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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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10
sec/part1/fermions.tex
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@@ -2843,7 +2843,7 @@ In similar way we can compute all correlators
\GGexcvacket
}{\braket{\GGexcvac}}
\\
& =
= &
\frac{%
\left\langle
\rR\qty[
@@ -2865,4 +2865,12 @@ using Wick's theorem since the algebra and the action of creation and annihilati
In particular taking one $\Psi(z)$ and one $\Psi^*(w)$ we get the Green function which is nothing else but the contraction in equation~\eqref{eq:gen_Radial_order}.
\subsubsection{Summary and Conclusions}
In a technical and direct way we showed the computation of amplitudes involving an arbitrary number of Abelian spin and matter fields.
The approach we introduced does not generally rely on \cft techniques and can be seen as an alternative to bosonization and old methods based on the Reggeon vertex.
Starting from this work the future direction may involve the generalisation to non Abelian spin fields and the application to twist fields.
In this sense this approach might be the only way to compute the amplitudes involving these complicated scenarios.
This analytical approach may also shed some light on the non existence of a technique similar to bosonisation for twist fields.
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\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.
What has gone unnoticed is that 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 Equation (6.16) of \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 $\ccA_{\text{closed}}( q ) \sim q^{4 - \ap \norm{\vb{p}_{\perp}}^2}$ and $\ccA_{\text{closed}}( q ) \sim q^{1 - \ap \norm{\vb{p}_{\perp}}^2} \tr\qty( \liebraket{T_1}{T_2}_+ \liebraket{T_3}{T_4}_+ )$ ($T_i$ 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 Generalized Null Boost Orbifold (\gnbo) as a generalization 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.
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{Summary and Conclusions}
In the previous analysis it seems that string theory cannot do better than field theory when the latter does not exist, at least at the perturbative level where one deals with particles.
Moreover when spacetime becomes singular, the string massive modes are not anymore spectators.
Everything seems to suggest that issues with spacetime singularities are hidden into contact terms and interactions with massive states.
This would explain in an intuitive way why the eikonal approach to gravitational scattering works well: the eikonal is indeed concerned with three point massless interactions.
In fact it appears that the classical and quantum scattering on an electromagnetic wave~\cite{Jackiw:1992:ElectromagneticFieldsMassless} or gravitational wave \cite{tHooft:1987:GravitonDominanceUltrahighenergy} in \bo and \nbo are well behaved.
From this point of view the ACV approach~\cite{Soldate:1987:PartialwaveUnitarityClosedstring,Amati:1987:SuperstringCollisionsPlanckian} may be more sensible, especially when considering massive external states~\cite{Black:2012:HighEnergyString}.
Finally it seems that all issues are related with the Laplacian associated with the space-like subspace with vanishing volume at the singularity.
If there is a discrete zero eigenvalue the theory develops divergences.
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In the previous part we mainly focused on the mathematical tools needed to compute amplitudes in a phenomenologically valid string theory framework of particle physics.
This ultimately led to the introduction of intersecting D-branes and point-like defects to perform the calculation of correlation functions involving twist and spin fields, inevitably necessary fields when considering chiral matter fields.
While this is indeed a good starting point to build an entire string phenomenology, the theory cannot be limited to the study of particle physics models.
String Theory is in fact considered to be one of the candidate theories for the description of quantum gravity alongside the nuclear interactions.
As a \emph{theory of everything} it is therefore fascinating to analyse cosmological implications as seen from its description.
In this part of the thesis we focus on the implications of the string theory when considering for instance the Big Bang singularity, or, broadly speaking, singularities which exist in one point in time.\footnotemark{}
\footnotetext{%
They are intended as distinct from time-like singularities such as black holes which are present for extended periods of time in one spatial point.
The space-like singularities we consider are the opposite: they exist in a given instant.
}
Among the different possible descriptions of such space-like singularities~\cite{Berkooz:2007:ShortReviewTime} we concentrate on string theory solutions on time-dependent orbifolds.
Before delving into the subject we briefly present their definition and the reason behind their relevance in what follows~\cite{CaramelloJr:2019:IntroductionOrbifolds,Bachas:2002:NullBraneIntersections,Bachas:2003:RelativisticStringPulse}.
\subsection{Quotient Spaces and Orbifolds}
First of all we recall the formal definition of orbifold to better introduce the idea of a manifold locally isomorphic to a quotient space.
Let therefore $M$ be a topological space and $G$ a group with an action $\ccG: G \times M \to M$ defined by $\ccG(g,\, p) = g p$ for $g \in G$ and $p \in M$.
Then the \emph{isotropy subgroup} (or \emph{stabiliser}) of $p \in M$ is $G_p = \qty{ g \in G \mid g p = p }$ such that $G_{gp} = g\, G_p\, g^{-1}$.
Given an element $p \in M$ its \emph{orbit} is $Gp = \qty{ gp \in M \mid g \in G }$.
The action of the group is said \emph{transitive} if $Gp = M$ and \emph{effective} if its $\ker{\ccG} = \qty{ \1 }$.
The \emph{orbit space} $M / G$ is the set of equivalence classes given by the orbital partitions and inherits the quotient topology from $M$.
