}}\mathrm{R}_{\quantity(t)})^{i}_{j}\uppsi_{+}^{j}}_{\upsigma=0})\neq
0,\end{split}$

while the corresponding condition for the Hamiltonian $\mathrm{H}$:

$\begin{split}&\int\limits_{\uptau_{i}}^{\uptau_{f}}\differential{\uptau}\evaluated{\quantity(\mathcal{T}_{++}(\uptau+\upsigma)-\mathcal{T}_{--}(\uptau-\upsigma))}_{\upsigma=0}^{\upsigma=\uppi}\\\
&=-i\frac{T}{4}\int\Updelta\uptau\quantity(\evaluated{\uppsi_{+,\,i}^{*}\quantity(\mathrm{R}_{\quantity(t)}^{\dagger}\overset{\leftrightarrow}{\partial_{\uptau}}\mathrm{R}_{\quantity(t)})^{i}_{j}\uppsi_{+}^{j}}_{\upsigma=0})=0\mbox{\quad
if\quad}\quantity(\uptau_{i},\uptau_{f})\in\quantity(\hat{\uptau}_{t},\hat{\uptau}_{t-1}).\end{split}$

In both cases we used the shorthand graphical notation

$\int\Updelta\uptau=\quantity(\int\limits_{\uptau_{i}}^{\hat{\uptau}_{t_{0}}}+\sum\limits_{t=t_{0}}^{t_{N}-1}\int\limits_{\hat{\uptau}_{t}}^{\hat{\uptau}_{t+1}}+\int\limits_{\hat{\uptau}_{N}}^{\uptau_{f}})\differential{\uptau}$

to simplify the written form of the (non-)conservation rules and to stress the
piecewise nature of the integration domain due to the presence of the defects.

These relations therefore prove that the generator of the
$\upsigma$-translations (3.6) is not conserved in time because of the boundary
conditions, while the time evolution operator $\mathrm{H}$ is only piecewise
conserved and therefore globally time dependent.

###  3.2 Basis of Solutions and Dual Modes

Suppose to have a complete basis of modes $\uppsi_{n,\,\pm}^{i}$ such that:

$\begin{cases}\uppsi_{n,\,+}^{i}(\uptau,0)=\quantity(\mathrm{R}_{\quantity(t)})^{i}_{j}\uppsi_{n,\,-}^{j}(\uptau,0)&\mbox{\quad
for\quad}\uptau\in\quantity(\hat{\uptau}_{t},\hat{\uptau}_{t-1})\\\
\uppsi_{n,\,+}^{i}(\uptau,\uppi)=-\uppsi_{n,\,-}^{i}(\uptau,\uppi)&\mbox{\quad
for\quad}\uptau\in\mathds{R}\end{cases},$

related to a complete basis of the modes of the “double field”
$\Uppsi_{n}^{i}$ as in (2.5). The modes $\uppsi_{n}$ (and their counterparts
$\Uppsi_{n}$) are a basis of solutions of the equations of motion and the
boundary conditions for
$\uptau\in\mathds{R}\setminus\quantity{\hat{\uptau}_{t}}_{0\leq t\leq N}$. The
fields $\uppsi^{i}$ (and the fields $\Uppsi^{i}$) are then a superposition of
such modes:

$\uppsi^{i}_{\pm}(\upxi_{\pm})=\sum\limits_{n\in\mathds{Z}}b_{n}\uppsi^{i}_{n,\,\pm}(\upxi_{\pm})\quad\Rightarrow\quad\Uppsi^{i}(\upxi)=\sum\limits_{n\in\mathds{Z}}b_{n}\Uppsi^{i}_{n}(\upxi).$
(3.7)

In order to extract the “coefficients” $b_{n}$ we first introduce the dual
basis ${}^{*}\uppsi_{n,\,\pm}$ (and ${}^{*}\Uppsi_{n}$) in an abstract sense
such that:

  * •

the dual fields ${}^{*}\uppsi_{n,\,\pm}$ (and ${}^{*}\Uppsi_{n}$) must be
solutions to the equations of motion,

  * •

the dual fields ${}^{*}\uppsi_{n,\,\pm}$ (and ${}^{*}\Uppsi_{n}$) can differ
from $\uppsi_{n,\,\pm}$ (and $\Uppsi_{n}$) in their behavior at the boundary,

  * •

the functional form of ${}^{*}\uppsi_{n,\,\pm}$ (and ${}^{*}\Uppsi_{n}$) is
fixed by the request of time invariance of the usual anti-commutation
relations $\quantity[b_{n},b_{m}^{\dagger}]_{+}$ (that is $b_{n}$ and
$b_{n}^{\dagger}$ can evolve in time, but their anti-commutation relations
must remain constant).

We then define the conserved product for the “double fields” (3.3) in such a
way that:

$\evaluated{\left\langle\left\langle{}^{*}\Uppsi_{n}\right.,\Uppsi_{m}\right\rangle}_{\uptau=\uptau_{0}}=\mathcal{N}\int\limits_{0}^{2\uppi}\differential{\upsigma}{}^{*}\Uppsi_{n,\,i}^{*}(\uptau+\upsigma)\Uppsi_{m}^{i}(\uptau+\upsigma)=\updelta_{n,m}.$
(3.8)

In the previous expression we changed the notation of the product in order to
stress that we are dealing with the space of solutions whose basis is
$\quantity{\Uppsi_{n}}$ and a dual space with basis
$\quantity{{}^{*}\Uppsi_{n}}$ 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 (3.9).

Given the previous product we can extract the operators as

$\displaystyle\left\langle\left\langle{}^{*}\Uppsi_{n}\right.,\Uppsi\right\rangle$
$\displaystyle=$ $\displaystyle b_{n},$
$\displaystyle\left\langle\left\langle{}^{*}\Uppsi_{n}^{*}\right.,\Uppsi^{*}\right\rangle$
$\displaystyle=$ $\displaystyle b_{n}^{\dagger}.$

As a consequence of the canonical anti-commutation relations

$\quantity[\Uppsi^{i}\quantity(\uptau,\upsigma),\Uppsi^{*}_{j}\quantity(\uptau,\upsigma^{\prime})]_{+}=\frac{2}{T}\updelta^{i}_{j}\updelta(\upsigma-\upsigma^{\prime}),$

we have then:

$\evaluated{\quantity[b_{n},b_{m}^{\dagger}]_{+}}_{\uptau=\uptau_{0}}=\frac{2}{T}\mathcal{N}\evaluated{\left\langle\left\langle{}^{*}\Uppsi_{n}\right.,{}^{*}\Uppsi_{m}\right\rangle}_{\uptau=\uptau_{0}}.$
(3.9)

As per its definition, the product (3.8) is time independent as long as the
integrand ${}^{*}\Uppsi_{n}^{*}\Uppsi_{m}$ is free of singularities at
$\uptau=\hat{\uptau}_{t}$ for $t=1,2,\dots,N$. Such request on the dual basis
automatically fixes its possible form. Clearly this does not exclude the
possibility to have singularities in $\Uppsi_{m}$ or ${}^{*}\Uppsi_{n}$
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 and defining the fields to
fulfill some basic requirements 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 easier to address in a Euclidean formulation and indeed
this is the way we will pursue.

##  4 Point-like Defect CFT: the Euclidean Formulation

The main motivation behind the Euclidean reformulation of the previous
sections is the fact that the solution to the equations of motion on the
Euclidean strip (or in the complex plane) might be easier to study than its
Lorentzian worldsheet form. This is specifically the case when
$\mathrm{R}_{\quantity(t)}\in\mathrm{U}(1)^{\mathrm{N}_{f}}\subset\mathrm{U}(\mathrm{N}_{f})$
on which we shall focus in this paper. The presence of a time dependent
Hamiltonian is however not completely standard and we can neither blindly
apply the usual Wick rotation nor the usual CFT techniques. We will then be a
bit pedantic in order not to miss anything.

In the following two subsections we focus on coordinate changes from the strip
to the upper plane not relying on the CFT properties since we have not shown
that the theory is a CFT. We then find the explicit expression of modes which
satisfy the equations of motion and the boundary conditions and compute the
dual modes. Finally we show the algebra of the creation and annihilation
operators. This step is conceptually separated from the definition of the Fock
space where this algebra is represented: we will in fact take care of it in
the following sections.

###  4.1 Fields on the Strip

Performing the Wick rotation as $\uptau_{E}=i\uptau$ such that
$e^{iS_{M}}=e^{-S_{E}}$ the Minkowskian action (2.1) becomes555 We define the
coordinates $\upxi=\uptau_{E}+i\upsigma$, $\bar{\upxi}=\uptau_{E}-i\upsigma$
such that $\overline{\upxi}=\upxi^{*}$ and:
$\partial_{\upxi}=\partialderivative{\upxi}=\frac{1}{2}\quantity(\partialderivative{\uptau_{E}}-i\partialderivative{\upsigma}),$
$\partial_{\overline{\upxi}}=\partialderivative{\overline{\upxi}}=\frac{1}{2}\quantity(\partialderivative{\uptau_{E}}+i\partialderivative{\upsigma}).$
:

$S_{E}=\frac{T}{2}\iint\differential{\upxi}\differential{\overline{\upxi}}\frac{1}{2}~{}\matrixquantity({\widehat{\uppsi}}_{E,\,+,\,i}^{*}\overset{\leftrightarrow}{\partial_{\overline{\upxi}}}{\widehat{\uppsi}}_{E,\,+}^{i}+{\widehat{\uppsi}}_{E,\,-,\,i}^{*}\overset{\leftrightarrow}{\partial_{\upxi}}{\widehat{\uppsi}}_{E,\,-}^{i}),$
(4.1)

where the Euclidean fermion on the strip is connected to the Minkowskian
formulation through

${\widehat{\uppsi}}_{E,\,\pm}^{i}(\upxi,\overline{\upxi})=\uppsi_{\pm}^{i}(-i\upxi,-i\overline{\upxi}).$

As a consequence, the Euclidean “complex conjugation” $\star$ (which can be
defined off-shell) acts as

$\quantity[{\widehat{\uppsi}}_{E,\,\pm}^{i}(\upxi,\overline{\upxi})]^{\star}={\widehat{\uppsi}}_{E,\,\pm
i}^{*}(-\overline{\upxi},-\upxi).$ (4.2)

The equations of motion are as usual

$\displaystyle\partial_{\upxi}{\widehat{\uppsi}}_{E,\,-}^{i}(\upxi,\overline{\upxi})=$
$\displaystyle\partial_{\overline{\upxi}}{\widehat{\uppsi}}_{E,\,+}^{i}(\upxi,\overline{\upxi})=$
$\displaystyle 0,$
$\displaystyle\partial_{\upxi}{\widehat{\uppsi}}_{E,\,-,\,i}^{*}(\upxi,\overline{\upxi})=$
$\displaystyle\partial_{\overline{\upxi}}{\widehat{\uppsi}}_{E,\,+,\,i}^{*}(\upxi,\overline{\upxi})=$
$\displaystyle 0,$

whose solutions are the holomorphic functions
${\widehat{\uppsi}}_{E,\,+}(\upxi)$ and
${\widehat{\uppsi}}_{E,\,-}(\overline{\upxi})$ (and
${\widehat{\uppsi}}_{E,\,+}^{*}(\upxi)$ and
${\widehat{\uppsi}}_{E,\,-}^{*}(\overline{\upxi})$). In these coordinates the
boundary conditions (2.4) translate to:

$\begin{cases}{\widehat{\uppsi}}_{E,\,-}^{i}(\uptau_{E}-i0^{+})&=\quantity(\mathrm{R}_{\quantity(t)})^{i}_{j}{\widehat{\uppsi}}_{E,\,+}^{j}(\uptau_{E}+i0^{+})\\\
{\widehat{\uppsi}}_{E,\,-,\,i}^{*}(\uptau_{E}-i0^{+})&=\quantity(\mathrm{R}_{\quantity(t)}^{*})_{i}^{j}{\widehat{\uppsi}}_{E,\,+,\,j}^{*}(\uptau_{E}+i0^{+})\end{cases}$
(4.3)

for $\uptau_{E}\in\quantity(\hat{\uptau}_{Et},\hat{\uptau}_{Et-1})$ and

$\begin{cases}{\widehat{\uppsi}}_{E,\,-}^{i}(\uptau_{E}-i\uppi)&=-{\widehat{\uppsi}}_{E,\,+}^{i}(\uptau_{E}+i\uppi)\\\
{\widehat{\uppsi}}_{E,\,-,\,i}^{*}(\uptau_{E}-i\uppi)&=-{\widehat{\uppsi}}_{E,\,+,\,i}^{*}(\uptau_{E}+i\uppi)\end{cases},$

where $t=1,2,\dots,N$ and $\hat{\uptau}_{E,\,t}$ are the Wick-rotated
locations of the $N$ zero-dimensional defects, analytically continued to a
real value.

The conserved product on the strip needs a slight change in the definition and
becomes:

$\left\langle\widehat{\upalpha}^{*}_{E},\widehat{\upbeta}_{E}\right\rangle=\mathcal{N}\int\limits_{0}^{\uppi}\differential{\upsigma}\quantity(\widehat{\upalpha}^{*}_{E,\,+,\,i}\widehat{\upbeta}_{E,\,+}^{i}+\widehat{\upalpha}^{*}_{E,\,-,\,i}\widehat{\upbeta}_{E,\,-}^{i}),$
(4.4)

where $\widehat{\upalpha}^{*}_{E}$ and $\widehat{\upbeta}_{E}$ are the
Euclidean counterparts of the generic solutions in the original definition of
the product in (3.2). In the Euclidean context we have to explicitly write
$\widehat{\upalpha}^{*}_{E}$ because it is no longer the “complex conjugate”
of $\widehat{\upalpha}_{E}$ in the traditional sense but the product is
conserved only when it couples two solutions which have different boundary
conditions as in (4.3).

The definition of the stress-energy tensor in (3.4) requires a change in the
numerical factor in order to use the usual CFT normalization666 The canonical
coefficient in front of the CFT stress-energy tensor is such that the
Euclidean Hamiltonian $\mathrm{L}_{0}$ is normalized such that
$\mathcal{T}_{\upzeta\upzeta}(\upzeta)=\sum_{n}\mathrm{L}_{n}e^{-n\upzeta}$
(we have anticipated the double strip notation defined in the next subsection
for simplicity) then
$\mathrm{H}_{E}=\mathrm{L}_{0}=\int\limits_{0}^{2\uppi}\frac{\differential{\upphi}}{2\uppi}\mathcal{T}_{\upzeta\upzeta}(\uptau_{E}+i\upphi)$
therefore
$\mathcal{T}_{\upzeta\upzeta}(\upzeta)=2\uppi\mathcal{T}^{(can)}_{\upzeta\upzeta}(\upzeta)$.
and becomes (introducing a spacetime variable central charge as well):

$\begin{split}\mathcal{T}_{\upxi\upxi}(\upxi)=&-\frac{\uppi
T}{2}{\widehat{\uppsi}}_{E,\,+,\,i}^{*}(\upxi)\overset{\leftrightarrow}{\partial_{\upxi}}{\widehat{\uppsi}}^{i}_{E,\,+}(\upxi)+\widehat{\mathcal{C}}(\upxi),\\\
\mathcal{T}_{\overline{\upxi}\overline{\upxi}}(\overline{\upxi})=&-\frac{\uppi
T}{2}{\widehat{\uppsi}}_{E,\,-,\,i}^{*}(\overline{\upxi})\overset{\leftrightarrow}{\partial_{\overline{\upxi}}}{\widehat{\uppsi}}^{i}_{E,\,-}(\overline{\upxi})+\widehat{\overline{\mathcal{C}}}(\overline{\upxi}),\end{split}$

where $\widehat{{\mathcal{C}}}$ and $\widehat{\overline{\mathcal{C}}}$ are the
leftover terms after the regularization of the singularities due to the normal
ordering.

