\subsection{Motivation}

Unfortunately and puzzlingly the first attempts to consider space-like~\cite{Craps:2002:StringPropagationPresence} or light-like singularities~\cite{Liu:2002:StringsTimeDependentOrbifold,Liu:2002:StringsTimeDependentOrbifolds} by means of orbifold  techniques yielded divergent four points \emph{closed string} amplitudes (see \cite{Cornalba:2004:TimedependentOrbifoldsString,Craps:2006:BigBangModels} for reviews).

These singularities are commonly assumed to be connected to a large backreaction of the incoming matter into the singularity due to the exchange of a single graviton~\cite{Berkooz:2003:CommentsCosmologicalSingularities,Horowitz:2002:InstabilitySpacelikeNull}.
This claim was already questioned in the literature where the $O$-plane orbifold was constructed.
This orbifold should in fact be stable against the gravitational collapse but it exhibits divergences in the amplitudes (see the discussion in \cite{Cornalba:2004:TimedependentOrbifoldsString}).
In what follows we show a direct computation showing that the presence of the divergence is not related to a gravitational response.

Unnoticed in the Null Boost Orbifold (\nbo) \cite{Liu:2002:StringsTimeDependentOrbifold}, even the four \emph{open string} tachyons amplitude is divergent.
Since we are working at tree level gravity is not an issue.
In fact in~\cite{Liu:2002:StringsTimeDependentOrbifold} the four tachyons amplitude in the divergent region reads
\begin{equation}
  A_4 \sim \int\limits_{q \sim \infty} \frac{\dd{q}}{\abs{q}} \ccA( q )
\end{equation}
where $\ccA_{\text{closed}}( q ) \sim q^{4 - \ap \norm{\vec{p}_{\perp}}^2}$ and $\ccA_{\text{closed}}( q ) \sim q^{1 - \ap \norm{\vec{p}_{\perp}}^2} \tr\qty( \liebraket{T_1}{T_2}_+ \liebraket{T_3}{T_4}_+ )$ ($T_i$ for $i = 1,\, 2,\, 3,\, 4$ are Chan-Paton matrices).
Moreover divergences in string amplitudes are not limited to four points: interestingly we show that the open string three point amplitude with two tachyons and the first massive state may be divergent when some \emph{physical} polarisations are chosen.
The true problem is therefore not related to a gravitational issue but to the non existence of the effective field theory.
In fact when we express the theory using the eigenmodes of the kinetic terms some coefficients do not exist, not even as a distribution.
This holds true for both open and closed string sectors since it manifests also in the four scalar contact term.
The issue can be roughly traced back to the vanishing volume of a subspace and the existence of a discrete zero mode of the Laplacian on this subspace.

As an introduction to the problem we first deal with singularities of the open string sector.
We try to build a consistent scalar \qed and show that the vertex with four scalar fields is ill defined.
Divergences in scalar QED are due to the behaviour of the eigenfunctions of the scalar d'Alembertian near the singularity but in a somehow unexpected way.
Near the singularity $u = 0$ in lightcone coordinates almost all eigenfunctions behave as $\frac{1}{\sqrt{\abs{u}}} e^{i \frac{\cA}{u}}$ with $\cA \neq 0$.
The product of $N$ eigenfunctions gives a singularity $\abs{u}^{-N/2}$ which is technically not integrable.
However the exponential term $e^{i \frac{\cA}{u}}$ allows for an interpretation as distribution when $\cA = 0$ is not an isolated point.
When $\cA = 0$ is isolated the singularity is definitely not integrable and there is no obvious interpretation as a distribution.
Specifically in the \nbo we find $\cA \sim \frac{l^2}{k_+}$ where $l$ is the momentum along the compact direction.
As a consequence we find the eigenfunction associated to the discrete momentum $l = 0$ along the orbifold compact direction with an isolated $\cA = 0$.
It is the eigenfunction which is constant along that direction and it is the root of all divergences.

We then check whether the most obvious ways of regularizing the theory by making $\cA$ not vanishing may work.
The first regularisation we try is to use a Wilson line along the compact direction even though the diverging three point string amplitude involves an anti-commutator of the Chan-Paton factor therefore it is divergent also for a neutral string, i.e.\ for a string with both ends attached to the same D-brane.
This kind of string does not feel Wilson lines.
Moreover anti-commutators are present in amplitudes with massive states in unoriented and supersymmetric strings and therefore neither worldsheet parity nor supersymmetry can help.
The second obvious regularisation is the introduction of higher derivatives couplings to the Ricci tensor which is the only non vanishing tensor associated to the (regularised) metric.
In any case it seems that a sensible regularisation must couple to all open string in the same way and this suggests a gravitational coupling.
We then give a cursory look to whether closed string winding modes could help~\cite{Berkooz:2003:StringsElectricField}, as already suggested in~\cite{Liu:2002:StringsTimeDependentOrbifolds,Craps:2002:StringPropagationPresence} in analogy to the resolution of static singularities.
Twisted closed strings become massless near the singularity and they should in some way be included.
They generate a background potential $B_{\mu\nu}$ which is equivalent to a electromagnetic background from the open string perspective. 
Under a plausible modification of the scalar action which is suggested by the two-tachyons---two-photons amplitude the problems seem to be solvable.

In any case the origin of the string divergence seems to originate from the lack of contact terms in the effective field theory.
Since these terms arise from string theory also through the exchange of massive string states we examine three point amplitudes with one massive state.
A deeper understanding of the subject requires the study of the polarisations of the massive state on the orbifold as seen from the covering Minkowski space before the computation of the overlap of the wave functions.
We then go back to string theory and we verify that in the \nbo the open string three points amplitude with two tachyons and one first level massive string state does indeed diverge when some physical polarisation are chosen.

We then introduce the generalised Null Boost Orbifold (\gnbo) as a generalisation of the \nbo which still has a light-like singularity and is generated by one Killing vector.
However in this model there are two directions associated with $\cA$, one compact and one non compact.
We can then construct the scalar \qed and the effective field theory which extends it with the inclusion of higher order terms since all terms have a distributional interpretation~\cite{Estrada:2012:GeneralIntegral}.
However if a second Killing vector is used to compactify the formerly non compact direction, the theory has again the same problems as in the \nbo.
In the literature there are however also other attempts at regularizing the \nbo such as the Null Brane.
This kind of orbifold was originally defined in \cite{Figueroa-OFarrill:2001:GeneralisedSupersymmetricFluxbranes,Cornalba:2004:TimedependentOrbifoldsString} and studied in perturbation theory in \cite{Liu:2002:StringsTimeDependentOrbifolds}.
The Null Brane shares with the \gnbo the existence of a non compact direction on the orbifold.
In this case it is indeed possible to build single particle wave functions which leads to the convergence of the smeared amplitudes.

We finally present also a brief examination of the Boost Orbifold (\bo) where the divergences are generally milder~\cite{Horowitz:1991:SingularStringSolutions,Khoury:2002:BigCrunchBig}.
The scalar eigenfunctions behave in time $t$ as $\abs{t}^{\pm i\, \frac{l}{\Delta}}$ near the singularity but there is one eigenfunction which behaves as $\log(\abs{t})$ and again it is the constant eigenfunction along the compact direction which is the origin of all divergences.
In particular the scalar \qed on the \bo can be defined and the first term which gives a divergent contribution is of the form $\abs{\phi~\dphi}^2$, i.e.\ divergences are hidden into the derivative expansion of the effective field theory.
Again three points open string amplitudes with one massive state diverge.

\subsection{Scalar QED on NBO and Divergences} 
\label{sec:NOscalarQED}

As discussed the four open string tachyons amplitude diverges in the \nbo.
The literature on the subject (see for instance~\cite{Cornalba:2004:TimedependentOrbifoldsString} and references therein) suggests that this can be cured by the eikonal resummation.
We therefore consider the scalar \qed on the \nbo as a first approach.
In this case all eigenmodes can be written using elementary functions thus making the issues even more evident.
Its action is given by
\begin{equation}
  \rS_{\text{s}\qed}
  =
  \int\limits_{\Omega} \dd[D]{x}\,
  \sqrt{- \det g}
  \qty(
    - \qty(D^{\mu} \phi)^*\, D_{\mu} \phi
    - M^2  \qty(\phi^*)\, \phi
    - \frac{1}{4} f^{\mu\nu}\, f_{\mu\nu}
    - \frac{g_4}{4} \abs{\phi}^4
  ),
\end{equation}
with
\begin{equation}
  D_{\mu} \phi
  =
  \qty(\ipd{\mu} -i\, e\,  a_{\mu}) \phi,
  \qquad
  f_{\mu\nu}
  =
  \ipd{\mu} a_{\nu} - \ipd{\nu} a_{\mu}.
\end{equation}
We reserve small letters for quantities defined on the orbifold and capital letters for those defined in flat space.
Moreover $\Omega$ denotes the orbifold.
We will construct directly both the scalar and the spin-1 eigenfunctions  which we can use as a starting point for the perturbative computations.


\subsubsection{Geometric Preliminaries}
\label{sec:geometric_preliminaries_nbo}

In Minkowski spacetime $\ccM^{1,D-1}$ with coordinates $\qty(x^{\mu}) = \qty(x^+,\, x^-,\, x^2,\, \vec{x})$
and metric
\begin{equation}
  \dss[2]{s}
  =
  - 2 \dd{x^+} \dd{x^-}
  + \qty(\dd{x^2})^2
  + \eta_{ij} \dd{x}^i \dd{x}^j,
\end{equation}
we consider the following change of coordinates to $\qty(x^{\alpha}) = (u,\, v,\, z,\, \vec{x})$
\begin{equation}
  \begin{cases}
    x^- & = u
    \\
    x^2 & = \Delta u z
    \\
    x^+ & = v + \frac{1}{2} \Delta^2 u z^2
  \end{cases}
  \qquad
  \Leftrightarrow
  \qquad
  \begin{cases}
    u & = x^-
    \\
    z & = \frac{x^2}{\Delta\, x^-}
    \\
    v & = x^+ - \frac{1}{2} \frac{(x^2)^2}{x^-}
  \end{cases}.
  \label{eq:NBO_coordinates}
\end{equation}
Then the metric becomes:
\begin{equation}
  \dss[2]{s}
  =
  - 2\, \dd{u}\, \dd{v}
  + \qty(\Delta u )^2 (\dd{z})^2
  + \eta_{ij} \dd{x}^i \dd{x}^j,
\end{equation}
along with the non vanishing geometrical quantities
\begin{equation}
  -\det g = \qty( \Delta u )^2,
\end{equation}
and
\begin{equation}
  \tensor{\Gamma}{_z^v_z} = \Delta^2 u,
  \qquad
  \tensor{\Gamma}{_u^z_z} = u^{-1}.
\end{equation}
Riemann and Ricci tensor components however vanish since at this stage we only performed a change of coordinates from the original Minkowski spacetime.
Locally it is the same as the \nbo and they must have the same local differential geometry.

The \nbo is introduced by identifying points along the orbits of the Killing vector:
\begin{equation}
  \begin{split}
    \kappa
    & =
    - i \qty(2 \pi \Delta) J_{+2}
    \\
    & =
    \qty(2 \pi \Delta)\, (x^2 \ipd{+} + x^- \ipd{2})
    \\
    & =
    2 \pi \ipd{z},
  \end{split}
  \label{eq:nbo_killing_vector}
\end{equation}
in such a way that
\begin{equation}
  x^{\mu} \equiv \cK^{n}\, x^{\mu},
  \qquad
  n \in \Z,
\end{equation}
where $\cK^{n}= e^{n\kappa}$, leads to the identifications
\begin{equation}
  x=
  \mqty( x^- \\  x^2 \\ x^+ \\ \vec{x} )
  \equiv
  \cK^{n} x
  =
  \mqty(%
    x^- \\
    x^2 + n \qty(2 \pi \Delta) x^- \\
    x^+ + n \qty(2 \pi \Delta) x^2 + \frac{1}{2} n^2 \qty(2 \pi \Delta)^2 x^- \\
   \vec{x}
  )
\end{equation}
or to 
\begin{equation}
  \qty( u,\, v,\, z,\, \vec{x} )
  \equiv
  \qty( u,\, v,\, z + 2 \pi n,\, \vec{x} )
\end{equation}
in coordinates $\qty(x^{\alpha})$ where $\kappa = 2 \pi \ipd{z}$ is a global Killing vector.

As a reference for the future, we notice that we could regularise the metric as
\begin{equation}
  \dss[2]{s}
  =
  - 2\, \dd{u}\, \dd{v}
  + \Delta^2 \qty(u^2 + \epsilon^2) (\dd{z})^2
  + \eta_{ij} \dd{x}^i \dd{x}^j.
\end{equation}
The non vanishing geometrical quantities are then:
\begin{equation}
  -\det g = \Delta^2 \qty(u^2 + \epsilon^2),
\end{equation}
and
\begin{equation}
  \tensor{\Gamma}{_z^v_z} = \Delta^2 u,
  \qquad
  \tensor{\Gamma}{_u^z_z} = \frac{u}{u^2 + \epsilon^2},
\end{equation}
which lead to the following Riemann and Ricci tensor components:
\begin{equation}
  \tensor{R}{^z_u_z_u} = - \frac{\epsilon^2}{\qty(u^2+ \epsilon^2)^2},
  \quad
  \tensor{R}{^v_z_z_u} = - \frac{\Delta^2 \epsilon^2}{u^2 + \epsilon^2},
  \quad
  \tensor{Ric}{_u_u} = -\frac{\epsilon^2}{\qty(u^2+ \epsilon^2)^2}.
\end{equation}
Since $\delta_{\text{reg}}(u) = \frac{1}{\pi} \frac{\epsilon}{u^2+ \epsilon^2}$ then $\tensor{R}{^z_u_z_u} = - \pi^2 \qty[ \delta_{\text{reg}}(u) ]^2$.


\subsubsection{Free Scalar Action} 
 
We study the eigenmodes of the Laplacian operator to diagonalize the scalar kinetic term given by:\footnotemark{}
\footnotetext{%
  The factor $-g^{\alpha\beta}$ is due to the choice of the East coast convention for the metric, namely:
  \begin{equation*}
    - g^{\alpha\beta}
    \ipd{\alpha} \phi^*\, \ipd{\beta} \phi
    -
    M^2 \phi^*\, \phi
    \sim
    \abs{\dot{\phi}}^2 - M^2 \abs{\phi}^2
    \sim
    \rE^2 - M^2.
    \end{equation*}
}
\begin{equation}
  \begin{split}
    \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \phi ]
    & =
    \int\limits_{\Omega} \dd[D]{x}\,
    \sqrt{- \det g}~
    \qty(%
      - g^{\alpha\beta} \ipd{\alpha} \phi^*\, \ipd{\beta} \phi
      - M^2  \phi^*\, \phi
    )
    \\
    & =
    \int\limits_{\R^{D-3}} \dd[D-3]{\vec{x}}\,
    \infinfint{u}\,
    \infinfint{v}\,
    \finiteint{z}{0}{2\pi}
    \abs{\Delta u}
    \\
    & \times
    \qty(%
      \ipd{u} \phi^*\, \ipd{v} \phi\,
      +
      \ipd{v} \phi^*\, \ipd{u} \phi\,
      -
      \frac{1}{\qty(\Delta u)^2} \ipd{z} \phi^*\, \ipd{z} \phi\,
      -
      \ipd{i} \phi^*\, \ipd{i} \phi
      -
      M^2 \phi^*\, \phi
    ).
  \end{split}
\end{equation}
The solution to the equation of motion is enough when we want to perform the canonical quantization.
Since we use Feynman diagrams we consider the path integral approach: we take off-shell modes and solve the eigenvalue problem $\square \phi_r = r \phi_r$.
Comparing with the flat case we see that $r$ is $2\, k_-\, k_+ - \norm{\vec{k}}^2$ when $k$ is the impulse in flat coordinates.
We therefore have
\begin{equation}
  -2 \ipd{u} \ipd{v} \phi_r
  -
  \frac{1}{u} \ipd{v} \phi_r
  +
  \frac{1}{\qty(\Delta u)^2} \ipd{z}^2 \phi_r
  +
  \ipd{i}^2 \phi_r
  =
  r \phi_r.
  \label{eq:nbo_eom}
\end{equation}
Using Fourier transforms it follows that the eigenmodes are
\begin{equation}
  \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x})
  =
  e^{i k_+ v + i l z + i \vec{k} \cdot \vec{x}}\,
  \tphi_{k_-kr}(u),
\end{equation}
with
\begin{equation}
  \tphi_{k_-kr}(u)
  =
  \frac{1}{\sqrt{\qty( 2 \pi )^D~ \abs{2 \Delta k_+\, u}}}
  e^{
    - i \frac{l^2}{2 \Delta^2 k_+} \frac{1}{u}
    + i \frac{\norm{\vec{k}}^2 + r}{2 k_+} u
  },
\end{equation}
and
\begin{equation}
  \phi^*_{k_-kr}\qty(u,\, v,\, z,\, \vec{x})
  =
  \phi_{\mkmkr}\qty(u,\, v,\, z,\, \vec{x}).
\end{equation}
We chose the numeric factor in order to get a canonical normalisation:
\begin{equation}
  \begin{split}
    &
    \qty( \phi_{k_-krN{1}},\,  \phi_{k_-krN{2}} )
    \\
    = &
    \int\limits_{\R^{D-3}} \dd[D-3]{\vec{x}}\,
    \infinfint{u}\,
    \infinfint{v}\,
    \finiteint{z}{0}{2\pi}
    \abs{\Delta u}\,
    \phi_{k_-krN{1}}\, \phi_{k_-krN{2}}
    \\
    = &
    \delta^{D-3}( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)})\,
    \delta( r_{\qty(1)} - r_{\qty(2)})\,
    \delta( k_{\qty(1)\, +} + k_{\qty(2)\, +})\,
    \delta_{l_{\qty(1)} + l_{\qty(2)},\, 0}.
  \end{split}
\end{equation} 
We can then perform the off-shell expansion
\begin{equation}
  \phi\qty(u,\, v,\, z,\, \vec{x})
  =
  \int\limits_{\R^{D-3}} \dd[D-3]{\vec{k}}
  \infinfint{k_+}
  \infinfint{r}
  \infinfsum{l}
  \cA_{k_-kr}\,
  \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}),
\end{equation}
such that the scalar kinetic term becomes 
\begin{equation}
  \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \cA ]
  =
  \int\limits_{\R^{D-3}} \dd[D-3]{\vec{k}}\,
  \infinfint{k_+} 
  \infinfint{r}
  \infinfsum{l}
  \qty(r - M^2)\,
  \cA_{k_-kr}\,
  \cA_{k_-kr}^*.
\end{equation}


\subsubsection{Free Photon Action}

The action of the free photon can be written as
\begin{align}
  \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ a ]
  =
  \int\limits_{\Omega} \dd[D]{x}\,
  \sqrt{-\det g}\,
  \qty(%
   - \frac{1}{2} g^{\alpha\beta} g^{\gamma\delta}
   D_{\alpha} a_{\gamma} \qty( D_{\beta} a_{\delta} - D_{\delta} a_{\beta})
  ).
\end{align}
We choose to enforce the Lorenz gauge:\footnotemark{}
\footnotetext{%
  Indeed it is exactly the usual Lorenz gauge since locally the space is Minkowski.
}
\begin{equation}
  D^{\alpha} a_{\alpha}
  =
  - \frac{1}{u} a_{v}
  - \ipd{u} a_{v}
  - \ipd{v} a_{u}
  + \frac{1}{\Delta^2 u^2} \ipd{z} a_z
  + \eta^{ij} \ipd{i} a_j
  =
  0.
  \label{eq:Lorenz_gauge}
\end{equation}
As covariant derivatives commute since we are locally flat, the \eom read $\qty(\square a)_{\alpha} = 0$.
Explicitly we have:
\begin{equation}
  \begin{split}
    \qty( \square a )_u
    & =
    \frac{1}{u^2} a_{v}
    -
    \frac{2}{\Delta^2 u^3} \ipd{z} a_z
    +
    \qty[
      -
      2 \ipd{u} \ipd{v}
      -
      \frac{1}{u} \ipd{v}
      +
      \frac{1}{\Delta^2 u^2} \ipd{z}^2
      +
      \eta^{ij} \ipd{i} \ipd{j}
    ]
    a_u,
    \\
    \qty( \square a )_v
    & =
    \qty[
    -
      2 \ipd{u} \ipd{v}
      -
      \frac{1}{u} \ipd{v}
      +
      \frac{1}{\Delta^2 u^2} \ipd{z}^2
      +
      \eta^{ij} \ipd{i} \ipd{j}
    ]
    a_v,
    \\
    \qty( \square a )_z
    & =
    -
    \frac{2}{u} \ipd{z} a_v
    +
    \qty[
      -
      2 \ipd{u} \ipd{v}
      +
      \frac{1}{u} \ipd{v}
      +
      \frac{1}{\Delta^2 u^2} \ipd{z}^2
      +
      \eta^{ij} \ipd{i} \ipd{j}
    ]
    a_z,
    \\
    \qty( \square a )_i
    & =
    \qty[
      -
      2 \ipd{u} \ipd{v}
      -
      \frac{1}{u} \ipd{v}
      +
      \frac{1}{\Delta^2 u^2} \ipd{z}^2
      +
      \eta^{ij} \ipd{i} \ipd{j}
    ]
    a_i.
  \end{split}
\end{equation}

As in the previous scalar case we are actually interested in solving
the eigenmodes problem $\qty(\square a_r)_\alpha= r \,a_{r\, \alpha}$.
We proceed hierarchically: first we solve for $a_v$ and $a_i$ whose equations are the same as in the scalar field, then we insert the solutions as a source in the equation for $a_z$ and eventually we solve for $a_u$.\footnotemark{}
\footnotetext{%
  Notice that inside the square brackets of the differential equation for $a_z$ there is a different sign for the term $\frac{1}{u} \ipd{v}$ with respect to the equation for the scalar field.
}
We get the solutions:
\begin{equation}
  \begin{split}
    \norm{\tildea_{k_-kr\, \alpha}(u)}
    \,=
    \mqty(%
      \tildea_u
      \\
      \tildea_v
      \\
      \tildea_z
      \\
      \tildea_i
    )
    & =
    \sum\limits_{%
      \underline{\alpha}
      \in 
      \qty{ \underu, \underv, \underz,\underi }
    }
    \pol{\alpha}
    \norm{\tildea^{\underline{\alpha}}_{k_-kr\, \alpha}(u)}
    \\
    & =
    \pol{u}
    \mqty(
      1
      \\
      0
      \\
      0
      \\
      0
    )\,
    \tphi_{k_-kr}(u)
    \\
    & +
    \pol{v}
    \mqty(
      \frac{i}{2 k_+ u}
      +
      \frac{1}{2} \qty( \frac{l}{\Delta k_+} )^2 \frac{1}{u^2}
      \\
      1
      \\
      \frac{l}{k_+}
      \\
      0
    )\,
    \tphi_{k_-kr}(u)
    \\
    & +
    \pol{z}
    \mqty(
      \frac{l}{\Delta k_+ \abs{u}}
      \\
      0
      \\
      \Delta \abs{u}
      \\
      0
    )\,
    \tphi_{k_-kr}(u)
    \\
    & +
    \pol{j}
    \mqty(
      0
      \\
      0
      \\
      0
      \\
      \delta_{\underline{ij}}
    )\,
    \tphi_{k_-kr}(u),
    \label{eq:Orbifold_spin1_pol}
  \end{split}
\end{equation}
then we can expand the off-shell fields as
\begin{equation}
  a_{\alpha}\qty(u,\, v,\, z,\, \vec{x} )
  =
  \int \ccD k
  \sum\limits_{%
    \underline{\alpha}
    \in
    \qty{ \underu, \underv, \underz,\underi }
  }
  \infinfsum{l}
  \pol{\alpha}\,
  {a}^{\underline{\alpha}}_{k_-kr\, \alpha}\qty(u,\, v,\, z,\, \vec{x} ),
\end{equation}
where
\begin{equation}
  a^{\underline{\alpha}}_{k_-kr\, \alpha}\qty(u,\, v,\, z,\, \vec{x})
  =
  \tildea^{\underline{\alpha}}_{k_-kr\, \alpha}(u)\,
  e^{i\, \qty( k_+ v + l z + \vec{k} \cdot \vec{x})}
\end{equation}
and $\int \ccD k = \int\limits_{\R^{D-3}} \dd[D-3]{\vec{k}} \infinfint{k_+} \infinfint{r}$.