Let now $M$ be a manifold and $G$ a Lie group acting continuously and transitively on $M$.
For every point $p \in M$ we can define a continuous bijection $\lambda_p \colon G / G_p \to Gp = M$.\footnotemark{}
\footnotetext{%
For any $U \subset M$ and a given $p \in M$ then $\lambda_p^{-1}\qty( U ) = \pi_p\qty( \qty{ g \in G \mid g p \in U } )$ where $\pi_p \colon G \to G / G_p$ is the projection map.
Thus $\lambda_p^{-1}\qty( U )$ is an open subset if $U$ is open: the bijection is continuous.
}
Such map is a diffeomorphism if $M$ and $G$ are locally compact spaces and $M / G$ is in turn a manifold itself.
If $G$ is a discrete or finite group the action is called \emph{properly discontinuous}, that is for every $U \subset M$ then $\qty{ g \in G \mid U \cap g U \neq \emptyset }$ is finite.
The definition of orbifold intuitively includes quotient manifolds such as $M / G$: analogously to manifold which are locally Euclidean, in the broad sense orbifolds are locally modelled by quotients with actions given by finite groups.
An \emph{orbifold chart} $\qty( \tildeU,\, G,\, \phi )$ of dimension $n \in \N$ for an open subset $U \in M$ is made of:
\begin{itemize}
\item a connected open subset $\tildeU \subset \R^n$,
\item a finite group $G$ acting acting on $\tildeU$,
\item a map $\phi \colon \tildeU \to M$ defined by the composition $\phi = \pi \circ \ccP$ where $\ccP \colon \tildeU \to \tildeU / G$ defines the orbits and $\pi \colon \tildeU / G \to M$.
\end{itemize}
An embedding $\eta \colon \qty( \tildeU_2,\, G_1,\, \phi_1 ) \hookrightarrow \qty( \tildeU_2,\, G_2,\, \phi_2 )$ between two charts is such that $\phi_2 \circ \eta = \phi_1$.
Suppose now $U_i = \phi_i\qty( \tildeU_i )$ for $i = 1,\, 2$ and take $p \in U_1 \cap U_2$.
The charts are \emph{compatible} if there exist an open subset $V$ such that $p \in V \subset U_1 \cap U_2$ and a chart $\qty( \tildeV,\, G,\, \phi )$ admitting two embeddings in the previous charts.
A $n$-dimensional \emph{orbifold atlas} is then a collection $\qty{ \qty( U_i,\, G_i,\, \phi_i ) }_{i \in I}$ of compatible $n$-dimensional orbifold charts covering $M$.
The $n$-dimensional \emph{orbifold} $\ccO$ is finally defined as a paracompact Hausdorff topological space together with a $n$-dimensional orbifold atlas.\footnotemark{}
\footnotetext{%
In this context paracompact refers to a topological space $M$ which admits open covers with a \emph{locally finite} refinement.
In other words let $U = \qty{ U_i }_{i \in I}$ be a cover and $V = \qty{ V_j }_{j \in J}$ be its refinement (i.e.\ $\forall j \in J$, $\exists i \in I \mid V_j \subset U_i$).
Then $U$ is locally finite if $\forall p \in M$ there is a neighbourhood $B( p )$ of $p$ such that $\qty{ i \in I \mid U_i \cap B(p) \neq \emptyset }$ is finite.
}
\subsection{Orbifolds and Strings}
In string theory the notion of orbifold has a more stringent characterisation with respect to pure mathematics.
Differently from the general definition, orbifolds in physics usually appear as a global orbit space $M / G$ where $M$ is a manifold and $G$ the group of its isometries, often leading to the presence of \emph{fixed points} (i.e.\ points in the manifold which are left invariant by the action of $G$) where singularities emerge due to the presence of additional degrees of freedom given by \emph{twisted states} of the string~\cite{Dixon:1985:StringsOrbifolds,Dixon:1986:StringsOrbifoldsII}.
They are commonly introduced as singular limits of \cy manifolds~\cite{Candelas:1985:VacuumConfigurationsSuperstrings}, which in turn can be recovered using algebraic geometry to smoothen the singular points.
However they can also be used to model peculiar time-dependent backgrounds~\cite{Horowitz:1991:SingularStringSolutions,Khoury:2002:BigCrunchBig,Figueroa-OFarrill:2001:GeneralisedSupersymmetricFluxbranes,Cornalba:2002:NewCosmologicalScenario,Cornalba:2004:TimedependentOrbifoldsString,Craps:2006:BigBangModels,Bachas:2002:NullBraneIntersections,Bachas:2003:RelativisticStringPulse}.
They are in fact good toy models to study Big Bang scenarios in string theory and we focus specifically on the study of such cosmological singularity in the framework of string theory.