The canonical anti-commutation relations are then

$\evaluated{\quantity[{\widehat{\uppsi}}_{E,\,\pm}^{i}(\upxi_{1},\overline{\upxi}_{1}),{\widehat{\uppsi}}_{E,\,\pm,\,j}^{*}(\upxi_{2},\overline{\upxi}_{2})]_{+}}_{\real\upxi_{1}=\real\upxi_{2}}=\frac{2}{T}\updelta^{i}_{j}\updelta\quantity(\imaginary\upxi_{1}-\imaginary\upxi_{2}).$

Given the Euclidean modes ${\widehat{\uppsi}}^{i}_{E,\,\pm,\,n}$ and
${\widehat{\uppsi}}^{*}_{E,\,\pm,\,n,\,i}$ (where $n\in\mathds{Z}$) we can
then define the dual modes ${}^{*}{\widehat{\uppsi}}^{i}_{E,\,n}$ and
${}^{*}{\widehat{\uppsi}}^{*}_{E,\,n,\,i}$ such that the conserved product
(4.4) between them gives:

$\left\langle\left\langle{}^{*}{\widehat{\uppsi}}^{*}_{E,\,n}\right.,{\widehat{\uppsi}}_{E,\,m}\right\rangle=\left\langle\left\langle{}^{*}{\widehat{\uppsi}}_{E,\,n}\right.,{\widehat{\uppsi}}^{*}_{E,\,m}\right\rangle=\updelta_{n,m}.$

We can then expand the fields as

$\begin{cases}{\widehat{\uppsi}}^{i}_{E,\,+}(\upxi)&=\sum\limits_{n\in\mathds{Z}}b_{n}{\widehat{\uppsi}}^{i}_{E,\,+,\,n}(\upxi)\\\
{\widehat{\uppsi}}^{i}_{E,\,-}(\overline{\upxi})&=\sum\limits_{n\in\mathds{Z}}b_{n}{\widehat{\uppsi}}^{i}_{E,\,-,\,n}(\overline{\upxi})\end{cases}$

and

$\begin{cases}{\widehat{\uppsi}}^{*}_{E,\,+,\,i}(\upxi)&=\sum\limits_{n\in\mathds{Z}}b^{*}_{n}{\widehat{\uppsi}}^{*}_{E,\,+,\,n,\,i}(\upxi)\\\
{\widehat{\uppsi}}^{*}_{E,\,-,\,i}(\overline{\upxi})&=\sum\limits_{n\in\mathds{Z}}b^{*}_{n}{\widehat{\uppsi}}^{*}_{E,\,-,\,n,\,i}(\overline{\upxi})\end{cases}$

in order to extract the operators through the conserved product

$b_{n}=\left\langle\left\langle{}^{*}{\widehat{\uppsi}}^{*}_{E,\,n}\right.,{\widehat{\uppsi}}_{E}\right\rangle,\qquad
b^{*}_{n}=\left\langle\left\langle{}^{*}{\widehat{\uppsi}}_{E,\,n}\right.,{\widehat{\uppsi}}^{*}_{E}\right\rangle,$

and get the anti-commutation relations at fixed Euclidean time as

$\evaluated{\quantity[b_{n},b^{*}_{m}]_{+}}_{\uptau_{E}=\uptau_{E,\,0}}=\frac{2\mathcal{N}}{T}\left\langle\left\langle{}^{*}{\widehat{\uppsi}}^{*}_{E,\,n}\right.,{}^{*}{\widehat{\uppsi}}_{E,\,m}\right\rangle.$

###  4.2 Double Strip Formalism and Doubling Trick

It is natural to use the usual doubling trick on the strip in order to
simplify the previous expressions by gluing the holomorphic and anti-
holomorphic fields along the $\upsigma=\uppi$ boundary. Define the coordinate
$\upzeta=\uptau_{E}+i\upphi$ with $0\leq\upphi\leq 2\uppi$, we then have

${{\widehat{\Uppsi}}}(\upzeta)=\begin{cases}{\widehat{\uppsi}}_{E,\,+}(\upzeta)&\mbox{\quad
for\quad}\upphi=\upsigma\in\quantity[0,\uppi]\\\
-{\widehat{\uppsi}}_{E,\,-}(\upzeta-2\uppi i)&\mbox{\quad
for\quad}\upphi=2\uppi-\upsigma\in\quantity[\uppi,2\uppi]\end{cases}$

on-shell (and similarly for ${{\widehat{\Uppsi}}}^{*}(\upzeta)$ with the
substitution
${{\widehat{\uppsi}}}_{E,\,\pm}\to{{\widehat{\uppsi}}}_{E,\,\pm}^{*}$). The
“complex conjugation” $\star$ acts on the off-shell double fields as

$\quantity[{\widehat{\Uppsi}}^{i}(\upzeta,\overline{\upzeta})]^{\star}={\widehat{\Uppsi}}_{i}^{*}(-\overline{\upzeta},-\upzeta),$

while the boundary conditions are translated into

$\begin{cases}{\widehat{\Uppsi}}^{i}(\uptau_{E}+2\uppi
i^{-})=-\quantity(\mathrm{R}_{\quantity(t)})^{i}_{j}{\widehat{\Uppsi}}^{j}(\uptau_{E}+i0^{+})\\\
{\widehat{\Uppsi}}^{*i}(\uptau_{E}+2\uppi
i^{-})=-\quantity(\mathrm{R}_{\quantity(t)}^{*})^{i}_{j}{\widehat{\Uppsi}}^{*j}(\uptau_{E}+i0^{+})\end{cases}$

for $\uptau_{E}\in\quantity(\hat{\uptau}_{E,\,t},\hat{\uptau}_{E,\,t-1})$. The
conserved product can then be defined as

$\left\langle\widehat{A}^{*},\widehat{B}\right\rangle=\mathcal{N}\int\limits_{0}^{2\uppi}\differential{\upphi}\widehat{A}^{*}_{i}(\uptau_{E}+i\upphi)\widehat{B}^{i}(\uptau_{E}+i\upphi),$

where $\widehat{A}^{*}$ and $\widehat{B}$ are the double fields connected to
$\widehat{\upalpha}^{*}_{E}$ and $\widehat{\upbeta}_{E}$ in the previous
definition on the strip. The holomorphic stress-energy tensor is then

$\mathcal{T}_{\upzeta\upzeta}(\upzeta)=-\frac{\uppi
T}{2}{\widehat{\Uppsi}}_{i}^{*}(\upzeta)\overset{\leftrightarrow}{\partial_{\upzeta}}{\widehat{\Uppsi}}^{i}(\upzeta)+\widehat{\mathcal{C}}(\upzeta)$

and the canonical anti-commutation relations are now

$\evaluated{\quantity[{\widehat{\Uppsi}}^{i}(\upzeta_{1}),{\widehat{\Uppsi}}_{j}^{*}(\upzeta_{2})]_{+}}_{\real\upzeta_{1}=\real\upzeta_{2}}=\frac{2}{T}\updelta^{i}_{j}\updelta\quantity(\imaginary\upzeta_{1}-\imaginary\upzeta_{2}).$

The advantage the double fields formulation is in the mode expansion of the
fields which clarifies that only one coefficient $b_{n}$ (or $b_{n}^{*}$) is
needed for both $\uppsi_{E,\,+}$ and $\uppsi_{E,\,-}$ (or for both
$\uppsi^{*}_{E,\,+}$ and $\uppsi^{*}_{E,\,-}$). In fact, given the Euclidean
modes ${\widehat{\Uppsi}}^{i}_{n}$ and ${\widehat{\Uppsi}}^{*}_{n,\,i}$ (where
$n\in\mathds{Z}$), we can define the dual modes
${}^{*}{\widehat{\Uppsi}}^{i}_{n}$ and ${}^{*}{\widehat{\Uppsi}}^{*}_{n,\,i}$
such that

$\left\langle\left\langle{}^{*}{\widehat{\Uppsi}}^{*}_{n}\right.,{\widehat{\Uppsi}}_{m}\right\rangle=\left\langle\left\langle{}^{*}{\widehat{\Uppsi}}_{n}\right.,{\widehat{\Uppsi}}^{*}_{m}\right\rangle=\updelta_{n,m},$

and expand the double fields as

${\widehat{\Uppsi}}^{i}(\upzeta)=\sum\limits_{n\in\mathds{Z}}b_{n}{\widehat{\Uppsi}}^{i}_{n}(\upzeta),\qquad{\widehat{\Uppsi}}^{*}_{i}(\upzeta)=\sum\limits_{n\in\mathds{Z}}b^{*}_{n}{\widehat{\Uppsi}}^{*}_{n}(\upzeta)$

We then extract the operators as

$b_{n}=\left\langle\left\langle{}^{*}{\widehat{\Uppsi}}^{*}_{n}\right.,{\widehat{\Uppsi}}\right\rangle,\qquad
b^{*}_{n}=\left\langle\left\langle{}^{*}{\widehat{\Uppsi}}_{n}\right.,{\widehat{\Uppsi}}^{*}\right\rangle$
(4.5)

and finally get the anti-commutation relations as

$\evaluated{\quantity[b_{n},b^{*}_{m}]_{+}}_{\uptau_{E}=\uptau_{E,\,0}}=\frac{2\mathcal{N}}{T}\left\langle\left\langle{}^{*}{\widehat{\Uppsi}}^{*}_{n}\right.,{}^{*}{\widehat{\Uppsi}}_{m}\right\rangle.$

###  4.3 Fields on the Upper Half Plane

Figure 4.1: Due to the conformal transformation from the (double) strip to the
complex plane, fields are glued on the $x<0$ semi-axis, while there are non
trivial discontinuities (in the figure they are represented by strips with
different values of opacity) for $x_{t}<x<x_{t-1}$ for $t=1,2,\dots,N$ and
where $x_{t}=\exp(\hat{\uptau}_{E,\,t})$.

To perform the actual computations we shall however consider another set of
coordinates on the upper half $\mathcal{H}$ of the complex plane:

$u=e^{\upxi}\in\mathcal{H}=\quantity{w\in\mathds{C}\mid\imaginary w\geq 0},$

where $\upxi=\uptau_{E}+i\upsigma$ and $\upsigma\in\quantity[0,\uppi]$ define
the usual strip, or on the entire complex plane:

$z=e^{\upzeta}\in\mathds{C},$

where $\upzeta=\uptau_{E}+i\upphi$ and $\upphi\in\quantity[0,2\uppi]$ define
the double strip. Under this change of coordinates the Euclidean action (4.1)
becomes

$\begin{split}S_{E}&=\frac{T}{2}\iint\differential{u}\differential{\overline{u}}\frac{1}{2}\quantity(\frac{1}{u}{\widehat{\uppsi}}_{E,\,+,\,i}^{*}\overset{\leftrightarrow}{\partial_{\overline{u}}}{\widehat{\uppsi}}_{E,\,+}^{i}+\frac{1}{\overline{u}}{\widehat{\uppsi}}_{E,\,-,\,i}^{*}\overset{\leftrightarrow}{\partial_{u}}{\widehat{\uppsi}}_{E,\,-}^{i})\\\
&=\frac{T}{2}\iint\differential{u}\differential{\overline{u}}\frac{1}{2}\quantity(\uppsi_{E,\,+,\,i}^{*}\overset{\leftrightarrow}{\partial_{\overline{u}}}\uppsi_{E,\,+}^{i}+\uppsi_{E,\,-,\,i}^{*}\overset{\leftrightarrow}{\partial_{u}}\uppsi_{E,\,-}^{i}),\end{split}$

where we have naturally introduced the off-shell field redefinitions

$\uppsi_{E,\,+}^{i}(u,\overline{u})=\frac{1}{\sqrt{u}}{\widehat{\uppsi}}_{E,\,+}^{i}(\upxi,\overline{\upxi}),\qquad\uppsi_{E,\,-}^{i}(u,\overline{u})=\frac{1}{\sqrt{\overline{u}}}{\widehat{\uppsi}}_{E,\,-}^{i}(\upxi,\overline{\upxi}).$
(4.6)

This way, in the Euclidean context, fields with the hat sign on top represent
strip and double strip definitions, while fields without the hat sign are
defined on $\mathcal{H}$ or $\mathds{C}$. We could have anticipated these
redefinitions from a CFT argument where

$\uppsi(u)=\evaluated{\quantity(\derivative{u}{\upxi})^{-\frac{1}{2}}{{\widehat{\uppsi}}}(\upxi)}_{\upxi=\ln(u)},$

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 (4.6), the “complex conjugation”
$\star$ then becomes

$\quantity[\uppsi_{E,\,+,\,i}(u,\overline{u})]^{\star}=\frac{1}{\overline{u}}\uppsi_{E,\,+,\,i}^{*}\quantity(\frac{1}{\overline{u}},\frac{1}{u}),\qquad\quantity[\uppsi_{E,\,-,\,i}(u,\overline{u})]^{\star}=\frac{1}{u}\uppsi_{E,\,-,\,i}^{*}\quantity(\frac{1}{\overline{u}},\frac{1}{u}).$

When we choose the cut of the square root on the real negative axis the
boundary conditions are translated into

$\begin{cases}\uppsi_{E,\,-}^{i}(x-i0^{+})=\quantity(\mathrm{R}_{\quantity(t)})^{i}_{j}\uppsi_{E,\,+}^{j}(x+i0^{+})\\\
\uppsi^{*}_{E,\,-,\,i}(x-i0^{+})=\quantity(\mathrm{R}_{\quantity(t)}^{*})_{i}^{j}\uppsi^{*}_{E,\,+,\,j}(x+i0^{+})\end{cases}$

for $x\in\quantity(x_{t},x_{t-1})$, where
$x_{t}=\exp(\hat{\uptau}_{E,\,t})>0$, and

$\uppsi_{E,\,-}^{i}(x-i0^{+})=\uppsi_{E,\,+}^{i}(x+i0^{+}),\qquad\uppsi^{*}_{E,\,-,\,i}(x-i0^{+})=\uppsi^{*}_{E,\,+,\,i}(x+i0^{+})$

for $x<0$.