We can also compute the normalisation as
\begin{equation}
  \begin{split}
    \qty(a_{\qty(1)},\, a_{\qty(2)})
    & =
    \int\limits_{\R^{D-3}} \dd[D-3]{\vec{x}}
    \infinfint{u}
    \infinfint{v}
    \finiteint{z}{0}{2\pi}
    \abs{\Delta u}
    \\
    & \times
    g^{\alpha\beta}\,
    a_{k_-krN{1}\, \alpha}\,  a_{k_-krN{2}\, \beta}
    \\
    & =
    \genpolN{1} \circ \genpolN{2}
    \\
    & \times
    \delta^{D-3}( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)})\,
    \delta( r_{\qty(1)} - r_{\qty(2)})\,
    \delta( k_{\qty(1)\, +} + k_{\qty(2)\, +})\,
    \delta_{l_{\qty(1)} + l_{\qty(2)},\, 0},
  \end{split}
\end{equation}
where:\footnotemark{}
\footnotetext{%
  We use a shortened version of the polarisations $\cE$ for the sake of readability.
  We write $\polabbrN{\alpha}{n} = \polN{\alpha}{n}$ thus hiding the understood dependence of the components of $\cE_{(n)}$ on the momenta.
}
\begin{equation}
  \begin{split}
    \cE_{\qty(1)} \circ \cE_{\qty(2)}
    =
    - \polabbrN{u}{1}\, \polabbrN{v}{2}
    - \polabbrN{v}{1}\, \polabbrN{u}{2}
    + \polabbrN{z}{1}\, \polabbrN{z}{2}
    +
    \eta^{\underline{ij}}\,
    \polabbrN{i}{1}\, \polabbrN{j}{2}.
  \end{split}
\end{equation}
Finally the Lorenz gauge reads
\begin{equation}
   \eta^{i \underj}\, k_i\, \pol{j}
   -
   k_+\, \pol{u}
   -
   \frac{\norm{\vec{k}}^2 + r}{2\, k_+} \pol{v}
   =
   0,
  \label{eq:explicit_orbifold_Lorenz}
\end{equation}
which does not impose any constraint on the transverse polarisation
$\pol{z}$.
The photon kinetic term becomes 
\begin{equation}
  \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \cE ]
  =
  \int\limits_{\R^{D-3}} \dd[D-3]{\vec{k}}
  \infinfint{k_+}
  \infinfint{r}
  \infinfsum{l}\,
  \frac{r}{2}\,
  \cE_{k_-kr}\,
  \circ
  \cE_{k_-kr}^*.
\end{equation}


\subsubsection{Cubic Interaction}

With the definition of the d'Alembertian eigenmodes we can now examine the cubic vertex which reads
\begin{equation}
  \rS_{\text{s}\qed}^{(\text{cubic})}\qty[\phi,\, a]
  =
  \int\limits_{\Omega} \dd[D]{x}\,
  \sqrt{- \det g}\,
  \qty(%
    -i\, e\,
    g^{\alpha\beta}
    a_{\alpha}
    \qty(%
      \phi^*\, \ipd{\beta} \phi
      -
      \ipd{\beta} \phi^*\, \phi
    )
  ).
\end{equation}
Its computation involves integrals such as
\begin{equation}
  \int \dd{u}\,
  \abs{\Delta u}\,
  \qty(\frac{l}{u})^2
  \finiteprod{i}{1}{3} \tphi_{k_-krN{i}}
  \sim
  \int\limits_{u \sim 0} \dd{u}\, 
  \qty(\frac{l^2}{\abs{u}^{\frac{5}{2}}})
  e^{%
    -i \finitesum{i}{1}{3} \frac{l_{\qty(i)}^2}{2\, \Delta^2 k_{\qty(i)\, +)}}
    \frac{1}{u}
  },
\end{equation}
and
\begin{equation}
  \int \dd{u}\,
  \abs{\Delta u}\,
  \qty(\frac{1}{u})
  \finiteprod{i}{1}{3} \tphi_{k_-krN{i}}
  \sim
  \int\limits_{u \sim 0} \dd{u}\, 
  \qty(\frac{1}{u\, \abs{u}^{\frac{1}{2}}})
  e^{%
    -i \finitesum{i}{1}{3} \frac{l_{\qty(i)}^2}{2\, \Delta^2 k_{\qty(i)\, +}}
    \frac{1}{u}
  },
  \label{eq:nbo_div_integral}
\end{equation}
which can be interpreted as hints that the theory is troublesome.
The first integral diverges if the exponential functions are all equal to unity.
Fortunately it happens when all factors $l_{\qty(i)}$ (where $i = 1,\, 2,\, 3$) vanish.
In this case however the integral vanishes if we set $l_{\qty(i)} = 0$ before its evaluation.
This however suggests that when all $l_{\qty(i)} = 0$, i.e.\ when the eigenfunctions are constant along the compact direction $z$, something suspicious is happening.
On the other side when at least one $l$ is different from zero we have an integral such as:
\begin{equation}
  \int\limits_{u \sim 0} \dd{u}\, 
  \abs{u}^{-\nu}\, e^{i \frac{\cA}{u}}
  \sim
  \int\limits_{t \sim \infty} \dd{t}\, 
  t^{\nu-2}\, e^{i \cA t}.
\end{equation}
All $l_{\qty(i)}$ are discrete but $k_{\qty(i)\, +}$ are not thus $\cA$ has an isolated zero.
Otherwise it has continuous value and may be given a distributional meaning, similar to a derivative of the Dirac delta function.
The second integral has the same issues when all $l_{\qty(*)} = 0$ but, since it is not proportional to any $l$ as it stands, it is divergent unless we consider a principal part regularization.

We can give in any case meaning to the cubic terms
and we get:\footnotemark{}
\footnotetext{%
  The notation $\qty(2) \rightarrow \qty(3)$ meaning is that all previous terms inside the curly brackets appear again in exactly the same structure but with momenta of particle $\qty(3)$ in place of those of particle $\qty(2)$.
}
\begin{equation}
  \begin{split}
    \rS_{\text{s}\qed}^{(\text{cubic})}\qty[ \cA,\, \cE ]
    & =
    \finiteprod{i}{1}{3}
    \qty[%
      \int\limits_{\R^{D-3}} \dd[D-3]{\vec{k}_{\qty(i)}}\,
      \infinfint{r_{\qty(i)}}\,
      \infinfint{k_{\qty(i)\, +}}
      \sum_{l_{\qty(i)}}
    ]\,
    \qty(2\pi)^{D-1}
    \\
    & \times
    e~
    \delta\qty(\finitesum{i}{1}{3} \vec{k}_{\qty(i)})\,
    \delta\qty(\finitesum{i}{1}{3} k_{\qty(i)\, +})\,
    \delta_{\finitesum{i}{1}{3} l_{\qty(i)},\, 0}\,
    \\
    & \times
    \qty(\cA_{\mkmkrN{2}})^*\, \cA_{k_-krN{3}}
    \\
    & \times
    \Biggl\lbrace
      \polN{u}{1}\,
      k_{\qty(2)\, +}\,
      \cI_{\qty{3}}^{\qty[0]}
      \\
      & +
      \polN{z}{1}\,
      \frac{%
        k_{\qty(2)\, +} l_{\qty(1)}
        -
        l_{\qty(2)} k_{\qty(1)\, +}
      }{\Delta k_{\qty(1)\, +}}\,
      \cJ_{\qty{3}}^{\qty[-1]}
      \\
      & +
      \polN{v}{1}\,
      \ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vec{k}_{\qty(2)})
      \\
      & -
      \eta^{\underi\, j}\,
      \polN{i}{1}\,
      k_{\qty(2)_j}\,
      \cI_{\qty{3}}^{\qty[0]}\,
      -
      \qty( \qty(2) \rightarrow \qty(3) )
    \Biggr\rbrace,
  \label{eq:sQED_cubic_final}   
  \end{split}
\end{equation}
where
\begin{equation}
  \begin{split}
    \ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vec{k}_{\qty(2)})
    & =
    \frac{\norm{\vec{k}_{\qty(2)}}^2 + r_{\qty(2)}}{2\, k_{\qty(2)\, +}} \cI_{\qty{3}}^{\qty[0]}
    +
    i
    \frac{k_{\qty(2)\, +}}{2\, k_{\qty(1)\, +}}
    \cI_{\qty{3}}^{\qty[-1]}
    \\
    & +
    \frac{1}{2} \frac{k_{\qty(2)\, +}}{\Delta^2}
    \qty(%
      \frac{l_{\qty(1)}}{k_{\qty(1)\, +}}
      -
      \frac{l_{\qty(2)}}{k_{\qty(2)\, +}}
    )^2
    \cI_{\qty{3}}^{\qty[-2]}.
  \end{split}
\end{equation}
In the previous expressions we also defined for future use:
\begin{eqnarray}
  \cI_{\qty(1) \dots (N)}^{\qty[\nu]}
  =
  \cI_{\qty{N}}^{\qty[\nu]}
  & = &
  \infinfint{u}\,
  \abs{\Delta u}\, u^{\nu}\,
  \finiteprod{i}{1}{N}
  \tphi_{k_-krN{i}}
  \\
  \cJ_{\qty{N}}^{\qty[\nu]}
  & = &
  \infinfint{u}\,
  \abs{\Delta}\, \abs{u}^{1 + \nu}
  \finiteprod{i}{1}{N} \tphi_{k_-krN{i}}.
\end{eqnarray}
For the sake of brevity from now on we use
\begin{eqnarray}
  \tphi_{\qty(i)} & = & \tphi_{k_-krN{i}},
  \\
  \tphi_{\qty(i)} & = & \tphi_{k_-krN{i}}
\end{eqnarray}
when not causing confusion.


\subsubsection{Quartic Interactions and Divergences}

The issue with the divergent vertex is even more visible when considering the quartic terms:
\begin{equation}
  \rS_{\text{s}\qed}^{(\text{quartic})}\qty[ \phi,\, a ]
  =
  \int\limits_{\Omega} \dd[D]{x}\,
  \sqrt{- \det g}\,
  \qty(%
    e^2\, g^{\mu\nu}\, a_{\mu} a_{\nu}\, \abs{\phi}^2
    -
    \frac{g_4}{4}\abs{\phi}^4
  ),
\end{equation}
which can be expressed using the modes as:
\begin{equation}
  \begin{split}
    \rS_{\text{s}\qed}^{(\text{quartic})}\qty[ \phi,\, a ]
    & =
    \finiteprod{i}{1}{4}
    \qty[%
      \int \dd[D-3]{\vec{k}_{\qty(i)}}
      \dd{k_{\qty(i)\, +}}
      \dd{r_{\qty(i)}}
      \sum_{l_{\qty(i)}}
    ]\,
    \qty(2\pi)^{D-1}
    \\
    & \times
    \delta\qty( \finitesum{i}{1}{4} \vec{k}_{\qty(i)} )\,
    \delta\qty( \finitesum{i}{1}{4} k_{\qty(i)\, +} )\,
    \delta_{\finitesum{i}{1}{4} l_{\qty(i)},\, 0}
    \\
    & \times
    \Biggl\lbrace
      e^2\,
      \qty(\cA_{\mkmkrN{3}})^* \cA_{k_-krN{4}}
      \\
      & \times
      \Biggl[
       \qty(\genpolN{1} \circ \genpolN{2})\,
       \cI_{\qty{4}}^{\qty[0]}
        \\
        & -
        \frac{i}{2}
        \polN{v}{1}\, \polN{v}{2}
        \qty(%
          \frac{1 }{k_{\qty(2)\, +}}
          +
          \frac{1}{k_{\qty(1)\, +}}
        )\,
        \cI_{\qty{4}}^{\qty[-1]}
        \\
        & +
        \frac{1}{2}\,
        \frac{\polN{v}{1} \polN{v}{2} }{\Delta^2}
        \qty(%
          \frac{l_{\qty(1)}}{k_{\qty(1)\, +}}
          -
          \frac{l_{\qty(2)}}{k_{\qty(2)\, +}}
        )^2\,
        \cI_{\qty{4}}^{\qty[-2]}
      \Biggr]
      \\
      & -
      \frac{g_4}{4}\,
      \ccA\qty(\qty{k_+,\, l,\, \vec{k},\, r})\,
      \cI_{\qty{4}}^{\qty[0]}
    \Biggr\rbrace,
  \end{split}
\end{equation}
where
\begin{equation}
  \begin{split}
    \ccA\qty(\qty{k_+,\, l,\, \vec{k},\, r})
    & =
    \qty(\cA_{\mkmkrN{1}})^*\,
    \qty(\cA_{\mkmkrN{2}})^*\,
    \\
    & \times
    \cA_{k_-krN{3}}\,
    \cA_{k_-krN{4}}.
  \end{split}
\end{equation}
When setting $l_{\qty(*)} = 0$ all the surviving terms are divergent.
The explicit behaviour is $\cI_{\qty{4}}^{\qty[0]} \sim \int \dd{u}\, \abs{u}^{1 -4 \times \frac{1}{2}}$ and $\cI_{\qty{4}}^{\qty[-1]} \sim \int \dd{u}\, u^{-1}\, \abs{u}^{1 - 4 \times \frac{1}{2}}$ since $\eval{\tphi}_{l = 0} \sim \abs{u}^{-\frac{1}{2}}$.
Higher order terms in the effective field theory have even worse behaviour.
This makes the theory ill defined and the string theory which should give this effective theory ill defined too.


\subsubsection{Failure of Obvious Divergence Regularizations}
\label{sec:saving}

From the discussion in the previous section the origin of the divergences is the sector $l = 0$.
When $l = 0$ the highest order singularity of the Fourier transformed d'Alembertian equation vanishes.
Explicitly we have:
\begin{equation}
  A\, \ipd{u} \tphi_{k_-kr}
  +
  B(u)\, \tphi_{k_-kr}
  =
  A\, e^{-\int^u \frac{B(u)}{A} du}\, 
  \ipd{u} \qty[ e^{+\int^u \frac{B(u)}{A} du} \tphi_{k_-kr} ]
  =
  0,
\end{equation}
with
\begin{equation}
  A = \qty(-2\, i\, k_+),
  \qquad
  B(u)
  =
  -\qty(\norm{\vec{k}}^2 + r)
  -
  i\, k_+\, \frac{1}{u}
  -
  \frac{l^2}{\Delta^2}\, \frac{1}{u^2}.
\end{equation}
This implies the absence of the oscillating factor $e^{i \frac{\cA}{u} }$ when $l$ vanishes.
It follows that any deformation which prevents the coefficient of the highest order singularity from vanishing will do the trick.

The first and easiest possibility is to add a Wilson line along $z$, i.e.\ $a = \theta \dd{z}$.
This shifts $l \rightarrow l - e\, \theta$ and regularises the scalar \qed.
Unfortunately this does not work in the string theory where Wilson lines on D25-branes are not felt by the neutral strings starting and ending on the same D-brane.
In fact not all interactions involve commutators of the Chan-Paton factors which vanish for neutral strings.
For instance the interaction of two tachyons with the first massive state involves an anti-commutator as we discuss later. 
The anti-commutators are present also in amplitudes of supersymmetric strings with massive states and therefore the issue is not solved by supersymmetry.

A second possibility is to include higher derivative couplings to curvature as natural in the string theory.
If we regularise the metric in a minimal way as shown at the end of~\Cref{sec:geometric_preliminaries_nbo}, only $\tensor{Ric}{_u_u}$ does not vanish.
We can introduce:
\begin{equation}
  \begin{split}
    &
    S_{\mathrm{HE}}^{(\text{higher R})}\qty[ \phi,\, g ]
    \\
    = &
    \int\limits_{\Omega} \dd[D]{x}\,
    \sqrt{- \det g}\,
    \qty(%
      \finitesum{k}{1}{+\infty}
      \qty(\ap)^{2 k-1}\,
      \finiteprod{j}{1}{k}\,
      g^{\mu_j \nu_j}\, g^{\rho_j\sigma_j}\,
      \tensor{Ric}{_{\mu_j}_{\rho_j}}\,
      \qty(%
        \finitesum{s}{0}{2k}
        c_{k s}\, \ipd{\nu_j}^{2k - s}\phi^*\, \ipd{\sigma_j}^s \phi
      )
    )
    \\
    = &
    \int\limits_{\Omega} \dd[D]{x}\,
    \sqrt{- \det g}\,
    \qty(%
      \ap\,
      g^{\mu\nu}\, g^{\rho\sigma}\,
      \tensor{Ric}{_{\mu}_{\rho}}\,
      \qty(%
        c_{12}
        \phi^*\, \ipd{\nu\sigma}^2 \phi
        +
        c_{11}
        \ipd{\nu} \phi^*\, \ipd{\sigma} \phi
        +
        c_{10}
        \ipd{\nu\sigma}^2 \phi^*\, \phi
      )
    ),
  \end{split}
\end{equation}
where $\ap$ has been introduced after dimensional analysis and in order to have all adimensional $c$ factors.
Since only $\tensor{Ric}{_u_u}$ is non vanishing and it depends only on $u$,
the regularised d'Alembertian eigenmode problem now reads:
\begin{equation}
  \begin{split}
    -
    2 \ipd{u} \ipd{v} \phi_r
    & -
    \frac{u}{u^2 + \epsilon^2} \ipd{v} \phi_r
    +
    \frac{1}{\Delta^2 (u^2+ \epsilon^2)} \ipd{z}^2 \phi_r
    \\
    & +
    \finitesum{k}{1}{+\infty} \qty(\ap)^{2k-1}\,
    C_k\,
    \tensor{Ric}{_u_u}^k\,
    \ipd{v}^{2k} \phi
    +
    \ipd{i}^2 \phi_r
    -
    r\, \phi_r
    =
    0,
  \end{split}
\end{equation}
with $C_k = \finitesum{s}{0}{2k} (-1)^s\, c_{k s}$.
We can perform the usual Fourier transform and the function $B(u)$ becomes
\begin{equation}
  \begin{split}
    B(u)
    & =
    -
    \qty(\norm{\vec{k}}^2 + r)
    -
    i\, k_+\, \frac{u}{u^2 + \epsilon^2}
    -
    \frac{l^2}{\Delta^2}\, \frac{1}{u^2+\epsilon^2}
    \\
    & +
    \finitesum{k}{1}{+\infty}
    \qty(\ap)^{2k-1}\,
    C_k\,
    \qty(\frac{\epsilon^2}{(u^2 + \epsilon^2)^2})^k
    (-i k_+)^{2k}.
  \end{split}
\end{equation}
When $u = 0$ we have:
\begin{equation}
  B(0)
  \sim
  - \frac{l^2}{\Delta^2}\, \frac{1}{\epsilon^2}
  +
  \finitesum{k}{1}{+\infty}
  \qty(\ap)^{2k-1}\,
  C_k\,
  \frac{(-i k_+)^{2k}}{\epsilon^{2 k}}.
\end{equation}
Though the correction seems to lead to a cure for the divergence, ff we consider $\ap$ and $\epsilon^2$ uncorrelated we lose predictability.
However if $\ap \sim \epsilon^2$ as natural in string theory we do not solve the problem since 
\begin{equation}
  B(0)
  \stackrel{\ap \sim \epsilon^2}{\sim}
  - \frac{l^2}{\Delta^2}\, \frac{1}{\epsilon^2}
  +
  \finitesum{k}{1}{+\infty}
  C_k\,
  (-i k_+)^{2k}
  \epsilon^{2k - 2}
\end{equation}
and the curvature terms are no longer singular.
    

\subsubsection{A Hope from Twisted State Background}

The issue with the divergences is associated with the dipole string and its charge neutral states since the charged ones can be cured rather trivially by a Wilson line.

On the other hand we know that the usual time-like orbifolds are well defined because of a presence of a $B_{\mu\nu}$ background and this field is sourced by strings.
We may switch on such a background in the open string.
For open strings $F$ is equivalent to such $B$ field so we can consider what happens to an open string in an electromagnetic background.

The choice of such a background is limited first of all by the request that it must be an exact string solution, i.e.\ it needs to obey the \eom derived from the Dirac--Born--Infeld action.
If a closed string winds the compact direction $z$ then it is coupled to $B_{z u}$, $B_{z v}$  and $B_{z i}$ but if we choose
\begin{equation}
  \frac{1}{2\pi\ap} B(u)
  =
  f(u) \dd{u} \wedge \dd{z}.
  \label{eq:F_bck}
\end{equation}
then
\begin{equation}
  \det( g + 2 \pi \ap f(u) ) = \det(g).
\end{equation}
It is therefore a solution of the open string \eom for any $f\qty(u,\, v,z,x^i)$.
As the two-tachyons---two-photons amplitude suggests, suppose that the action for a real neutral scalar $\phi$ is given by:
\begin{equation}
  \begin{split}
    S_{\text{scalar}}^{(\text{kinetic})}\qty[ \phi ]
    & =
    \int\limits_{\Omega} \dd[D]{x}\,
    \sqrt{- \det g}\,
    \frac{1}{2}
    \qty(
      - g^{\alpha\beta}
      \ipd{\alpha} \phi\, \ipd{\beta} \phi
      - M^2 \phi^2
      + c_1
      \qty(\ap)^2\, \ipd{\mu} \phi\, \ipd{\nu} \phi
      \tensor{f}{^{\mu}^\kappa} \tensor{f}{^{\nu}_\kappa}
    )
    \\
    & =
    \int\limits_{\R^{D-3}} \dd[D-3]{\vec{x}}\,
    \infinfint{u}\,
    \infinfint{v}\,
    \finiteint{z}{0}{2\pi}
    \abs{\Delta u}\,
    \frac{1}{2}\,
    \Biggl(
      2\, \ipd{u} \phi\,\ipd{v} \phi\,
    \\
      & -
      \frac{1}{\qty(\Delta u )^2} \qty(\ipd{z}  \phi)^2
      -
      \eta^{ij} \ipd{i}\phi\, \ipd{j} \phi
      -
      M^2 \phi^2
      +
      c_1 \qty(\ap)^2 \frac{1}{\qty(\Delta u)^2} \qty(\ipd{v} \phi)^2 f^2(u)
    \Biggr)
    ,
  \end{split}
\end{equation}
Performing the same steps as before we get
\begin{equation}
  B(u)
  =
  - \qty(\norm{\vec{k}}^2 + r)
  -
  i\, k_+\, \frac{1}{u}
  +
  \frac{\qty(c_1 \qty(\ap)^2 f(u)^2 k_+^2 - l^2)}{\Delta^2\, u^2},
\end{equation}
so even for a constant $f(u) = f_0$ we get a solution which solves the issues.
Notice however that the ``trivial'' solution $f = f_0 \dd{u} \wedge \dd{z}$ is not trivial in Minkowski coordinates where it reads $f = \frac{f_0}{x^-} \dd{x^-} \wedge \dd{x^2}$.
Though appealing, the study of the string in the presence of this non trivial background needs a deeper analysis and it surely is a direction to cover in the future.