% vim: ft=tex

406
thesis.bib
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@@ -78,6 +78,23 @@
number = {08}
}
@article{Amati:1987:SuperstringCollisionsPlanckian,
title = {Superstring Collisions at Planckian Energies},
author = {Amati, Daniele and Ciafaloni, Marcello and Veneziano, Gabriele},
date = {1987-10},
journaltitle = {Physics Letters B},
shortjournal = {Physics Letters B},
volume = {197},
pages = {81--88},
issn = {03702693},
doi = {10.1016/0370-2693(87)90346-7},
annotation = {http://web.archive.org/web/20201001155446/https://linkinghub.elsevier.com/retrieve/pii/0370269387903467},
file = {/home/riccardo/.local/share/zotero/files/amati_et_al_1987_superstring_collisions_at_planckian_energies3.pdf},
keywords = {archived},
langid = {english},
number = {1-2}
}
@article{Anastasopoulos:2011:ClosedstringTwistfieldCorrelators,
title = {On Closed-String Twist-Field Correlators and Their Open-String Descendants},
author = {Anastasopoulos, Pascal and Bianchi, Massimo and Richter, Robert},
@@ -167,6 +184,42 @@
file = {/home/riccardo/.local/share/zotero/files/angelantonj_sagnotti_2002_open_strings.pdf}
}
@article{Bachas:2002:NullBraneIntersections,
title = {Null {{Brane Intersections}}},
author = {Bachas, Constantin and Hull, Chris},
date = {2002-12-10},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2002},
pages = {035--035},
issn = {1029-8479},
doi = {10.1088/1126-6708/2002/12/035},
abstract = {We study pairs of planar D-branes intersecting on null hypersurfaces, and other related configurations. These are supersymmetric and have finite energy density. They provide open-string analogues of the parabolic orbifold and null-fluxbrane backgrounds for closed superstrings. We derive the spectrum of open strings, showing in particular that if the D-branes are shifted in a spectator dimension so that they do not intersect, the open strings joining them have no asymptotic states. As a result, a single non-BPS excitation can in this case catalyze a condensation of massless modes, changing significantly the underlying supersymmetric vacuum state. We argue that a similar phenomenon can modify the null cosmological singularity of the time-dependent orbifolds. This is a stringy mechanism, distinct from black-hole formation and other strong gravitational instabilities, and one that should dominate at weak string coupling. A by-product of our analysis is a new understanding of the appearance of 1/4 BPS threshold bound states, at special points in the moduli space of toroidally-compactified type-II string theory.},
archivePrefix = {arXiv},
eprint = {hep-th/0210269},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/bachas_hull_2002_null_brane_intersections.pdf;/home/riccardo/.local/share/zotero/storage/RG2NPWNH/0210269.html},
number = {12}
}
@article{Bachas:2003:RelativisticStringPulse,
title = {Relativistic {{String}} in a {{Pulse}}},
author = {Bachas, Constantin},
date = {2003-06},
journaltitle = {Annals of Physics},
shortjournal = {Annals of Physics},
volume = {305},
pages = {286--309},
issn = {00034916},
doi = {10.1016/S0003-4916(03)00065-4},
abstract = {I study a relativistic open string coupling through its endpoints to a plane wave with arbitrary temporal profile. The string's transverse oscillations respond linearly to the external field. This makes it possible to solve the classical equations, and to calculate the quantum-mechanical S-matrix in closed form. I analyze the dynamics of the string as the characteristic frequency and duration of the pulse are continuously varied. I derive, in particular, the multipole expansion in the adiabatic limit of very long wavelengths, and discuss also more violent phenomena such as shock waves, cusps and null brane intersections. Apart from their relevance to the study of time-dependence in superstring theory, these results could have other applications, such as the teleportation of gravitational wave bursts by cosmic strings.},
archivePrefix = {arXiv},
eprint = {hep-th/0212217},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/bachas_2003_relativistic_string_in_a_pulse.pdf},
number = {2}
}
@article{Berkooz:1996:BranesIntersectingAngles,
title = {Branes {{Intersecting}} at {{Angles}}},
author = {Berkooz, Micha and Douglas, Michael R. and Leigh, Robert G.},
@@ -185,6 +238,41 @@
number = {1-2}
}
@article{Berkooz:2003:CommentsCosmologicalSingularities,
title = {Comments on Cosmological Singularities in String Theory},
author = {Berkooz, Micha and Craps, Ben and Kutasov, David and Rajesh, Govindan},
date = {2003-03},
journaltitle = {Journal of High Energy Physics},
volume = {2003},
pages = {031--031},
issn = {1029-8479},
doi = {10.1088/1126-6708/2003/03/031},
abstract = {We compute string scattering amplitudes in an orbifold of Minkowski space by a boost, and show how certain divergences in the four point function are associated with graviton exchange near the singularity. These divergences reflect large tree-level backreaction of the gravitational field. Near the singularity, all excitations behave like massless fields on a 1+1 dimensional cylinder. For excitations that are chiral near the singularity, we show that divergences are avoided and that the backreaction is milder. We discuss the implications of this for some cosmological spacetimes. Finally, in order to gain some intuition about what happens when backreaction is taken into account, we study an open string rolling tachyon background as a toy model that shares some features with R\^\{1,1\}/Z.},