The product (4.4) is then

$\left\langle\upalpha^{*},\upbeta\right\rangle=-i\mathcal{N}\int\limits_{\begin{subarray}{c}\absolutevalue{u}=\exp(\hat{\uptau}_{E}),\\\
0\leq\imaginary
u\leq\uppi\end{subarray}}\quantity[\upalpha^{*}_{+,\,i}(u)\upbeta_{+}^{i}(u)\differential{u}-\upalpha^{*}_{-,\,i}(\overline{u})\upbeta_{-}^{i}(\overline{u})\differential{\overline{u}}],$
(4.7)

and the stress-energy tensor777 Rewriting the operator part of the stress-
energy tensor from the strip formulation into the coordinates on $\mathcal{H}$
we actually get
$\mathcal{T}_{\upxi\upxi}(\upxi(u))=u^{2}\mathcal{T}_{uu}(u).$ The reason of
the presence of $u^{2}$ can be understood in two ways. Using GR we know that
$\mathcal{T}_{\upxi\upxi}(\upxi)(d\upxi)^{2}=\mathcal{T}_{uu}(u)(du)^{2}$.
Another more physical way is to notice that a translation in $\upxi$ is a
dilatation of $u$: the infinitesimal generator of $\upxi$ translation must be
the infinitesimal generator of $u$ dilatation, i.e.  $P_{\upxi}\sim\int
d\upsigma~{}\mathcal{T}_{\upxi\upxi}\sim D_{u}\sim\int duu\mathcal{T}_{uu}.$
becomes:

$\begin{split}\mathcal{T}_{uu}(u)=&-\frac{\uppi
T}{2}\uppsi_{E,\,+,\,i}^{*}(u)\overset{\leftrightarrow}{\partial_{u}}\uppsi^{i}_{E,\,+}(u)+\widehat{\mathcal{C}}(u),\\\
\mathcal{T}_{\overline{u}\overline{u}}(\overline{u})=&-\frac{\uppi
T}{2}\uppsi_{E,\,-,\,i}^{*}(\overline{u})\overset{\leftrightarrow}{\partial_{\overline{u}}}\uppsi^{i}_{E,\,-}(\overline{u})+\widehat{\overline{\mathcal{C}}}(\overline{u}).\end{split}$

Finally the anti-commutation relations are

$\begin{cases}\evaluated{\quantity[\uppsi_{E,\,+}^{i}(u_{1},\overline{u}_{1}),\uppsi_{E,\,+,\,j}^{*}(u_{2},\overline{u}_{2})]_{+}}_{\absolutevalue{u_{1}}=\absolutevalue{u_{2}}}&=\frac{2}{Tu_{1}}\updelta^{i}_{j}\updelta(\arg(u_{1})-\arg(u_{2}))\\\
\evaluated{\quantity[\uppsi_{E,\,-}^{i}(u_{1},\overline{u}_{1}),\uppsi_{E,\,-,\,j}^{*}(u_{2},\overline{u}_{2})]_{+}}_{\absolutevalue{u_{1}}=\absolutevalue{u_{2}}}&=\frac{2}{T\overline{u}_{1}}\updelta^{i}_{j}\updelta(\arg(u_{1})-\arg(u_{2})),\end{cases}$

which despite the strange look of the expression are perfectly compatible with
the definition (4.5) leading to:

$\quantity[b_{n},b^{*}_{m}]_{+}=\frac{2\mathcal{N}}{T}\left\langle\left\langle{}^{*}{\widehat{\uppsi}}^{*}_{E,\,n}\right.,{}^{*}{\widehat{\uppsi}}_{E,\,m}\right\rangle=\frac{2\mathcal{N}}{T}\left\langle\left\langle{}^{*}\uppsi^{*}_{E,\,n}\right.,{}^{*}\uppsi_{E,\,m}\right\rangle$

when the product $\left\langle\left\langle\cdot\right.,\cdot\right\rangle$ is
defined according to (4.7), we expand the fields in modes as

$\begin{cases}\uppsi^{i}_{E,\,+}(u)=\sum\limits_{n\in\mathds{Z}}b_{n}\uppsi^{i}_{E,\,+,\,n}(u)\\\
\uppsi^{i}_{E,\,-}(\overline{u})=\sum\limits_{n\in\mathds{Z}}b_{n}\uppsi^{i}_{E,\,-,\,n}(\overline{u})\end{cases}$

and

$\begin{cases}\uppsi^{*}_{E,\,+,\,i}(u)=\sum\limits_{n\in\mathds{Z}}b^{*}_{n}\uppsi^{*}_{E,\,+,\,n,\,i}(u)\\\
\uppsi^{*}_{E,\,-,\,i}(\overline{u})=\sum\limits_{n\in\mathds{Z}}b^{*}_{n}\uppsi^{*}_{E,\,-,\,n,\,i}(\overline{u})\end{cases}$

and ${}^{*}\uppsi_{E,\,n}$ and ${}^{*}\uppsi^{*}_{E,\,n}$ are the
corresponding dual modes on the upper half plane.

###  4.4 Fields on the Complex Plane and the Doubling Trick

As in the double strip formulation, we can use the doubling trick in order to
define the fields on the subset $\mathds{C}\setminus\quantity[x_{N},x_{1}]$:

$\Uppsi(z)=\begin{cases}\uppsi_{E,\,+}(u)&\mbox{\quad
for\quad}z=u\in\mathcal{H}\setminus\quantity[x_{N},x_{1}]\\\
\uppsi_{E,\,-}(\overline{u})&\mbox{\quad
for\quad}z=\overline{u}\in\mathcal{H}^{*}\setminus\quantity[x_{N},x_{1}]\end{cases}$

where $z=\exp(\uptau_{E}+i\upphi)=x+iy$ and
$\mathcal{H}^{*}=\quantity{w\in\mathds{C}\mid\imaginary w\leq 0}$ (the same
goes for $\Uppsi^{*}$ with the exchange
$\uppsi_{E,\,\pm}\to\uppsi_{E,\,\pm}^{*}$).

In this case the “complex conjugation” $\star$ acts off-shell as

$\quantity[\Uppsi^{i}(z,\overline{z})]^{\star}=\frac{1}{\bar{z}}\Uppsi_{i}^{*}\quantity(\frac{1}{\overline{z}},\frac{1}{z})$
(4.8)

and the boundary conditions are

$\begin{cases}\Uppsi^{i}(x-i0^{+})&=\quantity(\mathrm{R}_{\quantity(t)})^{i}_{j}\Uppsi^{j}(x+i0^{+}),\\\
\Uppsi^{*\,i}(x-i0^{+})&=\quantity(\mathrm{R}_{\quantity(t)}^{*})^{i}_{j}\Uppsi^{*\,j}(x+i0^{+}),\end{cases}$
(4.9)

for $x\in\quantity(x_{t},x_{t-1})$, where $x_{t}=\exp(\hat{\uptau}_{Et})>0$
for $t\in\quantity{1,2,\dots,N}$. When $x<0$ we get

$\begin{cases}\Uppsi(x-i0^{+})&=\Uppsi(x+i0^{+}),\\\
\Uppsi^{*}(x-i0^{+})&=\Uppsi^{*}(x+i0^{+})\end{cases}$ (4.10)

instead.

Given the relations $\differential{z}=iz\differential{\upphi}$ we can write
the conserved product (4.7) as:

$\left\langle
A^{*},B\right\rangle=2\uppi\mathcal{N}\oint\limits_{\absolutevalue{z}=\exp(\uptau_{E})}\frac{\differential{z}}{2\uppi
i}A^{*}_{i}(z)B^{i}(z),$ (4.11)

where we explicitly stressed that the integral has to be performed at a fixed
Euclidean time $\uptau_{E}$: 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 (3.4) in the
new coordinates:

$\mathcal{T}(z)=-\frac{\uppi
T}{2}\Uppsi^{*}_{i}(z)\overset{\leftrightarrow}{\partial_{z}}\Uppsi^{i}(z)+\mathcal{C}(z),$

where $\mathcal{T}=\mathcal{T}_{zz}$ for simplicity.

Finally the canonical anti-commutation relations between the fields are:

$\evaluated{\quantity[\Uppsi^{i}(z_{1}),\Uppsi_{j}^{*}(z_{2})]_{+}}_{\absolutevalue{z_{1}}=\absolutevalue{z_{2}}}=\frac{2}{Tz_{1}}\updelta^{i}_{j}\updelta(\arg(z_{1})-\arg(z_{2})),$

the fields expansion in modes reads

$\Uppsi^{i}(z)=\sum\limits_{n\in\mathds{Z}}b_{n}\Uppsi^{i}_{n}(z),\qquad\Uppsi^{*}_{i}(z)=\sum\limits_{n\in\mathds{Z}}b^{*}_{n}\Uppsi^{*}_{n.i}(z),$
(4.12)

and the anti-commutation relations among the operators are

$\quantity[b_{n},b^{*}_{m}]_{+}=\frac{2\mathcal{N}}{T}\left\langle\left\langle{}^{*}\Uppsi^{*}_{n}\right.,{}^{*}\Uppsi_{m}\right\rangle,$

when we introduce the dual modes ${}^{*}\Uppsi_{n}(z)$ and
${}^{*}\Uppsi^{*}_{n}(z)$ whose normalization is

$\left\langle\left\langle{}^{*}\Uppsi^{*}_{n}\right.,{\Uppsi}_{m}\right\rangle=\left\langle\left\langle{}^{*}\Uppsi_{n}\right.,{\Uppsi}^{*}_{m}\right\rangle=\updelta_{m,n}.$

##  5 Algebra of Creation and Annihilation Operators

In this section we find the explicit expression of the modes which satisfy the
equations of motion and the boundary conditions. We then compute the dual
fields and finally the algebra of the creators and annihilators.

###  5.1 NS Complex Fermions

In order to check that this formalism agrees with known results we start from
the simplest case at hand: NS complex fermions. Consider the usual definition:

$\begin{cases}\uppsi_{-}^{i}(\uptau,0)&=\uppsi_{+}^{i}(\uptau,0),\\\
\uppsi_{-}^{i}(\uptau,\uppi)&=-\uppsi_{+}^{i}(\uptau,\uppi)\end{cases}$

for $\uptau\in\mathds{R}$, which can be recovered from $\eqref{eq:boundary-
conditions-solutions}$ setting $\mathrm{R}_{\quantity(t)}\equiv\mathds{1}$ .
In the Euclidean formulation, we use (4.9) and (4.10) to get:

$\begin{cases}\Uppsi(x-i0^{+})&=\Uppsi(x+i0^{+})\\\
\Uppsi^{*}(x-i0^{+})&=\Uppsi^{*}(x+i0^{+})\end{cases}$

for $x\in\mathds{R}$.

In order to recover the definition of the dual modes (3.8) using the Euclidean
conserved product (4.11), we define:

$\displaystyle\Uppsi^{i}_{(n,i_{0})}(z)$ $\displaystyle=$
$\displaystyle\mathcal{N}_{\Uppsi}\updelta^{i}_{i_{0}}z^{-n},$
$\displaystyle{}^{*}\Uppsi_{(m,j_{0}),\,j}(z)$ $\displaystyle=$
$\displaystyle\quantity(2\uppi\mathcal{N}\mathcal{N}_{\Uppsi})^{-1}\updelta_{j,j_{0}}z^{m-1},$

and similarly for $\Uppsi^{*}$, in such a way that

$\left\langle\left\langle{}^{*}\Uppsi_{(n,i_{0})}^{*}\right.,\Uppsi_{(m,j_{0})}\right\rangle=\left\langle\left\langle{}^{*}\Uppsi_{(m,j_{0})}\right.,\Uppsi^{*}_{(n,i_{0})}\right\rangle=\updelta_{n,m}\updelta_{i_{0},j_{0}}.$

As a consequence we find

$\left\langle\left\langle{}^{*}\Uppsi_{(n,i_{0})}^{*}\right.,{}^{*}\Uppsi_{(m,i_{1})}\right\rangle=\frac{1}{2\uppi\mathcal{N}\mathcal{N}^{2}_{\Uppsi}}\updelta_{i_{0},i_{1}}\updelta_{n+m,1}.$

Consider the NS expansion in modes of the double fields:

$\displaystyle\Uppsi^{i}(z)$ $\displaystyle=$
$\displaystyle\sum\limits_{n\in\mathds{Z}}\sum\limits_{i_{0}}b_{(n,i_{0})}\Uppsi^{i}_{(n,i_{0})}(z),$
$\displaystyle\Uppsi^{*}_{i}(z)$ $\displaystyle=$
$\displaystyle\sum\limits_{n\in\mathds{Z}}\sum\limits_{i_{0}}b^{*}_{(n,i_{0})}\Uppsi^{*}_{(n,i_{0}),\,i}(z),$

then

$\displaystyle b_{(n,i_{0})}$ $\displaystyle=$
$\displaystyle\left\langle\left\langle{}^{*}\Uppsi_{(n,i_{0})}^{*}\right.,\Uppsi\right\rangle,$
$\displaystyle b^{*}_{(n,i_{0})}$ $\displaystyle=$
$\displaystyle\left\langle\left\langle{}^{*}\Uppsi_{(n,i_{0})}\right.,\Uppsi^{*}\right\rangle,$

and

$\quantity[b_{(n,i_{0})},b^{*}_{(m,j_{0})}]_{+}=\frac{1}{\uppi
T\mathcal{N}_{\Uppsi}^{2}}\updelta_{i_{0},j_{0}}\updelta_{n+m,1}.$ (5.1)

###  5.2 Twisted Complex Fermions: preliminaries

We can now move to a more general discussion of $\mathrm{N}_{f}=1$ 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

$\begin{cases}\mathrm{R}_{\quantity(t)}&=e^{i\uppi\upalpha_{(t)}}\in\mathrm{U}(1)\\\
\mathrm{R}_{\quantity(t)}^{*}&=e^{-i\uppi\upalpha_{(t)}}\in\mathrm{U}(1)\end{cases}$

such that $0<\upalpha_{(t)}<2$. We have the boundary conditions:

$\begin{cases}\Uppsi(x-i0^{+})&=e^{i\uppi\upalpha_{(t)}}\Uppsi(x+i0^{+})\\\
\Uppsi^{*}(x-i0^{+})&=e^{-i\uppi\upalpha_{(t)}}\Uppsi^{*}(x+i0^{+})\end{cases},$

for $x\in(x_{t},x_{t-1})$, and

$\begin{cases}\Uppsi(x-i0^{+})&=\Uppsi(x+i0^{+})\\\
\Uppsi^{*}(x-i0^{+})&=\Uppsi^{*}(x+i0^{+})\end{cases},$

for $x<0$. Also in this case we can refer to Figure 4.1 to keep in mind the
intuitive picture. The boundary conditions can be recast in the form of
monodromy factors. Performing a loop around $x_{t}$ we find

$\Uppsi(x_{t}+\updelta
e^{i0^{+}})=e^{i\uppi\quantity(\upalpha_{(t)}-\upalpha_{(t+1)})}\Uppsi(x_{t}+\updelta
e^{2\uppi i}),$

where $\updelta\in\mathds{R}^{+}$ is small enough888 Technically,
$0<\updelta<\min\quantity(\absolutevalue{x_{t-1}-x_{t}},\absolutevalue{x_{t}-x_{t+1}})$.
and the $\pm$ in the phase represents the position relative to the real axis
($+$ is in the upper half plane, while $-$ in the lower half plane). Let us
define999 Notice that the choice of the range for $\upepsilon_{\quantity(t)}$
is not unique. We can choose $0<\upalpha_{(t)}<2$ leading to
$\upepsilon_{\quantity(t)}=\upalpha_{(t+1)}-\upalpha_{(t)}+2\uptheta(\upalpha_{(t)}-\upalpha_{(t+1)})$
Then in this case
$\upepsilon_{\quantity(t)}=2-\overline{\upepsilon}_{\quantity(t)}$ and
$\upepsilon_{\quantity(t)},\,\overline{\upepsilon}_{\quantity(t)}\in\quantity(0,2)$.
We will however stick to the first definition in the following sections since
it allows to consider the NS case as special.