\subsection{NBO Eigenfunction from the Covering Space}
\label{sec:Eigenmodes_from_Covering}

We recover the eigenfunctions from the covering Minkowski space in order to elucidate the connection between the polarisations in \nbo and in Minkowski.
Moreover we generalise the result to a symmetric two index tensor which is the polarisation of the first massive state to compute the two-tachyons--one-massive-state amplitude in the next section and to show that it diverges.


\subsubsection{Spin-0 Wave Function from Minkowski space}

We start with the usual plane wave in flat space and we express it in the new coordinates (we do not write the dependence on $\vec{x}$ since it is trivial):
\begin{equation}
  \begin{split}
    \psi_{k_+\, k_-\, k_2}\qty(x^+,\, x^-,\, x^2)
    & =
    e^{i\, \qty( k_+ x^+ + k_- x^- + k_2 x^2 )}
    \\
    & =
    e^{%
      i\, \qty[%
        k_+ v
        +
        \frac{2\, k_+ k_- - k_2^2}{2 k_+} u
        +
        \frac{1}{2} \Delta^2 k_+ u
        \qty( z + \frac{k_2}{\Delta k_+} )^2
      ]
    }
    \\
    & =
    \psi_{k_+\, k_-\, k_2}\qty(u,\, v,\, z).
  \end{split}
\end{equation}
The corresponding wave function on the \nbo is obtained by the periodicity of $z$.
This can be done in two ways either in $\qty(x^{\mu})$ coordinates or in $\qty(x^{\alpha}) = \qty(u\, v\, z)$.
From the first we study how the map to the orbifold gives the function a dependence on the equivalence class of momenta.
Implementing the projection on periodic $z$ functions we get:
\begin{equation}
  \begin{split}
    \Psi_{\qty[k_+\, k_-\, k_2]}\qty(\qty[x^+,\, x^-,\, x^2])
    & =
    \infinfsum{n}
    \psi_{k_+\, k_-\, k_2}\qty( \cK^n\qty(x^+,\, x^-,\, x^2) )
    \\
    & =
    \infinfsum{n}
    \psi_{\cK^{-n}\qty( k_+\, k_-\, k_2 )}\qty( x^+,\, x^-,\, x^2 ),
  \end{split}
\end{equation}
where we write $\qty[k_+\, k_-\, k_2]$ since the function depends on the equivalence class of $\qty(k_+\, k_-\, k_2)$ only.
The equivalence relation is given by
\begin{equation}
  k =
  \mqty(
     k_+\\ k_-\\ k_2
  )
  \equiv
  \cK^{-n} k
  =
  \mqty(
    k_+ 
    \\ 
    k_- + n \qty(2 \pi \Delta) k_2 + \frac{1}{2} n^2 \qty(2 \pi \Delta)^2 k_+
    \\
    k_2 + n \qty(2 \pi \Delta) k_+
  ).
  \end{equation}
It allows us to choose a representative with
\begin{equation}
  \begin{cases}
    0 \le \frac{k_2}{\Delta \abs{k_+}} < 2 \pi,
    & \qquad
    k_+ \neq 0
    \\
    0 \le \frac{k_-}{\Delta \abs{k_2}} < 2 \pi,
    & \qquad
    k_+ = 0, \quad k_2 \neq 0
  \end{cases}.
\end{equation}
  
If we perform the computation in $\qty(u,\, v,\, z)$ coordinates we get:
\begin{equation}
  \begin{split}
    \Psi_{\qty[k_+\, k_-\, k_2]}\qty(u,\, v,\, z)
    & =
    \infinfsum{n}
    \psi_{k_+\, k_-\, k_2}\qty(u,\, v,\, z + 2 \pi n)
    \\
    & =
    \infinfsum{n}
    e^{%
      i\, \qty[%
        k_+ v
        +
        \frac{r}{2 k_+} u
        +
        \frac{1}{2} \qty(2 \pi \Delta)^2 k_+ u
        \qty[ n + \frac{1}{2\pi} \qty( z + \frac{k_2}{\Delta k_+} ) ]^2
      ]
    },
  \end{split}
\end{equation}
with $r = 2\, k_+ k_- - k_2^2$ and $\Im(k_+ u) > 0$, i.e.\ $k_+ u = \abs{k_+ u} e^{i \epsilon}$ and $\pi > \epsilon > 0$.
There is no separate dependence on $z$ and on $\frac{k_2}{\Delta k_+}$: we could fix the range $0 \le z + \frac{k_2}{\Delta k_+} < 2\pi$.
However this symmetry is broken when considering the photon eigenfunction.

We can now use the Poisson resummation
\begin{equation}
  \infinfsum{n}
  e^{i\, a\, (n + b)^2}
  =
  \int \dd{s}\,
  \delta_P(s) e^{i\, a\, (s + b)^2}
  =
  \qty(2\pi)^2
  \frac{e^{-i\, \qty( \frac{\pi}{4} + \frac{1}{2} arg(a) ) }}{2 \sqrt{\pi \abs{a}}}
  \infinfsum{m} e^{-i\, \frac{\pi^2 m^2}{a} + i\, 2 \pi b m},
\end{equation}
to finally get:\footnotemark{}
\footnotetext{%
  In the expression we insert the variables $\vec{k}$ and $\vec{x}$ for completeness.
  We also set $r = 2\, k_+ k_- -k_2^2 - \norm{\vec{k}}^2$.
}
\begin{equation}
  \begin{split}
    \Psi_{\qty[k_+\, k_-\, k_2\, \vec{k}]}\qty(u,\, v,\, z,\, \vec{x})
    & =
    \sqrt{2\pi}~
    \frac{2 e^{-i \frac{\pi}{4}}}{\Delta}
    \\
    & \times
    \infinfsum{l}
    \qty[
      \frac{1}{\sqrt{\abs{k_+ u}}}
      e^{%
        i\, \qty[%
          k_+ v
          +
          l z
          -
          \frac{l^2}{2 \Delta^2 k_+}\, \frac{1}{u}
          +
          \frac{r + \norm{\vec{k}}^2}{2 k_+} u
          +
          \vec{k} \cdot \vec{x}
        ]
      }
    ]
    e^{i\, l\, \frac{k_2}{\Delta k_+}}
    \\
    & =
    \cN\,
    \infinfsum{l}
    \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x})    
    e^{i\, l\, \frac{k_2}{\Delta k_+}},
  \end{split}
\label{eq:Psi_phi}
\end{equation}
when $k_+ \neq 0$ and where
\begin{equation}
  \cN
  =
  \sqrt{\frac{\qty(2\pi)^D}{\pi \Delta}}
  \frac{e^{-i \frac{\pi}{4}}}{\pi}.
\end{equation}
The fact that $\Psi$ depends only on the equivalence class $\qty[k_+\, k_-\, k_2\, k]$ allows us to restrict $0 \le \frac{k_2}{\Delta\, \abs{k_+}} < 2 \pi$ so that we can invert the previous expression and get:
\begin{equation}
  \begin{split}
    \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x})    
    & =
    \frac{1}{\cN}\,
    \frac{1}{2 \pi \Delta \abs{k_+}}
    \finiteint{k_2}{0}{2 \pi \Delta \abs{k_+}}
    e^{-i\, l\, \frac{k_2}{\Delta k_+}}\,
    \Psi_{\qty[k_+\, k_-\, k_2\, k]}\qty(u,\, v,\, z,\, \vec{x}).
  \end{split}
\end{equation}


\subsubsection{Spin-1 Wave Function from Minkowski space}

We go through the steps in the previous case for an electromagnetic wave.
We concentrate on $x^+$, $x^-$ and $x^2$ coordinates and reinstate $\vec{x}$ at the end.
We start with the usual plane wave in flat space  $\psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}$ and we express it in both Minkowski and orbifold coordinates.
We use the notation $\psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}$ to stress that it is the eigenfunction and not the field which is obtained as
\begin{equation}
  A_{\mu}(x)\, \dd{x}^{\mu}
  =
  \int\limits_{\R^3} \dd[3]{k}\,
  \sum_{\qty{\epsilon_+,\, \epsilon_-,\, \epsilon_2}}
  \psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2},
\end{equation}
where the sum  is performed over $\epsilon_+$, $\epsilon_-$, $\epsilon_2$ independent and compatible with $k$.
The explicit expression for the eigenfunction with constant $\epsilon_+$, $\epsilon_-$ and $\epsilon_2$ is:\footnotemark{}
\footnotetext{%
  We introduce the normalisation factor $\cN$ in order to have a less cluttered relation between $\epsilon$ and $\cE$.
}
\begin{equation}
  \begin{split}
    \cN\,
    \psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}\qty(x^+,\, x^-,\, x^2)
    & =
    \qty(\epsilon_+ \dd{x^+} + \epsilon_- \dd{x^-} + \epsilon_2 \dd{x^2})\,
    e^{i\, \qty( k_+ x^+ + k_- x^- + k_2 x^2 )}
    \\
    & =
    \qty( \epsilon_u\, \dd{u} + \epsilon_z\, \dd{z} + \epsilon_v\, \dd{v})\,
    \\
    & \times
    e^{%
      i\, \qty[%
        k_+ v
        +
        \frac{2\, k_+ k_- - k_2^2}{2 k_+} u
        +
        \frac{1}{2} \Delta^2 k_+ u \qty( z + \frac{k_2}{\Delta k_+})^2
      ]
    }
    \\
    & =
    \cN
    \psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}\qty(u,\, v,\, z),
  \end{split}
\end{equation}
with
\begin{equation}
  \begin{split}
    \epsilon_v & = \epsilon_+,
    \\
    \epsilon_u(z)
    & =
    \epsilon_- + (\Delta z)\, \epsilon_2 + (\frac{1}{2} \Delta^2 z^2)\, \epsilon_+,
    \\
    \epsilon_z(u,\, z)
    & =
    \qty(\Delta u)\,
    \qty(\epsilon_2 + \Delta z\, \epsilon_+ ).
  \end{split}
\end{equation}
Notice that we are not imposing any gauge condition.
Moreover if $(\epsilon_+,\, \epsilon_-,\, \epsilon_2)$ are constant then $(\epsilon_u,\, \epsilon_v,\, \epsilon_z)$ are generic functions.
It is worth stressing that they are not the polarisations in the orbifold which are in any case constant: the fact that they depend on the coordinates is simply the statement that not all eigenfunctions of the vector d'Alembertian are equal.

Building the corresponding function on the orbifold amounts to summing the images created by the orbifold group:
\begin{equation}
  \begin{split}
    \cN\,
    \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x])
    =
    \infinfsum{n}
    \vec{\epsilon} \cdot \qty( \cK^n \dd{x})~
    \psi_{k}\qty( \cK^n x)
    =
    \infinfsum{n}
    \cK^{-n}
    \vec{\epsilon} \cdot \dd{x}~
    \psi_{\cK^{-n} k}\qty(x).
  \end{split}
\end{equation}
Under the action of the Killing vector $\epsilon$ transforms exactly as the $k$ since it is induced by $\epsilon \cdot \cK^n \dd{x} = \cK^{-n} \epsilon \cdot \dd{x}$, that is:
\begin{equation}
  \epsilon
  =
  \mqty( \epsilon_+ \\ \epsilon_2 \\ \epsilon_- )
  \equiv
  \cK^{-n} \epsilon
  =
  \mqty(%
    \epsilon_+ 
    \\
    \epsilon_2 + n\, \qty(2 \pi \Delta)\, \epsilon_+
    \\ 
    \epsilon_- + n\, \qty(2 \pi \Delta)\, \epsilon_2 + \frac{1}{2} n^2 \qty(2 \pi \Delta)^2 \epsilon_+
  ).
\end{equation}
However the pair $\qty(\vec{k},\, \vec{\epsilon})$ transforms with the same $n$ since both are ``dual'' to $x$, i.e.\ their transformation rules are dictated by $x$.
There is therefore only one equivalence class $\qty[\vec{k},\, \vec{\epsilon}]$ and not two separate classes $\qty[\vec{k}]$, $\qty[\vec{\epsilon}]$.
In other words, a representative of the combined equivalence class is the one with $0 \le k_2 < 2 \pi \Delta \abs{k_+}$ when $k_+ \neq 0$. 

In order to write the eigenfunctions on the orbifold in orbifold coordinates we notice that $\dd{u}, \dd{v}$ and $\dd{z}$ are invariant.
We write
\begin{equation}
  \begin{split}
    \cN\,
    \Psi^{[1]}_{\qty[\vec{k},\, \vec{\epsilon}]}\qty(\qty[x])
    & =
    \infinfsum{n}
    \epsilon \cdot \qty( \cK^n \dd{x})\,
    \psi_{k}( \cK^n x)
    \\
    & =
    \dd{v}\,
    \qty[ \epsilon_+\, \infinfsum{n} \psi_k\qty( \cK^n x) ]
    \\
    & +
    \dd{z}\,
    \qty(\Delta u)\,
    \qty[%
      \epsilon_2\, \infinfsum{n} \psi_k\qty( \cK^n x)
      +
      \epsilon_+\, \Delta\, \infinfsum{n} \qty(z + 2\pi n) \psi_k\qty(\cK^n x)
    ]
    \\
    & +
    \dd{u}\,
    \Biggl[
      \epsilon_-\, \infinfsum{n} \psi_k\qty(\cK^n x)
      +
      \epsilon_2\, \Delta\, \infinfsum{n} \qty(z + 2 \pi n) \psi_k\qty(\cK^n x)
      \\
      & +
      \frac{1}{2} \epsilon_+\, \Delta^2\, \infinfsum{n} \qty(z + 2 \pi n)^2 \psi_k(\cK^n x)
    \Biggr].
  \end{split}
\end{equation}
From a direct computation we get:\footnotemark{}
\footnotetext{%
  These expressions may be written using Hermite polynomials.
}
\begin{equation}
  \begin{split}
    \infinfsum{n} \qty(z + 2\pi n)\, \psi_k(\cK^n x)
    & =
    \qty(%
      \frac{1}{i\, \Delta u}
      \pdv{}{k_2}
      -
      \frac{k_2}{\Delta k_+}
    )
    \Psi_{\qty[k]}\qty(\qty[x])
    \\
    \infinfsum{n} \qty(z + 2\pi n)^2\, \psi_k(\cK^n x)
    & =
    \qty(%
      \frac{1}{i\, \Delta\, u}
      \pdv{}{k_2}
      -
      \frac{k_2}{\Delta k_+}
    )^2
    \Psi_{\qty[k]}\qty(\qty[x]).
  \end{split}
  \label{eq:sum_z_psi}
\end{equation}
Then it follows that
\begin{equation}
  \begin{split}
    \cN\,
    \Psi^{[1]}_{\qty[\vec{k},\, \vec{\epsilon}]}\qty(\qty[x])
    & =
    \dd{v}\,
    \qty[%
      \epsilon_+\, \Psi_{\qty[k]}\qty(\qty[x])
    ]
    \\
    & +
    \dd{z}\,
    \qty(\Delta u)\,
    \qty[%
      \frac{\epsilon_2 k_+ - \epsilon_+ k_2}{k_+}\,
      \Psi_{\qty[k]}\qty(\qty[x])
      - 
      \epsilon_+\, \frac{i}{u} \pdv{}{k_2} \Psi_{\qty[k]}\qty(\qty[x])
    ]
    \\
    & +
    \dd{u}\,
    \Biggl[
      \qty(%
        \epsilon_-
        -
        \epsilon_2 \frac{k_2}{k_+}
        +
        \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2
      )\,
      \Psi_{\qty[k]}\qty(\qty[x])
      + 
      \frac{i}{2 u} \frac{\epsilon_+}{k_+}\,
      \Psi_{\qty[k]}\qty(\qty[x])
      \\
      & -
      \frac{\epsilon_2 k_+ - \epsilon_+ k_2}{k_+}
      \frac{i}{u} \pdv{}{k_2} \Psi_{\qty[k]}\qty(\qty[x])
      -
      \frac{1}{2} \epsilon_+\,
      \frac{1}{u^2}\,
      \pdv[2]{}{k_2}
      \Psi_{\qty[k]}\qty(\qty[x])
    \Biggr].
  \end{split}
  \label{eq:a_uvz_from_covering}
\end{equation}
Many coefficients of $\Psi$ or its derivatives contain $k_2$. 
They cannot be expressed using the quantum numbers $k_-kr$ of the orbifold but are invariant on it.
They are new orbifold quantities we interpret as orbifold polarisations.
Using~\eqref{eq:Psi_phi} we can finally write
\begin{equation}
  \begin{split}
    \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x])
    & =
    \infinfsum{l}
    \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x})    
    e^{i\, l \frac{k_2}{\Delta k_+}}
    \\
    & \times
    \Biggl\lbrace
      \dd{v}\, \epsilon_+
      \\
      & +
      \dd{z}\, \qty(\Delta u)
      \qty[ %
        \frac{\epsilon_2 k_+ - \epsilon_+ k_2}{k_+}
         +
        \epsilon_+ \frac{1}{\Delta u}\, \frac{l}{k_+}
      ]
      \\
      & +
      \dd{u}\,
      \Biggl[
        \qty(%
          \epsilon_-
          -
          \epsilon_2\, \frac{k_2}{k_+}
          +
          \frac{1}{2} \epsilon_+\, \qty( \frac{k_2}{k_+} )^2
        )
        + 
        \frac{i}{2 u}\, \frac{\epsilon_+}{k_+}
        \\
        & +
        \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+}
        \frac{1}{u}\, \frac{l}{\Delta  k_+}
        +
        \epsilon_+\, \frac{1}{2 u^2}\, \qty(\frac{l}{ \Delta k_+})^2
      \Biggr]
    \Biggr\rbrace.
  \end{split}
  \label{eq:spin1_from_covering}
\end{equation}

If we compare the last expression with~\eqref{eq:Orbifold_spin1_pol} we find:
\begin{equation}
  \begin{split}
    \pol{v}
    & =
    \epsilon_+
    \\
    \pol{z}
    & =
    \sgn(u)
    \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+}
    \\
    \pol{u}
    & =
    \epsilon_-
    -
    \epsilon_2\, \frac{k_2}{k_+}
    +
    \frac{1}{2} \epsilon_+\, \qty( \frac{k_2}{k_+} )^2,
  \end{split}
  \label{eq:eps_calE}
\end{equation}
which implies that the true polarisations $\qty(\epsilon_+,\, \epsilon_-,\, \epsilon_2)$  and $\pol{*}$ are constant as it turns out from direct computation.
A different way of reading the previous result is that the polarisations on the orbifold are the coefficients of the highest power of $u$.

We can also invert the previous relations to get:
\begin{equation}
  \begin{split}
    \epsilon_+
    & =
    \pol{v} 
    \\
    \epsilon_2
    & =
    \pol{z}\, \sgn(u)
    +
    \pol{v}\, \frac{k_2}{k_+}
    \\
    \epsilon_-
    & =
    \pol{u}
    +
    \pol{z}\, \sgn(u)\, \frac{k_2}{k_+}
    +
    \pol{v}\, \frac{1}{2} \qty( \frac{k_2}{k_+} )^2,
  \end{split}
  \label{eq:calE_eps}
\end{equation}
and use them in Lorenz gauge $\vec{k} \cdot \vec{\epsilon} = 0$ in order to get the gauge conditions expressed with the orbifold polarisations.
If the definition of orbifold polarisations is right the result cannot depend on $k_2$ since it is not a quantum number of orbifold eigenfunctions.
Taking into account $k_- = \frac{\norm{\vec{k}}^2+ k_2^2 + r}{2 k_+}$ in $\vec{k} \cdot \vec{\epsilon} = 0$ we get exactly the expression for the Lorenz gauge for orbifold polarisations~\eqref{eq:Lorenz_gauge}. 


\subsubsection{Tensor Wave Function (Spin-2) from Minkowski space}

We can use the analysis of the previous section in the case of a second order symmetric tensor wave function.
Again we suppress the dependence on $\vec{x}$ and $\vec{k}$ with a caveat: the Minkowskian polarisations $S_{+\, i}$, $S_{-\, i}$ and $S_{2\, i}$ transform non trivially, therefore we give the full expressions in \Cref{sec:NO_tensor_wave} even if these components behave effectively as a vector of the orbifold.