annotation = {ZSCC: 0000117},
file = {/home/riccardo/.local/share/zotero/files/berkooz_et_al_2003_comments_on_cosmological_singularities_in_string_theory.pdf},
number = {03}
}
@article{Berkooz:2003:StringsElectricField,
ids = {Pioline:2003:StringsElectricField},
title = {Strings in an Electric Field, and the {{Milne Universe}}},
author = {Berkooz, Micha and Pioline, Boris},
date = {2003-11-17},
journaltitle = {Journal of Cosmology and Astroparticle Physics},
shortjournal = {J. Cosmol. Astropart. Phys.},
volume = {2003},
pages = {007--007},
issn = {1475-7516},
doi = {10.1088/1475-7516/2003/11/007},
abstract = {Arguably the simplest model of a cosmological singularity in string theory, the Lorentzian orbifold \$\textbackslash Real\^\{1,1\}/boost\$ is known to lead to severe divergences in scattering amplitudes of untwisted states, indicating a large backreaction toward the singularity. In this work we take a first step in investigating whether condensation of twisted states may remedy this problem and resolve the spacelike singularity. By using the formal analogy with charged open strings in an electric field, we argue that, contrary to earlier claims, twisted sectors do contain physical scattering states, which can be viewed as charged particles in an electric field. Correlated pairs of twisted states will therefore be produced, by the ordinary Schwinger mechanism. For open strings in an electric field, on-shell wave functions for the zero-modes are determined, and shown to analytically continue to non-normalizable modes of the usual Landau harmonic oscillator in Euclidean space. Closed strings scattering states of the Milne orbifold continue to non-normalizable modes in an unusual Euclidean orbifold of \$\textbackslash Real\^2\$ by a rotation by an irrational angle. Irrespective of the formal analogy with the Milne Universe, open strings in a constant electric field, or colliding D-branes, may also serve as a useful laboratory to study time-dependence in string theory.},
archivePrefix = {arXiv},
eprint = {hep-th/0307280},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/berkooz_pioline_2003_strings_in_an_electric_field,_and_the_milne_universe.pdf},
issue = {11},
number = {LPTHE-03-21, WIS-20-03-DPP}
}
@article{Berkooz:2004:ClosedStringsMisner,
title = {Closed {{Strings}} in {{Misner Space}}: {{Stringy Fuzziness}} with a {{Twist}}},
shorttitle = {Closed {{Strings}} in {{Misner Space}}},
@@ -204,6 +292,23 @@
number = {10}
}
@article{Berkooz:2007:ShortReviewTime,
title = {A {{Short Review}} of {{Time Dependent Solutions}} and {{Space}}-like {{Singularities}} in {{String Theory}}},
author = {Berkooz, Micha and Reichmann, Dori},
date = {2007-09},
journaltitle = {Nuclear Physics B - Proceedings Supplements},
shortjournal = {Nuclear Physics B - Proceedings Supplements},
volume = {171},
pages = {69--87},
issn = {09205632},
doi = {10.1016/j.nuclphysbps.2007.06.008},
abstract = {These lecture notes provide a short review of the status of time dependent backgrounds in String theory, and in particular those that contain space-like singularities. Despite considerable efforts, we do not have yet a full and compelling picture of such backgrounds. We review some of the various attempts to understand these singularities via generalizations of the BKL dynamics, using worldsheet methods and using non-perturbative tools such as the AdS/CFT correspondence and M(atrix) theory. These lecture notes are based on talks given at Cargese 06 and the dead-sea conference 06.},
archivePrefix = {arXiv},
eprint = {0705.2146},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/berkooz_reichmann_2007_a_short_review_of_time_dependent_solutions_and_space-like_singularities_in.pdf;/home/riccardo/.local/share/zotero/storage/6E7HYPT8/0705.html}
}
@article{Bertolini:2006:BraneWorldEffective,
title = {Brane World Effective Actions for {{D}}-Branes with Fluxes},
author = {Bertolini, Matteo and Billo, Marco and Lerda, Alberto and Morales, Jose F. and Russo, Rodolfo},
@@ -240,6 +345,22 @@
number = {08}
}
@article{Black:2012:HighEnergyString,
title = {High Energy String–Brane Scattering for Massive States},
author = {Black, William and Monni, Cristina},
date = {2012-06},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {859},
pages = {299--320},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2012.02.009},
annotation = {http://web.archive.org/web/20201001155554/https://linkinghub.elsevier.com/retrieve/pii/S0550321312000958},
keywords = {archived},
langid = {english},
number = {3}
}
@article{Blumenhagen:2007:FourdimensionalStringCompactifications,
title = {Four-Dimensional String Compactifications with {{D}}-Branes, Orientifolds and Fluxes},