$\upepsilon_{\quantity(t)}=\upalpha_{(t+1)}-\upalpha_{(t)}+\uptheta(\upalpha_{(t)}-\upalpha_{(t+1)}-1)-\uptheta(\upalpha_{(t+1)}-\upalpha_{(t)}-1)$

such that

$-1<\upepsilon_{\quantity(t)}<1\qquad\forall t=1,2,\dots,N,$

then the previous loop around $x_{t}$ induces a monodromy

$\begin{cases}\Uppsi(x_{t}+\updelta
e^{i0^{+}})&=e^{-i\uppi\upepsilon_{\quantity(t)}}\Uppsi(x_{t}+\updelta
e^{2i\uppi^{+}})\\\ \Uppsi^{*}(x_{t}+\updelta
e^{i0^{+}})&=e^{-i\uppi\overline{\upepsilon}_{\quantity(t)}}\Uppsi^{*}(x_{t}+\updelta
e^{2i\uppi^{+}}),\end{cases}$ (5.2)

where
$\overline{\upepsilon}_{\quantity(t)}=-\upepsilon_{\quantity(t)}\Rightarrow-1<\overline{\upepsilon}_{\quantity(t)}<1$
thus showing a symmetry under the exchange of:

$\Uppsi\longleftrightarrow\Uppsi^{*}\qquad\Rightarrow\qquad\upepsilon_{\quantity(t)}\longleftrightarrow\overline{\upepsilon}_{\quantity(t)}.$

####  5.2.1 Usual Twisted Fermions

As it is useful in the discussion of the meaning of the defects, 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 $\upepsilon$ the
argument of the monodromy factor arising from the presence of the cut on the
interval $(0,+\infty)$.

In order to fulfill the requests (5.2) we can write the modes as:

$\begin{split}\Uppsi_{n}^{(\mathrm{E})}&=\mathcal{N}_{\Uppsi}z^{-n+\mathrm{E}},\\\
\Uppsi_{n}^{*\,(\overline{\mathrm{E}})}&=\mathcal{N}_{\Uppsi}z^{-n+\overline{\mathrm{E}}},\end{split}$
(5.3)

such that

$\begin{split}\mathrm{E}=n_{\mathrm{E}}+\frac{\upepsilon}{2},&\quad
n_{\mathrm{E}}\in\mathds{Z},\\\
\overline{\mathrm{E}}=n_{\overline{\mathrm{E}}}+\frac{\overline{\upepsilon}}{2},&\quad
n_{\overline{\mathrm{E}}}\in\mathds{Z}.\end{split}$

Together with the integer factor $n_{\mathrm{E}}$ and
$n_{\overline{\mathrm{E}}}$ we also define a third integer for later
convenience101010 The choice discussed in footnote 9 gives
$\mathrm{L}=n_{\mathrm{E}}+n_{\overline{\mathrm{E}}}+1$. We can easily
exchange the definitions using
$\overline{\upepsilon}_{\quantity(t)}^{2nd}=\overline{\upepsilon}_{\quantity(t)}^{1st}+2$
and $n^{2nd}_{\overline{\mathrm{E}}}=n^{1st}_{\overline{\mathrm{E}}}-1$. :

$\mathrm{L}=\mathrm{E}+\overline{\mathrm{E}}=n_{\mathrm{E}}+n_{\overline{\mathrm{E}}}\in\mathds{Z}.$

In order to extract the creators and annihilators from the conserved product
(4.11), we define the dual basis as:

$\displaystyle{}^{*}\Uppsi_{n}^{(\overline{\mathrm{E}})}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2\uppi\mathcal{N}\mathcal{N}_{\Uppsi}}z^{n-1-\overline{\mathrm{E}}},$
$\displaystyle{}^{*}\Uppsi_{n}^{*\,(\mathrm{E})}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2\uppi\mathcal{N}\mathcal{N}_{\Uppsi}}z^{n-1-\mathrm{E}}.$

This way we compute the usual anti-commutation relations as

$\left\langle\left\langle{}^{*}\Uppsi_{n}^{*\,(\mathrm{E})}\right.,{}^{*}\Uppsi_{m}^{(\overline{\mathrm{E}})}\right\rangle=\frac{\updelta_{n+m,1+\mathrm{L}}}{2\uppi\mathcal{N}\mathcal{N}_{\Uppsi}^{2}}\quad\Rightarrow\quad\quantity[b_{n},b_{m}^{*}]_{+}=\frac{1}{\uppi
T\mathcal{N}_{\Uppsi}^{2}}\updelta_{n+m,1+\mathrm{L}},$ (5.4)

which are constant in time independently of $\mathrm{E}$ or
$\overline{\mathrm{E}}$ since the only possible singularities are at $z=0$ and
$z=\infty$. We can then expand the fields $\Uppsi(z)$ and $\Uppsi^{*}(z)$
using this basis or the more conventional one as

$\displaystyle\Uppsi(z)$ $\displaystyle=$
$\displaystyle\sum\limits_{n\in\mathds{Z}}b^{(\mathrm{E})}_{n}\Uppsi_{n}^{(\mathrm{E})}(z)=\sum\limits_{n\in\mathds{Z}}b_{n+n_{\mathrm{E}}}\Uppsi_{n}^{(\frac{\upepsilon}{2})}(z),$
(5.5) $\displaystyle\Uppsi^{*}(z)$ $\displaystyle=$
$\displaystyle\sum\limits_{n\in\mathds{Z}}b^{*\,(\overline{\mathrm{E}})}_{n}\Uppsi_{n}^{*\,(\overline{\mathrm{E}})}(z)=\sum\limits_{n\in\mathds{Z}}b^{*}_{n+n_{\overline{\mathrm{E}}}}\Uppsi_{n}^{*\,(-\frac{\upepsilon}{2})}(z),$
(5.6)

where we have used the shorter notation $b=b^{(\frac{\upepsilon}{2})}$ and
$b^{*}=b^{*\,(\frac{\upepsilon}{2})}$.

####  5.2.2 Generic Case With Defects

We now consider one complex fermion in the presence of $N$ defects such that
the modes satisfy:

$\Uppsi_{n}(x_{t}+\updelta e^{2\uppi
i^{+}})=e^{i\uppi\upepsilon_{\quantity(t)}}\Uppsi_{n}(x_{t}+\updelta
e^{i0^{+}})$

for $t=1,2,\dots,N$ and $\updelta>0$. We define the basis of solutions as:

$\displaystyle\Uppsi_{n}(z;\\{x_{t},\mathrm{E}_{\quantity(t)}\\})$
$\displaystyle=$
$\displaystyle\mathcal{N}_{\Uppsi}z^{-n}\prod\limits_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{\mathrm{E}_{\quantity(t)}},$
(5.7)
$\displaystyle\Uppsi^{*}_{n}(z;\\{x_{t},\overline{\mathrm{E}}_{\quantity(t)}\\})$
$\displaystyle=$
$\displaystyle\mathcal{N}_{\Uppsi}z^{-n}\prod\limits_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{\overline{\mathrm{E}}_{\quantity(t)}},$
(5.8)

where we generalise the definition of

$\displaystyle\mathrm{E}_{\quantity(t)}=$ $\displaystyle
n_{\mathrm{E}_{\quantity(t)}}+\frac{\upepsilon_{\quantity(t)}}{2},$
$\displaystyle\quad n_{\mathrm{E}_{\quantity(t)}}\in\mathds{Z},$
$\displaystyle\overline{\mathrm{E}}_{\quantity(t)}=$ $\displaystyle
n_{\overline{\mathrm{E}}_{\quantity(t)}}+\frac{\overline{\upepsilon}_{(t)}}{2}$
$\displaystyle\quad n_{\overline{\mathrm{E}}_{\quantity(t)}}\in\mathds{Z}$

and we define $N$ integer factors:

$\mathrm{L}_{\quantity(t)}=\mathrm{E}_{\quantity(t)}+\overline{\mathrm{E}}_{\quantity(t)}=n_{\mathrm{E}_{\quantity(t)}}+n_{\overline{\mathrm{E}}_{\quantity(t)}}\in\mathds{Z},$

for $t=1,2,\dots,N$, in analogy to (4.11). From the definition of the
conserved product (4.11), we compute the dual basis:

$\displaystyle{}^{*}\Uppsi_{n}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2\uppi\mathcal{N}\mathcal{N}_{\Uppsi}}z^{n-1}\prod\limits_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{-\overline{\mathrm{E}}_{\quantity(t)}},$
$\displaystyle{}^{*}\Uppsi^{*}_{n}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2\uppi\mathcal{N}\mathcal{N}_{\Uppsi}}z^{n-1}\prod\limits_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{-\mathrm{E}_{\quantity(t)}},$

and the conserved products between dual modes:

$\left\langle\left\langle{}^{*}\Uppsi_{n}^{*}\right.,{}^{*}\Uppsi_{m}\right\rangle=\frac{1}{2\uppi\mathcal{N}\mathcal{N}_{\Uppsi}^{2}}\oint\frac{\differential{z}}{2\uppi
i}z^{n+m-2}\prod\limits_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{-\mathrm{L}_{\quantity(t)}}.$

Notice that the products are radially invariant only if

$\mathrm{L}_{\quantity(t)}\leq 0\qquad\forall t\in\quantity{1,2,\dots,N},$
(5.9)

since the integrand must not present time dependent singularities on the
integration path, thus

$\begin{split}\left\langle\left\langle{}^{*}\Uppsi_{n}^{*}\right.,{}^{*}\Uppsi_{m}\right\rangle&=\frac{1}{2\uppi\mathcal{N}\mathcal{N}_{\Uppsi}^{2}}\oint\frac{\differential{z}}{2\uppi
i}\prod\limits_{t=1}^{N}\sum\limits_{k_{t}=0}^{\absolutevalue{\mathrm{L}_{\quantity(t)}}}\binom{\absolutevalue{\mathrm{L}_{\quantity(t)}}}{k_{t}}\quantity(-\frac{1}{x_{t}})^{k_{t}}z^{k_{t}+n+m-2}\\\
&=\frac{1}{2\uppi\mathcal{N}\mathcal{N}_{\Uppsi}^{2}}p_{1-n-m},\end{split}$

where we defined

$p_{k}=\prod\limits_{t=1}^{N}\sum\limits_{k_{t}=0}^{\absolutevalue{\mathrm{L}_{\quantity(t)}}}\binom{\absolutevalue{\mathrm{L}_{\quantity(t)}}}{k_{t}}\quantity(-\frac{1}{x_{t}})^{k_{t}}\updelta_{\sum\limits_{t=1}^{N}k_{t},k}$
(5.10)

such that

$\displaystyle p_{0\leq
k\leq\sum\limits_{t=1}^{N}\absolutevalue{\mathrm{L}_{\quantity(t)}}}$
$\displaystyle\neq$ $\displaystyle 0,$ $\displaystyle
p_{k\leq-1}=p_{k\geq\sum\limits_{t=1}^{N}\absolutevalue{\mathrm{L}_{\quantity(t)}}+1}$
$\displaystyle=$ $\displaystyle 0.$

We can finally write

$\quantity[b_{n},b^{*}_{m}]_{+}=\frac{1}{\uppi
T\mathcal{N}_{\Uppsi}^{2}}p_{1-n-m},\qquad
1-\sum\limits_{t=1}^{N}\absolutevalue{\mathrm{L}_{\quantity(t)}}\leq n+m\leq
1.$ (5.11)

##  6 Representation of the Algebra: Definition of the In-Vacuum

In the previous section we computed the algebra of the operators for different
theories. We now define in-vacua where they are represented. This is the first
step to define the building blocks for computing correlation functions. We
will show how to recover the usual NS vacuum and the usual twisted vacuum with
a slightly different twist from the usual. Finally we will discuss the vacuum
in the presence of a generic number of defects.

In the previous section we have seen that given the monodromies of the defects
we can have many different singularities. Since we want to identify the
defects as (excited) spin fields we want to understand what is the local
singularity associated with excited twisted vacua.

###  6.1 NS Fermions

The case of NS fermions is trivial since there are no defects. The in-vacuum
can be correctly obtained either by requiring $\Uppsi(z)$ and $\Uppsi^{*}(z)$
to be non singular as $z\to 0$ when applied on the vacuum or by the same
request on $\hat{\Uppsi}(\upxi)$ and $\hat{\Uppsi}^{*}(\upxi)$. In both cases
we get the same vacuum which turns out to be
${\mathrm{SL}_{2}\quantity(\mathds{R})}$ invariant:

$b_{(n,i_{0})}\ket{0}_{{\mathrm{SL}_{2}\quantity(\mathds{R})}}=b^{*}_{(n,i_{0})}\ket{0}_{{\mathrm{SL}_{2}\quantity(\mathds{R})}}=0,\quad
n\geq 1.$ (6.1)

The spectrum of the theory can be constructed acting with operators
$b_{(n,i_{0})}$ and $b^{*}_{(n,i_{0})}$ with $n\leq 0$.

###  6.2 Twisted Fermion

Consider the case of the usual twisted fermion in Section 5.2.1. We start from
the definition of the excited vacuum and work out the way to the minimum
energy configuration. We will then discuss the result.

####  6.2.1 Excited vacuum

Define the excited vacuum $\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}$ as:

$b^{(\mathrm{E})}_{n}\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}=b^{*\,(\overline{\mathrm{E}})}_{n}\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}=0,\quad
n\geq 1.$ (6.2)

The reason for the introduction of $E$ and $\bar{E}$ is to be able to define
this vacuum as above, i.e. with a $n$ range independent on them and, at the
same time, to have a non trivial singularity as $z\rightarrow 0$ which does
depend on them, explicitly

$\Uppsi(z)\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}\sim
z^{\mathrm{E}}(\dots),\qquad\Uppsi^{*}(z)\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}\sim
z^{\overline{\mathrm{E}}}(\dots).$ (6.3)

By comparison with (5.7) and (5.8) this behavior suggests that in the point
$x_{t}$ there is a hidden operator which creates
$\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}$ with
$\mathrm{E}=\mathrm{E}_{\quantity(t)}$ and
$\overline{\mathrm{E}}=\overline{\mathrm{E}}_{\quantity(t)}$.