We start with the usual wave in flat space and we express either in the Minkowskian coordinates
\begin{equation}
  \begin{split}
    \cN\, \psi^{[2]}_{k\, S}\qty( x^+,\, x^-,\, x^2 )
    & =
    S_{\mu\nu}\,
    \psi_k( x )\, \dd{x}^{\mu}\, \dd{x}^{\nu}
    \\
    & =
    \Bigl(%
      S_{++}\, \dd{x}^+\, \dd{x}^+
      +
      2\, S_{+\, x}\, \dd{x}^+\, \dd{x}^2
      +
      2\, S_{+\, -}\, \dd{x}^+\, \dd{x}^-
      \\
      & +
      2\, S_{2\, 2}\, \dd{x}^2\, \dd{x}^2
      +
      2\, S_{2\, -}\, \dd{x}^2\, \dd{x}^-
      \\
      & +
      2\, S_{-\, -}\, \dd{x}^-\, \dd{x}^-
    \Bigr)
    e^{i\, \qty( k_+ x^+ + k_- x^- + k_2 x^2 )},
  \end{split}
\end{equation}
or in orbifold coordinates
\begin{equation}
  \begin{split}
    \cN\,
    \psi^{[2]}_{k\, S}(x)
    & =
    S_{\alpha\, \beta}\, \psi_k(x)\, \dd{x}^{\alpha}\, \dd{x}^{\beta}
    \\
    & =
    \Biggl\lbrace
      \dss[2]{v}\,
      S_{+\, +}
      \\
      & +
      \dd{v}\, \dd{z}\, \Delta u
      \qty[%
        2\, S_{+\, 2} 
        +
        S_{+\, +} \Delta z
      ]
      \\
      & +
      \dd{v}\, \dd{u}\,
      \qty[%
        2\, S_{+\, -}
        +
        2\, S_{+\, 2} \Delta z
        +
        S_{+\, +} \Delta^2 z^2
      ]
      \\
      & +
      \dss[2]{z}\, \Delta^2 u^2\,
      \qty[%
        S_{2\, 2} 
        +
        2\, S_{+\, 2} \Delta  z
        +
        S_{+\, +} \Delta^2 z^2
      ]
      \\
      & +
      \dd{z}\, \dd{v}\, \Delta u\,
      \qty[%
        2\, S_{-\, 2} 
        +
        2\, \qty(S_{2\, 2}
        +
        S_{+\, -})\, \Delta  z
        +
        3\, S_{+\, 2} \Delta^2  z^2
        +
        S_{+\, +}\, \Delta^3  z^3
      ]
      \\
      & +
      \dd{u}^2\,
      \qty[%
        S_{-\, -}
        +
        2\, S_{-\, 2} \Delta z
        +
        \qty(S_{2\, 2} + S_{+\, -})\, \Delta^2 z^2
        +
        S_{+\, 2} \Delta^3 z^3
        +
        \frac{1}{4} S_{+\, +} \Delta^4 z^4
      ]
    \Biggr\rbrace
    \\
    & \times
    e^{%
      i\, \qty[%
        k_+ v
        +
        \frac{2\, k_+ k_- - k_2^2}{2 k_+} u
        +
        \frac{1}{2} \Delta^2 k_+ u \qty( z + \frac{k_2}{\Delta k_+})^2
      ]
    }.
  \end{split}
\end{equation}
Now we define the tensor on the orbifold as a sum over all images as
\begin{equation}
  \begin{split}
    \cN\,
    \Psi^{[2]}_{[k\, S]}\qty(\qty[x])
    & =
    \infinfsum{n}
    \qty( \cK^n \dd{x}) \cdot S \cdot ( \cK^n \dd{x})~
    \psi_{k}( \cK^n x)
    \\
    & =
    \infinfsum{n}
    \dd{x} \cdot ( \cK^{-n}\, S ) \cdot \dd{x}~
    \psi_{\cK^{-n} k}\qty(x).
  \end{split}
\end{equation}
In the last line we have defined the induced action of the Killing vector on $\qty(\vec{k}, S)$ which can be explicitely written as:
\begin{equation}
  \cK^{-n}
  \mqty(%
    S_{ +\, + }
    \\
    S_{ +\, 2 }
    \\
    S_{ +\, - }
    \\
    S_{ 2\, 2 }
    \\
    S_{ 2\, - }
    \\
    S_{ -\, - }
  )
  =
  \mqty(%
    S_{ +\, + }
    \\
    S_{ +\, 2 } + n \Delta S_{ +\, + }
    \\
    S_{ +\, - } + n \Delta S_{ +\, 2 } + \frac{1}{2} n^2 \Delta^2 S_{ +\, + }
    \\
    S_{ 2\, 2 } + 2 n \Delta S_{ +\, 2 } + n^2 \Delta^2 S_{ +\, + }
    \\
    S_{ 2\, - } + n \Delta \qty(S_{ 2\, 2 } + S_{ +\, - }) + \frac{3}{2} n^2 \Delta^2 S_{ +\, 2 } + \frac{1}{2} n^3 \Delta^3 S_{ +\, + }
    \\
    S_{ -\, - } + 2 n \Delta S_{ -\, 2 } + n^2 \Delta^2 \qty(S_{ 2\, 2 } + S_{ +\, - } ) + n^3 \Delta^3 S_{ +\, 2 } + \frac{1}{4} n^4 \Delta^4 S_{ +\, + }
  ).
\end{equation}
  
Computing the tensor on the orbifold in its own coordinates is equivalent to summing over all the shifts $z \rightarrow \qty(z + 2 \pi n)$ and the use of a generalisation of~\eqref{eq:sum_z_psi}, i.e.\, to substitute $\qty(\Delta\, z)^j \psi_k \rightarrow \qty( \frac{1}{i\, u} \pdv{}{k_2} - \frac{k_2}{\Delta\, k_+} )^j \Psi_{\qty[k]}\qty(\qty[x])$.
When expressing all in the $\phi$ basis, the last step is equivalent to $\qty(\Delta\, z)^j \psi_k \rightarrow \qty(\frac{l}{\Delta\, u\, k_+})^j + \dots$.
We identify the basic polarisations on the orbifold by considering the highest power in $u$:
\begin{equation}
  \begin{split}
    \cS_{u\,u}
    & =
    \frac{1}{4}  K^4\, S_{+\,+}
    +
    K^2\, S_{+\,-}
    -
    K^3\, S_{+\,2}
    +
    S_{-\,-}
    -
    2\, K\, S_{-\,2}
    +
    S_{2\,2}\, K^2
    \\
    \cS_{u\,v}
    & =
    \frac{1}{2} K^2\, S_{+\,+}
    +
    S_{+\,-}
    -
    K\, S_{+\,2}
    \\
    \cS_{u\,z}
    & =
    -\frac{1}{2} K^3\,S_{+\,+}
    -
    K\, S_{+\,-}
    +
    \frac{3}{2} K^2\, S_{+\,2}
    +
    S_{-\,2}
    -
    K\, S_{2\,2}
    \\
    \cS_{v\,v}
    & =
    S_{+\,+}
    \\
    \cS_{v\,z}
    & =
    S_{+\,2} - K\, S_{+\,+}
    \\
    \cS_{z\,z}
    & =
    K^2\, S_{+\,+} - 2\, K\, S_{+\,2} + S_{2\,2}.
  \end{split}
\end{equation}
where $K = \frac{k_2}{k_+}$.
The previous equations can be inverted to get:
\begin{equation}
  \begin{split}
    S_{-\,-}
    & =
    K^2\, \qty(\cS_{z\,z} + \cS_{u\,v})
    +
    K^3\, \cS_{v\,z}
    +
    \frac{1}{4} K^4\, \cS_{v\,v}
    +
    2\,
    K\,
    \cS_{u\,z}
    +
    \cS_{u\,u}
    \\
    S_{+\,-}
    & =
    K\, \cS_{v\,z}
    +
    \frac{1}{2} K^2\, \cS_{v\,v}
    +
    \cS_{u\,v}
    \\
    S_{-\,2}
    & =
    K\, \qty(\cS_{z\,z} + \cS_{u\,v})
    +
    \frac{3}{2}\, K^2\, \cS_{v\,z}
    +
    \frac{1}{2} K^3\, \cS_{v\,v}
    +
    \cS_{u\,z}
    \\
    S_{+\,+}
    & =
    \cS_{v\,v}
    \\
    S_{+\,2}
    & =
    \cS_{v\,z}
    +
    K\, \cS_{v\,v}
    \\
    S_{2\,2}
    & =
    \cS_{z\,z}
    +
    2\, K\, \cS_{v\,z}
    +
    K^2\, \cS_{v\,v}.
  \end{split}
\end{equation}
Since we plan to use the previous quantities in the case of the first massive string state we compute the relevant quantities.
In particular we have the trace:
\begin{equation}
  \tr(S) = \cS_{z\,z} - 2\, \cS_{u\,v}
\end{equation}
and the transversality conditions
\begin{equation}
  \begin{split}
    \text{trans}~
    \cS_{v}
    & =
    \qty(\vec{k} \cdot S)_{+}
    =    
    -\frac{\qty(r + \norm{\vec{k}}^2)}{2\, k_+}\,
    \cS_{v\,v}
    -
    k_{+}\, \cS_{u\,v},
    \\
    \text{trans}~
    \cS_{z}
    & =
    \qty(\vec{k} \cdot S)_{2}
    -
    K~ \qty(\vec{k} \cdot S)_{+}
    =
    -\frac{\qty(r + \norm{\vec{k}}^2)}{2\, k_+}\,
    \cS_{v\,z}
    -
    k_{+}\, \cS_{u\,z},
    \\
    \text{trans}~
    \cS_{u}
    & =
    \qty(\vec{k} \cdot S)_{-}
    -
    K~ \qty(\vec{k} \cdot S)_{2}
    +
    \frac{1}{2} K^2 \qty(\vec{k} \cdot S)_{+}
    =
    -\frac{\qty(r + \norm{\vec{k}}^2)}{2\, k_+}\,
    \cS_{u\,v}
    -
    k_{+}\, \cS_{u\,u}
  \end{split}
\end{equation}
where we used $k_- = \frac{\qty(r + \norm{\vec{k}}^2 + k_2^2)}{\qty(2 k_+)}$.
These conditions do not depend on $K$ since $k_2$ is not an orbifold quantum number.

The final expression for the orbifold symmetric tensor is
\begin{equation}
  \begin{split}
    \Psi^{[2]}_{\qty[\vec{k},\, S]}\qty(\qty[x])
    & =
    \infinfsum{l}
    \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x})    
    e^{i\, l \frac{k_2}{\Delta k_+}}
    \\
    & \times
    \Biggl\lbrace
      \dss[2]{v}\, \cS_{v\, v}
      \\
      & +
      2\, \Delta\, u\, \dd{v}\, \dd{z}\,
      \qty[%
        \cS_{v\,z}
        +
        \qty(\frac{L \cS_{v\,v}}{\Delta})
        \frac{1}{u}
      ]
      \\
      & +
      2\, \dd{v}\, \dd{u}\,
      \qty[%
        \cS_{u\,v}
        +
        \qty(\frac{L\, \cS_{v\,z}}{\Delta} + \frac{i\, \cS_{v\,v}}{2\, k_{+}})
        \frac{1}{u}
        +
        \qty(\frac{L^2\, \cS_{v\,v}}{2\, \Delta^2})
        \frac{1}{u^2}
      ]
      \\
      & +
      \qty(\Delta\, u)^2 \dd{z}^2\,
      \qty[%
        \cS_{z\,z}
        +
        \qty(\frac{2\, L\, \cS_{v\,z}}{\Delta} + \frac{i\, \cS_{v\,v}}{k_{+}})
        \frac{1}{u}
        +
        \qty(\frac{L^2\, \cS_{v\,v}}{\Delta^2})
        \frac{1}{u^2}
      ]
      \\
      & +
      2\, \Delta\, u\, \dd{z}\, \dd{u}\,
      \Bigl[%
        \cS_{u\,z}
        +
        \qty(%
          \frac{L\, \cS_{z\,z}}{\Delta}
          +
          \frac{3\, i\, \cS_{v\,z}}{2\, k_{+}}
          +
          \frac{L\, \cS_{u\,v}}{\Delta}
        )
        \frac{1}{u}
        +
        \qty(%
          \frac{3\, L^2\, \cS_{v\,z}}{2\, \Delta^2}
          +
          \frac{3\, i\, L\, \cS_{v\,v}}{2\, \Delta\, k_{+}}
        )
        \frac{1}{u^2}
        \\
        & +
        \qty(\frac{L^3\,\cS_{v\,v}}{2\,\Delta^3})
        \frac{1}{u^3}
      \Bigr]
      \\
      & +
      \dd{u}^2\,
      \Bigl[
        \cS_{u\,u}
        +
        \qty(%
          \frac{i\, \cS_{z\,z}}{k_{+}}
          +
          \frac{2\, L\, \cS_{u\,z}}{\Delta}
          +
          \frac{i\, \cS_{u\,v}}{k_{+}}
        )
        \frac{1}{u}
        \\
        & +
        \qty(%
          \frac{L^2\, \cS_{z\,z}}{\Delta^2}
          +
          \frac{3\, i\, L\, \cS_{v\,z}}{\Delta\, k_{+}}
          -
          \frac{3\, \cS_{v\,v}}{4\, k_{+}^2}
          +
          \frac{L^2\, \cS_{u\,v}}{\Delta^2}
        )
        \frac{1}{u^2}
        \\
        & +
        \qty(%
          \frac{L^3\, \cS_{v\,z}}{\Delta^3}
          +
          \frac{3\, i\, L^2\, \cS_{v\,v}}{2\, \Delta^2\, k_{+}}
        )
        \frac{1}{u^3}
        +
        \qty(\frac{L^4 \cS_{v\,v}}{4 \Delta^4})
        \frac{1}{u^4}
      \Bigr]
    \Biggr\rbrace,
  \end{split}
\end{equation}
where $L = \frac{l}{k_+}$.


\subsection{Overlaps of Wave Functions and Their Derivatives}

In this section we compute overlaps of wave functions.
We give their expressions using both integrals over the eigenfunctions and sums of products of delta functions.
The latter is the expression which is naturally obtained by computing tree level string amplitudes on the orbifold when one starts with Minkowski amplitudes and adds the images.
This is equivalent to computing emission vertices on the orbifold and then their correlation functions since this amounts to transfer the sum over the spacetime images to the sum of the polarisations images.


\subsubsection{Overlaps Without Derivatives}

We start from the simplest case of the overlap of $N$ scalar wave functions.
We compute the overlap of orbifold wave functions and then we express it as sum of images of the corresponding Minkowski overlap thus establishing a dictionary between Minkowski and orbifold spaces.
Explicitly we consider the following overlap where all the polarisations $\cA_{(i)}$ have been set to one
\begin{equation}
  \begin{split}
    I^{(N)}
    & =
    \int\limits_{\Omega} \dd[3]{x}\,
    \sqrt{-\det g}~
    \finiteprod{i}{1}{N} \Psi_{\qty[k_{\qty(i)\, +}\, k_{\qty(i)\, -}\, k_{\qty(i)\, 2}]}\qty(\qty[x^+,\, x^-,\, x^2]))
    \\
    & =
    \int\limits_{\ccM^{1,2}} \dd[3]{x}\,
    \sqrt{-\det g}~
    \psi_{k_{\qty(1)\, +}\, k_{\qty(1)\, -}\, k_{\qty(1)\, 2}}\qty(x^+,\, x^-,\, x^2))\,
    \\
    & \times
    \finiteprod{i}{2}{N} \infinfsum{m_{(i)}} 
    \psi_{k_{\qty(i)\, +}\, k_{\qty(i)\, -}\, k_{\qty(i)\, 2}}( \cK^{m_{(i)} }\qty(x^+,\, x^-,\, x^2) )      
    \\
    & =    
    \int\limits_{\ccM^{1,2}} \dd[3]{x}\,
    \sqrt{-\det g}~
    \psi_{k_{\qty(1)\, +}\, k_{\qty(1)\, -}\, k_{\qty(1)\, 2}}\qty(x^+,\, x^-,\, x^2))
    \\
    & \times
    \finiteprod{i}{2}{N}  \infinfsum{m_{(i)}} 
    \psi_{\cK^{m_{(i)}}\qty(k_{\qty(i)\, +}\, k_{\qty(i)\, -}\, k_{\qty(i)\, 2}) }\qty(x^+,\, x^-,\, x^2)
    \\
    & =
    \qty(2\pi)^3
    \delta\qty(\infinfsum{i} k_{\qty(i)\, +})\,
    \\
    & \times
    \eval{%
      \finiteprod{i}{2}{N} \infinfsum{m_{(i)}} 
      \delta\qty(\infinfsum{i} \cK^{m_{(i)}}~ k_{\qty(i)\, 2} )\,
      \delta\qty(\infinfsum{i} \cK^{m_{(i)}}~ k_{\qty(i)\, -} )\,
    }_{m_{\qty(1)} = 0},
  \end{split}
\end{equation}
where $\Omega = \ccM^{1,2} / \Gamma$ is the fundamental region identifying the orbifold.
We used the unfolding trick to rewrite the integral as an integral over $\ccM^{1,2}$ thus dropping the sum over the images of particle $\qty(1)$.
We then moved the action of the Killing vector from $x$ to $k$ and finally we used the usual $\delta$ definition.
The previous integral can be expressed as:
\begin{equation}
  \begin{split}
    I^{(N)}
    & =
    \cN^N
    \sum_{ \qty{ l_{\qty(i)} } \in \Z^N } 
    e^{%
      i \finitesum{i}{1}{N} l_{\qty(i)} \frac{k_{\qty(i)\, 2}}{\Delta k_{\qty(i)\, +}}
    }
    \int\limits_{\Omega} \dd[3]{x}\,
    \sqrt{-\det g}\,
    \finiteprod{i}{1}{N}\phi_{k_-rN{i}}\qty(\qty[x])
    \\
    & =
    \cN^N
    \sum_{ \qty{ l_{\qty(i)} } \in \Z^N } 
    e^{%
      i \finitesum{i}{1}{N} l_{\qty(i)} \frac{k_{\qty(i)\, 2}}{\Delta k_{\qty(i)\, +}}
    }\,
    \qty(2\pi)^2
    \delta\qty( \finitesum{i}{1}{N} k_{\qty(i)\, +} )
    \delta_{\finitesum{i}{1}{N} l_{\qty(i)},\, 0}\,
    \cI_{\qty{N}}^{\qty[0]}.
  \end{split}
\end{equation}
From the last expression we recover the overlap of the wave functions as:
\begin{equation}
  \begin{split}
    & \int\limits_{\Omega} \dd[3]{x}\,
    \finiteprod{i}{1}{N}   
    \phi_{k_-rN{i}}\qty( \qty[x] )
    \\
    = &
    \frac{1}{\cN^N}\,
    \finiteprod{i}{1}{N}   
    \int\limits_0^{2\pi \Delta \abs{k_{\qty(i)\, +}}}
    \frac{\dd{k_{\qty(i)\, 2}}}{2 \pi \Delta \abs{k_{\qty(i)\, +}}}\,
    e^{%
      -i\, l_{\qty(i)} \frac{k_{\qty(i)\, 2}}{\Delta k_{\qty(i)\, i}}
    }\,
    I^{(N)}
    \\
    = &
    \qty(2\pi)^3\,
    \delta\qty(\infinfsum{i} k_{\qty(i)\, +} )\,
    \frac{1}{\cN^N}\,
    \finiteprod{i}{1}{N}   
    \int\limits_0^{2\pi \Delta \abs{k_{\qty(i)\, +}}}
    \frac{\dd{k_{\qty(i)\, 2}}}{2 \pi \Delta \abs{k_{\qty(i)\, +}}}\,
    e^{%
      -i\, l_{\qty(i)} \frac{k_{\qty(i)\, 2}}{\Delta k_{\qty(i)\, +}}
    }\,
    \\
    \times &
    \finiteprod{j}{2}{N} \infinfsum{m_{(j)}} 
    \delta\qty( \finitesum{j}{2}{N} \cK^{m_{(j)} } k_{\qty(j)\, 2} )\,
    \delta\qty( \finitesum{j}{2}{N} \cK^{m_{(j)} } k_{\qty(j)\, -} ).
  \end{split}
\end{equation}
It follows from the explicit expression of $\cI_{\qty{n}}^{\qty[0]}$ that all overlaps $I^{(N)}$ for $N \ge 4$ diverge. 

Intuitively we are in fact summing over infinite distributions with accumulation points of their support.
Nevertheless the existence of the accumulation point is not sufficient since the three scalars overlap, i.e.\ the three tachyons amplitude, converges.


\subsubsection{An Overlap With One Derivative}

Since we will also compute the amplitude involving two tachyons and one photon, as a preliminary step we consider the overlap in Minkowski space:
\begin{equation}
  J_{Mink}
  =
  i\,
  \qty(2\pi)^3\,
  \qty(\epsilon_{\qty(1)} \cdot k_{\qty(2)\, 2})\,
  \delta\qty( \infinfsum{i} k_{\qty(i)\, +} )\,
  \delta\qty( \infinfsum{i} k_{\qty(i)\, 2} )\,
  \delta\qty( \infinfsum{i} k_{\qty(i)\, -} ).
\end{equation}
Summing over momenta and polarisations we then get to an expression which depends on equivalence classes as:
\begin{equation}
  \begin{split}
    J\qty(%
      \qty[k_{\qty(1)}, \epsilon_{\qty(1)}],\,
      \qty[k_{\qty(2)}],\,
      \qty[k_{\qty(3)}]
    )
    & =
    i\, \qty(2\pi)^3 \delta\qty( \infinfsum{i} k_{\qty(i)\, +} )\,
    \\
    & \times
    \sum_{\qty{ m_{(i)} } \in \Z^3}
    \delta_{m_{\qty(1)},\, 1}\,
    \qty(%
      \cK^{m_{\qty(1)}} \epsilon_{\qty(1)}
      \cdot
      \cK^{m_{\qty(2)}} k_{\qty(2)\, 2}
    )
    \\
    & \times
    \delta\qty( \infinfsum{i} \cK^{m_{(i)} } k_{\qty(i)\, 2} )\,
    \delta\qty( \infinfsum{i} \cK^{m_{(i)} } k_{\qty(i)\, -} ).
  \end{split}
  \label{eq:Spin_001_overlap_from_covering}
\end{equation}
The expression depends only on equivalence classes.\footnotemark{}
\footnotetext{%
  In order to prove it, under $\qty(k_{\qty(1)},\, \epsilon_{\qty(1)}) \rightarrow \cK^{s}\qty(k_{\qty(1)},\, \epsilon_{\qty(1)})$ we can use $\cK^{s} a \cdot b = a \cdot \cK^{-s} b$ and the invariance of deltas $\delta^3(\cK^{s}a) = \delta^3(a)$.
}

The previous expression can be written as
\begin{equation}
  J
  =
  \int\limits_{\Omega} \dd[3]{x}\,
  \eta^{\mu\nu}\,
  \Psi^{[1]}_{\qty[k_{\qty(1)}, \epsilon_{\qty(1)}]\, \mu}\qty(\qty[x])\,
  \ipd{\nu} \Psi_{ \qty[k_{\qty(2)}] }\qty(\qty[x])\,
  \Psi_{ \qty[k_{\qty(3)}] }\qty(\qty[x])
\end{equation}
where we performed the unfolding using $a_{\qty[k_{\qty(1)},\, \epsilon_{\qty(1)}]\, \mu}\qty(\qty[x])$.\footnotemark{}
\footnotetext{%
  Clearly we can perform the unfolding using whichever other field and this amount to keep the corresponding $m_{(i)}$ fixed in place of $m_{\qty(1)}$.
}
Notice that the previous expression is invariant despite the fact that the derivatives $\ipd{\mu}$ are not well defined on the orbifold.
The fact that $\Psi^{[1]}_{\mu}$ is not invariant in turns helps in recovering the required invariance.

We can then evaluate the previous expression with Minkowskian polarisations using~\eqref{eq:spin1_from_covering} which is nothing else but a rearrangement of terms of~\eqref{eq:Spin_001_overlap_from_covering}.
We have:
\begin{equation}
  \begin{split}
    J
    & =
    i\, \cN^2
    \sum_{ \qty{ l_{\qty(i)} } \in \Z^3 }
    e^{%
      i\, \finitesum{i}{1}{3} l_{\qty(i)} \frac{k_{\qty(i)\, 2}}{\Delta k_{\qty(i)\, +}}
    }\,
    \qty(2\pi)^2\,
    \delta\qty( \finitesum{i}{1}{3} k_{\qty(i)\, +} )
    \delta_{\finitesum{i}{1}{3} l_{\qty(i)}}
    \\
    & \times
    \int\limits_{\Omega} \dd[3]{x}\,
    \finiteprod{i}{1}{3}
    \phi_{k_-rN{i}}\qty(\qty[x]))
    \Biggl\lbrace
      \epsilon_{\qty(1)\, +}\,
      \qty[
        \frac{i}{2 u}
        +
        \frac{l_{\qty(2)}^2}{k_{\qty(2)\, +}} \frac{1}{2 \Delta^2\, u^2}
        +
        \frac{r_{\qty(i)}}{2 k_{\qty(2)\, +}}
      ]
      \\
      & +
      \frac{1}{\Delta\, u}
      \qty[%
        \epsilon_{\qty(1)\, 2}\,
        +
        \frac{1}{\Delta u} \epsilon_{\qty(1)\, +} \frac{l_{\qty(1)}}{k_{\qty(1)\, +}}
      ]\,
      l_{\qty(2)}
      \\
      & +
      \qty[%
        \epsilon_{\qty(1)\, -}
        +
        \epsilon_{\qty(1)\, 2} \frac{1}{\Delta u} \frac{l_{\qty(1)}}{k_{\qty(1)\, +}}
        +
        \epsilon_{\qty(1)\, +} \frac{1}{2 (\Delta u)^2} \frac{l_{\qty(1)}^2}{k_{\qty(1)\, +}^2}
      ]\,
      k_{\qty(2)\, +}
    \Biggr\rbrace.
  \end{split}
  \label{eq:divergence_overlap_spin1}
\end{equation}
Divergences occur when $l = 0$ because of the absence of the factor $e^{i \frac{A}{u}}$.
However all explicit factors $\frac{1}{u}$ come always with $l$: when $l = 0$ they do not give any contribution.
The divergence in this case comes actually only from the contribution of the first line $\eval{\ipd{u}\phi}_{l = 0} = -\frac{1}{2u} \eval{\phi}_{l = 0}$.
Since we still have to subtract the contribution of the exchange $\qty(2) \leftrightarrow \qty(3)$ the contribution is cancelled in scalar \qed or with Abelian tachyons.
It does not cancel when considering the non Abelian case and the related colour factors unless one uses a kind of principal part regularisation since replacing $\finiteint{u}{-\abs{a}}{\abs{b}}\, \frac{\sgn(u)}{\abs{u}^{\frac{3}{2}}}$ with $\lim\limits_{\delta \rightarrow 0} \qty[\finiteint{u}{-\abs{a}}{-\abs{\delta}} + \finiteint{u}{-\abs{\delta}}{\abs{b}}]\, \frac{\sgn(u)}{\abs{u}^{\frac{3}{2}}}$ gives a finite result.