author = {Blumenhagen, Ralph and Körs, Boris and Lüst, Dieter and Stieberger, Stephan},
@@ -363,6 +484,19 @@
langid = {english}
}
@article{CaramelloJr:2019:IntroductionOrbifolds,
title = {Introduction to Orbifolds},
author = {Caramello Jr, Francisco C.},
date = {2019-11-17},
abstract = {We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology -- such as the fundamental group, coverings and Euler characteristic -- Differential Topology/Geometry -- including orbibundles, differential forms, integration and (equivariant) De Rham cohomology -- and Riemannian Geometry, surveying generalizations of classical theorems to this setting.},
archivePrefix = {arXiv},
eprint = {1909.08699},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/caramello_jr_2019_introduction_to_orbifolds.pdf;/home/riccardo/.local/share/zotero/storage/N6FSCMLT/1909.html},
keywords = {⛔ No DOI found},
primaryClass = {math}
}
@article{Chamoun:2004:FermionMassesMixing,
title = {Fermion Masses and Mixing in Intersecting Brane Scenarios},
author = {Chamoun, Nidal and Khalil, Shaaban and Lashin, Elsayed},
@@ -429,6 +563,82 @@
file = {/home/riccardo/.local/share/zotero/files/cleaver_2007_in_search_of_the_(minimal_supersymmetric)_standard_model_string.pdf}
}
@article{Cornalba:2002:NewCosmologicalScenario,
title = {A {{New Cosmological Scenario}} in {{String Theory}}},
author = {Cornalba, Lorenzo and Costa, Miguel S.},
date = {2002-09-03},
journaltitle = {Physical Review D},
shortjournal = {Phys. Rev. D},
volume = {66},
pages = {066001},
issn = {0556-2821, 1089-4918},
doi = {10.1103/PhysRevD.66.066001},
abstract = {We consider new cosmological solutions with a collapsing, an intermediate and an expanding phase. The boundary between the expanding (collapsing) phase and the intermediate phase is seen by comoving observers as a cosmological past (future) horizon. The solutions are naturally embedded in string and M-theory. In the particular case of a two-dimensional cosmology, space-time is flat with an identification under boost and translation transformations. We consider the corresponding string theory orbifold and calculate the modular invariant one-loop partition function. In this case there is a strong parallel with the BTZ black hole. The higher dimensional cosmologies have a time-like curvature singularity in the intermediate region. In some cases the string coupling can be made small throughout all of space-time but string corrections become important at the singularity. This happens where string winding modes become light which could resolve the singularity. The new proposed space-time casual structure could have implications for cosmology, independently of string theory.},
archivePrefix = {arXiv},
eprint = {hep-th/0203031},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/cornalba_costa_2002_a_new_cosmological_scenario_in_string_theory2.pdf;/home/riccardo/.local/share/zotero/storage/D7EJLIH7/0203031.html},
number = {6}
}
@article{Cornalba:2004:TimedependentOrbifoldsString,
ids = {Cornalba:2004:TimeDependentOrbifolds},
title = {Time-Dependent Orbifolds and String Cosmology},
author = {Cornalba, Lorenzo and Costa, Miguel S.},
date = {2004-02},
journaltitle = {Fortschritte der Physik},
volume = {52},
pages = {145--199},
issn = {00158208},
doi = {10.1002/prop.200310123},
abstract = {In these lectures, we review the physics of time-dependent orbifolds of string theory, with particular attention to orbifolds of three-dimensional Minkowski space. We discuss the propagation of free particles in the orbifold geometries, together with their interactions. We address the issue of stability of these string vacua and the difficulties in defining a consistent perturbation theory, pointing to possible solutions. In particular, it is shown that resumming part of the perturbative expansion gives finite amplitudes. Finally we discuss the duality of some orbifold models with the physics of orientifold planes, and we describe cosmological models based on the dynamics of these orientifolds.},
annotation = {ZSCC: 0000143},
archivePrefix = {arXiv},
eprint = {hep-th/0310099},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/cornalba_costa_2004_time-dependent_orbifolds_and_string_cosmology.pdf},
number = {2-3}
}
@article{Craps:2002:StringPropagationPresence,
ids = {Craps:2002:StringPropagationPresencea},
title = {String {{Propagation}} in the {{Presence}} of {{Cosmological Singularities}}},
author = {Craps, Ben and Kutasov, David and Rajesh, Govindan},
date = {2002-06-26},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2002},
pages = {053--053},
issn = {1029-8479},
doi = {10.1088/1126-6708/2002/06/053},
