From ddce7cdffcdcd2c5c4c8cfe8bbc58b5dd969d188 Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Wed, 21 Oct 2020 11:17:08 +0200 Subject: [PATCH] Few corrections to spelling in the introductions Signed-off-by: Riccardo Finotello --- sciencestuff.sty | 1 + sec/abstract.tex | 8 ++--- sec/outline.tex | 21 +++++++----- sec/part1/introduction.tex | 70 ++++++++++++++++++-------------------- sec/part2/conclusion.tex | 2 +- sec/part2/divergences.tex | 3 +- sec/part2/introduction.tex | 4 +-- thesis.bib | 17 --------- 8 files changed, 54 insertions(+), 72 deletions(-) diff --git a/sciencestuff.sty b/sciencestuff.sty index b5538d7..71919b0 100644 --- a/sciencestuff.sty +++ b/sciencestuff.sty @@ -1405,6 +1405,7 @@ \providecommand{\OO}[1]{\ensuremath{\mathrm{O}(#1)}\xspace} \providecommand{\SO}[1]{\ensuremath{\mathrm{SO}(#1)}\xspace} +\providecommand{\ISO}[1]{\ensuremath{\mathrm{ISO}(#1)}\xspace} \providecommand{\U}[1]{\ensuremath{\mathrm{U}(#1)}\xspace} \providecommand{\SU}[1]{\ensuremath{\mathrm{SU}(#1)}\xspace} \providecommand{\SL}[2]{\ensuremath{\mathrm{SL}_{#1}(#2)}\xspace} diff --git a/sec/abstract.tex b/sec/abstract.tex index aa9e681..3ba230f 100644 --- a/sec/abstract.tex +++ b/sec/abstract.tex @@ -1,18 +1,18 @@ -We present topics in phenomenology of string theory ranging from particle physics amplitudes and Big Bang-like singularities to the study of state-of-the-art deep learning techniques for string compactifications based on recent advancements in artificial intelligence. +We present topics of phenomenological relevance in string theory ranging from particle physics amplitudes and Big Bang-like singularities to the study of state-of-the-art deep learning techniques for string compactifications based on recent advancements in artificial intelligence. We show the computation of the leading contribution to amplitudes in the presence of non Abelian twist fields in intersecting D-branes scenarios in non factorised tori. This is a generalisation to the current literature which mainly covers factorised internal spaces. We also study a new method to compute amplitudes in the presence of an arbitrary number of spin fields introducing point-like defects on the string worldsheet. -This method can then be treated as an alternative computation with respect to bosonization and approaches based on the Reggeon vertex. +The procedure can then be treated as an alternative computation with respect to bosonization and approaches based on the Reggeon vertex. We then present an analysis of Big Bang-like cosmological divergences in string theory on time-dependent orbifolds. -We show that the nature of the divergences are not due to gravitational feedback but to the lack of an underlying effective field theory. +We show that divergences are not due to gravitational feedback but to the lack of an underlying effective field theory. We also introduce a new orbifold structure capable of fixing the issue and reinstate a distributional interpretation to field theory amplitudes. We finally present a new artificial intelligence approach to algebraic geometry and string compactifications. We compute the Hodge numbers of Complete Intersection Calabi--Yau $3$-folds using deep learning techniques based on computer vision and object recognition techniques. We also include a methodological study of machine learning applied to data in string theory: as in most applications machine learning almost never relies on the blind application of algorithms to the data but it requires a careful exploratory analysis and feature engineering. We thus show how such an approach can help in improving results by processing the data before using it. -We then show that the deep learning approach can reach the highest accuracy in the task with smaller networks and less parameters. +We then show that the deep learning approach can reach the highest accuracy in the task with smaller networks, less parameters and less data. This is a novel approach to the task: differently from previous attempts we focus on using convolutional neural networks capable of reaching higher accuracy on the predictions and ensuring phenomenological relevance to results. In fact parameter sharing and concurrent scans of the configuration matrix retain better generalisation properties and adapt better to the task than fully connected networks. diff --git a/sec/outline.tex b/sec/outline.tex index b7a6bea..9b03ffa 100644 --- a/sec/outline.tex +++ b/sec/outline.tex @@ -1,4 +1,4 @@ -This thesis follows my research work as a Ph.D.