These relations are subject to consistency conditions since

$\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}=\uppi
T\mathcal{N}_{\Uppsi}^{2}\quantity[b^{(\mathrm{E})}_{n},b^{*\,(\overline{\mathrm{E}})}_{\mathrm{L}+1-n}]_{+}\ket{T_{\mathrm{E},\overline{\mathrm{E}}}},$

that is we cannot have two in-annihilators (namely both $b^{(\mathrm{E})}_{n}$
and $b^{*\,(\overline{\mathrm{E}})}_{\mathrm{L}+1-n}$) with non vanishing
anti-commutation relations. Specifically we have that (see Figure 6.1 for a
graphic description):

$1\leq n\leq\mathrm{L}\quad\Rightarrow\quad
b^{(\mathrm{E})}_{n}\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}=0,\qquad
b^{*\,(\overline{\mathrm{E}})}_{\mathrm{L}+1-n}\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}=0,$

that is

$\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}=\uppi
T\mathcal{N}_{\Uppsi}^{2}\quantity[b^{(\mathrm{E})}_{n},b^{*\,(\overline{\mathrm{E}})}_{\mathrm{L}+1-n}]_{+}\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}=0,$

which is not consistent: the theory does not exist. We shall therefore
consider only cases such that

$\mathrm{L}\leq 0,$

analogously to (5.9).

Moreover notice that for $\mathrm{L}\leq-1$ both
$b^{(\mathrm{E})}_{\mathrm{L}\leq n\leq 0}$ and
$b^{*\,(\overline{\mathrm{E}})}_{\mathrm{L}\leq n\leq 0}$ are in- and out-
creation operators: in the next section we will show that this case is not
acceptable.

Figure 6.1: As a consistency condition, we have to exclude the values of
$\mathrm{L}$ for which both $b^{(\mathrm{E})}_{n}$ and
$b^{*\,(\overline{\mathrm{E}})}_{\mathrm{L}+1-n}$ are in-annihilators with a
non vanishing anti-commutation relation.

####  6.2.2 Minimum Energy Vacuum

The vacuum $\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}$ defined in the
previous section is not however associated to the lowest energy. In fact the
usual way to build the vacuum would be to require $\Uppsi(z)$ and
$\Uppsi^{*}(z)$ to be non singular as $z\to 0$ for the in-vacuum so that
$b^{(\mathrm{E})}_{n}\ket{\mathrm{T}}=0\mbox{\quad for\quad}n>\mathrm{E},$ and
$b^{*\,(\overline{\mathrm{E}})}_{n}\ket{\mathrm{T}}=0\mbox{\quad
for\quad}n>\overline{\mathrm{E}}.$ However this procedure almost always fails
to give a good definition of the vacuum (it works only for NS fermions). For
example when $\upepsilon>0$ we have:

$0=\uppi
T\mathcal{N}_{\Uppsi}^{2}\quantity[b^{(\mathrm{E})}_{1+n_{\mathrm{E}}},b^{*\,(\overline{\mathrm{E}})}_{n_{\overline{\mathrm{E}}}}]_{+}\ket{\mathrm{T}}=\ket{\mathrm{T}},$

which is not consistent since both $b^{(\mathrm{E})}_{1+n_{\mathrm{E}}}$ and
$b^{*\,(\overline{\mathrm{E}})}_{n_{\overline{\mathrm{E}}}}$ are annihilators
as $1+n_{\mathrm{E}}>E$ and $n_{\overline{\mathrm{E}}}>\bar{E}$.

The minimum energy vacuum is instead defined in a proper way on the strip.
Requiring that the action of $\hat{\Uppsi}(\upxi)$ and
$\hat{\Uppsi}^{*}(\upxi)$ for $\upxi\to-\infty$ on the vacuum is well defined
we get

$\displaystyle b^{(\mathrm{E})}_{n}\ket{\mathrm{T}}$ $\displaystyle=$
$\displaystyle 0,\quad n>\mathrm{E}+\frac{1}{2},$ $\displaystyle
b^{*\,(\overline{\mathrm{E}})}_{m}\ket{\mathrm{T}}$ $\displaystyle=$
$\displaystyle 0,\quad m>\overline{\mathrm{E}}+\frac{1}{2}.$

This is a good definition of the vacuum since
$-\frac{1}{2}<\frac{\upepsilon}{2}=-\frac{\overline{\upepsilon}}{2}<\frac{1}{2}$
implies that $b^{(\mathrm{E})}_{n}$ and $b^{*\,(\overline{\mathrm{E}})}_{m}$
are annihilation operators for $n\geq n_{\mathrm{E}}+1>\mathrm{E}+\frac{1}{2}$
and $m\geq n_{\overline{\mathrm{E}}}+1>\overline{\mathrm{E}}+\frac{1}{2}$ so
that

$0=\uppi
T\mathcal{N}_{\Uppsi}^{2}\quantity[b^{(\mathrm{E})}_{n},b^{*\,(\overline{\mathrm{E}})}_{m}]_{+}\ket{\mathrm{T}}=\updelta_{n+m,\mathrm{E}+\overline{\mathrm{E}}+1}\ket{\mathrm{T}}=0$

This way we get a consistent definition of the twisted vacuum111111 Notice
that the second choice of $\upepsilon$ interval discussed in footnote 9 needs
to distinguish between two cases: $0<\frac{\upepsilon}{2}<\frac{1}{2}$ (and
$\frac{1}{2}<\frac{\overline{\upepsilon}}{2}<1$) and
$\frac{1}{2}<\frac{\upepsilon}{2}<1$ (and
$0<\frac{\overline{\upepsilon}}{2}<\frac{1}{2}$).  which however is not in
general ${\mathrm{SL}_{2}\quantity(\mathds{R})}$ invariant as we will show
after the construction of the stress-energy tensor.

####  6.2.3 Relation Between Vacua

The two vacua $\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}$ and
$\ket{\mathrm{T}}$ are related. Consider for example, the case
$n_{\mathrm{E}}\geq 1$ and the definition of the vacua:

$\displaystyle b^{(\mathrm{E})}_{n}\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}$
$\displaystyle=$ $\displaystyle 0,\quad n\geq 1,$ $\displaystyle
b^{(\mathrm{E})}_{n}\ket{\mathrm{T}}$ $\displaystyle=$ $\displaystyle 0,\quad
n\geq 1+n_{\mathrm{E}}.$

Then for $1\leq n\leq n_{\mathrm{E}}$ the modes $b^{(\mathrm{E})}_{n}$ act as
a annihilation operator on $\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}$ and as
a creation operator on $\ket{\mathrm{T}}$:

$\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}\propto
b^{(\mathrm{E})}_{n_{\mathrm{E}}}b^{(\mathrm{E})}_{n_{\mathrm{E}}-1}\dots
b^{(\mathrm{E})}_{1}\ket{\mathrm{T}}.$ (6.4)

Moreover, since $\mathrm{L}=n_{\mathrm{E}}+n_{\overline{\mathrm{E}}}\leq
0\Rightarrow n_{\overline{\mathrm{E}}}\leq-1$, we have:

$\displaystyle
b^{*\,(\overline{\mathrm{E}})}_{m}\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}$
$\displaystyle=$ $\displaystyle 0,\quad m\geq 1,$ $\displaystyle
b^{*\,(\overline{\mathrm{E}})}_{n}\ket{\mathrm{T}}$ $\displaystyle=$
$\displaystyle 0,\quad m\geq 1-\absolutevalue{n_{\overline{\mathrm{E}}}},$

which leads for the same argument to:

$\ket{\mathrm{T}}\propto
b^{*\,(\overline{\mathrm{E}})}_{0}b^{*\,(\overline{\mathrm{E}})}_{1}\dots
b^{*\,(\overline{\mathrm{E}})}_{1-\absolutevalue{n_{\overline{\mathrm{E}}}}}\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}.$
(6.5)

In order to check the consistency of the definition, we require that:

$\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}=\quantity(\uppi
T\mathcal{N}_{\Uppsi}^{2})^{n_{\mathrm{E}}}b^{(\mathrm{E})}_{n_{\mathrm{E}}}b^{(\mathrm{E})}_{n_{\mathrm{E}}-1}\dots
b^{(\mathrm{E})}_{1}b^{*\,(\overline{\mathrm{E}})}_{0}b^{*\,(\overline{\mathrm{E}})}_{1}\dots
b^{*\,(\overline{\mathrm{E}})}_{1-\absolutevalue{n_{\overline{\mathrm{E}}}}}\ket{T_{\mathrm{E},\overline{\mathrm{E}}}},$

where the number of $b$ operators has to match the number of $b^{*}$
operators:

$n_{\mathrm{E}}+n_{\overline{\mathrm{E}}}=\mathrm{E}+\overline{\mathrm{E}}=\mathrm{L}=0.$
(6.6)

The same procedure applies also in the case $n_{\mathrm{E}}\leq 0$, leading to
the same result.

As a consequence of (6.6), we can express the twisted vacuum as:

$\displaystyle b^{(\mathrm{E})}_{n}\ket{\mathrm{T}}$ $\displaystyle=$
$\displaystyle 0,\quad n\geq 1+n_{\mathrm{E}},$ $\displaystyle
b^{*\,(\overline{\mathrm{E}})}_{m}\ket{\mathrm{T}}$ $\displaystyle=$
$\displaystyle 0,\quad m\geq 1-n_{\mathrm{E}}.$

###  6.3 Generic Case with Defects

Since the fields in presence of defects behave as NS fields in the limit $z\to
0$, we can define the vacuum in the usual fashion by requiring a finite limit
$\lim\limits_{z\to
0}\Uppsi(z){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}$.
We get as in the NS case :

$b_{n}{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}=b^{*}_{n}{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}=0,\quad
n\geq 1.$ (6.7)

##  7 Asymptotic Fields and Relation Between Asymptotic Fields Vacua and the
Vacuum

In this section we define the asymptotic in-field and out-field and discuss
how their vacua are related to that of the theory with defects. The relation
is “radial time dependent” thus explicitly showing that an interaction is
hidden in the defects. In particular the vacuum for the theory with defects
can be identified with ${\mathrm{SL}_{2}\quantity(\mathds{R})}$ in-field
vacuum while it is connected by a Bogoliubov transformation to the
${\mathrm{SL}_{2}\quantity(\mathds{R})}$ out-field vacuum.

In the following we use the expansion of

$P(z;\\{x_{t},\mathrm{E}_{\quantity(t)}\\})=\prod_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{E_{t}},$

around the origin and infinity with coefficients

$\displaystyle\mathfrak{C}_{k}\quantity(0,\quantity{x_{t},\mathrm{E}_{\quantity(t)}})$
$\displaystyle=$
$\displaystyle\sum_{\\{k_{t}\\}\in\mathds{N}^{N}}\prod\limits_{t=1}^{N}\quantity[\binom{\mathrm{E}_{\quantity(t)}}{k_{t}}\quantity(-\frac{1}{x_{t}})^{k_{t}}]\updelta_{\sum\limits_{t=1}^{N}k_{t},k}$
$\displaystyle\mathfrak{C}_{k}\quantity(\infty,\quantity{x_{t},\mathrm{E}_{\quantity(t)}})$
$\displaystyle=$
$\displaystyle\sum_{\\{k_{t}\\}\in\mathds{N}^{N}}\prod\limits_{t=1}^{N}\quantity[\binom{\mathrm{E}_{\quantity(t)}}{k_{t}}\quantity(-x_{t})^{k_{t}-\mathrm{E}_{\quantity(t)}}]\updelta_{\sum\limits_{t=1}^{N}k_{t},k},$

so that we can write

$\begin{split}P(z;\\{x_{t},\mathrm{E}_{\quantity(t)}\\})&\underset{\absolutevalue{z}<x_{N}}{=}\sum_{k=0}^{\infty}\mathfrak{C}_{k}\quantity(0,\quantity{x_{t},\mathrm{E}_{\quantity(t)}})z^{k}\\\
&\underset{\absolutevalue{z}>x_{1}}{=}\sum_{k=0}^{\infty}\mathfrak{C}_{k}\quantity(\infty,\quantity{x_{t},\mathrm{E}_{\quantity(t)}})z^{-k+\sum\limits_{t=1}^{N}\mathrm{E}_{\quantity(t)}}.\end{split}$

We do not discuss intermediate fields, i.e. expansions for
$x_{t}<\absolutevalue{z}<x_{t-1}$, as it is not possible to clearly
disentangle the effects of defects before and after this range since, as we
will argue, the vacuum in presence of defects is related to the radial
ordering of the operators associated with the defects as in (8.3).

###  7.1 Asymptotic in-field and relation between its vacuum and generic case
vacuum

Consider the definitions of the basis of solutions (5.7) and (5.8) and expand
around $z=0$. Let us concentrate on the first case since analogous relations
can be written for $b_{n}^{*\,(0)}$ with the substitutions of
$\mathrm{E}_{\quantity(t)}$ with $\overline{\mathrm{E}}_{\quantity(t)}$. We
get for $0\leq\absolutevalue{z}<x_{N}$

$\displaystyle\Uppsi_{n}(z)$
$\displaystyle\underset{\absolutevalue{z}<x_{N}}{=}$
$\displaystyle\sum\limits_{k=0}^{+\infty}\mathfrak{C}_{k}\quantity(0,\quantity{x_{t},\mathrm{E}_{\quantity(t)}})\Uppsi^{(0)}_{n-k}(z),$
(7.1)

and $\Uppsi^{(0)}_{n}(z)=\mathcal{N}_{\Uppsi}z^{-n}$ as in (5.3) with
$\mathrm{E}=0$ 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 which can be seen
close to the origin.

We can now relate the operators of the system with defects with those of the
asymptotic in-field. To this purpose we can then substitute the expansion
(7.1) into the usual expression of the modes (4.12):

$\Uppsi(z)=\sum\limits_{n\in\mathds{Z}}b_{n}\Uppsi_{n}(z)\underset{\absolutevalue{z}<x_{N}}{=}\Uppsi^{(in)}(z)=\sum\limits_{n\in\mathds{Z}}b_{n}^{(0)}\Uppsi_{n}^{(0)}(z)$

thus leading to

$b_{n}^{(0)}=\sum\limits_{k=0}^{+\infty}b_{n+k}\mathfrak{C}_{k}\quantity(0,\quantity{x_{t},\mathrm{E}_{\quantity(t)}}).$

By writing
$\Uppsi^{(0)}_{n}(z)=\Uppsi_{n}(z)\,P(z;\\{x_{t},-\mathrm{E}_{\quantity(t)}\\})$
these expressions can also be inverted:

$\begin{split}b_{n}&=\sum\limits_{k=0}^{+\infty}\mathfrak{C}_{k}\quantity(0,\quantity{x_{t},-\mathrm{E}_{\quantity(t)}})b^{(0)}_{n+k},\end{split}$

The important point is that annihilation operators of the asymptotic theory,
i.e. operators with positive index, are expressed only using annihilation
operators of the theory with defects, this means that we can set

${\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}=\ket{0_{(in)}}_{\mathrm{SL}_{2}\quantity(\mathds{R})}.$

###  7.2 Relation between generic case vacuum and asymptotic out-field vacuum

As done in the previous section we can also explicitly compute the expansion
for $\absolutevalue{z}>x_{1}$ (define for simplicity
$\mathrm{M}=\sum\limits_{t=1}^{N}\mathrm{E}_{\quantity(t)}$):

$\displaystyle\Uppsi_{n}(z)$
$\displaystyle\underset{\absolutevalue{z}>x_{1}}{=}$
$\displaystyle\sum\limits_{k=0}^{+\infty}\mathfrak{C}_{k}\quantity(\infty,\quantity{x_{t},\mathrm{E}_{\quantity(t)}})\Uppsi^{(0)}_{n+k-\mathrm{M}}(z),$

which connects the asymptotic behavior of the modes of the fermion with
defects to the modes of a NS fermion which can be seen close to the infinity.