\subsubsection{An Overlap With Two Derivatives}
\label{sec:overlap}

We can generalise the previous expressions to more general cases.
Since we use the results from~\Cref{sec:Eigenmodes_from_Covering} we miss some non trivial contributions from polarisations like $\cS_{v\, i}$.
These contributions do not alter the final result.
However for completeness we give the lengthy full expression in~\Cref{sec:NO_full_TTS}.
  
We consider:\footnotemark{}
\footnotetext{%
  The underlying idea is to compute the amplitudes involving two tachyons and one massive state.
}
\begin{equation}
  K
  =
  \int\limits_{\Omega} \dd[3]{x}\,
  \sqrt{-\det g}~
  \eta^{\mu\nu}\, \eta^{\rho\sigma}\,
  \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, \mu\rho}\qty(\qty[x])\,
  \ipd{\nu}\ipd{\sigma} \Psi_{ \qty[k_{\qty(2)}] }\qty(\qty[x])\,
  \Psi_{ \qty[k_{\qty(1)}] }\qty(\qty[x]),
\end{equation}
in Minkowskian coordinates or
\begin{equation}
  K
  =
  \int\limits_{\Omega} \dd[3]{x}\,
  \sqrt{-\det g}~
  g^{\alpha\beta}\, g^{\gamma\delta}\,
  \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, \alpha\gamma}\qty(\qty[x])\,
  D_{\beta} \ipd{\delta} \Psi_{ \qty[k_{\qty(2)}] }\qty(\qty[x])\,
  \Psi_{ \qty[k_{\qty(1)}] }\qty(\qty[x])
\end{equation}
in orbifold coordinates where we need to use covariant derivatives.
Using the unfolding trick over $\qty(3)$ we get
\begin{equation}
  \begin{split}
    K
    & =
    \qty(2\pi)^3\,
    \delta\qty( \infinfsum{i} k_{\qty(i)\, +} )\,
    \finiteprod{i}{2}{N} \infinfsum{m_{(i)}}
    S_{\qty(3) \mu \rho}\,
    \qty(\cK^{m_{\qty(2)}} k_{\qty(2)\, 2})^{\mu}
    \qty(\cK^{m_{\qty(2)}} k_{\qty(2)\, 2})^{\rho}\,
    \\
    & \times
    \delta\qty( \infinfsum{i} \cK^{m_{(i)}} k_{\qty(i)\, 2} )\,
    \delta\qty( \infinfsum{i} \cK^{m_{(i)}} k_{\qty(i)\, -} ).
  \end{split}
  \label{eq:Spin_002_overlap_from_covering}
\end{equation}
Explicitly in orbifold coordinates we can write
\begin{equation}
  \begin{split}
    K
    & =
    \int\limits_{\Omega} \dd[3]{x}\,
    \sqrt{-\det g}~
    \Biggl[
      \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, u u }\,
      \ipd{v}^2 \Psi_{ \qty[k_{\qty(2)}] }
      -
      \frac{2}{ \qty(\Delta u)^2 }
      \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, u z }\,
      \ipd{v} \ipd{z} \Psi_{ \qty[k_{\qty(2)}] }
      \\
      & +
      2\, \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, u v }\,
      \ipd{v} \ipd{u} \Psi_{ \qty[k_{\qty(2)}] }
      +
      \frac{1}{ \qty(\Delta u)^4 }
      \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, z z }\,
      \qty(%
        \ipd{z}^2 \Psi_{ \qty[k_{\qty(2)}] }
        -
        \Delta^2 u\, \ipd{v} \Psi_{ \qty[k_{\qty(2)}] }
      ) 
      \\
      & -
      \frac{2}{ \qty(\Delta u)^2 }
      \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, z v }\,
      \qty(%
        \ipd{z} \ipd{u} \Psi_{ \qty[k_{\qty(2)}] }
        -
        \frac{1}{u}\, \ipd{z} \Psi_{\qty[k_{\qty(2)}] }
      )
      +
      \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, v v }\,
      \ipd{u}^2 \Psi_{ \qty[k_{\qty(2)}] }
    \Biggr]
    \Psi_{ \qty[k_{\qty(1)}] }.
  \end{split}
\end{equation}
Keeping the terms which do not vanish when all $l = 0$ and considering only the leading order in $\frac{1}{u}$ we get
\begin{equation}
  K
  \sim
  \int \dd{u}\, \abs{u}\,
  \frac{3}{4}
  \frac{\qty(k_{\qty(2)\, +} + k_{\qty(3)\, +})^2}{k_{\qty(3)\, +}^2}\,
  \cS_{\qty(3)\, v v}~
  \frac{1}{u^2}
  \eval{\finiteprod{i}{1}{3} \phi_{\qty(i)}}_{l_{\qty(*)} = 0},
  \label{eq:divergence_overlap_spin2}
\end{equation}
which is divergent as $\abs{u}^{-\frac{5}{2}}$.


\subsection{Three Points Amplitudes with One Massive State in String Theory}
\label{sec:NO3ptsMassive}

We consider string amplitudes including massive states.
They are obtained using the inheritance principle and therefore they are connected to the integrals and relations derived in~\Cref{sec:overlap}.
In particular we want to use the inheritance principle on the momenta and polarisations, i.e.\ we start form amplitudes in Minkowski expressed with momenta and polarisations and then we implement on them the projection on the orbifold.
In particular it is worth stressing that, as there is one Killing vector acting on the spacetime coordinates, there is only one common Killing vector action on all the momenta and polarisations of each field as discussed for spin-1 and spin-2 cases.
Moreover this approach gives the complete answer only for tree level amplitudes since inside the loops twisted states may be created in pairs.
The final result is that the open string amplitude with two tachyons and the first massive (level 2) state diverges and there is no obvious way of curing it since the divergence is also present in the Abelian sector.
The open string expansion we use is
\begin{equation}
  X\qty(u,\, \baru)
  =
  x_0
  -
  i\, 2\ap\, p\, \ln(\abs{u})
  +
  i\, \sqrt{\frac{\ap}{2}}
  \sum_{n \in \Z \setminus \qty{0}} \frac{\alpha_n}{n}
  \qty( u^{-n} + \baru^{-n} ).
\end{equation}


\subsubsection{First Massive State in String Theory}

Before computing the amplitude we would like to review the possible polarisations of the first massive state in open string.
The first massive vertex is:
\begin{equation}
  \begin{split}
  V_M\qty(x;\, k,\, S,\, \xi)
  & =
  \colon
    \qty(%
      \frac{i}{\sqrt{2 \ap}}
      \xi \cdot \ipd{x}^2 X\qty(x,\, x)
      +
      \qty( \frac{i}{\sqrt{2 \ap}} )^2
      S_{\mu\nu}\, \ipd{x} X^{\mu}\qty(x,\, x)\, \ipd{x} X^{\nu}\qty(x,\, x)
    )
  \\
  & \times
  e^{i\, k \cdot X\qty(x,\, x)}
  \colon.
  \end{split}
\end{equation}
The corresponding state is:
\begin{equation}
  \lim\limits_{x \rightarrow 0}
  V_M\qty(x;\, k,\, S,\, \xi) \ket{0}
  =
  \ket{k,\, S,\, \xi}
  =
  \qty( \xi \cdot \alpha_{-2} + \alpha_{-1} \cdot S \cdot \alpha_{-1} )
  \ket{k}.
\end{equation}
For the state to be physical we require:
\begin{align}
  \begin{split}
    \qty( L_0 - 1)
    \ket{k,\, S,\, \xi}
    & =
    0
    \qquad \Rightarrow \qquad
    \ap k^2 = -1
    \\
    L_1
    \ket{k,\, S,\, \xi}
    & =
    0
    \qquad \Rightarrow \qquad
    S \cdot k + \xi = 0
    \\
    L_2
    \ket{k,\, S,\, \xi}
    & =
    0
    \qquad \Rightarrow \qquad
    k \cdot \xi + \tr S = 0.
  \end{split}
\end{align}
String gauge invariance allows us to add:
\begin{equation}
  L_{-1}
  \qty( \chi \cdot \alpha_{-1} \ket{k} )
  =
  \qty( \chi \cdot \alpha_{-2} + \chi \cdot \alpha_{-1}\, k \cdot \alpha_{-1} )
  \ket{k},
\end{equation}
subject to the physical constraints $\ap k^2 + 1 = 0$ and $\chi \cdot k = 0$.
In critical string theory there is another gauge invariance generated by $L_{-2} + \frac{3}{2} L_{-1}^2$.
We can add a multiple of
\begin{equation}
  \qty( L_{-2} + \frac{3}{2} L_{-1}^2 )
  \ket{k} 
  =
  \qty(%
    \frac{5}{2} k \cdot \alpha_{-2}
    +
    \frac{3}{2} \qty(  k \cdot \alpha_{-1} )^2
    +
    \frac{1}{2} \alpha_{-1}^2
  )\,
  \ket{k},
\end{equation}
to set $a = 0$.
Therefore the only non trivial \dof refer to $S^{TT}$, that is:
\begin{equation}
  \tr S^{TT} = k \cdot S^{TT} = \xi = 0.
\end{equation}
  
We check that, given $k = \qty(k_+,\, k_-,\, k_2,\, \vec{k})$ such that $-2\, k_+ k_- + k_2^2 + \norm{\vec{k}}^2 = -1$, we can find a non trivial $S^{TT}$ with non vanishing components in the directions $\pm,\, 2$ only.
In fact we find a two parameters family of solutions.
The parameters may be taken to be $S_{+\, +}$ and $S_{+\, 2}$.
Explicitly we have
\begin{equation}
  \mqty(%
    S_{+\, +} \\
    S_{+\, -} \\
    S_{+\, 2} \\
    S_{-\, -} \\
    S_{-\, 2} \\
    S_{2\, 2}
  )
  =
  \mqty(%
    1 \\
    -\frac{k_-}{k_+} \\
    0 \\
    \frac{k_- \qty(k_- k_+ -2 k_2^2)}{k_+^3} \\
    -2 \frac{k_- k_2}{k_+^2} \\
    -2 \frac{k_-}{k_+}
  )\,
  S_{+\, +}
  +
  \mqty(%
    0 \\
    \frac{k_2}{k_+} \\
    1\\
    \frac{2 k_2 \qty(-k_- k_+ + k_2^2)} {k_+^3} \\
    \frac{k_- k_+ -2 k_2^2} {k_+^2} \\
    2 \frac{k_2}{k_+}
  )\,
  S_{+\, 2}
\end{equation}
There is even a non trivial solution for the special case $k = \qty(k_+,\, k_- = \frac{1}{k_+},\, k_2 = 0, \vec{k} = \vec{0})$.

Using the expressions for $S^{T T}$ in orbifold coordinates, we check that there are two possible indepdendent polarisations $\cS_{v\, v}$ and $\cS_{v\, z}$ which correspond to the those used above.
The non trivial solution is:
\begin{equation}
  \mqty(%
    \cS_{v\, v} \\
    \cS_{u\, v} \\
    \cS_{v\, z} \\
    \cS_{u\, u} \\
    \cS_{u\, z} \\
    \cS_{z\, z}       
  )
  =
  \mqty(%
    1 \\
    - \frac{r + \norm{\vec{k}}^2}{2 k_+^2} \\
    0 \\
    \qty( \frac{r + \norm{\vec{k}}^2}{2 k_+^2} )^2 \\
    0 \\
    -2 \frac{r + \norm{\vec{k}}^2}{2 k_+^2}
  )\,
  \cS_{v\, v}
  +
  \mqty(%
    0 \\
    -\frac{r + \norm{\vec{k}}^2}{2 k_+^2} \\
    1 \\
    0 \\ 0\\ 0\\
  )\,
  \cS_{v\, z}.
\end{equation}


\subsubsection{Two Tachyons and the First Massive State}

This Minkowskian amplitude is given by the sum of two colour ordered sub-parts as:
\begin{equation}
  \cA_{TTM}
  =
  A_{T_{\qty(1)} T_{\qty(2)} M_{\qty(3)}}\,
  \tr(T_{\qty(1)} T_{\qty(2)} T_{\qty(3)})
  +
  A_{T_{\qty(2)} T_{\qty(1)} M_{\qty(3)}}\,
  \tr(T_{\qty(2)} T_{\qty(1)} T_{\qty(3)}).
\end{equation}
We find:
\begin{equation}
  \begin{split}
    A_{T_{\qty(1)} T_{\qty(2)} M_{3)}}
    & =
    \left\langle\left\langle k_{\qty(1)} \right.\right|\,
    V_T\qty(1;\, k_{\qty(2)})\,
    \qty(%
      \alpha_{-1} \cdot S_{\qty(3)}^{TT} \cdot \alpha_{-1} \ket{k_{\qty(3)}}
    )
    \\
    & =
    \left\langle\left\langle k_{\qty(1)} \right.\right|\,
    e^{i\, k_{\qty(2)} \cdot x_0}\,
    e^{-\sqrt{2 \ap} k_{\qty(2)} \cdot \alpha_{1}}\,
    \qty(%
      \alpha_{-1} \cdot S_{\qty(3)}^{TT} \cdot \alpha_{-1} \ket{k_{\qty(3)}}
    )
    \\
    & =
    \qty(2\pi)^D\,
    \qty(\sqrt{2 \ap})^2\,
    \delta^D\qty(\finitesum{i}{1}{3} k_{\qty(i)} )\,
    k_{\qty(2)} \cdot S_{\qty(3)}^{TT} \cdot k_{\qty(2)}.
  \end{split}
\end{equation}
The transversality of $S_{\qty(3)}^{TT}$ finally leads to:
\begin{equation}
  \cA_{TTM}
  =
  2\,
  \qty(2\pi)^D\,
  \qty(\sqrt{2 \ap})^2\,
  \delta^D\qty( \finitesum{i}{1}{3} k_{\qty(i)} )\,
  k_{\qty(2)} \cdot S_{\qty(3)}^{TT} \cdot k_{\qty(2)}\,
  \tr\qty( \liebraket{T_{\qty(1)}}{T_{\qty(2)}}_+\, T_{\qty(3)} ).
\end{equation}
Then we can compute the orbifold amplitude as:
\begin{equation}
  \begin{split}
    \cA_{TTM}
    & =
    \qty(2\pi)^{D-2}
    \delta^{D-3}\qty( \finitesum{i}{1}{3} \vec{k}_{\qty(i)} )
    \delta\qty( \finitesum{i}{1}{3} k_{\qty(i)\, +} )
    \\
    & \times
    2 \qty(\sqrt{2 \ap})^2\,
    \sum_{ \qty{m_{\qty(1)},\, m_{\qty(2)},\, m_{\qty(3)}} \in \Z^3 }\,
    \delta_{m_{\qty(3)},\, 1}\,
    \qty(\cK^{m_{\qty(2)}} k_{\qty(2)})
    \cdot
    S_{\qty(3)}^{TT}
    \cdot
    \qty(\cK^{m_{\qty(2)}} k_{\qty(2)})
    \\
    & \times
    \delta\qty( \finitesum{i}{1}{3} \qty(\cK^{m_{\qty(i)}} k_{\qty(i)\, 2}) )\,
    \delta\qty( \finitesum{i}{1}{3} \qty(\cK^{m_{\qty(i)}} k_{\qty(i)\, -}) )\,
    \tr\qty( \liebraket{T_{\qty(1)}}{T_{\qty(2)}}_+\, T_{\qty(3)} ).
  \end{split}
\end{equation}
Such amplitude can then be expressed using an overlap:
\begin{equation}
  \begin{split}
    \cA_{TTM}
    & =
    2\, \qty(-i \sqrt{2 \ap} )^2\,
    \int\limits_{\Omega} \dd^3x\,
    g^{\mu\nu}\, g^{\rho\sigma}\,
    \Psi^{[2]}_{\qty[k_{\qty(3)},\, S_{\qty(3)}]\, \mu\rho}\qty(\qty[x])\,
    \ipd{\nu}\ipd{\sigma} \Psi_{\qty[k_{\qty(2)}] }\qty(\qty[x])\,
    \Psi_{\qty[k_{\qty(1)}] }\qty(\qty[x])
    \\
    & \times
    \tr\qty( \liebraket{T_{\qty(1)}}{T_{\qty(2)}}_+\, T_{\qty(3)}),
    \\
    & =
    2\, \qty(-i \sqrt{2 \ap} )^2\,
    \int\limits_{\Omega} \dd^3x\,
    g^{\alpha\beta}\, g^{\gamma\delta}\,
    \Psi^{[2]}_{\qty[k_{\qty(3)},\, S_{\qty(3)}]\, \alpha\gamma}\qty(\qty[x])\,
    D_{\beta} \ipd{\delta} \Psi_{\qty[k_{\qty(2)}] }\qty(\qty[x])\,
    \Psi_{\qty[k_{\qty(1)}] }\qty(\qty[x])
    \\
    & \times
    \tr\qty( \liebraket{T_{\qty(1)}}{T_{\qty(2)}}_+\, T_{\qty(3)} ).
  \end{split}
\end{equation}
As discussed in~\Cref{sec:overlap} the integral is divergent when $S_{+\, +} =\cS_{v\, v} \neq 0$ and the divergence cannot be avoided even introducing a Wilson line around $z$ since the amplitude involves an anticommutator which does not vanish in the Abelian sector.


\subsection{Scalar QED on the Generalised NBO and Divergences}
\label{sect:genNOscalarQED}

The issues related to the vanishing volume of the compact directions lead to incurable divergences.
We introduce the \gnbo by inserting one additional non compact direction with respect to the \nbo and show that divergences no longer occur.
As for the \nbo, we first present the geometry of the \gnbo and study scalar and spin-1 eigenfunctions to build the scalar \qed on the orbifold.
We then show how the presence of a non compact direction can cure the theory when considering amplitudes and overlaps.


\subsubsection{Geometric Preliminaries}

Consider Minkowski spacetime $\ccM^{1,D-1}$ and the change of coordinates from the lightcone set $( x^{\mu} ) = ( x^+, x^-, x^2, x^3, \vec{x} )$ to $( x^{\alpha} ) = ( u, v, w, z, \vec{x} )$:
\begin{equation}
  \begin{split}
    &\begin{cases}
      x^- & = u
      \\
      x^+ & = v + \frac{\Delta_2^2}{2} u ( z + w )^2 + \frac{\Delta_3^2}{2} u ( z - w )^2
      \\
      x^2 & = \Delta_2 u ( z + w )
      \\
      x^3 & = \Delta_3 u ( z - w )
    \end{cases}
    \\
    \Leftrightarrow
    &\begin{cases}
      u & = x^-
      \\
      v & = x^+ - \frac{1}{2 x^-} \left( (x^2)^2 + (x^3)^2 \right)
      \\
      w & = \frac{1}{2x^-} \left( \frac{x^2}{\Delta_2} - 
                                  \frac{x^3}{\Delta_3} \right)
      \\
      z & = \frac{1}{2x^-} \left( \frac{x^2}{\Delta_2} +
                                  \frac{x^3}{\Delta_3} \right)
    \end{cases}
  \end{split}
  \label{eq:orbifold_coordinates}
\end{equation}
where we do not perform any change on the transverse coordinates $\vec{x}$.
The metric in these coordinates is non diagonal:
\begin{equation}
  \dd{s}^2 = - 2 \dd{u}\dd{v} 
           + ( \Delta_2^2 + \Delta_3^2 ) u^2 ( \dd{w}^2 + \dd{z}^2 )
           + 2 ( \Delta_2^2 - \Delta_3^2 ) u^2 \dd{w}\dd{z}
           + \eta_{ij}\, \dd{x}^i \dd{x}^j,
  \label{eq:orbifold_metric}
\end{equation}
and its determinant is:
\begin{equation}
  - \det g  = 4\, \Delta_2^2 \Delta_3^2\, u^4.
\end{equation}
From the previous expressions we can also derive the non vanishing Christoffel symbols:
\begin{equation}
  \begin{split}
    \tensor{\Gamma}{_w^v_w}
    =
    \tensor{\Gamma}{_z^v_z} & = ( \Delta_2^2 + \Delta_3^2 ) u,
    \\
    \tensor{\Gamma}{_w^v_z} & = ( \Delta_2^2 - \Delta_3^2 ) u,
    \\
    \tensor{\Gamma}{_u^w_w}
    =
    \tensor{\Gamma}{_u^z_z} & = u^{-1},
  \end{split}
\end{equation}
which however produce a vanishing Ricci tensor and curvature scalar since we are considering Minkowski spacetime anyway and~\eqref{eq:orbifold_coordinates} is just a map from $\ccM^{1,D-1}$ to the \gnbo.

We introduce the \gnbo by identifying points in space along the orbits of the Killing vector:
\begin{equation}
  \begin{split}
    \kappa & = - 2 \pi i\, ( \Delta_2 J_{+2} + \Delta_3 J_{+3} )
    \\
           & = 2 \pi\, ( \Delta_2 x^2 + \Delta_3 x^3 ) \ipd{+}
             + 2 \pi \Delta_2 x^- \ipd{2} 
             + 2 \pi \Delta_3 x^- \ipd{3}
    \\
           & = 2 \pi\, \ipd{z}
  \end{split}
\label{eq:gnbo_killing_vector}
\end{equation}
in such a way that
\begin{equation}
  x^{\mu} \sim e^{n\kappa} x^{\mu}, \qquad n \in \Z
\end{equation}
leads to the identifications
\begin{equation}
  x
  =
  \mqty( x^- \\ x^2 \\ x^3 \\ x^+ \\ \vec{x} )
  \equiv
  \cK^{n} x
  =
  \mqty(%
    x^-
    \\
    x^2 + 2 \pi n \Delta_3^-
    \\
    x^3 + 2 \pi n \Delta_3x^-
    \\
    x^+ + 2 \pi n \Delta_3^2
    + 2 \pi n \Delta_3x^3
    + (2 \pi n)^2 \frac{\Delta_2^2+\Delta_3^2}{2} x^-
    \\
    \vec{x}
  ),
\end{equation}
or to the simpler
\begin{equation}
  \qty( u,\, v,\, w,\, z ) \sim \qty( u,\, v,\, w,\, z + 2 \pi n )
\label{eq:orbifold_identifications}
\end{equation}
using the map to the orbifold coordinates~\eqref{eq:orbifold_coordinates} where the Killing vector $\kappa = 2 \pi\, \ipd{z}$ does not depend on the local spacetime configuration.
As in the previous case, the difference between Minkowski spacetime and the \gnbo is therefore global.