\ student and candidate at the \emph{Università degli Studi di Torino}, Italy. +This thesis follows my research work as a Ph.D.\ student and candidate at the \emph{Università degli Studi di Torino} in Italy. During my programme I mainly dealt with the topic of string theory and its relation with a viable formulation of phenomenology in this framework. I tried to cover mathematical aspects related to amplitudes in intersecting D-branes scenarios and in the presence of defects on the worldsheet, but I also worked on computational issues such as the application of recent deep learning and machine learning techniques to the compactification of the extra-dimensions of superstrings. In this manuscript I present the original results obtained over the course of my Ph.D.\ programme. @@ -12,26 +12,29 @@ Then the analysis of a specific setup involving angled D6-branes intersecting in \footnotetext{% For instance this is a generalisation of the typical setup involving D6-branes filling entirely the $4$-dimensional spacetime and embedded as lines in $T^2 \times T^2 \times T^2$, where the possible rotations performed by the D-branes are parametrised by Abelian $\SO{2} \simeq \U{1}$ rotations. } -Here a general framework to deal with \SO{4} rotated D-branes is presented alongside the computation of the leading term of amplitudes involving an arbitrary number of non Abelian twist fields located at their intersection, that is the exponential contribution of the classical bosonic string in this geometry. +Here a general framework to deal with \SO{4} rotated D-branes is introduced alongside the computation of the leading term of amplitudes involving an arbitrary number of non Abelian twist fields located at their intersection, that is the exponential contribution of the classical bosonic string in this geometry. Finally point-like defects along the time direction of the (super)string worldsheet are introduced and the propagation of fermions on such surface studied in detail. -In this setup the stress-energy tensor presents a time dependence but it still respects the usual operator product expansion. +In this setup the stress-energy tensor has a time dependence but it still respects the usual operator product expansion. Thus the theory is still conformal though time dependence is due to the defects where spin fields are located. -Through the study of the operator algebra the computation of amplitudes in the presence of spin fields and matter fields are computed with a method alternative to the usual bosonization, but which may be expanded also to twist fields and to more general configurations. +Through the study of the operator algebra the computation of amplitudes in the presence of spin fields and matter fields is eventually displayed by means of a method alternative to the usual bosonization, but which might be expanded also to twist fields and to more general configurations (e.g.\ non Abelian spin fields). \Cref{part:cosmo} deals with cosmological singularities in string and field theory. -The main focus is on time-dependent orbifolds as simple models of Big Bang-like singularities in string theory: after a brief introduction on the concept of orbifold from the mathematical and the physical point of views, the Null Boost Orbifold is introduced as first example. -Differently from what usually referred, the divergences appearing in the amplitudes are not a consequence of gravitational feedback, but they appear also at the tree level of open string amplitudes. +The main focus is on time-dependent orbifolds as simple models of Big Bang-like singularities in string theory: after a brief introduction on the concept of orbifold from the mathematical and the physical point of views, the Null Boost Orbifold is introduced as a first example. +Differently from what usually referred in the literature, the divergences appearing in the amplitudes are not a consequence of gravitational feedback, but they appear also at the