This relation can be used to link out-operators with the operators of the
theory with defects as

$\Uppsi(z)=\sum\limits_{n\in\mathds{Z}}b_{n}\Uppsi_{n}(z)\underset{|z|>x_{1}}{=}\Uppsi^{(out)}(z)=\sum\limits_{n\in\mathds{Z}}b_{n}^{(\infty)}\Uppsi_{n}^{(0)}(z)$

thus getting

$b_{n}^{(\infty)}=\sum\limits_{k=0}^{+\infty}b_{n+\mathrm{M}-k}\mathfrak{C}_{k}\quantity(\infty,\quantity{x_{t},\mathrm{E}_{\quantity(t)}}).$
(7.2)

These expressions can also be inverted as

$\begin{split}b_{n}&=\sum\limits_{k=0}^{+\infty}\mathfrak{C}_{k}\quantity(\infty,\quantity{x_{t},-\mathrm{E}_{\quantity(t)}})b^{(\infty)}_{n+\mathrm{M}-k}.\end{split}$

As we will show later, we must take $\mathrm{M}=0$. Then the important point
is that annihilation operators of the asymptotic theory, i.e. operators with
positive index, are expressed using both annihilation and creator operators of
the theory with defects while creators, i.e. operators with non negative
index, are expressed connected with creators only. It follows from the vacuum
definition that

$\begin{split}&\quantity(\mathfrak{C}_{0}\quantity(\infty,\quantity{x_{t},-\mathrm{E}_{\quantity(t)}})b^{(\infty)}_{1}+\mbox{creators}){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}=0\\\
&\quantity(\mathfrak{C}_{0}\quantity(\infty,\quantity{x_{t},-\mathrm{E}_{\quantity(t)}})b^{(\infty)}_{2}+\mathfrak{C}_{1}\quantity(\infty,\quantity{x_{t},-\mathrm{E}_{\quantity(t)}})b^{(\infty)}_{1}+\mbox{creators}){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}=0\\\
&\vdots\end{split},$

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 get the
relation

${\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}={\cal
N}_{(out)}(\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\})e^{\sum_{m,n\leq
0}{\cal
M}_{mn}^{(out)}(\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\})b^{(\infty)*}_{m}b^{(\infty)}_{n}}\ket{0_{(out)}}_{\mathrm{SL}_{2}\quantity(\mathds{R})},$

so that the two ${\mathrm{SL}_{2}\quantity(\mathds{R})}$ vacua are connected
by a Bogoliubov transformation. More precisely we get (see appendix A for
details)

$\displaystyle\quantity{\Uppsi^{(out,+)}(z)+\oint_{|z|,|w|>x_{1}}\frac{dw}{2\uppi
i}\frac{P(z;\\{x_{t},\mathrm{E}_{\quantity(t)}\\})P(w;\\{x_{t},-\mathrm{E}_{\quantity(t)}\\})-1}{z-w}\Uppsi^{(out,-)}(w)}{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}=0,$
(7.3)

and the corresponding equation for $\Uppsi^{(out)*}(z)$ with the substitution
$\mathrm{E}\rightarrow\overline{\mathrm{E}}$. Notice that the kernel of the
integral is nothing else (up to a multiplicative constant) but the regularised
propagator, i.e. the propagator in the presence of defects (8.2) to which the
NS propagator has been subtracted. The previous equation can be solved
explicitly by

$\begin{split}{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}=&{\cal
N}(\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\})\\\
&e^{\oint_{|z|,|w|>x_{1}}\frac{dz}{2\uppi i}\frac{dw}{2\uppi
i}\Uppsi^{(out,-)*}(z)\frac{P(z;\\{x_{t},\mathrm{E}_{\quantity(t)}\\})P(w;\\{x_{t},-\mathrm{E}_{\quantity(t)}\\})-1}{z-w}\Uppsi^{(out,-)}(w)}\ket{0_{(out)}}_{\mathrm{SL}_{2}\quantity(\mathds{R})}.\end{split}$

In the previous equation there is no need to specify whether
$\absolutevalue{z}$ is greater or less than $\absolutevalue{w}$ since
$\Uppsi^{(out,-)*}(z)$ and $\Uppsi^{(out,-)}(w)$ anti-commute. Finally
deriving the same expression using $\Uppsi^{(out,-)*}(z)$ and comparing with
the previous one we deduce that
$\overline{\mathrm{E}}_{\quantity(t)}=-\mathrm{E}_{\quantity(t)}$.

##  8 Contractions and Stress-Energy Tensor

Given the definitions of the in-vacuum of the theory and the algebra of
operators, we can finally define the normal ordering operation and proceed to
compute the contractions and OPEs 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.

###  8.1 NS Complex Fermion

First of all we deal with the simple case of NS fermions and using the algebra
(5.1) we compute the OPE of fermion fields

$\Uppsi^{i}(z)\Uppsi^{*}_{j}(w)=\;:\Uppsi^{i}(z)\Uppsi^{*}_{j}(z):+\frac{1}{\uppi
T}\frac{\updelta^{i}_{j}}{z-w},\quad\absolutevalue{w}<\absolutevalue{z},$

where the operation $\;:\cdot:$ is the normal ordering with respect to the
${\mathrm{SL}_{2}\quantity(\mathds{R})}$ vacuum defined in (6.1) .

Secondly we get to the expression of the stress-energy tensor:

$\begin{split}\mathcal{T}(z)&=\lim\limits_{w\to z}\quantity[-\frac{\uppi
T}{2}\quantity(\Uppsi^{*}_{i}(z)\partial_{w}\Uppsi^{i}(w)-\partial_{z}\Uppsi^{*}_{i}(z)\Uppsi^{i}(w))+\frac{\mathrm{N}_{f}}{(z-w)^{2}}]\\\
&=-\frac{\uppi
T}{2}\;:\Uppsi^{*}_{i}(z)\overset{\leftrightarrow}{\partial_{z}}\Uppsi^{i}(z):\end{split}$

so we are now able to derive the necessary minimal subtraction

$\mathfrak{h}(z-w)=\frac{\mathrm{N}_{f}}{(z-w)^{2}},$

to get the stress-energy tensor.

###  8.2 Twisted Fermion

We can now go back to $\mathrm{N}_{f}=1$ theories. First of all we consider
the simplest case of the usual twisted fermion with the mode expansion (5.5)
and (5.6). We do not implement beforehand the constraint (6.6) but we recover
it in a different way. Both excited and twisted vacua can be treated on the
same footing since the their difference amount to choose $n_{\mathrm{E}}$ and
$n_{\overline{\mathrm{E}}}$.

####  8.2.1 OPE and Stress-Energy Tensor

Using the anti-commutation relations (5.4) we can compute the OPE

$\Uppsi(z)\Uppsi^{*}(w)=\;N_{E,\bar{E}}\left[\Uppsi(z)\Uppsi^{*}(w)\right]+\frac{1}{\uppi
T}\quantity(\frac{z}{w})^{\mathrm{E}}\frac{1}{z-w},\quad\absolutevalue{w}<\absolutevalue{z},$

and

$\Uppsi^{*}(w)\Uppsi(z)=\;N_{E,\bar{E}}\left[\Uppsi^{*}(w)\Uppsi(z)\right]+\frac{1}{\uppi
T}\quantity(\frac{w}{z})^{\overline{\mathrm{E}}}\frac{1}{w-z},\quad\absolutevalue{w}>\absolutevalue{z}.$

If we require that the previous results can be assembled in a well defined
continuous radial ordering $R\quantity[\Uppsi(z)\Uppsi^{*}(w)]$ we need to set
$\mathrm{E}=-\overline{\mathrm{E}}$ so we can write

$R\quantity[\Uppsi(z)\Uppsi^{*}(w)]=\;N_{E,\bar{E}}\left[\Uppsi(z)\Uppsi^{*}(w)\right]+\frac{1}{\uppi
T}\quantity(\frac{z}{w})^{\mathrm{E}}\frac{1}{z-w}.$

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. Either as

$\begin{split}\mathcal{T}(z)&=\lim\limits_{\begin{subarray}{c}w\to z\\\
\absolutevalue{w}<\absolutevalue{z}\end{subarray}}\quantity[-\frac{\uppi
T}{2}\quantity(\Uppsi^{*}(z)\partial_{w}\Uppsi(w)-\partial_{z}\Uppsi^{*}(z)\Uppsi(w))+\frac{1}{(z-w)^{2}}]\\\
&=-\frac{\uppi
T}{2}\;:\Uppsi^{*}(z)\overset{\leftrightarrow}{\partial_{z}}\Uppsi(z):+\frac{\mathrm{E}^{2}}{2z^{2}},\end{split}$
(8.1)

or

$\begin{split}\mathcal{T}(z)&=\lim\limits_{\begin{subarray}{c}w\to z\\\
\absolutevalue{w}<\absolutevalue{z}\end{subarray}}\quantity[-\frac{\uppi
T}{2}\quantity(-\partial_{z}\Uppsi(z)\Uppsi^{*}(w)+\Uppsi(z)\partial_{w}\Uppsi^{*}(w))+\frac{1}{(z-w)^{2}}]\\\
&=-\frac{\uppi
T}{2}\;:\Uppsi^{*}(z)\overset{\leftrightarrow}{\partial_{z}}\Uppsi(z):+\frac{\overline{\mathrm{E}}^{2}}{2z^{2}},\end{split}$

which however must coincide for consistency. Since

$\;:\Uppsi(z)\overset{\leftrightarrow}{\partial_{z}}\Uppsi^{*}(z):=\;:\Uppsi^{*}(z)\overset{\leftrightarrow}{\partial_{z}}\Uppsi(z):\,,$

then we must then require $\mathrm{E}^{2}=\overline{\mathrm{E}}^{2}$.

We can get a stronger constraint by computing the OPE
$\mathcal{T}(z)\mathcal{T}(w)$. In fact the cancellation of the cubic
divergence requires $\mathrm{E}+\overline{\mathrm{E}}=0$.

It the follows that the vacuum $\ket{T_{\mathrm{E},\overline{\mathrm{E}}}}$ is
actually $\ket{T_{\mathrm{E}}}$, notation we will use from now on.

####  8.2.2 Virasoro Operators and Conformal Dimensions

From the usual definition of the stress-energy tensor in terms of the Virasoro
generators $\mathcal{T}(z)=\sum\limits_{k\in\mathds{Z}}L_{k}z^{-k-2},$ we can
extract the operators $L_{k}$ from any of the previous definitions:

$\begin{split}L_{(\mathrm{E})k}&=-\frac{\uppi
T}{2}\mathcal{N}_{\Uppsi}^{2}\sum\limits_{n\in\mathds{Z}}\;N_{E,\bar{E}}\left[b^{*\,(\overline{\mathrm{E}})}_{n}b^{(\mathrm{E})}_{k+1-n}\right](2n-k+2\mathrm{E}-1)+\frac{\mathrm{E}^{2}}{2}\updelta_{k,0}\\\
&=\frac{\uppi
T}{2}\mathcal{N}_{\Uppsi}^{2}\sum\limits_{n=1}^{\infty}\Big{[}(2n-k+2\mathrm{E}-1)\;N_{E,\bar{E}}\left[b^{(\mathrm{E})}_{k+1-n}b^{*\,(\overline{\mathrm{E}})}_{n}\right]\\\
&+(2n-k-2\mathrm{E}-1)\;N_{E,\bar{E}}\left[b^{*\,(\overline{\mathrm{E}})}_{k+1-n}b^{(\mathrm{E})}_{n}\right]\Big{]}+\frac{\mathrm{E}^{2}}{2}\updelta_{k,0}\end{split}.$

Looking back at the analysis of the excited and twisted vacua, we already
hinted to the fact that they are not in general
${\mathrm{SL}_{2}\quantity(\mathds{R})}$ invariant. In particular we can see
that that the excited vacua $\ket{T_{\mathrm{E}}}$ is a primary field

$L_{(\mathrm{E})k>0}\ket{T_{\mathrm{E}}}=0,\qquad
L_{(\mathrm{E})0}\ket{T_{\mathrm{E}}}=\frac{\mathrm{E}^{2}}{2}\ket{T_{\mathrm{E}}},$

with non trivial conformal dimensions
$\Updelta\quantity(\ket{T_{\mathrm{E}}})=\frac{\mathrm{E}^{2}}{2}$. This
operator is an excited spin field
$\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x)$ inserted at $x=0$ whose
bosonized expression is given by

$\mathrm{S}_{\mathrm{E}}\quantity(x)=e^{i\mathrm{E}\upphi(x)},$

where $\upphi$ is such that

$\left\langle\upphi(z)\upphi(w)\right\rangle=-\frac{1}{(z-w)^{2}}.$

In fact the minimal conformal dimension is achieved for
$n_{\mathrm{E}}=n_{\overline{\mathrm{E}}}=0$, i.e.
$\Updelta\quantity(\ket{\mathrm{T}})=\frac{\upepsilon^{2}}{8}$ and we know
this is the basic spin field. We can further check this idea by showing that
the conformal dimensions are consistent. Using (6.5) we get

$\begin{split}L_{(\mathrm{E})0}\ket{\mathrm{T}}&=L_{0}\quantity(b^{*\,(\overline{\mathrm{E}})}_{0}b^{*\,(\overline{\mathrm{E}})}_{-1}\dots
b^{*\,(\overline{\mathrm{E}})}_{2-n_{\mathrm{E}}}\ket{T_{\mathrm{E}}})\\\
&=\left[\sum\limits^{n_{\mathrm{E}}}_{n=1}(n-\frac{\mathrm{E}+1}{2})+\frac{\mathrm{E}^{2}}{2}\right]\ket{\mathrm{T}}=+\frac{1}{8}\upepsilon^{2}\ket{\mathrm{T}}.\end{split}$

###  8.3 Generic Case With Defects

We will now apply the same procedure to 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 (6.7). We will consider the mode expansion (5.7)
and (5.8) as well as the anti-commutation relations (5.11).