abstract = {We study string propagation in a spacetime with positive cosmological constant, which includes a circle whose radius approaches a finite value as |t|\textbackslash to\textbackslash infty, and goes to zero at t=0. Near this cosmological singularity, the spacetime looks like R\^\{1,1\}/Z. In string theory, this spacetime must be extended by including four additional regions, two of which are compact. The other two introduce new asymptotic regions, corresponding to early and late times, respectively. States of quantum fields in this spacetime are defined in the tensor product of the two Hilbert spaces corresponding to the early time asymptotic regions, and the S-matrix describes the evolution of such states to states in the tensor product of the two late time asymptotic regions. We show that string theory provides a unique continuation of wavefunctions past the cosmological singularities, and allows one to compute the S-matrix. The incoming vacuum evolves into an outgoing state with particles. We also discuss instabilities of asymptotically timelike linear dilaton spacetimes, and the question of holography in such spaces. Finally, we briefly comment on the relation of our results to recent discussions of de Sitter space.},
archivePrefix = {arXiv},
eprint = {hep-th/0205101},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/craps_et_al_2002_string_propagation_in_the_presence_of_cosmological_singularities.pdf},
issue = {06},
number = {EFI-02-77}
}
@article{Craps:2006:BigBangModels,
ids = {Craps:2006:BigBangModelsa},
title = {Big {{Bang Models}} in {{String Theory}}},
author = {Craps, Ben},
date = {2006-11-07},
journaltitle = {Classical and Quantum Gravity},
shortjournal = {Class. Quantum Grav.},
volume = {23},
pages = {S849-S881},
issn = {0264-9381, 1361-6382},
doi = {10.1088/0264-9381/23/21/S01},
abstract = {These proceedings are based on lectures delivered at the "RTN Winter School on Strings, Supergravity and Gauge Theories", CERN, January 16 - January 20, 2006. The school was mainly aimed at Ph.D. students and young postdocs. The lectures start with a brief introduction to spacetime singularities and the string theory resolution of certain static singularities. Then they discuss attempts to resolve cosmological singularities in string theory, mainly focusing on two specific examples: the Milne orbifold and the matrix big bang.},
archivePrefix = {arXiv},
eprint = {hep-th/0605199},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/craps_2006_big_bang_models_in_string_theory.pdf},
number = {21}
}
@article{Cremades:2003:YukawaCouplingsIntersecting,
title = {Yukawa Couplings in Intersecting {{D}}-Brane Models},
author = {Cremades, Daniel and Ibanez, Luis E. and Marchesano, Fernando},
@@ -710,6 +920,37 @@
number = {24}
}
@article{Dixon:1985:StringsOrbifolds,
title = {Strings on Orbifolds},
author = {Dixon, Lance J. and Harvey, Jeffrey A. and Vafa, Cumrun and Witten, Edward},
date = {1985-01},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {261},
pages = {678--686},
issn = {05503213},
doi = {10.1016/0550-3213(85)90593-0},
annotation = {http://web.archive.org/web/20201001140216/https://linkinghub.elsevier.com/retrieve/pii/0550321385905930},
keywords = {archived},
langid = {english}
}
@article{Dixon:1986:StringsOrbifoldsII,
title = {Strings on Orbifolds ({{II}})},
author = {Dixon, Lance J. and Harvey, Jeffrey A. and Vafa, Cumrun and Witten, Edward},
date = {1986-09},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {274},
pages = {285--314},
issn = {05503213},
doi = {10.1016/0550-3213(86)90287-7},
annotation = {http://web.archive.org/web/20201001140250/https://linkinghub.elsevier.com/retrieve/pii/0550321386902877},
keywords = {archived},
langid = {english},
number = {2}
}
@article{Duo:2007:NewTwistField,
title = {New Twist Field Couplings from the Partition Function for Multiply Wrapped {{D}}-Branes},
author = {Duo, Dario and Russo, Rodolfo and Sciuto, Stefano},
@@ -762,6 +1003,24 @@
number = {1-2}
}
@article{Figueroa-OFarrill:2001:GeneralisedSupersymmetricFluxbranes,
title = {Generalised Supersymmetric Fluxbranes},
author = {Figueroa-O'Farrill, José and Simón, Joan},
date = {2001-12-10},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2001},
pages = {011--011},
issn = {1029-8479},
doi = {10.1088/1126-6708/2001/12/011},
abstract = {We classify generalised supersymmetric fluxbranes in type II string theory obtained as Kaluza-Klein reductions of the Minkowski space vacuum of eleven-dimensional supergravity. We obtain two families of smooth solutions which contains all the known solutions, new solutions called nullbranes, and solutions interpolating between them. We explicitly construct all the solutions and we study the U-duality orbits of some of these backgrounds.},
archivePrefix = {arXiv},
eprint = {hep-th/0110170},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/figueroa-o'farrill_simón_2001_generalised_supersymmetric_fluxbranes.pdf;/home/riccardo/.local/share/zotero/storage/JTHMFQBY/0110170.html},
number = {12}
}
@article{Finotello:2019:ClassicalSolutionBosonic,
ids = {Finotello:2019:ClassicalSolutionBosonica},
title = {The {{Classical Solution}} for the {{Bosonic String}} in the {{Presence}} of {{Three D}}-Branes {{Rotated}} by {{Arbitrary SO}}(4) {{Elements}}},
@@ -1018,6 +1277,39 @@
number = {9-10}
}
@article{Horowitz:1991:SingularStringSolutions,
title = {Singular String Solutions with Nonsingular Initial Data},
author = {Horowitz, Gary T. and Steif, Alan R.},
date = {1991-04},