tree level of open string amplitudes. The source of the divergences are shown in string and field theory amplitudes due to the presence of the compact dimension and its conjugated momentum which prevents the interpretation of the amplitudes even as a distribution. -In fact the introduction of a non compact direction of motion on the orbifold restores the physical interpretation of the amplitude, hence the origin of the divergences comes from geometrical aspects of the orbifold models. +In fact the introduction of a non compact direction of motion on the orbifold restores the physical significance of the amplitude, hence the origin of the divergences comes from geometrical aspects of the orbifold models. Namely it is hidden in contact terms and interaction with massive string states which are no longer spectators, thus invalidating the usual approach with the inheritance principle. \Cref{part:deeplearning} is dedicated to state-of-the-art application of deep learning techniques to the field of string theory compactifications. The Hodge numbers of Complete Intersection Calabi--Yau $3$-folds are computed through a rigorous data science and machine learning analysis. -In fact the blind application of neural networks to the configuration matrix of the manifolds can be improved by exploratory data analysis and feature engineering, from which to infer behaviour and relations of topological quantities invisibly hidden in the configuration matrix. +In fact the blind application of neural networks to the configuration matrix of the manifolds can be improved by exploratory data analysis and feature engineering, from which it is possible to infer behaviour and relations of topological quantities invisibly hidden in the raw data.\footnotemark{} +\footnotetext{% + Through accurate data analysis it is possible to reach the same performance of neural networks using simpler algorithms (in some cases even trivial linear models can reach the same accuracy in the predictions). +} Deep learning techniques are then applied to the configuration matrix of the manifolds to obtain the Hodge numbers as a regression task.\footnotemark{} \footnotetext{% Many previous approaches have proposed classification tasks to get the best performance out of machine learning models. - This however implies specific knowledge of the definition interval of the Hodge numbers and does not generalise well to unknown examples of Complete Intersection Calabi--Yau manifolds. + This however implies specific knowledge of the definition interval of the Hodge numbers and does not generalise well to unknown samples. } A new neural network architecture based on recent computer vision advancements in the field of computer science is eventually introduced: it utilises parallel convolutional kernels to extract the Hodge numbers from the configuration matrix and it reaches near perfect accuracy on the prediction of \hodge{1}{1}. Such model also leads to good preliminary results for \hodge{2}{1} which has been mostly ignored by previous attempts. diff --git a/sec/part1/introduction.tex b/sec/part1/introduction.tex index 9aa5b2e..5ef5765 100644 --- a/sec/part1/introduction.tex +++ b/sec/part1/introduction.tex @@ -12,7 +12,7 @@ in order to reproduce known results. For instance a good string theory could provide a unified framework by predicting the existence of a larger gauge group containing the \sm as a subset. In what follows we deal with the definition of mathematical tools to compute amplitudes to be used in phenomenological calculations related to the study of particles. -In this introduction we present instruments and frameworks used throughout the manuscript as many other aspects are strongly connected and their definitions are interdependent. +In this introduction we present instruments and frameworks used throughout the manuscript as many aspects are strongly connected and their definitions are interdependent. In particular we recall some results on the symmetries of string theory and how to recover a realistic description of $4$-dimensional physics. @@ -58,7 +58,7 @@ The \eom for the string $X^{\mu}\qty(\tau, \sigma)$ is therefore: \qquad \alpha,\, \beta = 0, 1. \end{equation} -In this formulation $\gamma_{\alpha\beta}$ are the components of the worldsheet metric with Lorentzian signature $\qty(-,\, +)$. +In this formulation $\gamma_{\alpha\beta}$ are the components of the worldsheet metric. As there are no derivatives of $\gamma_{\alpha\beta}$, the \eom of the metric is a constraint ensuring the equivalence of Polyakov's and Nambu and Goto's formulations. In fact \begin{equation} @@ -89,7 +89,7 @@ implies = S_{NG}[X], \end{equation} -where $S_{NG}[X]$ is the Nambu--Goto action of the classical string, $\dotX = \ipd{\tau} X$ and $\pX = \ipd{\sigma} X$. +where $S_{NG}[X]$ is the Nambu--Goto action of the classical string, and $\dotX = \ipd{\tau} X$ and $\pX = \ipd{\sigma} X$. The symmetries of $S_P\qty[\gamma,\, X]$ are keys to the success of the string theory framework~\cite{Polchinski:1998:StringTheoryIntroduction}. Specifically~\eqref{eq:conf:polyakov} displays symmetries under: @@ -155,8 +155,8 @@ In fact the classical constraint on the tensor is simply \qty( \ipd{\alpha} X \cdot \ipd{\beta} X - - \frac{1}{2} \eta_{\alpha\beta}\, - \eta^{\lambda\rho}\, + \frac{1}{2} \gamma_{\alpha\beta}\, + \gamma^{\lambda\rho}\, \ipd{\lambda} X \cdot \ipd{\rho} X ) = @@ -285,7 +285,7 @@ The transformation on a field $\phi_{\omega, \bomega}$ of weight $(\omega, \bome \cint{\barw} \ddbz \bepsilon(\barz)\, \rR\qty( \overline{\cT}(\barz)\, \phi_{\omega, \bomega}( w, \barw ) ), \end{split} \end{equation} -where in the last passage we used the fact that the difference of ordered integrals becomes the contour integral of the radially ordered product computed as a infinitesimally small anti-clockwise loop around $w$. +where in the last passage we used the fact that the difference of ordered integrals becomes the contour integral of the radially ordered product computed as a infinitesimally small anti-clockwise loop around $w$ and $\barw$. Equating the result with the expected variation \begin{equation} \begin{split} @@ -324,7 +324,7 @@ we find the short distance singularities of the components of the stress-energy \end{equation} where we drop the radial ordering symbol $\rR$ for simplicity. Since the contour $\ccC_{w}$ is infinitely small around $w$, the conformal properties of $\phi_{\omega, \bomega}( w, \barw )$ are entirely defined by these relations. -In fact $\phi_{\omega, \bomega}( w, \barw )$ is a \emph{primary field} if its short distance behaviour with the stress-energy tensor is as such. +In fact $\phi_{\omega, \bomega}( w, \barw )$ is a \emph{primary field} if its short distance behaviour with the stress-energy tensor is as in~\eqref{eq:conf:primary}. This is an example of an \emph{operator product expansion} (\ope) \begin{equation} \phi^{(i)}_{\omega_i, \bomega_i}( z, \barz )\, \phi^{(j)}_{\omega_j, \bomega_j}( w, \barw ) @@ -336,9 +336,9 @@ This is an example of an \emph{operator product expansion} (\ope) \phi^{(k)}_{\omega_k, \bomega_k}( w, \barw ) \label{eq:conf:ope} \end{equation} -which is an asymptotic expansion containing the full information on the singularities.\footnotemark{} +which is an asymptotic expansion containing full information on the singularities.\footnotemark{} \footnotetext{% - The expression \eqref{eq:conf:ope} is valid assuming the normalisation of the 2-points function + Expression~\eqref{eq:conf:ope} is valid assuming the normalisation of the 2-points function \begin{equation*} \left\langle \phi^{(i)}_{\omega_i, \bomega_i}( z, \barz )\, \phi^{(j)}_{\omega_j, \bomega_j}( w, \barw ) \right\rangle = @@ -400,7 +400,7 @@ This ultimately leads to the quantum algebra \label{eq:conf:virasoro} \end{equation} known as Virasoro algebra, unique central extension of the classical de Witt algebra, with central charge $c$. -Operators $L_n$ and $\barL_n$ are called Virasoro operators.\footnotemark{} +Coefficients $L_n$ and $\barL_n$ are called Virasoro operators.