As in the usual twisted case, we will first consider the contraction of the
field $\Uppsi$ and $\Uppsi^{*}$ and then move to the stress-energy tensor.
Using the anti-commutation relations and
$\sum\limits_{k\in\mathds{Z}}p_{k}z^{k}=\prod\limits_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{-\mathrm{L}_{\quantity(t)}}$
where $p_{k}$ is defined in (5.10). We have:

$\Uppsi(z)\Uppsi^{*}(w)=\;:\Uppsi(z)\Uppsi^{*}(w):+\frac{1}{\uppi
T}\frac{1}{z-w}\prod\limits_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{\mathrm{E}_{\quantity(t)}}\quantity(1-\frac{w}{x_{t}})^{-\mathrm{E}_{\quantity(t)}},$

as well as

$\Uppsi^{*}(z)\Uppsi(w)=\;:\Uppsi^{*}(z)\Uppsi(w):+\frac{1}{\uppi
T}\frac{1}{z-w}\prod\limits_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{\overline{\mathrm{E}}_{\quantity(t)}}\quantity(1-\frac{w}{x_{t}})^{-\overline{\mathrm{E}}_{\quantity(t)}},$

both for $\absolutevalue{w}<\absolutevalue{z}$. If we require that the
previous results can be assembled in a well defined continuous radial ordering
$R\quantity[\Uppsi(z)\Uppsi^{*}(w)]$ we need to set
$\mathrm{E}_{\quantity(t)}=-\overline{\mathrm{E}}_{\quantity(t)}$ so we can
write

$R\quantity[\Uppsi(z)\Uppsi^{*}(w)]=\;:\Uppsi(z)\Uppsi^{*}(w):+\frac{1}{\uppi
T}\frac{1}{z-w}\prod\limits_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{\mathrm{E}_{\quantity(t)}}\quantity(1-\frac{w}{x_{t}})^{-\mathrm{E}_{\quantity(t)}}.$
(8.2)

We can then expand the results around $z$:

$\displaystyle\begin{aligned}
R\quantity[\Uppsi(z)\Uppsi^{*}(w)]&=\;:\quantity(\Uppsi\Uppsi^{*})(z):+\;:\quantity(\Uppsi\partial\Uppsi^{*})(z):\,(w-z)\\\
&+\frac{1}{\uppi
T}\left[\frac{-1}{w-z}+\sum\limits_{t=1}^{N}\frac{\mathrm{E}_{\quantity(t)}}{z-x_{t}}\right.\\\
&\left.-\frac{1}{2}\quantity(\sum\limits_{t=1}^{N}\sum\limits_{u\neq
t}\frac{\mathrm{E}_{\quantity(t)}\mathrm{E}_{\quantity(u)}}{(z-x_{t})(z-x_{u})}+\sum\limits_{t=1}^{N}\frac{\mathrm{E}_{\quantity(t)}\quantity(\mathrm{E}_{\quantity(t)}-1)}{(z-x_{t})^{2}})(w-z)\right]\\\
&+\order{(w-z)^{2}},\end{aligned}$

and around $w$

$\displaystyle\begin{aligned}
R\quantity[\Uppsi(z)\Uppsi^{*}(w)]&=\;:\quantity(\Uppsi\Uppsi^{*})(w):+\;:\quantity(\partial\Uppsi\Uppsi^{*})(w):\,(z-w)\\\
&+\frac{1}{\uppi
T}\left[\frac{1}{z-w}+\sum\limits_{t=1}^{N}\frac{\mathrm{E}_{\quantity(t)}}{w-x_{t}}\right.\\\
&\left.+\frac{1}{2}\quantity(\sum\limits_{t=1}^{N}\sum\limits_{u\neq
t}\frac{\mathrm{E}_{\quantity(t)}\mathrm{E}_{\quantity(u)}}{(w-x_{t})(w-x_{u})}+\sum\limits_{t=1}^{N}\frac{\mathrm{E}_{\quantity(t)}\quantity(\mathrm{E}_{\quantity(t)}-1)}{(w-x_{t})^{2}})(z-w)\right]\\\
&+\order{(z-w)^{2}},\end{aligned}$

so that the stress-energy tensor becomes:

$\displaystyle\mathcal{T}(z)$ $\displaystyle=-\frac{\uppi
T}{2}\;:\Uppsi(z)\overset{\leftrightarrow}{\partial_{z}}\Uppsi^{*}(z):+\frac{1}{2}\quantity(\sum\limits_{t=1}^{N}\frac{\mathrm{E}_{\quantity(t)}}{z-x_{t}})^{2}$
$\displaystyle=\frac{\uppi
T}{2}\mathcal{N}_{\Uppsi}^{2}\sum_{n,m}:b_{n}b^{*}_{m}:z^{-n-m}\quantity[\frac{m-n}{z}+2\sum\limits_{t=1}^{N}\frac{\mathrm{E}_{\quantity(t)}}{z-x_{t}}]+\frac{1}{2}\quantity(\sum\limits_{t=1}^{N}\frac{\mathrm{E}_{\quantity(t)}}{z-x_{t}})^{2}.$

The last expression shows that the energy momentum tensor $\mathcal{T}(z)$ is
radial time dependent but it satisfies the usual OPE.

Notice first of all that the vacuum
${\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}$
is actually ${\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}$, i.e.
it depends only on $x_{t}$ and $\mathrm{E}_{\quantity(t)}$. Then 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 (6.3) that the
singularity in the modes (5.7) and (5.8) at the point $x_{t}$ is associated
with a primary conformal operator which creates $\ket{T_{\mathrm{E}}}$ with
$\mathrm{E}=\mathrm{E}_{\quantity(t)}$. In fact by comparison with the stress
energy tensor of a excited vacuum (8.1), we can read from the second order
singularity that at the points $x_{t}$ there is an operator which creates the
excited vacuum ${\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}$ from
the ${\mathrm{SL}_{2}\quantity(\mathds{R})}$ vacuum
${\ket{0}}_{\mathrm{SL}_{2}\quantity(\mathds{R})}$. Given the discussion in
the previous section this is an excited spin field
$\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})=e^{i\mathrm{E}_{\quantity(t)}\upphi(x_{t})}$.
The first order singularities in $x_{u}-x_{t}$ are then the result of the
interaction between two of the previous excited spin fields. We can try to be
more precise. Using the usual CFT operatorial approach we can suppose that the
following identification holds

$\displaystyle{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}$
$\displaystyle={\cal
N}(\\{x_{t},\mathrm{E}_{\quantity(t)}\\})~{}\mathrm{S}_{\mathrm{E}_{\quantity(1)}}\quantity(x_{1})\dots\mathrm{S}_{\mathrm{E}_{\quantity(N)}}\quantity(x_{N}){\ket{0}}_{\mathrm{SL}_{2}\quantity(\mathds{R})}$
$\displaystyle={\cal
N}(\\{x_{t},\mathrm{E}_{\quantity(t)}\\})~{}R\quantity[\prod\limits_{t=1}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]{\ket{0}}_{\mathrm{SL}_{2}\quantity(\mathds{R})},$
(8.3)

then we get

$\displaystyle\mathcal{T}(z){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}$
$\displaystyle={\cal
N}(\\{x_{t},\mathrm{E}_{\quantity(t)}\\})~{}R\quantity[\mathcal{T}(z)\prod\limits_{t=1}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]{\ket{0}}_{\mathrm{SL}_{2}\quantity(\mathds{R})}.$

The fact that $\mathcal{T}(z)$ 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 the one with the radial ordering. In
fact an operatorial expression like
$\mathcal{T}(z)R\quantity[\partial\upphi(x_{1})\partial\upphi(x_{2})]{\ket{0}}_{\mathrm{SL}_{2}\quantity(\mathds{R})}$
is only defined for $|z|>x_{1,2}$. It then follows that

$\mathcal{T}(z){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}=\sum\limits_{t=1}^{N}\quantity(\frac{\mathrm{E}_{\quantity(t)}^{2}/2}{(z-x_{t})^{2}}+\frac{\
\partial_{x_{t}}-\partial_{x_{t}}\log{\cal
N}}{z-x_{t}}){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}+\mbox{regular
terms in $z$},$

which allows to write

$\displaystyle{\cal N}(\\{x_{t},\mathrm{E}_{\quantity(t)}\\})~{}$
$\displaystyle
R\quantity[\partial_{x_{t}}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]\prod_{u\neq
t}\mathrm{S}_{\mathrm{E}_{\quantity(u)}}\quantity(x_{u})]{\ket{0}}_{\mathrm{SL}_{2}\quantity(\mathds{R})}$
$\displaystyle=\mathrm{E}_{\quantity(t)}\,\quantity[\uppi
T\mathcal{N}_{\Uppsi}^{2}\,\sum_{n,m=0}^{\infty}\frac{b_{n}b^{*}_{m}}{x_{t}^{n+m}}+\sum_{u\neq
t}\frac{\mathrm{E}_{\quantity(u)}}{x_{t}-x_{u}}]{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}.$

This result shows the way non primary operators are reresentend in this
formalism and is consistent with the computation of the excited spin fields
correlator performed in section 11.

##  9 Hermitian Conjugation

In this section we focus on the operation of “Hermitian conjugation”. We write
“Hermitian conjugation” between quotes because the Hermitian conjugation
requires the existence of an inner product which is not yet available since we
have not defined the out-vacuum. What we are going to discuss is actually more
like the involutive $\star$ operator of $C^{\star}$ algebras with the catch
that 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
$z\rightarrow\frac{1}{\bar{z}}$, i.e. in evolving from $\uptau=+\infty$ to
$\uptau=-\infty$ so that the order of boundary singularities is reversed. In
the next section we use this correspondence between the $\star$ operator we
defined and the Hermitian conjugation to define the out-vacuum. The previous
warning does not apply to the usual twisted fermions with which we start.

###  9.1 Usual Twisted Fermions

In general for a chiral primary conformal operator of dimension $\Delta$ in
$z$ coordinates the Euclidean Hermitian conjugation is

$\quantity[O(z)]^{\dagger}=\evaluated{\quantity(w^{2\Delta}O(w))}_{w=1/\bar{z}}.$

As discussed above we cannot use the words “Euclidean Hermitian conjugation”
and define it since we do not have an inner product but we can define the
operation $\star$ which mimics its behavior. Therefore we define

$\quantity[\Uppsi(z;\mathrm{E})]^{\star}=\evaluated{\quantity[w\,\widetilde{\Uppsi}^{*}(w;-\widetilde{\mathrm{E}})]}_{w=1/\bar{z}},\quad\quantity[\Uppsi^{*}(z;\mathrm{E})]^{\star}=\evaluated{\quantity(w\,\widetilde{\Uppsi}(w;\widetilde{\mathrm{E}}))}_{w=1/\bar{z}},$
(9.1)

where we have not assumed that the action of $\star$ is a map between the same
space and we have written for example $\Uppsi(z;\mathrm{E})$ to make explicit
the dependence on the parameter $\mathrm{E}$ which enters in the modes. The
previous action agrees with (4.8). In terms of the basis (5.3), we can
write121212 The other possibility
$\quantity[\Uppsi_{n}^{(\mathrm{E})}(z)]^{\star}=\evaluated{\quantity[w\,\Uppsi_{-n}^{*\,(-\mathrm{E}-1)}\quantity(w)]}_{w=1/\bar{z}}$
is inconsistent with the anti-commutation relations.

$\quantity[\Uppsi_{n}^{(\mathrm{E})}(z)]^{\star}=\evaluated{\quantity[w\,\Uppsi_{1-n}^{(-\mathrm{E})\,*}\quantity(w)]}_{w=1/\bar{z}},\qquad\quantity[\Uppsi_{n}^{(-\mathrm{E})}(z)]^{\star}=\evaluated{\quantity[w\,\Uppsi_{1-n}^{(\mathrm{E})\,*}\quantity(w)]}_{w=1/\bar{z}},$

which shows that in this case the image of the $\star$ operator is the same of
the support. Using the mode expansion of (9.1) it follows that

$\quantity[b^{(\mathrm{E})}_{n}]^{\star}=b^{*\,(\overline{\mathrm{E}})}_{1-n},\qquad\quantity[b^{*\,(\overline{\mathrm{E}})}_{n}]^{\star}=b^{(\mathrm{E})}_{1-n}.$
(9.2)

The $\star$ action is compatible with the anti-commutation relations as we can
show by explicitly computing them:

$\quantity(\quantity[b^{(\mathrm{E})}_{n},b^{*\,(\overline{\mathrm{E}})}_{m}]_{+})^{\star}=\quantity[b^{*\,(\overline{\mathrm{E}})}_{1-n},b^{(\mathrm{E})}_{1-m}]_{+}=\frac{1}{\uppi
T\mathcal{N}_{\Uppsi}^{2}}\updelta_{n+m,1}.$

Furthermore $\star$ is involutive since:

$\quantity[\Uppsi_{n}^{(\mathrm{E})}(z)]^{\star\star}=\Uppsi_{n}^{(\mathrm{E})}(z)\quad\Rightarrow\quad\quantity[b^{(\mathrm{E})}_{n}]^{\star\star}=b^{(\mathrm{E})}_{n}.$

###  9.2 Generic Case With Defects

The situation in the generic case is more complex. Consider the modes given in
(5.7) then it is natural to define the action of the $\star$ operator on them
as:

$\begin{split}\quantity[\Uppsi_{n}(z;\\{x_{t},\mathrm{E}_{\quantity(t)}\\})]^{\star}&=\mathcal{N}_{\Uppsi}\,\overline{z}^{-n}\prod\limits_{t=1}^{N}\quantity(1-\frac{\overline{z}}{x_{t}})^{\mathrm{E}_{\quantity(t)}}\\\
&=\evaluated{\quantity(w\,\prod\limits_{t=1}^{N}\quantity(-\frac{1}{x_{t}})^{\mathrm{E}_{\quantity(t)}}\,\mathcal{N}_{\Uppsi}\,w^{-(M+1-n)}\prod\limits_{t=1}^{N}\quantity(1-\frac{w}{1/x_{t}})^{\overline{\mathrm{E}}_{\quantity(t)}})}_{w=1/\bar{z}}\\\
&=\evaluated{\quantity(w\,\prod\limits_{t=1}^{N}\quantity(-\frac{1}{x_{t}})^{\mathrm{E}_{\quantity(t)}}\,\widetilde{\Uppsi}_{M+1-n}^{*}\quantity(w;\\{\widetilde{x}_{t},\overline{\widetilde{\mathrm{E}}}_{\quantity(t)}\\}))}_{w=1/\bar{z}}\end{split}$

where we used $\mathrm{M}=\sum\limits_{t=1}^{N}\mathrm{E}_{\quantity(t)}$. In
this case the image of the $\star$ operator is a different space where the
defects are located in $\widetilde{x}_{t}$ and the singularities are
$\widetilde{\mathrm{E}}_{\quantity(t)}$ and
$\overline{\widetilde{\mathrm{E}}}_{\quantity(t)}$ with

$\widetilde{x}_{t}=\frac{1}{x_{t}},\qquad\widetilde{\mathrm{E}}_{\quantity(t)}=-\mathrm{E}_{\quantity(t)},\qquad\overline{\widetilde{\mathrm{E}}}_{\quantity(t)}=\mathrm{E}_{\quantity(t)},$

where we used
$\mathrm{E}_{\quantity(t)}+\overline{\mathrm{E}}_{\quantity(t)}=0$.