journaltitle = {Physics Letters B},
shortjournal = {Physics Letters B},
volume = {258},
pages = {91--96},
issn = {03702693},
doi = {10.1016/0370-2693(91)91214-G},
annotation = {http://web.archive.org/web/20201001140650/https://linkinghub.elsevier.com/retrieve/pii/037026939191214G},
keywords = {archived},
langid = {english},
number = {1-2}
}
@article{Horowitz:2002:InstabilitySpacelikeNull,
title = {Instability of Spacelike and Null Orbifold Singularities},
author = {Horowitz, Gary T. and Polchinski, Joseph},
date = {2002-11-25},
journaltitle = {Physical Review D},
shortjournal = {Phys. Rev. D},
volume = {66},
pages = {103512},
issn = {0556-2821, 1089-4918},
doi = {10.1103/PhysRevD.66.103512},
annotation = {http://web.archive.org/web/20201001150603/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.66.103512},
file = {/home/riccardo/.local/share/zotero/files/horowitz_polchinski_2002_instability_of_spacelike_and_null_orbifold_singularities.pdf},
keywords = {archived},
langid = {english},
number = {10}
}
@book{Hubsch:1992:CalabiyauManifoldsBestiary,
title = {Calabi-Yau Manifolds: {{A}} Bestiary for Physicists},
author = {Hubsch, Tristan},
@@ -1078,6 +1370,27 @@
number = {4}
}
@article{Jackiw:1992:ElectromagneticFieldsMassless,
ids = {Jackiw:1992:ElectromagneticFieldsMasslessa},
title = {Electromagnetic Fields of a Massless Particle and the Eikonal},
author = {Jackiw, R. and Kabat, D. and Ortiz, M.},
date = {1992-02-27},
journaltitle = {Physics Letters B},
shortjournal = {Physics Letters B},
volume = {277},
pages = {148--152},
issn = {0370-2693},
doi = {10.1016/0370-2693(92)90971-6},
abstract = {Electromagnetic fields of a massless charged particle are described by a gauge potential that is almost everywhere a pure gauge. Solution of quantum mechanical wave equations in the presence of such fields is therefore immediate and leads to a new derivation of the quantum electrodynamical eikonal approximation. The electromagnetic action in the eikonal limit is localized on a contour in a two-dimensional Minkowski subspace of four-dimensional space-time. The exact S-matrix of this reduced theory reproduces the eikonal approximation. In this way, we apply the recent gravitational consideration of't Hooft as well and Verlinde and Verlinde to electromagnetism.},
annotation = {ZSCC: 0000091},
archivePrefix = {arXiv},
eprint = {hep-th/9112020},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/jackiw_et_al_1992_electromagnetic_fields_of_a_massless_particle_and_the_eikonal.pdf},
issue = {1},
number = {MIT-CTP-2033}
}
@article{Johnson:2000:DBranePrimer,
title = {D-{{Brane Primer}}},
author = {Johnson, Clifford V.},
@@ -1131,6 +1444,24 @@
number = {4}
}
@article{Khoury:2002:BigCrunchBig,
title = {From {{Big Crunch}} to {{Big Bang}}},
author = {Khoury, Justin and Ovrut, Burt A. and Seiberg, Nathan and Steinhardt, Paul J. and Turok, Neil},
date = {2002-04-09},
journaltitle = {Physical Review D},
shortjournal = {Phys. Rev. D},
volume = {65},
pages = {086007},
issn = {0556-2821, 1089-4918},
doi = {10.1103/PhysRevD.65.086007},
abstract = {We consider conditions under which a universe contracting towards a big crunch can make a transition to an expanding big bang universe. A promising example is 11-dimensional M-theory in which the eleventh dimension collapses, bounces, and re-expands. At the bounce, the model can reduce to a weakly coupled heterotic string theory and, we conjecture, it may be possible to follow the transition from contraction to expansion. The possibility opens the door to new classes of cosmological models. For example, we discuss how it suggests a major simplification and modification of the recently proposed ekpyrotic scenario.},
archivePrefix = {arXiv},
eprint = {hep-th/0108187},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/khoury_et_al_2002_from_big_crunch_to_big_bang2.pdf;/home/riccardo/.local/share/zotero/storage/ZR347STA/0108187.html},
number = {8}
}
@article{Kiritsis:1994:StringPropagationGravitational,
title = {String {{Propagation}} in {{Gravitational Wave Backgrounds}}},
author = {Kiritsis, Elias and Kounnas, Costas},
@@ -1161,6 +1492,50 @@
langid = {english}
}
@article{Liu:2002:StringsTimeDependentOrbifold,
ids = {Liu:2002:StringsTimeDependent},
title = {Strings in a {{Time}}-{{Dependent Orbifold}}},
author = {Liu, Hong and Moore, Gregory and Seiberg, Nathan},
date = {2002-06},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2002},
pages = {045--045},
issn = {1126-6708},
doi = {10.1088/1126-6708/2002/06/045},
abstract = {We consider string theory in a time dependent orbifold with a null singularity. The singularity separates a contracting universe from an expanding universe, thus constituting a big crunch followed by a big bang. We quantize the theory both in light-cone gauge and covariantly. We also compute some tree and one loop amplitudes which exhibit interesting behavior near the singularity. Our results are compatible with the possibility that strings can pass through the singularity from the contracting to the expanding universe, but they also indicate the need for further study of certain divergent scattering amplitudes.},