The geodesic distance between the n-th copy and the base point on the orbifold can be computed in any set of coordinates and is:
\begin{equation}
  \Delta s^2_{(n)}
  =
  \qty( \Delta_2^2 + \Delta_3^2 )\,
  \qty( 2 \pi n x^- )^2
  \ge 0.
\end{equation}
Closed time-like curves are therefore avoided on the \gnbo, but there are closed null curves on the surface $x^- = u = 0$ where the Killing vector $\kappa$ vanishes.


\subsubsection{Free Scalar Field}

In order to build a quantum theory on the \gnbo using Feynman's approach to quantization, we first solve the eigenvalue equations for the fields and then build their off-shell expansion.
We start from a complex scalar field and then consider the free photon before moving to the scalar \qed interactions on the \gnbo.

Consider the action for a complex scalar field:
\begin{equation}
  \begin{split}
    \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \phi ]
    & =
    \int\limits_{\Omega} \dd[D]{x}
    \sqrt{-\det g }
    \qty(%
      -g^{\mu\nu} \ipd{\mu} \phi^* \ipd{\nu} \phi - M^2 \phi^* \phi
    )
    \\
    & =
    \int\limits_{\R^{D-4}} \dd[D-4]{\vec{x}}
    \infinfint{u} \infinfint{v} \infinfint{w}
    \int_0^{2\pi} \dd z\,
    2\, \abs{\Delta_2 \Delta_3}\, u^2
    \\
    & \times
    \Biggl[%
      \ipd{u} \phi^*\, \ipd{v} \phi
      +
      \ipd{v} \phi^*\, \ipd{u} \phi
      -
      \frac{1}{4 u^2}
      \Biggl(
        \qty( \frac{1}{\Delta_2^2} + \frac{1}{\Delta_3^2} )
        \qty( \ipd{w} \phi^*\, \ipd{w} \phi + \ipd{z} \phi^*\, \ipd{z} \phi )
    \\
        & +
        \qty( \frac{1}{\Delta_2^2} - \frac{1}{\Delta_3^2} )
        \qty( \ipd{w} \phi^*\, \ipd{z} \phi + \ipd{z} \phi^*\, \ipd{w} \phi )
      \Biggr)
      -
      \eta^{ij}\, \ipd{i} \phi^*\, \ipd{j} \phi
      -
      M^2 \phi^* \phi
   \Biggr].
  \end{split}
\end{equation}
As in the case of the \nbo, the solutions to the \eom are necessary to provide the modes of the quantum fields.
We study the eigenvalue equation $\square \phi_r = r \phi_r$, where $r$ is $2\, k_+ k_- - \vec{k}$ by comparison with the flat case ($k$ is the momentum associated to the flat coordinates).
We therefore need solve:
\begin{equation}
  \begin{split}
    \Biggl\lbrace
        &
        -2\, \ipd{u} \ipd{v}
        - \frac{2}{u} \ipd{v}
        + \frac{1}{4 u^2}
        \Biggl[
            \qty( \frac{1}{\Delta_2^2} + \frac{1}{\Delta_3^2} )
            \qty( \ipd{w}^2 + \ipd{z}^2 )
    \\
            & +
            2\, \qty( \frac{1}{\Delta_2^2} - \frac{1}{\Delta_3^2} )
            \ipd{w} \ipd{z}          
        \Biggr]
        +
        \eta^{ij}\, \ipd{i} \ipd{j}
        - r
    \Bigg\rbrace
    \phi_r
    =
    0.
  \end{split}
  \label{eq:scalar_eom}
\end{equation}
To this purpose, we introduce a Fourier transformation over $v,\, w,\, z,\, \vec{x}$:
\begin{equation}
  \phi_r\qty( u,\, v,\, w,\, z,\, \vec{x})
  =
  \infinfsum{l}\,
  \int\limits_{\R^{D-4}} \dd[D-4]{\vec{k}}
  \infinfint{k_+}
  \infinfint{p}\,
  e^{i\, \qty( k_+ v + p w + l z + \vec{k} \cdot \vec{x} )}
  \tphi_{k_-krgen}(u),
\end{equation}
where we defined $k_+,\, p,\, l,\, \vec{k}$ as associated momenta to $v,\, w,\, z,\, \vec{x}$ respectively.
We find:
\begin{equation}
  \phi_{k_-krgen}\qty( u,\, v,\, w,\, z,\, \vec{x} )
  =
  e^{i\, \qty( k_+ v + p w + l z + \vec{k} \cdot \vec{x} )}
  \tphi_{k_-krgen}( u ).
\end{equation}
where
\begin{equation}
  \tphi_{k_-krgen}( u )
  =
  \frac{1}{2 \sqrt{\qty(2 \pi)^D \abs{\Delta_2 \Delta_3 k_+}}}\,
  \frac{1}{\abs{u}}
  e^{%
    -i\, \qty(%
      \frac{1}{8 k_+ u}
      \qty[ \frac{(l + p)^2}{\Delta_2^2} + \frac{(l - p)^2}{\Delta_3^2} ]
      -
      \frac{\vec{k}^2 + r}{2 k_+} u
    )
  }.
  \label{eq:GNBO_reg_wave_functions}
\end{equation}
These solutions present the right normalisation, as we can verify through the product:
\begin{equation}
  \begin{split}
    & \left( \phi_{k_-krgenN{1}},\, \phi_{k_-krgenN{2}} \right)
    \\
    & =
    \int\limits_{\R^{D-4}} \dd[D-4]{\vec{x}}
    \infinfint{u}
    \infinfint{v}
    \infinfint{w}
    \finiteint{z}{0}{2\pi}\,
    2 \abs{\Delta_2 \Delta_3} u^2
    \\
    & \times
    \phi_{k_-krgenN{1}}~
    \phi_{k_-krgenN{2}}
    \\
    & =
    \delta^{D - 4}\qty( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)} )\,
    \delta\qty( k_{\qty(1)\, +} + k_{\qty(2)\, +} )\,
    \delta\qty( p_{\qty(1)} + p_{\qty(2)} )
    \delta\qty( r_{\qty(i)} + r_{\qty(i)} )\,
    \delta_{l_{\qty(1)},\, l_{\qty(2)}}.
  \end{split}
\end{equation}

Then we have the off-shell expansion:
\begin{equation}
  \begin{split}
    \phi_r\qty( u,\, v,\, w,\, z,\, \vec{x} )
    & =
    \frac{1}{2 \sqrt{\qty( 2 \pi )^D \abs{\Delta_2 \Delta_3 k_+}}}
    \infinfsum{l}\,
    \int\limits_{\R^{D-4}} \dd[D-4]{\vec{k}}
    \infinfint{k_+}
    \infinfint{p}
    \infinfint{r}
    \\
    & \times
    \frac{\cA_{k_-krgen}}{\abs{u}}
    e^{%
      i\, \qty(%
        k_+ v + p w + l z + \vec{k} \cdot \vec{x}
        -
        \frac{1}{8 k_+ u}
        \qty[ \frac{(l + p)^2}{\Delta_2^2} + \frac{(l - p)^2}{\Delta_3^2} ]
        +
        \frac{\vec{k}^2 + r}{2 k_+} u
      )
    }.
  \end{split}
\end{equation}


\subsubsection{Free Photon Action}

We then study the action of the free photon field $a$ using the Lorenz gauge which in the orbifold coordinates it reads:
\begin{equation}
  \begin{split}
    D^{\alpha} a_{\alpha}
    &
    =
    - \frac{2}{u} a_{v}
    - \ipd{v} a_u
    - \ipd{u} a_v
    \\
    &
    + \frac{1}{4 u^2}
    \qty(%
      \qty( \frac{1}{\Delta_2^2} + \frac{1}{\Delta_3^2} )\,
      \qty( \ipd{w} a_w + \ipd{z} a_z )
      +
      \qty( \frac{1}{\Delta_2^2} - \frac{1}{\Delta_3^2} )\,
      \qty( \ipd{w} a_z + \ipd{z} a_w )
    )
    \\
    &
    + \eta^{ij}\, \ipd{i} a_j
    =
    0.
  \end{split}
\end{equation}

We then solve the eigenvalue equations $\qty( \square a_r )_{\nu} = r a_{r\,\nu}$, which in components read:
\begin{equation}
  \begin{split}
    \qty( \square a_r )_u
    & =
    \frac{2}{u^2} a_{r\,v}
    \\
    & -
    \frac{1}{2 u^3}
    \qty[%
      \qty( \frac{1}{\Delta_2^2} + \frac{1}{\Delta_3^2} )
      \qty( \ipd{w} a_{r\,w} + \ipd{z} a_{r\,z} )
      +
      \qty( \frac{1}{\Delta_2^2} - \frac{1}{\Delta_3^2} )
      \qty( \ipd{w} a_{r\,z} + \ipd{z} a_{r\,w} )
    ]
    \\
    & +
    \Biggl\lbrace
      - 2 \ipd{u} \ipd{v}
      - \frac{2}{u} \ipd{v}
      \\
      & +
      \frac{1}{4 u^2}
      \qty[
        \qty( \frac{1}{\Delta_2^2} + \frac{1}{\Delta_3^2} )
        \qty( \ipd{w}^2 + \ipd{z}^2 )
        +
        \qty( \frac{1}{\Delta_2^2} - \frac{1}{\Delta_3^2} )
        2 \ipd{w} \ipd{z}
      ]
      + \nabla^2_T
    \Biggr\rbrace
    a_{r\,u},
    \\
    \qty( \square a_r )_v
    & =
    \Biggl\lbrace
      - 2 \ipd{u} \ipd{v}
      - \frac{2}{u} \ipd{v}
      \\
      & +
      \frac{1}{4 u^2}
      \qty[%
        \qty( \frac{1}{\Delta_2^2} + \frac{1}{\Delta_3^2} )
        \qty( \ipd{w}^2 + \ipd{z}^2 )
        +
        \qty( \frac{1}{\Delta_2^2} - \frac{1}{\Delta_3^2} )
        2 \ipd{w} \ipd{z}
      ]
      + \nabla^2_T
    \Biggr\rbrace
    a_{r\,v},
    \\
    \qty( \square a_r )_w
    & =
    - \frac{2}{u} \ipd{w} a_{r\,v}
    \\
    & +
    \Biggl\lbrace
      - 2 \ipd{u} \ipd{v} 
      \\
      & +
      \frac{1}{4 u^2}
      \qty[%
        \qty( \frac{1}{\Delta_2^2} + \frac{1}{\Delta_3^2} )
        \qty( \ipd{w}^2 + \ipd{z}^2 )
        + 
        \qty( \frac{1}{\Delta_2^2} - \frac{1}{\Delta_3^2} )
        2\, \ipd{w} \ipd{z}
      ]
      + \nabla^2_T
    \Biggr\rbrace
    a_{r\,w},
    \\
    \qty( \square a )_z
    & =
    - \frac{2}{u} \ipd{z} a_{r\,v}
    \\
    & +
    \Biggl\lbrace
      - 2 \ipd{u} \ipd{v}
      \\
      & +
      \frac{1}{4 u^2}
      \qty[%
        \qty( \frac{1}{\Delta_2^2} + \frac{1}{\Delta_3^2} )
        \qty( \ipd{w}^2 + \ipd{z}^2 )
        + 
        \qty( \frac{1}{\Delta_2^2} - \frac{1}{\Delta_3^2} )
        2 \ipd{w} \ipd{z}
      ]
      + \nabla^2_T
    \Biggr\rbrace
    a_{r\,z},
    \\
    \qty( \square a )_i
    & =
    \Biggl\lbrace
      - 2 \ipd{u} \ipd{v}
      - \frac{2}{u} \ipd{v}
      \\
      & +
      \frac{1}{4 u^2}
      \qty[%
        \qty( \frac{1}{\Delta_2^2} + \frac{1}{\Delta_3^2} )
        \qty( \ipd{w}^2 + \ipd{z}^2 )
        +
        \qty( \frac{1}{\Delta_2^2} - \frac{1}{\Delta_3^2} )
        2 \ipd{w} \ipd{z}
      ]
      + \nabla^2_T
    \Biggr\rbrace
    a_{r\,i},
  \end{split}
\end{equation}
where $\nabla^2_T = \eta^{ij}\, \ipd{i} \ipd{j}$ is the Laplace operator in the transverse coordinates $\vec{x}$.
These equations can be solved using standard techniques through a Fourier transform:
\begin{equation}
  \begin{split}
    a_{r\, \alpha}\qty( u,\, v,\, w,\, z,\, \vec{x})
    & =
    \infinfsum{l}\,
    \int\limits_{\R^{D-4}} \dd[D-4]{\vec{k}}
    \infinfint{k_+}
    \infinfint{p}\,
    \\
    & \times
    e^{i\, \qty( k_+ v + p w + l z + \vec{k} \cdot \vec{x} )}
    \tildea_{k_-krgen\, \alpha}(u).
  \end{split}
\end{equation}
We first solve the equations for $\tildea_{k_-krgen\, v}$ and $\tildea_{k_-krgen\, i}$ since they are identical to the scalar equation~\eqref{eq:scalar_eom}.
We then insert their solutions as sources for the equations for $\tildea_{k_-krgen\, u}$, $\tildea_{k_-krgen\, w}$ and $\tildea_{k_-krgen\, z}$.
The solutions can be written as the expansion:
\begin{equation}
  \begin{split}
    \norm{\tildea_{k_-krgen\, \alpha}(u)}
    & =
    \mqty(%
      \tildea_u
      \\
      \tildea_v
      \\
      \tildea_w
      \\
      \tildea_z
      \\
      \tildea_i
    )
    \\
    & =
    \sum\limits_{\ualpha \in \qty{\underu, \underv, \underw, \underz, \underi}}
    \cE_{k_-krgen\, \ualpha}\,
    \norm{\tildea^{\ualpha}_{k_-krgen\, \alpha}(u)}
    \\
    & =
    \cE_{k_-krgen\, \underu}\,
    \mqty( 1 \\ 0 \\ 0 \\ 0 \\ 0 )\,
    \tphi_{k_-krgen}
    \\
    & +
    \cE_{k_-krgen\, \underv}\,
    \mqty(%
      \frac{i}{2 k_+ u}
      +
      \frac{1}{8 k_+^2 u^2}
      \qty( \frac{(l + p)^2}{\Delta_2^2} + \frac{(l - p)^2}{\Delta_3^2} )
      \\
      1
      \\
      \frac{p}{k_+}
      \\
      \frac{l}{k_+}
      \\
      0
    )\,
    \tphi_{k_-krgen}
    \\
    & +
    \cE_{k_-krgen\, \underw}\,
    \mqty(
      \frac{1}{4 k_+ \abs{u}}
      \qty( \frac{l + p}{\Delta_2^2} - \frac{l - p}{\Delta_3^2} )
      \\
      0
      \\
      \abs{u}
      \\
      0
      \\
      0
    )\,
    \tphi_{k_-krgen}
    \\
    & +
    \cE_{k_-krgen\, \underz}\,
    \mqty(
      \frac{1}{4 k_+ \abs{u}}
      \qty( \frac{l + p}{\Delta_2^2} + \frac{l - p}{\Delta_3^2} )
      \\
      0
      \\
      0
      \\
      \abs{u}
      \\
      0
    )\,
    \tphi_{k_-krgen}
    \\
    & +
    \cE_{k_-krgen\, \underj}\,
    \mqty( 0 \\ 0 \\ 0 \\ 0 \\ \delta_{\underline{i j}} )\,
    \tphi_{k_-krgen}
  \end{split}
\end{equation}
Consider the Fourier transformed functions:
\begin{equation}
  a^{\ualpha}_{k_-krgen\, \alpha}\qty( u,\, v,\, w,\, z,\, \vec{x} )
  =
  e^{i\, \qty(k_+ v + p w + l z + \vec{k} \cdot \vec{x})}
  \tildea^{\ualpha}_{k_-krgen\, \alpha}( u ),
\end{equation}
then we can expand the off shell fields as
\begin{equation}
  \begin{split}
    a_{\alpha}(x)
    & =
    \infinfsum{l}\,
    \int\limits_{\R^{D-4}} \dd[D-4]{\vec{k}}\,
    \infinfint{k_+}
    \infinfint{p}
    \infinfint{r}
    \\
    & \times
    \sum_{\ualpha \in \qty{\underu, \underv, \underw, \underz, \underi}}
    \cE_{k_-krgen\, \alpha}\,
    a^{\ualpha}_{k_-krgen\, \alpha}(x).
  \end{split}
\end{equation}

We can compute the normalisation as:
\begin{equation}
  \begin{split}
    \qty( a_{(1)},\, a_{(2)} )
    & =
    \int\limits_{\R^{D-4}} \dd[D-4]{\vec{x}}
    \infinfint{u}
    \infinfint{v}
    \infinfint{w}
    \finiteint{z}{0}{2\pi}\,
    2 \abs{\Delta_2 \Delta_3} u^2
    \\
    & \times
    \qty(g^{\alpha\beta}\, a_{k_-krgenN{1}\, \alpha}\, a_{k_-krgenN{2}\, \beta})
    \\
    & =
    \delta^{D-4}\qty( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)} )\,
    \delta\qty( p_{(1)} + p_{(2)} )\,
    \delta\qty( k_{\qty(1)\, +} + k_{\qty(2)\, +} )\,
    \delta_{l_{(1)} + l_{(2)}, 0}
    \delta\qty( r_1 - r_2 )
    \\
    & \times
    \cE_{k_-krgenN{1}} \circ \cE_{k_-krgenN{2}},
  \end{split}
\end{equation}
where
\begin{equation}
  \begin{split}
    \cE_{(1)} \circ \cE_{(2)}
    & =
    -\cE_{(1)\, \underu}\, \cE_{(2)\, \underv}
    -\cE_{(1)\, \underv}\, \cE_{(2)\, \underu}
    \\
    &
    +\frac{1}{4}
    \Biggl[
        \qty( \frac{1}{\Delta_2^2} + \frac{1}{\Delta_3^2} )
        \qty(%
          \cE_{(1)\, \underw}\, \cE_{(2)\, \underw}
          +
          \cE_{(1)\, \underz}\, \cE_{(2)\, \underz}
        )
    \\
        & +
        \qty( \frac{1}{\Delta_2^2} - \frac{1}{\Delta_3^2} )
        \qty(%
          \cE_{(1)\, \underw}\, \cE_{(2)\, \underz}
          +
          \cE_{(1)\, \underz}\, \cE_{(2)\, \underw}
        )
    \Biggr]
  \end{split}    
\end{equation}
is independent of the coordinates.
The Lorenz gauge now reads:
\begin{equation}
  \eta^{i\underj}\, k_i \, \cE_{{k_-krgen} \underj}
  -
  k_+
  \cE_{k_-krgen\, \underu}
  -
  \frac{\vec{k}^2 + r}{2 k_+}
  \cE_{k_-krgen\, \underv}
  = 0.
\end{equation}
As in the previous case, the constraint equation does not pose any condition on the transverse polarisations $\cE_{k_-krgen\, \underw}$ and $\cE_{k_-krgen\, \underz}$.


\subsubsection{Cubic Interaction}

As previously studied on the \nbo, we show the scalar \qed 3-points vertex computation using the previously computed eigenmodes.
The presence of a continuous momentum in the non compact direction plays a major role in saving the convergence of the integrals.
In the case of the \gnbo we find:
\begin{equation}
  \begin{split}
    \rS_{\text{s}\qed}^{(\text{cubic})}\qty[\phi,\, a]
    & =
    \int\limits_{\Omega} \dd[D]{x} \sqrt{-\det g}\,
    \qty(%
      - i e g^{\mu\nu}\,
      a_{\mu}
      \qty( \phi^*\, \ipd{\nu} \phi - \ipd{\nu} \phi^*\, \phi )
    )
    \\
    & =
    \finiteprod{i}{1}{3}
    \infinfsum{l_{\qty(i)}}\,
    \int\limits_{\R^{D-4}} \dd[D-4]{\vec{k}_{\qty(i)}}
    \infinfint{k_{\qty(i)\, +}}
    \infinfint{p_{(i)}}
    \infinfint{r_{(i)}}
    \\
    & \times
    \qty( 2 \pi )^{D-1}\,
    \delta^{D-4}\qty( \finitesum{i}{1}{3} \vec{k}_{\qty(i)} )\,
    \delta\qty( \finitesum{i}{1}{3} p_{(i)} )\,
    \delta\qty( \finitesum{i}{1}{3} k_{\qty(i)\, +} )\,
    \delta_{\finitesum{i}{1}{3} l_{(i)},\, 0}
    \\
    & \times
    e~
    \cA^*_{\mkmkrgenN{2}}
    \cA_{k_-krgenN{3}}
    \\
    & \times
    \Biggl\lbrace
      \cE_{k_-krgenN{1}\, \underu}~
      k_{\qty(2)\, +}~
      \cI_{\qty{3}}^{\qty[0]}
      \\
      & +
      \cE_{k_-krgenN{1}\, \underv}~
      \Biggl[
        \qty( \frac{\vec{k}_{\qty(2)}^2 + r_{(2)}}{2 k_{\qty(2)\, +}} )\,
        \cI_{\qty{3}}^{\qty[0]}
        +
        i \frac{k_{\qty(2)\, +}}{k_{\qty(1)\, +}}\,
        \cI_{\qty{3}}^{\qty[-1]}
        \\
        & +
        \frac{k_{\qty(2)\, +}}{8}
        \Biggl[%
            \frac{1}{\Delta_2^2}
            \qty(%
              \frac{l_{(1)} + p_{(1)}}{k_{\qty(1)\, +}}
              +
              \frac{l_{(2)} + p_{(2)}}{k_{\qty(2)\, +}}
            )^2
            \\
            & +
            \frac{1}{\Delta_3^2}
            \qty(%
              \frac{l_{(1)} - p_{(1)}}{k_{\qty(1)\, +}}
              +
              \frac{l_{(2)} - p_{(2)}}{k_{\qty(2)\, +}}
            )^2
        \Biggr]\,
        \cI_{\qty{3}}^{\qty[-2]}
      \Biggr]
      \\
      & +
      \qty( \cE_{k_-krgenN{1}\, \underw} - \cE_{k_-krgenN{1}\, \underz} )
      \\
      & \times
      \Biggl[
          \frac{1}{\Delta_2^2}
          \qty(%
            \frac{k_{\qty(1)\, +} \qty( l_{(2)} + p_{(2)} )
                  +
                  k_{\qty(2)\, +} \qty( l_{(1)} + p_{(1)} )}{k_{\qty(1)\, +}}
          )
          \\
          & -
          \frac{1}{\Delta_3^2}
          \qty(%
            \frac{k_{\qty(1)\, +} \qty( l_{(2)} - p_{(2)} )
                  +
                  k_{\qty(2)\, +} \qty( l_{(1)} - p_{(1)} )} {k_{\qty(1)\, +}}
          )
      \Biggr]\,
      \cJ_{\qty{3}}^{\qty[-1]}
      \\
      & +
      \qty( (2) \leftrightarrow (3) )
    \Biggr\rbrace
  \end{split}
\end{equation}
where we defined:
\begin{equation}
  \begin{split}
    \cI_{\qty{N}}^{\qty[\nu]}
    & =
    \infinfint{u}
    2\, \abs{\Delta_2 \Delta_3} u^2\, u^{\nu}\, \finiteprod{i}{1}{N} \tphi_{k_-krgenN{i}},
    \\
    \cJ_{\qty{N}}^{\qty[\nu]}
    & =
    \infinfint{u}
    2\, \abs{\Delta_2 \Delta_3} u^2\, \abs{u}^{\nu}\, \finiteprod{i}{1}{N} \tphi_{k_-krgenN{i}}.
  \end{split}
\end{equation}


While in the \nbo case we need to regularise the integrals at least taking their principal part when all $l_{(*)} = 0$ in~\eqref{eq:nbo_div_integral}, the \gnbo does not need any specific manipulation.
In fact the form of $\tphi_{k_-krgenN{i}}$ in~\eqref{eq:GNBO_reg_wave_functions} prevents the formation of isolated zeros in the phase factor proportional to $u^{-1}$: the presence of the continuous momentum $p$, contrary to the \nbo where all momenta are discrete, gives the integrals a distributional interpretation, similar to a derivative of a Dirac $\delta$ function.