\footnotemark{} \footnotetext{% Notice that the subset of Virasoro operators $\qty{ L_{-1},\, L_0,\, L_1 }$ forms a closed sub-algebra generating the group $\SL{2}{\R}$. } @@ -499,14 +499,14 @@ and the components of the stress-energy tensor~\eqref{eq:conf:stringT} are \label{eq:conf:bosonicstringT} \end{equation} Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \barz )\, X^{\nu}( w, \barw ) \right\rangle = - \frac{1}{2} \eta^{\mu\nu} \ln\left| z - w \right|$ and the Wick theorem, we can prove that $c = D$ in~\eqref{eq:conf:TTexpansion}, where $D$ is the dimensions of spacetime (or equivalently the number of scalar fields $X^{\mu}$ in the action). -It can be shown that in order to cancel the central charge in bosonic string theory we need to introduce a pair of conformal ghosts $b(z)$ and $c(z)$ with conformal weights $(2, 0)$ and $(-1, 0)$ respectively, together with their anti-holomorphic counterparts $\barb(z)$ and $\barc(z)$. +It can be shown that in order to cancel the central charge in bosonic string theory we need to introduce a pair of conformal ghosts $b(z)$ and $c(z)$ with conformal weights $(2, 0)$ and $(-1, 0)$ respectively, together with their anti-holomorphic counterparts $\barb(z)$ and $\barc(z)$, as a first order Lagrangian theory. The non vanishing components of their stress-energy tensor can be computed as:\footnotemark{} \footnotetext{% In general a system of ghosts $b( z )$ and $c( z )$ with weight $(\lambda,\, 0)$ and $(1 - \lambda,\, 0)$ can be introduced as a standalone \cft with action~\cite{Friedan:1986:ConformalInvarianceSupersymmetry} \begin{equation*} S = \frac{1}{2 \pi} \iint \dd{z} \dd{\barz}\, b( z )\, \ipd{\barz} c( z ). \end{equation*} - The equations of motion are $\ipd{\barz} c( z ) = \ipd{\barz} b( z ) = 0$. + The \eom are $\ipd{\barz} c( z ) = \ipd{\barz} b( z ) = 0$. The \ope is \begin{equation*} b( z )\, c( z ) = \frac{\varepsilon}{z - w} + \order{1}, @@ -592,7 +592,7 @@ $\cT_{\text{full}}$ and $\overline{\cT}_{\text{full}}$ are then primary fields w \subsection{Superstrings} As bosonic string theory deals with commuting fields $X^{\mu}$, it is impossible to build spacetime fermions and consequently a consistent phenomenology. -It is in fact necessary to introduce worldsheet fermions (i.e.\ anti-commuting variables on the string worldsheet) as an extension to the bosonic coordinates. +It is in fact necessary to introduce \emph{worldsheet fermions} (i.e.\ anti-commuting variables on the string worldsheet) as an extension to the bosonic coordinates. We schematically and briefly recall some results due to the extension of bosonic string theory to the superstring as they will be used in what follows and mainly descend from the previous discussion. The superstring action is built as an addition to the bosonic equivalent~\eqref{eq:conf:polyakov}. @@ -651,7 +651,7 @@ The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmet - \bepsilon( \barz )\, \ipd{\barz} \barX^{\mu}( \barz ) \end{split} \end{equation} -generated by the currents $J( z ) = \epsilon( z )\, \cT_F( z )$ and $\barJ( \barz ) = \bepsilon( \barz )\, \overline{\cT}_F( \barz )$, where $\epsilon( z )$ and $\bepsilon( \barz ) = \qty( \epsilon( z ) )^*$ are anti-commuting fermions and +generated by $J( z ) = \epsilon( z )\, \cT_F( z )$ and $\barJ( \barz ) = \bepsilon( \barz )\, \overline{\cT}_F( \barz )$, where $\epsilon( z )$ and $\bepsilon( \barz ) = \qty( \epsilon( z ) )^*$ are anti-commuting fermions and \begin{equation} \begin{split} \cT_F( z ) @@ -722,7 +722,7 @@ In what follows we thus consider the superstring formulation in $D = 10$ dimensi It is however clear that low energy phenomena need to be explained by a $4$-dimensional theory in order to be comparable with other theoretical frameworks and experimental evidence. In this section we briefly review for completeness the necessary tools to be able to reproduce consistent models capable of describing particle physics and beyond. These results represent the background knowledge necessary to better understand more complicated scenarios involving strings. -As we will never deal directly with $4$-dimensional physics this is not a complete review and we refer to \cite{Anderson:2018:TASILecturesGeometric,Blumenhagen:2007:FourdimensionalStringCompactifications,Grana:2006:FluxCompactificationsString,Grana:2017:StringTheoryCompactifications,Uranga:2005:TASILecturesString} for more in-depth explanations. +As we will never deal directly with $4$-dimensional physics this is not a complete review and we refer to \cite{Anderson:2018:TASILecturesGeometric,Grana:2006:FluxCompactificationsString,Grana:2017:StringTheoryCompactifications,Uranga:2005:TASILecturesString} for