annotation = {ZSCC: 0000251},
archivePrefix = {arXiv},
eprint = {hep-th/0204168},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/liu_et_al_2002_strings_in_a_time-dependent_orbifold.pdf},
issue = {06},
langid = {english},
number = {RUNHETC-2002-11}
}
@article{Liu:2002:StringsTimeDependentOrbifolds,
ids = {Liu:2002:StringsTimeDependenta},
title = {Strings in {{Time}}-{{Dependent Orbifolds}}},
author = {Liu, Hong and Moore, Gregory and Seiberg, Nathan},
date = {2002-10},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2002},
pages = {031--031},
issn = {1126-6708},
doi = {10.1088/1126-6708/2002/10/031},
abstract = {We continue and extend our earlier investigation “Strings in a Time-Dependent Orbifold” (hep-th/0204168). We formulate conditions for an orbifold to be amenable to perturbative string analysis and classify the low dimensional orbifolds satisfying these conditions. We analyze the tree and torus amplitudes of some of these orbifolds. The tree amplitudes exhibit a new kind of infrared divergences which are a result of some ultraviolet effects. These UV enhanced IR divergences can be interpreted as due to back reaction of the geometry. We argue that for this reason the three dimensional parabolic orbifold is not amenable to perturbation theory. Similarly, the smooth four dimensional null-brane tensored with sufficiently few noncompact dimensions also appears problematic. However, when the number of noncompact dimensions is sufficiently large perturbation theory in these time dependent backgrounds seems consistent.},
annotation = {ZSCC: 0000208},
archivePrefix = {arXiv},
eprint = {hep-th/0206182},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/liu_et_al_2002_strings_in_time_dependent_orbifolds.pdf;/home/riccardo/.local/share/zotero/files/liu_et_al_2002_strings_in_time-dependent_orbifolds.pdf},
issue = {10},
langid = {english},
number = {RUNHETC-2002-19, NI-02014-MTH}
}
@article{Lust:2009:LHCStringHunter,
title = {The {{LHC String Hunter}}'s {{Companion}}},
author = {Lust, Dieter and Stieberger, Stephan and Taylor, Tomasz R.},
@@ -1501,6 +1876,22 @@
number = {3-4}
}
@article{Soldate:1987:PartialwaveUnitarityClosedstring,
title = {Partial-Wave Unitarity and Closed-String Amplitudes},
author = {Soldate, Mark},
date = {1987-03},
journaltitle = {Physics Letters B},
shortjournal = {Physics Letters B},
volume = {186},
pages = {321--327},
issn = {03702693},
doi = {10.1016/0370-2693(87)90302-9},
annotation = {http://web.archive.org/web/20201001155315/https://linkinghub.elsevier.com/retrieve/pii/0370269387903029},
keywords = {archived},
langid = {english},
number = {3-4}
}
@article{Stieberger:1992:YukawaCouplingsBosonic,
title = {Yukawa {{Couplings}} for {{Bosonic ZN Orbifolds}}: {{Their Moduli}} and {{Twisted Sector Dependence}}},
shorttitle = {Yukawa {{Couplings}} for {{Bosonic}} \${{Z}}\_{{N}}\$ {{Orbifolds}}},
@@ -1556,6 +1947,21 @@
file = {/home/riccardo/.local/share/zotero/files/taylor_zwiebach_2004_d-branes,_tachyons,_and_string_field_theory.pdf}
}
@article{tHooft:1987:GravitonDominanceUltrahighenergy,
title = {Graviton Dominance in Ultra-High-Energy Scattering},
author = {'t Hooft, Gerardus},
date = {1987-11},
journaltitle = {Physics Letters B},
volume = {198},
pages = {61--63},
doi = {10.1016/0370-2693(87)90159-6},
abstract = {The scattering process of two pointlike particles at CM energies in the order of Planck units or beyond, is very well calculable using known laws of physics, because graviton exchange dominates over all other interaction processes. At energies much higher than the Planck mass black hole production sets in, accompanied by coherent emission of real gravitons.},
annotation = {ZSCC: 0000000[s0]},
file = {/home/riccardo/.local/share/zotero/files/'t_hooft_1987_graviton_dominance_in_ultra-high-energy_scattering.pdf},
number = {1},
options = {useprefix=true}
}
@article{tHooft:2009:DimensionalReductionQuantum,
title = {Dimensional {{Reduction}} in {{Quantum Gravity}}},
author = {'t Hooft, Gerard},

11
thesis.tex
View File

@@ -23,6 +23,11 @@
pdfauthor={Riccardo Finotello}
}
%---- abbreviations
\newcommand{\bo}{\textsc{bo}\xspace}
\newcommand{\nbo}{\textsc{nbo}\xspace}
\newcommand{\gnbo}{\textsc{gnbo}\xspace}
%---- functions
\newcommand{\hyp}[4]{\ensuremath{\mathrm{F}\left( #1,\, #2;\, #3;\, #4 \right)}}
\newcommand{\poch}[2]{\ensuremath{\left( #1 \right)_{#2}}}
@@ -96,7 +101,7 @@
\input{sec/outline.tex}
%---- PARTICLE PHYSICS
\thesispart{Conformal Symmetry and Geometry of the Wordlsheet}
\thesispart{Conformal Symmetry and Geometry of the Worldsheet}
\section{Introduction}
\input{sec/part1/introduction.tex}
\section{D-branes Intersecting at Angles}
@@ -105,9 +110,11 @@
\input{sec/part1/fermions.tex}
%---- COSMOLOGY
\thesispart{Cosmology and Time Dependent Divergences}
\thesispart{Cosmological Backgrounds and Divergences}
\section{Introduction}
\input{sec/part2/introduction.tex}
\section{Time Dependent Orbifolds}
\input{sec/part2/divergences.tex}
%---- DEEP LEARNING
\thesispart{Deep Learning the Geometry of String Theory}