\subsubsection{Quartic Interactions}

As for the \nbo, we consider the quartic interaction for the scalar \qed action:
\begin{equation}
  \begin{split}
    \rS_{\text{s}\qed}^{(\text{quartic})}\qty[ \phi,\, a ]
    & =
    \int\limits_{\Omega} \dd[D]{x} \sqrt{-\det g}\,
    \qty(%
      e^2\, g^{\mu\nu}\, a_{\mu} a_{\nu} \abs{\phi}^2
      -
      \frac{g_4}{4} \abs{\phi}^4
    )
    \\
    & =
    \finiteprod{i}{1}{3}
    \qty(%
      \frac{1}{4\pi \sqrt{\qty(2\pi)^D \abs{\Delta_2\Delta_3 k_{\qty(i)\, +}}}}
    )
    \\
    & \times
    \infinfsum{l_{(i)}}\,
    \int\limits_{\R^{D-4}} \dd[D-4]{\vec{k}_{\qty(i)}}
    \infinfint{k_{\qty(i)\, +}}
    \infinfint{p_{(i)}}
    \infinfint{r_{(i)}}
    \\
    & \times
    \qty( 2 \pi )^{D-1}\,
    \delta^{D-4}\qty( \finitesum{i}{1}{3} \vec{k}_{\qty(i)} )\,
    \delta\qty( \finitesum{i}{1}{3} p_{(i)} )\,
    \delta\qty( \finitesum{i}{1}{3} k_{\qty(i)\, +} )\,
    \delta_{\finitesum{i}{1}{3} l_{(i)},\, 0}
    \\
    & \times
    \Biggl\lbrace
      e^2
      \cA^*_{\mkmkrgenN{3}}
      \cA_{k_-krgenN{4}}
      \\
      & \times
      \Biggl[
        \cE_{k_-krgenN{1}} \circ \cE_{k_-krgenN{2}}\,
        \cI_{\qty{4}}^{\qty[0]}
        \\
        & - i
        \cE_{k_-krgenN{1}\, \underv}\,
        \cE_{k_-krgenN{2}\, \underv}
        \\
        & \times
        \Biggl(
          \qty( \frac{1}{k_{\qty(1)\, +}} + \frac{1}{k_{\qty(2)\, +}} )\,
          \cI_{\qty{4}}^{\qty[-1]}
          \\
          & - i
          \qty(
              \frac{\cG_{+\,(1,2)}}{\Delta_2^2}
              +
              \frac{\cG_{-\,(1,2)}}{\Delta_3^2}
          )\,
          \cI_{\qty{4}}^{\qty[-2]}
        \Biggr)
        \\
        & +
        \frac{1}{4}
        \Biggl(
          \tilde{\cE}_{+\,(1,2)}\,
          \frac{\cG_{+\,(1,2)}}{\Delta_2^2}
          -
          \tilde{\cE}_{-\,(1,2)}\,
          \frac{\cG_{-\,(1,2)}}{\Delta_2^2}
        \Biggr)\,
        \cJ_{\qty{4}}^{\qty[-1]}
      \Biggr]
      \\
      & -
      \frac{g_4}{4}
      \cA^*_{\mkmkrgenN{1}}
      \cA^*_{\mkmkrgenN{2}}
      \\
      & \times
      \cA_{k_-krgenN{3}}
      \cA_{k_-krgenN{4}}
      \cI_{\qty{4}}^{\qty[0]}
    \Biggr\rbrace,
  \end{split}
\end{equation}
where we defined:
\begin{equation}
  \begin{split}
    \cG_{\pm\,\left( a, b \right)}
    & =
    \frac{l_{(a)} \pm p_{(a)}}{k_{\qty(a)\, +}}
    -
    \frac{l_{(b)} \pm p_{(b)}}{k_{\qty(b)\, +}},
    \\
    \tilde{\cE}_{\pm\,\left( a, b \right)}
    & =
    \cE_{k_-krgenN{a}\, \underv}
    \\
    & \times
    \qty( \cE_{k_-krgenN{b}\, \underw} \pm \cE_{k_-krgenN{b}\, \underz} )
    \\
    & -
    \cE_{k_-krgenN{b}\, \underv}
    \\
    & \times
    \qty( \cE_{k_-krgenN{a}\, \underw} \pm \cE_{k_-krgenN{a}\, \underz} )
  \end{split}
\end{equation}
for simplicity.

As the four points function in the \nbo case shows with clear evidence the presence of divergences when all $l_{(*)} = 0$, the \gnbo allows a distributional interpretation of the integrals $\cI_{\qty{N}}^{\qty[\nu]}$ and $\cJ_{\qty{N}}^{\qty[\nu]}$ in the previous expression.
In fact the regularization occurs in the same way as in the three points function in the \gnbo: the phase factor proportional to $u^{-1}$ has a continuous value due to the continous momentum $p$ and it does not present isolated zeros which would prevent the interpretation as distribution.


\subsubsection{Resurgence of Divergences and Null Brane Regularisation}

Looking back at the metric~\eqref{eq:orbifold_metric} and at the identifications~\eqref{eq:orbifold_identifications} it seems reasonable to wonder what would happen if we acted in the same way over $w$, since $2 \pi \ipd{w}$ is a Killing vector as well and it commutes with $2 \pi \ipd{z}$.
However from the analysis of \nbo and \gnbo, in the absence of at least one continuous transverse direction it is not possible to avoid the divergences associated with discrete zero energy modes and this is exactly what happens.

As mentioned in the introductory section, there have been attempts to regularise the \nbo using the Null Brane.
Differently from the \nbo, in this case the orbifold generator~\eqref{eq:nbo_killing_vector} includes an additional translation along an extra spatial dimension, namely:
\begin{equation}
  \begin{split}
    \kappa & = - 2 \pi i \Delta\, J_{+2} - 2 \pi i R P_3
    \\
    & =
    2 \pi\, (\Delta\, \ipd{z} + R,\ \ipd{3}).
  \end{split}
\end{equation}
with metric
\begin{equation}
  \dd{s}^2
  =
  -2 \dd{u} \dd{v}
  +
  \Delta^2 u^2 \qty(\dd{z})^2
  +
  \qty(\dd{x^3})^2
  +
  \eta_{ij}\, \dd{x}^i \dd{x}^j.
\end{equation}
Even though similar in appearance to the \gnbo Killing vector, this Killing vector is substantially different from~\eqref{eq:gnbo_killing_vector}.

The scalar field satisfies the same equation of motion as in the \nbo:
\begin{equation}
  \qty(%
    -2 \ipd{u} \ipd{v}
    - \frac{1}{u} \ipd{v}
    + \frac{1}{\qty(u \Delta)^2} \ipd{z}^2
    + \ipd{x^3}^2
    + \eta^{ij}\, \ipd{i} \ipd{j}
  )
  \phi_r
  =
  r \phi_r,
\end{equation}
where $i,\, j = 4,\, 5,\, \dots D - 1$.
The solution is:
\begin{equation}
  \tilde{\phi}_{\qty{ k_+\, k_z\, {k}_3\, \vec{k}\, r}}( u )
  \propto
  \frac{1}{\sqrt{\abs{u}}}
  e^{-i \frac{k_z^2}{2 k_+ } \frac{1}{u}
     + i \frac{{k}_3^2 + \vec{k}^2 + r}{2 k_+} u}.
\end{equation}
but with different periodicity conditions:
\begin{equation}
  e^{i 2 \pi n ( \Delta k_z + R k_3)}
  =
  1.
\end{equation}
This obscures the issue of the presence of a non compact direction.
To show the non compact direction hidden in this system we define the coordinates $\hatz = \frac{1}{2} \qty( \frac{x^3}{R} + \frac{z}{\Delta} )$ and $\hatx^3 = \frac{1}{2} \qty( \frac{x^3}{R} - \frac{z}{\Delta} )$ such that $\kappa = 2 \pi \ipd{\hatz}$ and
\begin{equation}
  \mqty( \hatz \\ \hatx^3 )
  \equiv
  \mqty( \hatz + 2 \pi n \\ \hatx^3 )
\end{equation}
upon the orbifold identification.
Then the momenta are $\hatk_{\hatz} = \hatl \in \Z$ and $\hatk_3\in \R$ and they are related to the momenta of the other coordinates as:
\begin{equation}
  k_3
  =
  \frac{\hatl + \hatk_3}{2 R},
  \qquad
  k_z
  =
  \frac{\hatl - \hatk_3}{2 \Delta},
\end{equation}
so that the solution can be written as
\begin{equation}
    \tilde{\phi}_{\qty{ k_+\, \hatl\, \hatk_3\, \vec{k}\, r}}( u )
    \propto
    \frac{1}{\sqrt{\abs{u}}}
    e^{-i \frac{(\hatl - \hatk_3)^2}{8 \Delta^2 k_+ } \frac{1}{u}
      + i \frac{(2 R)^{-2}(\hatk_3-\hat l)^2 + \vec{k}^2 + r}{2 k_+} u},
\end{equation}
which shows in a clear way that there is a non compact direction which allows a distributional interpretation as discussed in~\cite{Liu:2002:StringsTimeDependentOrbifolds}.
However this direction cannot be easily decoupled from the compact one.


\subsection{Comments on the BO}
\label{sec:BO}

In this section we would like to quickly show the analysis performed in the previous sections for the \nbo but in the case of the \bo.
The results are not very different apart from the fact that divergences are milder.
It is in fact possible to construct the full scalar \qed but nevertheless it is impossible to consider higher derivative terms in the effective theory.
Moreover some three point amplitudes with a massive state diverge.


\subsubsection{Geometric Preliminaries}

In $\ccM^{1,1}$ we consider the change of coordinates:
\begin{equation}
  \begin{cases}
    x^+ = t\, e^{+ \Delta \varphi}
    \\
    x^- = \sigma_-\, t\, e^{- \Delta \varphi}
  \end{cases}
  \qquad
  \Leftrightarrow
  \qquad
  \begin{cases}
    t = \sgn(x^+)\, \sqrt{ \abs{x^+ x^-} }
    \\
    \varphi = \frac{ 1 }{ 2 \Delta} \log \abs{\frac{x^+}{x^-}}
    \\
    \sigma_- = \sgn( x^+ x^-)
  \end{cases}
\end{equation}
where $\sigma_- = \pm 1$ and $t,\, \varphi \in \R$.
The metric reads:
\begin{equation}
  \begin{split}
    \dss[2]{s}
    & =
    -2\, \dd{x^+} \dd{x^+}
    \\
    & = -2 \sigma_- \qty( \dss[2]{t} - \qty(\Delta t)^2\, \dss[2]{\varphi} ).
  \end{split}
\end{equation}
Its determinant is:
\begin{equation}
  -\det g = 4 \Delta^2 t^2.
\end{equation}
In orbifold coordinates the non vanishing Christoffel symbols are:
\begin{equation}
  \tensor{\Gamma}{_\varphi^t_\varphi} = \Delta^2 t,
  \qquad
  \tensor{\Gamma}{_t^\varphi_\varphi} = t^{-1}.
\end{equation}

Using the orbifold coordinates $\qty(t,\, \varphi)$, the \bo is obtained by requiring the identification $\varphi \equiv \varphi + 2 \pi$ along the orbit of the global Killing vector $\kappa_{\varphi} = 2 \pi \ipd{\varphi}$.
We will therefore use the recurrent parameter $\Lambda = e^{2\pi \Delta}$ as shorthand notation.


\subsubsection{Free Scalar Action}

The action for a complex scalar $\phi$ is given by
\begin{equation}
  \begin{split}
    \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \phi ]
    & =
    \int\limits_{\Omega} \dd[D]{x}\,
    \sqrt{- \det g}
    \qty(
      -g^{\mu\nu}\, \ipd{\mu} \phi^*\, \ipd{\nu} \phi
      -M^2  \phi^* \phi
    )
    \\
    & =
    \sum_{\sigma_- \in \qty{\pm 1}}\,
    \int\limits_{\R^{D-2}} \dd[D-2]{\vec{x}}
    \infinfint{t}
    \finiteint{\varphi}{0}{2\pi}\,
    \Delta \abs{t}
    \\
    & \times
    \qty(
      \frac{1}{2} \sigma_-\,
      \ipd{t} \phi^*\, \ipd{t} \phi\,
      +
      \frac{1}{2}\, \frac{\sigma_-}{\qty(\Delta t)^2}
      \ipd{\varphi} \phi^*\, \ipd{\varphi}  \phi\,
      -
      \eta^{ij}\, \ipd{i} \phi^* \ipd{j} \phi
      -
      M^2 \phi^* \phi
    ).
  \end{split}
\end{equation}
As before we solve the associated eigenfunction problem for the d'Alembertian operator
\begin{align}
  \qty(
    -
    \frac{1}{2} \sigma_-\, \ipd{t}^2
    -
    \frac{1}{2} \sigma_-\, \frac{1}{t} \ipd{t}
    +
    \frac{1}{2} \sigma_-\, \frac{1}{\qty(\Delta\, t)^2} \ipd{\varphi}^2
    +
    \ipd{i}^2
  )
  \phi_r
  =
  r \phi_r
  .
\end{align}
with
\begin{equation}
  r = 2\, k_+ k_- - \vec{k}^2 = 2 \zeta_- m^2  - \vec{k}^2      
\end{equation}
where for later convenience (see the transformation of $k$ under the induced action of the Killing vector~\eqref{eq:BO_kpkp_equivalence}) we parameterise the momenta as:
\begin{equation}
  \begin{cases}
    k_+ & = m\, e^{+ \Delta \beta}
    \\
    k_- & = \zeta_- m\, e^{- \Delta \beta}
  \end{cases}
  \qquad
  \Leftrightarrow
  \qquad
  \begin{cases}
    m & = \sgn(k_+)\, \sqrt{ \abs{k_+ k_-} }
    \\
    \beta & = \frac{1}{2 \Delta} \log\abs{\frac{k_+}{k_-}}
    \\
    \zeta_- & = \sgn(k_+ k_-)
  \end{cases}
  \label{eq:kpkm_parametrization}
\end{equation}
where $\zeta_- = \pm 1$ and $m,\, \beta \in \R$.
To solve the problem we use standard techniques and perform the Fourier transform with respect to $\vec{x}$ and $\phi$ as :
\begin{equation}
  \phi\qty(t,\, \varphi,\, \vec{x})
  =
  \infinfsum{l}\,
  \int\limits_{\R^{D-2}} \dd[D-2]{\vec{x}}\,
  e^{i\, \vec{k} \cdot \vec x}
  e^{i\, l \varphi}
  H_{\lkrsi}(t),
\end{equation}
so that the new function $H_{\lkrsi}$ satisfies
\begin{align}
  \ipd{t}^2 H_{\lkrsi}
  +
  \frac{1}{t} \ipd{t} H_{\lkrsi}
  +
  \qty[
    \frac{l^2}{\qty(\Delta\, t)^2} 
    +
    2 \sigma_-
    \qty( r + \vec{k}^2 )
  ]
  H_{\lkrsi}
  =
  0,
\end{align}
which, upon the introduction of the natural quantities (see also~\eqref{eq:BO_PSI0_tau_lambda} for an explanation of the naturalness of $\lambda$)
\begin{equation}
  \tau = m\, t,
  \qquad
  \lambda = e^{\Delta\qty(\varphi + \beta)},
  \qquad
  \hsigma_- = \sigma_-\, \zeta_-,
\end{equation}
shows that the actual dependence on parameters is
\begin{equation}
  H_{\lkrsi}(t) = \tphi_{\lsi}\qty(\tau),
\end{equation}
so that
\begin{align}
  \ipd{\tau}^2 \tphi_{\lsi}
  +
  \frac{1}{\tau} \ipd{\tau} \tphi_{\lsi}
  +
  \qty[
    \frac{l^2}{\qty(\Delta\, \tau)^2} 
    +
    4 \hsigma_-
  ]
  \tphi_{\lsi}
  =
  0.
  \label{eq:BO_eq_diff_tilde_phi}
\end{align}
The asymptotic behaviour of the solutions is:
\begin{equation}
  \tphi_{\lsi} \sim
  \begin{cases}
    A_+ \abs{\tau}^{i \frac{l}{\Delta}}
    +
    A_- \abs{\tau}^{-i \frac{l}{\Delta}}
    & \qfor
    l \neq 0
    \\
    A_+ \log\abs{\tau} + A_-
    & \qfor
    l = 0
  \end{cases}.
  \label{eq:BO_asymtotics}
\end{equation}
    
\subsubsection{Eigenmodes on BO from Covering Space}

We now repeat the essential part of the analysis performed in the \nbo case.
As on the \nbo we say ``wave function'' and not eigenfunction since eigenfunctions for non scalar states require some  constraints on polarisations which we do not impose.


\paragraph{Scalar Wave Function}

We start as usual from the Minkowskian wave function and we write only the dependence on $x^+$ and $x^-$ since all the other coordinates are spectators
\begin{equation}
  \begin{split}
    \psi_{k_+\, k_-}\qty(x^+,\, x^-)
    & =
    e^{i\, \qty( k_+ x^+ + k_- x^- )}
    \\
    & =
    e^{%
      i\, m\, t\, \qty[%
        e^{\Delta \qty( \varphi + \beta )} + \hsigma_-\, t\, e^{\Delta \qty( \varphi - \beta )}
      ]
    }
    \\
    & =
    \psi_{k_+\, k_-}\qty(t,\, \varphi,\, \sigma_-).
  \end{split}
\end{equation}
We can compute the wave function on the orbifold by summing over all images:
\begin{equation}
  \begin{split}
    \Psi_{\qty[k_+\, k_-]}\qty(\qty[x^+,\, x^-])
    & =
    \infinfsum{n}
    \psi_{k_+\, k_-}\qty( \cK^n \qty(x^+,\, x^-))
    \\
    & =
    \infinfsum{n}
    \psi_{k_+\, k_-}\qty(x^+ e^{2\pi\Delta\,n},\, x^- e^{-2\pi\Delta\,n})
    \\
    & =
    \infinfsum{n}
    e^{%
      i \qty{%
        \qty[ k_+ e^{2\pi\Delta\,n} ] x^+ + \qty[ k_- e^{-2\pi\Delta\,n} ] x^-
      }
    }
    \\
    & =
    \infinfsum{n}
    \psi_{\cK^{-n}\qty( k_+\, k_-)}\qty( x^+,\, x^- ),
  \end{split}
\end{equation}
where we write $\qty[k_+\, k_-]$ because the function depends on the equivalence class of $k_+ k_-$ only.
The equivalence relation is given by
\begin{equation}
  k
  =
  \mqty( k_+ \\ k_- )
  \equiv
  \cK^{-n} k
  =
  \mqty( k_+ e^{2\pi\Delta\,n} \\ k_- e^{-2\pi\Delta\,n} ).
  \label{eq:BO_kpkp_equivalence}
\end{equation}
The previous equation explains the rationale for the parametrization~\eqref{eq:kpkm_parametrization} so that we can always choose a representative 
\begin{equation}
  0 \le \beta < 2 \pi, \qquad m \neq 0,
\end{equation}
or differently said $\beta \equiv \beta + 2 \pi$ and therefore we can use the dual quantum number $l$ using a Fourier transform.
Using the well adapted set of coordinates we can write the spin-0 wave function in a way to show the natural variables as
\begin{equation}
  \begin{split}
    \Psi_{\qty[k_+\, k_-]}\qty(\qty[x^+,\, x^-])
    & =
    \infinfsum{n}
    e^{%
      i\, \tau \qty[%
        \lambda e^{+2\pi\Delta\,n}
        +
        \hsigma_- \lambda^{-1} e^{ -2\pi\Delta\,n} 
      ]
    }  
    =
    \hPsi\qty(\tau,\, \lambda,\, \hsigma_-).
  \end{split}
  \label{eq:BO_PSI0_tau_lambda}
\end{equation}
Again the scalar eigenfunction has a unique equivalence class which mixes
coordinates and momenta.

Now we use the basic trick used in Poisson resummation
\begin{equation}
  \begin{split}
    \Psi_{\qty[k_+\, k_-]}\qty(\qty[x^+,\, x^-])
    & =
    \infinfint{s}
    \delta_P(s)\,
    e^{%
      i\, \qty{%
        k_+  x^+ \Lambda^s
        +
        k_-  x^- \Lambda^{-s}
      }
    }
    \\
    & =
    \frac{1}{2\pi}
    \infinfsum{l}
    \abs{\frac{k_+ x^+}{k_- x^-}}^{-i \frac{l}{2 \Delta} }\,
    \infinfint{s}
    e^{i\, 2 \pi\, l\, s}
    e^{i\, \sgn(k_+\, x^+) \sqrt{\abs{k_+ k_- x^+ x^-}}
      \qty{
        \Lambda^s
        +
        \sigma_- \zeta_- \Lambda^{-s}
      }
    }
    \\
    & =
    \frac{1}{2\pi}
    \infinfsum{l}
    \qty( e^{\Delta \qty( \varphi + \beta )} )^{-i \frac{l}{\Delta}}\,
    \infinfint{s}
    e^{i\, 2 \pi\, l\, s}
    e^{%
      i\, m t\, \qty{
        \Lambda^s
        +
        \sigma_- \zeta_- \Lambda^{-s}
      }
    }
    \\
    & =
    \frac{1}{2\pi}
    \infinfsum{l}
    e^{i\, l \beta}
    \qty[%
      e^{i\, l \varphi}\,
      \infinfint{s}
      e^{-i\, 2\pi l s}
      e^{%
        i\, m t\, \qty{
          \Lambda^s
          +
          \sigma_- \zeta_- \Lambda^{-s}
        }
      }
    ],    
  \end{split}
\end{equation}
where the last line represents the change of quantum number from $m\, \beta$ to $m\, l$ and allows us to identify
\begin{equation}
  \cN_{\text{BO}}
  \tphi_{\lsi}(\tau)
  =
  \frac{1}{2\pi}\,
  \infinfint{s}
  e^{-i\, 2 \pi\, l\, s}
  e^{%
    i\, \tau\, \qty{
      \Lambda^s
      +
      \hsigma_- \Lambda^{-s}
    }
  },    
\end{equation}
where $\cN_{\text{BO}}$ is a constant which depends on the normalization chosen for $\tphi_{\lsi}$. 
This expression gives an integral representation of the \ode solutions.