more in-depth explanations. In general we consider Minkowski space in $10$ dimensions $\ccM^{1,9}$. To recover $4$-dimensional spacetime we let it be defined as a product @@ -747,22 +747,18 @@ More on this topic is also presented in~\Cref{part:deeplearning} of this thesis In general an \emph{almost complex structure} $J$ is a tensor such that $\tensor{J}{^a_b}\, \tensor{J}{^b_c} = - \tensor{\delta}{^a_c}$. For any vector field $v_p \in \rT_p M$ defined in $p \in M$ we then define $(J v)^a = \tensor{J}{^a_b} v^b$, thus the tangent space $\rT_p M$ has the structure of a \emph{complex vector space}. -The tensor $J$ is called \emph{complex structure} if there exist a tensor $N$ such that +The tensor $J$ is called \emph{complex structure} if \begin{equation} - \tensor{N}{^a_{bc}}\, v_p^b\, w_p^c - = - \qty( - \liebraket{v_p}{w_p} - + - J - \qty( \liebraket{J\, v_p}{w_p} + \liebraket{v_p}{J\, w_p} ) - - - \liebraket{J\, v_p}{J\, w_p} - )^a + \liebraket{v_p}{w_p} + + + J + \qty( \liebraket{J\, v_p}{w_p} + \liebraket{v_p}{J\, w_p} ) + - + \liebraket{J\, v_p}{J\, w_p} = 0 \end{equation} -for any $v_p,\, w_p \in \rT_p M$, where $\liebraket{\cdot}{\cdot}\colon\, \rT_p M \times \rT_p M \to \rT_p M$ is the Lie braket of vector fields. +for any $v_p,\, w_p \in \rT_p M$, where $\liebraket{\cdot}{\cdot}\colon\, \rT_p M \times \rT_p M \to \rT_p M$ is the usual Lie braket of vector fields. A manifold $M$ is a \emph{complex} manifold if it is possible to define a complex structure $J$ on it.\footnotemark{} \footnotetext{% Notice that a smooth function $f\colon\, M \to \C$ whose pushforward of $v_p \in \rT_p M$ is $f_{*\, p}\colon\, \rT_p M \to \rT_{f(p)} \C \simeq \C$ is called holomorphic if $(J\, f_{*\, p}( v_p ))^a = i ( f_{*\, p}( v_p ) )^a$ as such expression encodes the Cauchy-Riemann equations. @@ -788,7 +784,7 @@ A manifold $M$ is a \emph{complex} manifold if it is possible to define a comple \quad \Rightarrow \quad - f\qty( z, \barz ) = f( z ). + f\qty( z, \barz ) \equiv f( z ). \end{equation*} } @@ -1207,7 +1203,7 @@ Reproducing the \sm or beyond \sm spectra are however strong constraints on the Many of the physical requests usually involve the introduction of D-branes and the study of open strings in order to be able to define chiral fermions and realist gauge groups. As seen in the previous section, D-branes introduce preferred directions of motion by restricting the hypersurface on which the open string endpoints live. -Specifically a Dp-brane breaks the original \SO{1,\, D-1} symmetry to $\SO{1,\, p} \otimes \SO{D - 1 - p}$.\footnotemark{} +Specifically a Dp-brane breaks the original \ISO{1,\, D-1} symmetry to $\ISO{1,\, p} \otimes \SO{D - 1 - p}$.\footnotemark{} \footnotetext{% Notice that usually $D = 10$ in the superstring formulation ($D = 26$ for purely bosonic strings), but we keep a generic indication of the spacetime dimensions when possible. } @@ -1242,7 +1238,7 @@ we find that at the massless level we have a single \U{1} gauge field in the rep \qquad \alpha_{-1}^i \regvacuum. \end{equation} -The introduction of a Dp-brane however breaks the Lorentz invariance down to $\SO{1,\, p} \otimes \SO{D - 1 - p}$. +The introduction of a Dp-brane however breaks the Lorentz invariance. Thus the gauge field in the original theory is split into \begin{equation} \begin{split} @@ -1293,7 +1289,7 @@ Matrices $\tensor{\lambda}{^a_{ij}}$ thus form a basis for expanding wave functi \end{equation} In general Chan-Paton factors label the D-brane on which the endpoint of the string lives as in the left of~\Cref{fig:dbranes:chanpaton}. Notice that strings stretching across different D-branes present an additional term in the mass shell condition proportional to the distance between the hypersurfaces: fields built using strings with Chan-Paton factors $\tensor{\lambda}{^a_i_j}$ for which $i \neq j$ will therefore be massive. -However when $N$ D-branes coincide in space and form a stack their mass vanishes again: it then possible to organise the $N^2$ resulting massless fields in a representation of the gauge group \U{N}, thus promoting the symmetry $\bigotimes\limits_{a = 1}^N \rU_a( 1 )$ of $N$ separate D-branes to a larger gauge group. +However when $N$ D-branes coincide