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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-10-20 19:29:13 +02:00
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sec/part2/divergences.tex
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@@ -49,14 +49,14 @@ We then go back to string theory and we verify that in the \nbo the open string
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}.
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.
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}.
We finally present also a brief examination of the Boost Orbifold (\bo) where the divergences are generally milder~\cite{Horowitz:1991:SingularStringSolutions}.
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.