\paragraph{Tensor Wave Function (Spin-2)}

We consider the tensor wave function in Minkowski space.
We focus on $x^+,\, x^-$ and $x^2$ since all other directions behave as $x^2$.
Differently from scalar function we need to keep the dependence on $x^2$ since it is needed for non trivial physical polarisations and it enters in the transversality conditions.
Explicitly we find
\begin{equation}
  \begin{split}
    \cN_{\text{BO}}
    \psi^{[2]}_{k\, S}\qty(x^+,\, x^-,\, x^2)
    & =
    S_{\mu\nu}\,
    \dd{x}^{\mu} \dd{x}^{\nu}\,
    \psi_k(x)
    \\
    & =
    \Biggl[
      S_{++}\, \qty(\dd{x^+})^2  
      +
      2\, S_{+-}\, \dd{x^+} \dd{x^-}
      +
      2\, S_{+2}\, \dd{x^+}  \dd{x^2}
      \\
      & +
      S_{--}\, (\dd{x^-})^2
      +
      2\, S_{-2}\, \dd{x^2} \dd{x^2}
      \\
      & +
      S_{22}\, (\dd{x^2})^2
    \Biggr]\,
    e^{i\, \qty( k_+ x^+ + k_- x^- + k_2 x^2 )},
  \end{split}
\end{equation}
which we rewrite in orbifold coordinates 
\begin{equation}
  \begin{split}
    \cN_{\text{BO}}\,
    \psi^{[2]}_{k\,S}\qty(t,\, \varphi,\, x^2,\, \sigma_-)
    & =
    S_{\alpha\beta}\,
    \dd{x}^{\alpha} \dd{x}^{\beta}\,
    \psi_k(x)
    \\
    & \times
    \Biggl[
      \dss[2]{t}\,
      \qty(%
        2\, S_{+\,-}\, \sigma_-
        +
        S_{+\,+}\, e^{2\,\Delta\, \varphi}
        +
        S_{-\,-}\, e^{- 2\,\Delta\, \varphi}
      )
      \\
      & +
      2\, \Delta\, t\,
      \dd{t}\, \dd{\varphi}\,
      \qty(%
        S_{+\,+}\, e^{2\,\Delta\, \varphi}
        -
        S_{-\,-}\, e^{-2\,\Delta\, \varphi}
      )
      \\
      & +
      \Delta^2\, t^2
      \dss[2]{\varphi}\,
      \qty(%
        -2\, S_{+\,-}\, \sigma_-
        +
        S_{+\,+}\, e^{2\,\Delta\, \varphi}
        +
        S_{-\,-}\, e^{- 2\, \Delta\, \varphi}
      )
      \\
      & +
      2\, \dd{t}\, \dd{x^2}\,
      \qty(%
        S_{-\,2}\,e^ {- \Delta\,\varphi }\,\sigma_-
        +
        S_{+\,2}\, e^{\Delta\,\varphi}
      )
      \\
      & +
      2 \Delta\, t\,
      \dd{x^2}\, \dd{\varphi}\,
      \qty(%
        S_{+\,2}\, e^{\Delta\, \varphi}
        -
        S_{-\,2}\, e^{-\Delta\, \varphi}\, \sigma_-
      )
      \\
      & +
      \qty(\dd{x^2})^2\, S_{2\,2}
    \Biggr]
    e^{%
      i\, m\, t\, \qty[%
        e^{\Delta \qty( \varphi + \beta )}
        +
        \hsigma_- e^{\Delta \qty( \varphi - \beta )}
      ]
      + i\, k_2 x^2
    }.
  \end{split}
\end{equation}

Now we define the tensor wave on the orbifold as a sum over all images as
\begin{equation}
  \begin{split}
    \cN_{\text{BO}}\,
    \Psi^{[2]}_{\qty[k\, S]}\qty(\qty[x])
    & =
    \infinfsum{n}
    \qty( \cK^n\, \dd{x} ) \cdot S \cdot \qty( \cK^n\, \dd{x} )\,
    \psi_{k}( \cK^n x)
    \\
    & =
    \infinfsum{n}
    \dd{x} \cdot \qty( \cK^{-n}\, S ) \cdot \dd{x}\,
    \psi_{ \cK^{-n}\, k}(x).
  \end{split}
\end{equation}
In the last line we have defined the induced action of the Killing vector on $\qty(k,\, S)$ which can be explicitly written as:
\begin{equation}
  \cK^{-n}
  \mqty(%
    S_{ +\, + } \\
    S_{ +\, - } \\
    S_{ -\, - } \\
    S_{ +\, 2 } \\
    S_{ -\,  2} \\
    S_{ 2\, 2 }
  )
  =
  \mqty(
    e^{2 n \Delta \varphi}\, S_{ +\, + } \\
    S_{ +\, - } \\
    e^{-2 n \Delta \varphi}\, S_{ -\, - } \\
    e^{n \Delta \varphi}\, S_{ +\, 2 } \\
    e^{-n \Delta \varphi}\, S_{-\, 2} \\
    S_{ 2\, 2 } 
  )
  ,
\end{equation}
and it amounts to a trivial scaling.
In orbifold coordinates computing the tensor wave simply amounts to sum over all the shifts $\varphi \rightarrow \varphi + 2 \pi n$.
Then we have to give a close  expression for the sum involving powers $e^{2 \pi \Delta n}$.
Explicitly we find:
\begin{equation}
  \begin{split}
    & \infinfsum{n}
    \qty( e^{2 \pi \Delta n} )^N
    e^{%
      i\, \tau \qty[%
        \lambda e^{+2\pi\Delta\,n}
        +
        \hsigma_- \frac{1}{\lambda} e^{-2\pi\Delta\,n} 
      ]
    }
    \\
    & =
    \begin{cases}
      \qty[%
        \frac{1}{2}
        \qty(%
          \frac{1}{\lambda} \ipd{\tau}
          +
          \frac{1}{\tau} \ipd{\lambda}
        )
      ]^N\,
      \hPsi\qty(\tau,\, \lambda,\, \hsigma_-)
      & \qfor
      N > 0
      \\
      \qty[%
        \frac{1}{2}
        \qty(%
          \lambda \ipd{\tau}
          -
          \frac{\lambda^2}{\tau} \ipd{\lambda}
        )
      ]^N\,
      \hPsi\qty(\tau,\, \lambda,\, \hsigma_-)
      & \qfor
      N < 0
    \end{cases},
  \end{split}
\end{equation}
where $\tau$ derivatives of $\tphi_{\lsi}$ of order higher than $2$  can be reduced with the help of the differential equation~\eqref{eq:BO_eq_diff_tilde_phi}.
  
We now have to identify the basic polaritazions on the orbifold.
However the quantum number $\beta$ is no longer a good quantum number on the orbifold and it is replaced by $l$.
The relations among orbifold polarisations and Minkowski polarisations may depend on $\beta$ as long as the traceless and transversality conditions on the orbifold are independent of it.\footnotemark{}
\footnotetext{%
  These conditions may be a linear combinations of the ones in Minkowski.
}
Finally it seems reasonable to use the natural variable $\lambda = e^{\Delta \qty( \varphi + \beta )}$.
Therefore we have:
\begin{equation}
  \begin{split}
    \cS_{t\,t} & = e^ {- 2\,\Delta\,\beta }\,S_{+\,+},
    \\
    \cS_{t\,\varphi} & = S_{+\,-},
    \\
    \cS_{t\,2} & = e^ {- \Delta\,\beta }\,S_{+\,2},
    \\
    \cS_{\varphi\,\varphi} & = e^{2\,\Delta\,\beta}\,S_{-\,-},
    \\
    \cS_{\varphi\,2} & = e^{\Delta\,\beta}\,S_{-\,2},
    \\
    \cS_{2\,2} & = S_{2\,2},
  \end{split}
\end{equation}
which can be trivially inverted as
\begin{equation}
  \begin{split}
    S_{+\,+} & = e^{2\,\Delta\,\beta}\,\cS_{t\,t},
    \\
    S_{+\,-} & = \cS_{t\,\varphi},
    \\
    S_{+\,2} & = e^{\Delta\,\beta}\,\cS_{t\,2},
    \\
    S_{-\,-} & = e^ {- 2\,\Delta\,\beta }\,\cS_{\varphi\,\varphi},
    \\
    S_{-\,2} & = e^ {- \Delta\,\beta }\,\cS_{\varphi\,2},
    \\
    S_{2\,2} & = \cS_{2\,2}.
  \end{split}
\end{equation}
We can then compute the trace:
\begin{equation}
  \tr(S) = -2\, \cS_{t\,\varphi} + \cS_{2\,2},
\end{equation}
while the transversality conditions become
\begin{equation}
  \begin{split}
    \qty(k \cdot S)_+
    & =
    -
    e^{\Delta\, \beta}\,
    \qty(%
      m\, \hsigma_-\, \sigma_-\, \cS_{t\,t}
      +
      m\, \cS_{t\,\varphi}
      -
      k_{2}\, \cS_{t\,2}
    )
    \\
    \qty(k \cdot S)_-
    & =
    -
    e^{-\Delta\, \beta}\,
    \qty(%
      m\, \hsigma_-\, \sigma_-\, \cS_{t\,\varphi}
      +
      m\, \cS_{\varphi\,\varphi}
      -
      k_{2}\, \cS_{\varphi\,2}
    )
    \\
    \qty(k \cdot S)_2
    & =
    -
    \qty(%
      m\, \hsigma_-\, \sigma_-\, \cS_{t\,2}
      +
      m\, \cS_{\varphi\,2}
      -
      k_{2}\, \cS_{2\,2}
    ),
  \end{split}
\end{equation}
which are independent from $\beta$ when it is set to zero.

The final expression of the wave function for the symmetric tensor on the orbifold is:
\begin{equation}
  \begin{split}
    \Psi^{[2]}_{\qty[k\, S]}\qty(\qty[x])
    & =
    \infinfsum{l} e^{i\, l \beta}
    \Biggl[
      S_{m\, l,\, t t}\, \dss[2]{t}  
      +
      2\, S_{m\, l,\, t \varphi}\, \dd{t} \dd{\varphi}
      +
      2\, S_{m\, l,\, t 2}\, \dd{t} \dd{x^2}
      \\
      & +
      S_{m\, l,\, \varphi \varphi}\, \dss[2]\varphi
      +
      2\, S_{m\, l,\, \varphi 2}\, \dd{\varphi} \dd{x^2}
      \\
      & +
      S_{m\, l,\, 22}\, \dd{x^2}^2
    \Biggr],
  \end{split}
\end{equation}
where the explicit expressions for the components are
\begin{equation}
  \begin{split}
    S_{m\, l,\, tt}
    & =
    \qty[%
      -
      \frac{\tphi_{\lsi}(\tau)\, l\, \lambda^{\frac{i\,l}{\Delta}}\,
      \qty(%
        l\,\cS_{t\,t}
        +
        i\,\Delta\,\cS_{t\,t}
        +
        l\,\cS_{\varphi\,\varphi}
        -
        i\,\Delta\,\cS_{\varphi\,\varphi}
      )
      }{2\,\Delta^2}
    ]
    \frac{1}{\tau^2}
    \\
    & +
    \qty[%
      \frac{1}{2\,\Delta}
      \dv{\tau}\, \tphi_{\lsi}(\tau)\,\lambda^\frac{i\,l}{\Delta}\,
      \qty(%
        i\,l\,\cS_{t\,t}
        -i\,l\,\cS_{\varphi\,\varphi}
        -\Delta\,\cS_{t\,t}
        -\Delta\,\cS_{\varphi\,\varphi}
      )
    ]
    \frac{1}{\tau}
    \\
    & +
    \qty[%
      \tphi_{\lsi}(\tau)\,\lambda^\frac{{i\,l}{\Delta}}\,
      \qty(%
        \hsigma_-\,\cS_{t\,t}
        +
        2\,\sigma_-\,\cS_{t\,\varphi}
        +
        \hsigma_-\,\cS_{\varphi\,\varphi}
      )
    ],              
  \end{split}
\end{equation}
\begin{equation}
  \begin{split}
    S_{m\, l,\, t \varphi}
    & =
    \qty[
      -\frac{\tphi_{\lsi}(\tau)\,l\,\lambda^\frac{{i\,l}{\Delta}}\,\qty(l\,\cS_{t\,t}+i\,\Delta\,\cS_{t\,t}-l\,\cS_{\varphi\,\varphi}+i\,\Delta\,\cS_{\varphi\,\varphi})}
      {2\,\Delta\,{m}}
    ]
    \frac{1}{\tau}
    \\
    & +
    \qty[
    \frac{\dv{\tau}\,\tphi_{\lsi}(\tau)\,
      \lambda^\frac{{i\,l}{\Delta}}\,
      \qty(
      i\,l\,\cS_{t\,t}
      -\Delta\,\cS_{t\,t}+
      i\,l\,\cS_{\varphi\,\varphi}
      +\Delta\,\cS_{\varphi\,\varphi}
      )}{2\,{m}}
    ]
    \\
    & +
    \qty[
      \frac{\Delta\, \hsigma_-\, \tphi_{\lsi}(\tau)\,
      \lambda^\frac{{i\,l}{\Delta}}\, \qty(\cS_{t\,t}-\cS_{\varphi\,\varphi})}{m}
    ]
    \tau,
  \end{split}
\end{equation}
\begin{equation}
  \begin{split}
    S_{m\, l,\, \varphi\varphi}
    & =
    \qty[
      -
      \frac{1}{2\,m^2}
      \tphi_{\lsi}(\tau)\,
      l\,
      \lambda^\frac{{i\,l}{\Delta}}\,
      \qty(
      l\, ( \cS_{t\,t} +\cS_{\varphi\,\varphi} )
      +i\,\Delta\, ( \cS_{t\,t} - \cS_{\varphi\,\varphi} )
      )
    ]
    \\
    & +
    \qty[
      \frac{1}{2\,m^2}
      \Delta\,
      \qty(\dv{\tau}\,\tphi_{\lsi}(\tau))\,
      \lambda^\frac{{i\,l}{\Delta}}\,
      \qty(
      i\,l\,\cS_{t\,t}
      -i\,l\,\cS_{\varphi\,\varphi}
      -\Delta\,\cS_{t\,t}
      -\Delta\,\cS_{\varphi\,\varphi}
      )
    ]
    \tau
    \\
    & +
    \qty[
      \frac{1}{m^2}
      \Delta^2\,
      \tphi_{\lsi}(\tau)\,
      \lambda^\frac{{i\,l}{\Delta}}\,
      \qty(
      \hsigma_-\,\cS_{t\,t}
      +\hsigma_-\,\cS_{\varphi\,\varphi}
      -2\,\sigma_-\,\cS_{t\,\varphi}
      )
    ]
    \tau^2,
  \end{split}
\end{equation}
together with the effectively vector components in the orbifold directions:
\begin{equation}
  \begin{split}
    S_{m\, l,\, t2}
    & =
    \qty[
      \frac{i}{2\,\Delta}
      \tphi_{\lsi}(\tau)\,
      l\,
      \lambda^\frac{{i\,l}{\Delta}}\,
      \qty(\cS_{t\,2}-\cS_{\varphi\,2}\,\sigma_-)
    ]
    \frac{1}{\tau}
    \\
    & +
    \qty[
      \frac{1}{2}
      \dv{\tau}\,\tphi_{\lsi}(\tau)\,
      \lambda^\frac{{i\,l}{\Delta}}\,
      \qty(\cS_{t\,2}+\cS_{\varphi\,2}\,\sigma_-)
    ],
  \end{split}
\end{equation}
and
\begin{equation}
  \begin{split}
    S_{m\, l,\, \varphi 2}
    & =
    \qty[
      \frac{i}{2\,m}
      \tphi_{\lsi}(\tau)\,
      l\,
      \lambda^\frac{{i\,l}{\Delta}}\,
      \qty(\cS_{t\,2}+\cS_{\varphi\,2}\,\sigma_-)
    ]
    \\
    & +
    \qty[
      \frac{1}{2\,m}
      \Delta\,
      \qty(\frac{d}{d\,\tau}\,\tphi_{\lsi}(\tau))\,
      \lambda^\frac{{i\,l}{\Delta}}\,
      \qty(\cS_{t\,2}-\cS_{\varphi\,2}\,\sigma_-)
    ]
    \tau,
  \end{split}
\end{equation}
and the effectively scalar component:
\begin{equation}
  S_{m\, l,\, 22}
  =
  \cS_{2\,2}\,
  \tphi_{\lsi}(\tau)\,
  \lambda^\frac{i\,l}{\Delta}.
\end{equation}


\subsection{Overlaps and Divergent Three Points String Amplitudes}
\label{sec:BOoverlap}

We consider some overlaps as done for the \nbo.
The connection between the overlaps on the orbifold and the sums of images remains unchanged when we change the Killing vector $\cK$, hence we can limit ourselves to discuss the integrals on the orbifold space.  


\subsubsection{Overlaps Without Derivatives}

Let us start with the simplest case of the overlap of $N$ scalar wave functions:
\begin{equation}
  \begin{split}
    I^{(N)}
    & =
    \int\limits_{\Omega} \dd[3]{x}\,
    \sqrt{- \det g}
    \finiteprod{i}{1}{N}
    \Psi_{\qty[k_{\qty(i)\, +}\, k_{\qty(i)\, -}]}(\qty[x^+,\, x^-,\, x^2]))
    \\
    & =
    \cN_{\text{BO}}^N
    \sum_{ \qty{l_{\qty(i)} } \in \Z^N } 
    e^{i \finitesum{i}{1}{N} l_{\qty(i)} \beta_{(i)} }
    \int\limits_{\Omega} \dd[3]{x}\,
    \sqrt{-\det g}
    \finiteprod{i}{1}{N} \phi_{\lsiN{i}}.
  \end{split}
\end{equation}
This is always a distribution since the problematic $l_{\qty(*)} = 0$ sector gives a divergence $\qty(\log\abs{t})^N$ when $t \sim 0$.
All other sectors have no issues because of the asymptotic behaviours~\eqref{eq:BO_asymtotics}. 


\subsubsection{An Overlap With Two Derivatives}

We consider in orbifold coordinates the overlap needed for the amplitude involving two tachyons and one massive state, i.e.:
\begin{equation}
  K
  =
  \int\limits_{\Omega} \dd[3]{x}\,
  \sqrt{-\det g}\,
  g^{\alpha\beta}\, g^{\gamma\delta}\,
  \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, \alpha\gamma}\qty(\qty[x])\,
  D_{\beta} \ipd{\delta} \Psi_{\qty[k_{\qty(2)}]}\qty(\qty[x])\,
  \Psi_{\qty[k_{\qty(1)}]}\qty(\qty[x]).
\end{equation}
Since we use the traceless condition we need to keep all momenta and polarisations.
We write:
\begin{equation}
  \begin{split}
    K
    & =
    \int\limits_{\Omega} \dd[3]{x}\,
    \sqrt{-\det g}
    \Biggl[
      \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\,  t t }\,
      \ipd{t}^2 \Psi_{ \qty[k_{\qty(2)}] }
      \\
      & -
      2\, \qty(\frac{1}{\Delta t})^2
      \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\,  t \varphi }\,
      \qty(%
        \ipd{t} \ipd{\varphi} \Psi_{ \qty[k_{\qty(2)}] }
        -
        \frac{1}{t} \ipd{\varphi} \Psi_{ \qty[k_{\qty(2)}] }
      )
      \\
      & +
      \qty(\frac{1}{\Delta t})^4
      \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\,  \varphi \varphi }\,
      \qty(%
        \ipd{\varphi}^2 \Psi_{ \qty[k_{\qty(2)}] }
        -
        \Delta^2 t \ipd{t} \Psi_{ \qty[k_{\qty(2)}] }
      )
      \\
      & -
      2\, \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\,  t 2 }\,
      \ipd{t} \ipd{2} \Psi_{ \qty[k_{\qty(2)}] }
      \\
      & +
      2\, \qty(\frac{1}{\Delta t})^2
      \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\,  \varphi 2 }\,
      \ipd{\varphi} \ipd{2} \Psi_{ \qty[k_{\qty(2)}] }
      \\
      & +
      \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\,  2 2 }\,
      \ipd{2}^2 \Psi_{ \qty[k_{\qty(2)}] }
    \Biggr]
    \Psi_{ \qty[k_{\qty(1)}] }.
  \end{split}
\end{equation}

Now consider the behaviour for $l_{\qty(*)} = 0$ for small $t$.
All the $\ipd{\varphi}$ can be dropped since they lower a $l_{\qty(2)}$.
The leading contributions from spin-2 components are $S_{m\, l\, t t} \sim \frac{1}{t^2}$, $S_{m\, l\, \varphi \varphi}, S_{m\, l\, 2\,2} \sim 1$ and $S_{m\, l\, t 2} \sim \frac{1}{t}$.
The leading $\frac{1}{t^4}$ reads:
\begin{equation}
  \begin{split}
    K
    & \sim
    \int\limits_{t \sim 0} \dd{t}\,
    \abs{t}\,
    \Biggl[
      -
      \frac{1}{2}
      \dv{\tau}\,
      \tphi_{\lsi}\,
      \qty( \cS_{t\,t} + \cS_{\varphi\,\varphi} )
      \frac{1}{\tau}\,
      \ipd{t}^2 \Psi_{ [k_{\qty(2)}] }
      \\
      & +
      \qty(\frac{1}{\Delta t})^4\,
      \frac{-\Delta^2}{2\,m^2}
      \dv{\tau}\,
      \tphi_{\lsi}\,
      \qty( \cS_{t\,t} + \cS_{\varphi\,\varphi} )
      \tau\,
      \qty( - \Delta^2 t \ipd{t} \Psi_{ [k_{\qty(2)}] })
    \Biggr]
    \Psi_{ \qty[k_{\qty(3)}] }
  \end{split}
\end{equation}
In the limit of our interest $\eval{\Psi_{ \qty[k] }}_{l = 0} \sim \eval {\tphi_{\lsi}}_{l = 0} \sim \log\abs{t}$.
The two terms add together because of sign of the covariant derivative to give:
\begin{equation}
  K
  \sim
  \int\limits_{t \sim 0} \dd{t}\, \abs{t}\,
  \qty[
    \qty( \frac{1}{2} + \frac{1}{2} )
    \frac{\cS_{t\,t} + \cS_{\varphi\,\varphi}}{ m^4}
    \frac{\log\abs{t}}{t^4}
    +
    \order{\frac{\qty(\log\abs{t})}{t}}
  ],
\end{equation}
which is divergent for the physical polarisation $\cS_{t\,t} = \cS_{\varphi\,\varphi} = -\hsigma_- \sigma_- \cS_{t\,\varphi} = -\frac{1}{2} \hsigma_- \sigma_- \cS_{2 2}$.
  

% vim: ft=tex