in space and form a stack their mass vanishes again: it is then possible to organise the $N^2$ resulting massless fields in a representation of the gauge group \U{N}, thus promoting the symmetry $\bigotimes\limits_{a = 1}^N \rU_a( 1 )$ of $N$ separate D-branes to a larger gauge group. It is also possible to show that in the field theory limit the resulting gauge theory is a Super Yang-Mills gauge theory. Eventually the massless spectrum of $N$ coincident Dp-branes is formed by \U{N} gauge bosons in the adjoint representation, $N^2 \times (D - 1 - p)$ scalars and $N^2$ sets of $(p+1)$-dimensional fermions~\cite{Uranga:2005:TASILecturesString}. @@ -1304,7 +1300,7 @@ These are the basic building blocks for a consistent string phenomenology involv Being able to describe gauge bosons and fermions is not enough. Physics as we test it in experiments poses stringent constraints on what kind of string models we can use. -For instance there is no way to describe chirality by simply using parallel D-branes and strings stretching among them, while requiring the existence of fermions transforming in different representations of the gauge group is necessary to reproduce \sm results~\cite{Aldazabal:2000:DBranesSingularitiesBottomUp, Ibanez:2012:StringTheoryParticle}. +For instance there is no way to describe chirality by simply using parallel D-branes and strings stretching among them, while requiring the existence of fermions transforming in different representations of the gauge group is necessary to reproduce \sm results~\cite{Ibanez:2012:StringTheoryParticle,Aldazabal:2000:DBranesSingularitiesBottomUp}. For instance, in the low energy limit it is possible to build a gauge theory of the strong force using a stack of $3$ coincident D-branes and an electroweak sector using $2$ D-branes. These stacks would separately lead to a $\U{3} \times \U{2}$ gauge theory. @@ -1322,8 +1318,8 @@ We therefore need to introduce more D-branes to account for all the possible com An additional issue comes from the requirement of chirality. Strings stretched across D-branes are naturally massive but, in the field theory limit, a mass term would mix different chiralities. We thus need to include a symmetry preserving mechanism for generating the mass of fermions. -In string theory there are ways to deal with the requirement~\cite{Uranga:2003:ChiralFourdimensionalString,Aldazabal:2000:DBranesSingularitiesBottomUp,Zwiebach:2009:FirstCourseString}. -These range from D-branes located at singular points of orbifolds to D-branes intersecting at angles. +In string theory there are ways to deal with the requirement. +These range from D-branes located at singular points of orbifolds to D-branes intersecting at angles~\cite{Uranga:2003:ChiralFourdimensionalString,Aldazabal:2000:DBranesSingularitiesBottomUp,Zwiebach:2009:FirstCourseString}. In this manuscript we focus on intersecting D6-branes filling the $4$-dimensional spacetime and whose additional $3$ dimensions are embedded in a \cy 3-fold (e.g.\ as lines in a factorised torus $T^6 = T^2 \times T^2 \times T^2$). This D-brane geometry supports chiral fermion states at their intersection: while some of the modes of the stretched string become indeed massive, the spectrum of the fields is proportional to combinations of the angles and some of the modes can remain massless. The light spectrum is thus composed of the desired matter content alongside with other particles arising from the string compactification. @@ -1343,17 +1339,17 @@ For instance quarks are localised at the intersection of the \emph{baryonic} sta The same applies to leptons created by strings attached to the \emph{leptonic} stack. Combinations of the additional \U{1} factors in the resulting gauge group finally lead to the definition of the hypercharge $Y$. -Physics in $4$ dimensions is eventually recovered by compactifying the extra-dimensions of the superstring.\footnotemark{} +Physics in four dimensions is eventually recovered by compactifying the extra-dimensions of the superstring.