We can therefore compute the action of the $\star$ operator on the creation
and annihilation operators as done previously and get:

$b_{n}^{\star}=\prod\limits_{t=1}^{N}\quantity(-\frac{1}{x_{t}})^{-\mathrm{E}_{\quantity(t)}}\,\widetilde{b}^{*}_{{M}+1-n},\qquad(b_{n}^{*})^{\star}=\prod\limits_{t=1}^{N}\quantity(-\frac{1}{x_{t}})^{\mathrm{E}_{\quantity(t)}}\,\widetilde{b}_{-M+1-n}.$

As in the previous situation, the anti-commutation relations are preserved by
the $\star$ operator. Explicitly we have:

$\quantity(\quantity[b_{n},b_{m}^{*}]_{+})^{\star}=\quantity[\widetilde{b}_{-M+1-m},\widetilde{b}_{{M}+1-n}^{*}]_{+}=\frac{1}{\uppi
T\mathcal{N}_{\Uppsi}^{2}}\updelta_{n+m,1}.$

Finally the $\star$ operator is involutive.

##  10 Definition of the Out-Vacuum

With the definition of the $\star$ operator we can now proceed to define the
out-vacuum such that it acts as the Hermitian conjugation in the usual cases.
It is conceptually separated from the definitions of the algebra of operators
and their representation on the in-vacuum. This is the last step before we can
compute any correlation function. 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.

###  10.1 Usual Twisted Fermions

Consider the definition of the in-vacuum (6.2) for the fields image of the
$\star$ operator, i.e. $\widetilde{\Uppsi}(w;\widetilde{\mathrm{E}})$ and
$\widetilde{\Uppsi}^{*}(w;\overline{\widetilde{\mathrm{E}}})$. It is defined
as

${\widetilde{b}}^{(\widetilde{\mathrm{E}})}_{n}\ket{\widetilde{T}_{\widetilde{\mathrm{E}},\overline{\widetilde{\mathrm{E}}}}}={\widetilde{b}}^{(\overline{\widetilde{\mathrm{E}}})*}_{n}\ket{\widetilde{T}_{\widetilde{\mathrm{E}},\overline{\widetilde{\mathrm{E}}}}}=0,\quad
n\geq 1.$

Then the usual Hermitian conjugation gives

$\bra{\widetilde{T}_{\widetilde{\mathrm{E}},\overline{\widetilde{\mathrm{E}}}}}\quantity({\widetilde{b}}^{(\widetilde{\mathrm{E}})}_{n})^{\dagger}=\bra{\widetilde{T}_{\widetilde{\mathrm{E}},\overline{\widetilde{\mathrm{E}}}}}\quantity({\widetilde{b}}^{(\overline{\widetilde{\mathrm{E}}})*}_{n})^{\dagger}=0,\quad
n\geq 1.$

Given the action of the $\star$ operator (9.2), if we want to identify it with
the Hermitian conjugate we are led to write

$\bra{T_{\mathrm{E}}}b^{(\mathrm{E})}_{n}=\bra{T_{\mathrm{E}}}b^{*\,(\overline{\mathrm{E}})}_{n}=0,\quad
n\leq 0.$

###  10.2 Generic Case With Defects

We can now analyze the case of an arbitrary number of defects using previous
relations. Following the steps of the previous section we can define the in-
vacuum for the tilded theory as

${\widetilde{b}}_{n}{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}={\widetilde{b}}_{n}{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}=0,\quad
n\geq 1,$

and then interpret it as the out-vacuum for the initial theory. The definition
of the out-vacuum is therefore:

$\displaystyle{\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}b_{n}$
$\displaystyle=$ $\displaystyle 0,\quad n\leq\mathrm{M},$
$\displaystyle{\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}b^{*}_{n}$
$\displaystyle=$ $\displaystyle 0,\quad n\leq-\mathrm{M}.$

Since the action of the $\star$ operator is compatible with the anti-
commutation relations this definition is consistent as definition for the out-
states. The consistency between the in-vacuum and out-vacuum is however not
granted and must be checked. If we assume that
$\innerproduct{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}\neq
0$ then using the anti-commutation relations we get

$\begin{split}\frac{1}{\uppi
T\mathcal{N}_{\Uppsi}^{2}}\innerproduct{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}&={\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}\quantity[b_{\mathrm{M}},b_{-\mathrm{M}+1}^{*}]_{+}{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}\\\
&={\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}b_{-\mathrm{M}+1}^{*}b_{\mathrm{M}}{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}\neq
0,\end{split}$

which requires
$b_{\mathrm{M}}{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}\neq 0$
There is a similar condition for the $b^{*}_{-\mathrm{M}}$, therefore we must
require $\mathrm{M}\leq 0$ and $-\mathrm{M}\leq 0$, thus

$\mathrm{M}=\sum\limits_{t=1}^{N}\mathrm{E}_{\quantity(t)}=0.$

The situation is therefore analogous to the case depicted in Figure 6.1 where
$\mathrm{M}$ and $\overline{\mathrm{M}}$ have the same role of $\mathrm{L}$
for the twisted fermion.

###  10.3 Asymptotic vacua

The discussion is essentially the same as in section 7 with the role of
asymptotic in- and out-fields exchanged. In particular we get

${\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}={}_{\mathrm{SL}_{2}\quantity(\mathds{R})}\bra{0_{(out)}},$

and

${\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}={\cal
N}_{(in)}(\\{x_{t},\mathrm{E}_{\quantity(t)}\\})~{}~{}{}_{\mathrm{SL}_{2}\quantity(\mathds{R})}\bra{0_{(in)}}e^{\sum_{m,n\geq
1}{\cal
M}_{mn}(\\{x_{t},\mathrm{E}_{\quantity(t)}\\})b^{(0)*}_{m}b^{(0)}_{n}}.$

##  11 Spin Field 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 (8.3) we can finally compute the spin field correlators

$\innerproduct{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}={\cal
N}(\\{x_{t},\mathrm{E}_{\quantity(t)}\\})\left\langle
R\quantity[\prod\limits_{t=1}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]\right\rangle.$

At first sight this expression might look incorrect since both
${\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}$ and
${\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}$ seem to contain
$R\quantity[\prod\limits_{t=1}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]$
as if were squaring the previous radial ordering. That it is not the case and
it can be seen in different ways. The simplest is to realize that such a
square would be divergent while the product seems to be perfectly finite. A
more sophisticated and rigorous way is to consider what the previous product
is from the point of view of asymptotic out field. In this case
${\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}={\cal
N}_{(out)}~{}R\quantity[\prod\limits_{t=1}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]\ket{0_{(out)}}_{\mathrm{SL}_{2}\quantity(\mathds{R})}$
and
${\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}={}_{\mathrm{SL}_{2}\quantity(\mathds{R})}\bra{0_{(out)}}$
so that ${\cal N}_{(out)}={\cal N}$. Moreover
$T(z)\underset{\absolutevalue{z}>x_{1}}{=}T_{(out)}(z)$ when the two energy
momentum tensors are normal ordered with respect to their different sets of
operators which are related as in (7.2). Hence all the expressions are surely
valid for $\absolutevalue{z}>x_{1}$ 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 it is not completely clear how to fix the normalization.
Moreover the result depends on the normalization chosen for the single spin
field and this normalization shows only up when we relate the $N$ points
correlators to $N-1$ points ones and these recursively down to two points
correlators. Therefore we need to consider quantities where the normalization
cancels. In particular we can consider

$\begin{split}&\partialderivative{x_{t}}\ln\left\langle
R\quantity[\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})\prod\limits_{u=1,u\neq
t}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(u)}}\quantity(x_{u})]\right\rangle\\\
&=\oint\limits_{\absolutevalue{z}=x_{t}}\frac{\differential{z}}{2\uppi
i}\frac{\left\langle
R\quantity[\mathcal{T}(z)\prod\limits_{t=1}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]\right\rangle}{\left\langle
R\quantity[\prod\limits_{t=1}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]\right\rangle}\\\
&=\quantity(\oint\limits_{\absolutevalue{z}>x_{t}}\frac{\differential{z}}{2\uppi
i}-\oint\limits_{\absolutevalue{z}<x_{t}}\frac{\differential{z}}{2\uppi
i})\frac{{\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}\mathcal{T}(z){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}}{\innerproduct{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}\\\
&=\frac{{\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}\quantity(L_{-1}^{x_{t}^{+}}-L_{-1}^{x_{t}^{-}}){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}}{\innerproduct{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}\end{split},$

since $\quantity[L_{-1},\mathcal{O}_{h}(z)]=\partial_{z}\mathcal{O}_{h}(z)$
for a quasi-primary operator $\mathcal{O}_{h}$. From the definition of
$\mathcal{T}(z)$ it follows that:

$L_{-1}^{x_{t}^{+}}-L_{-1}^{x_{t}^{-}}=\oint\limits_{\mathcal{C}_{x_{t}}}\frac{\differential{z}}{2\uppi
i}\mathcal{T}(z)=\uppi
T\,\mathcal{N}_{\Uppsi}^{2}\,\mathrm{E}_{\quantity(t)}\,\sum_{n,m}:b_{n}b^{*}_{m}:x_{t}^{-m-n}+\sum\limits_{u=1,u\neq
t}^{N}\frac{\mathrm{E}_{\quantity(u)}\mathrm{E}_{\quantity(t)}}{x_{t}-x_{u}},$

where $\mathcal{C}_{x_{t}}$ is a small path circling $x_{t}$. Therefore

$\partialderivative{x_{t}}\ln\left\langle
R\quantity[\prod\limits_{u}\mathrm{S}_{\mathrm{E}_{\quantity(u)}}\quantity(x_{u})]\right\rangle=\sum\limits_{u\neq
t}\frac{\mathrm{E}_{\quantity(u)}\mathrm{E}_{\quantity(t)}}{x_{t}-x_{u}},$

which can be solved by

$\left\langle
R\quantity[\prod\limits_{t=1}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]\right\rangle=\mathcal{N}_{0}\quantity(\\{\mathrm{E}_{\quantity(t)}\\})\prod\limits_{t=1,t>u}^{N}\quantity(x_{u}-x_{t})^{\mathrm{E}_{\quantity(u)}\mathrm{E}_{\quantity(t)}}.$

The constant $\mathcal{N}_{0}\quantity(\\{\mathrm{E}_{\quantity(t)}\\})$ which
depends on the $\mathrm{E}_{\quantity(t)}$ only can then be fixed by using the
OPE. The last equation reproduces the usual bosonization procedure.

In a similar way we can compute all the correlators as

$\displaystyle\frac{{\bra{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}R\quantity[\prod\limits_{i}\Uppsi(x_{i})\prod\limits_{j}\Uppsi^{*}(x_{j})]{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}}{\innerproduct{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)}\\}}}}}$
$\displaystyle=\frac{\left\langle
R\quantity[\prod\limits_{i}\Uppsi(x_{i})\prod\limits_{j}\Uppsi^{*}(x_{j})\prod\limits_{t=1}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]\right\rangle}{\left\langle
R\quantity[\prod\limits_{t=1}^{N}\mathrm{S}_{\mathrm{E}_{\quantity(t)}}\quantity(x_{t})]\right\rangle},$

by using Wick theorem since the algebra and the action of creators and
annhilators is the usual. In particular taking one $\Uppsi(z)$ and one
$\Uppsi^{*}(w)$ we get the Green function which is nothing else but the
contraction in equation (8.2) exactly as in the usual case.

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##  Appendix A  Details on Reflection Condition on the Vacuum with an
Arbitrary Number of Defects for Asymptotic Field

We would like to provide some details on how (7.3) can be derived. First we
introduce the projector of positive frequency and negative frequency modes for
the NS fermion as

$\displaystyle P^{(+,0)}(z,w)$ $\displaystyle=$
$\displaystyle\frac{+1}{z-w},\quad\absolutevalue{z}>\absolutevalue{w}$
$\displaystyle P^{(-,0)}(z,w)$ $\displaystyle=$
$\displaystyle\frac{-1}{z-w},\quad\absolutevalue{z}<\absolutevalue{w},$

so that for example

$\oint\limits_{\absolutevalue{z}>\absolutevalue{w}}\frac{\differential{w}}{2\uppi
i}P^{(+,0)}(z,w)\Uppsi^{(0)}(0)=\Uppsi^{(0,+)}(z),$

and similarly for the negative frequency modes.

Likewise we introduce the projectors for the field with defects as

$\displaystyle P^{(+)}(z,w)$ $\displaystyle=$
$\displaystyle\frac{P(z;\\{x_{t},\mathrm{E}_{\quantity(t)}\\})P(w;\\{x_{t},-\mathrm{E}_{\quantity(t)}\\})}{z-w},~{}~{}\quad\absolutevalue{z}>\absolutevalue{w}$
$\displaystyle P^{(-)}(z,w)$ $\displaystyle=$
$\displaystyle\frac{-P(z;\\{x_{t},\mathrm{E}_{\quantity(t)}\\})P(w;\\{x_{t},-\mathrm{E}_{\quantity(t)}\\})}{z-w},\quad\absolutevalue{z}<\absolutevalue{w},$

with
$P(z;\\{x_{t},\mathrm{E}_{\quantity(t)}\\})=\prod_{t=1}^{N}\quantity(1-\frac{z}{x_{t}})^{\mathrm{E}_{\quantity(t)}}$
as in the main text.

It is then immediate to compute

$\displaystyle\quantity(P^{(+)}P^{(+,0)})(z,w)$
$\displaystyle=\oint\limits_{\absolutevalue{z}>\absolutevalue{\upzeta}>\absolutevalue{w}}\frac{d\upzeta}{2\uppi
i}P^{(+)}(z,\upzeta)P^{(+,0)}(\upzeta,w)=P^{(+,0)}(z,w)$
$\displaystyle\quantity(P^{(+)}P^{(-,0)})(z,w)$
$\displaystyle=\frac{P(z;\\{x_{t},\mathrm{E}_{\quantity(t)}\\})P(w;\\{x_{t},-\mathrm{E}_{\quantity(t)}\\})-1}{z-w}.$

The last equation is valid when
$\mathrm{M}=\sum\limits_{t=1}^{N}\mathrm{E}_{\quantity(t)}\leq 0$ and for
$\absolutevalue{z}$ and $\absolutevalue{w}$ arbitrary.

Specializing the previous expressions to the $\Uppsi^{(out)}(z)$ case we need
to add the constraints that $\absolutevalue{z}>x_{1}$ and
$\absolutevalue{w}>x_{1}$.

Finally the vacuum in presence of defects can be described by

$\displaystyle\Uppsi^{(+)}(z){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}$
$\displaystyle=\quantity(P^{(+)}\Uppsi)(z){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}$
$\displaystyle=\quantity(P^{(+)}\Uppsi^{(out)})(z){\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}$
$\displaystyle=\quantity[\quantity(P^{(+)}P^{(+,0)}\Uppsi^{(out)})(z)+\quantity(P^{(+)}P^{(-,0)}\Uppsi^{(out)})(z)]{\ket{{\Omega_{\\{x_{t},\mathrm{E}_{\quantity(t)},\overline{\mathrm{E}}_{\quantity(t)}\\}}}}}$
$\displaystyle=0,$

where we assumed $\absolutevalue{z}>x_{1}$ and which immediately becomes
(7.3).