\footnotemark{} \footnotetext{% We specifically reviewed particle physics interactions. - Gravitational interactions in general remain untouched by these constructions and still propagate in $10$-dimensional spacetime. + Gravitational interactions in general remain untouched by D-branes constructions and still propagate in $10$-dimensional spacetime unless the internal space is compact. } Fermions localised at the intersection of the D-branes are however naturally $4$-dimensional as they only propagate in the non compact Minkowski space $\ccM^{1,3}$. The presence of compactified dimensions however leads to phenomena such as \emph{family replications} of the fermions. With accurate calibration it is in fact possible to recover the quark and lepton families in the \sm. Consider for example the simple \cy factorised manifold $T^6 = T^2 \times T^2 \times T^2$ and introduce stacks of D6-branes as lines in each of the bi-tori. Even though the lines might never intersect on a plane, they can have points in common on a torus due to the identifications~\cite{Zwiebach:2009:FirstCourseString}. -Since each intersections supports a different set of fermions with different spectrum, the angles of the intersecting branes can be calibrated to reproduce the separation in mass of the families of quarks and leptons in the \sm. +Since each intersections supports a different set of fermions with different spectrum, the angles of the intersecting branes can be calibrated to reproduce the mass separation of the families of quarks and leptons in the \sm. % vim: ft=tex diff --git a/sec/part2/conclusion.tex b/sec/part2/conclusion.tex index f36295d..daf68df 100644 --- a/sec/part2/conclusion.tex +++ b/sec/part2/conclusion.tex @@ -1,5 +1,5 @@ From 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 spectators anymore. +Moreover when spacetime becomes singular, the string massive modes are not spectators anymore. 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: it 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. diff --git a/sec/part2/divergences.tex b/sec/part2/divergences.tex index 6dcff17..c3e777c 100644 --- a/sec/part2/divergences.tex +++ b/sec/part2/divergences.tex @@ -1,7 +1,6 @@ \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). - +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}). diff --git a/sec/part2/introduction.tex b/sec/part2/introduction.tex index 94fec36..da5e13a 100644 --- a/sec/part2/introduction.tex +++ b/sec/part2/introduction.tex @@ -18,13 +18,13 @@ First of all we recall the formal definition of orbifold to better introduce the 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 $G_{gp} = g^{-1}\, G_p\, g$. 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 kernel is trivial, i.e.\ $\ker{\ccG} = \qty{ \1 }$. +The action of the group is said \emph{transitive} if $Gp = M$ and \emph{effective} if its kernel is trivial, i.e.\ $\ker{\ccG} = \qty{ \1 }_{\ccM}$. 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. + 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 a 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. diff --git a/thesis.bib b/thesis.bib index a7aeb00..6e249a3 100644 --- a/thesis.bib +++ b/thesis.bib @@ -549,23 +549,6 @@ number = {1-3} } -@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}, - date = {2007}, - journaltitle = {Physics Reports}, - volume = {445}, - pages = {1--193}, - issn = {03701573}, - doi = {10/dthp8q}, - abstract = {This review article provides a pedagogical introduction into various classes of chiral string compactifications to four dimensions with D-branes and fluxes. The main concern is to provide all necessary technical tools to explicitly construct four-dimensional orientifold vacua, with the final aim to come as close as possible to the supersymmetric Standard Model. Furthermore, we outline the available methods to derive the resulting four-dimensional effective action. Finally, we summarize recent attempts to address the string vacuum problem via the statistical approach to D-brane models.}, - archivePrefix = {arXiv}, - eprint = {hep-th/0610327}, - eprinttype = {arxiv}, - file = {/home/riccardo/.local/share/zotero/files/blumenhagen_et_al_2007_four-dimensional_string_compactifications_with_d-branes,_orientifolds_and_fluxes.pdf}, - number = {1-6} -} - @book{Blumenhagen:2009:IntroductionConformalField, title = {Introduction to {{Conformal Field Theory}}}, author = {Blumenhagen, Ralph and Plauschinn, Erik},