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Add Chan-Paton factors and SM-like scenario building

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-09-04 18:58:06 +02:00
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2
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\subsection{Motivation}
% vim ft=tex

141
sec/part1/introduction.tex
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@@ -6,6 +6,7 @@ As a first test of validity, the string theory should properly extend the known
In particular its description in terms of fundamental strings should be able to include a gauge algebra isomorphic to that of
\begin{equation}
\SU{3}_{\rC} \otimes \SU{2}_{\rL} \otimes \U{1}_{\rY}
\label{eq:intro:smgroup}
\end{equation}
in order to reproduce known results.
For instance, string theory could provide a unified framework by predicting the existence of a larger gauge group containing the \sm{} as a subset.
@@ -217,14 +218,14 @@ In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$
\begin{figure}[tbp]
\centering
\begin{subfigure}[c]{0.45\linewidth}
\begin{subfigure}[b]{0.45\linewidth}
\centering
\def\svgwidth{\linewidth}
\import{img}{complex_plane.pdf_tex}
\caption{Radial ordering.}
\end{subfigure}
\hfill
\begin{subfigure}[c]{0.45\linewidth}
\begin{subfigure}[b]{0.45\linewidth}
\centering
\def\svgwidth{\linewidth}
\import{img}{radial_ordering.pdf_tex}
@@ -949,7 +950,7 @@ The diamond in this case is
& & & 1 & & &
},
\end{equation}
where we used $h^{r,s} = h^{d-r, d-s}$ to stress the fact that the only independent Hodge numbers are $h^{1,1}$ and $h^{2,1}$ for $m = 3$.
where we used $h^{r,s} = h^{m-r, m-s}$ to stress the fact that the only independent Hodge numbers are $h^{1,1}$ and $h^{2,1}$ for $m = 3$.
These results will also be the starting point of~\Cref{part:deeplearning} in which the ability to predict the values of the Hodge numbers using \emph{artificial intelligence} is tested.
@@ -1159,7 +1160,7 @@ In fact the original Neumann boundary condition~\eqref{eq:tduality:bc} becomes a
0.
\end{split}
\end{equation}
The coordinate of the endpoint in the compact direction is therefore fixed and constrained on a hypersurface called \emph{Dp-brane}, where $p$ stands for the dimension of the surface (in this case $p = D - 1$):
The coordinate of the endpoint in the compact direction is therefore fixed and constrained on a hypersurface called \emph{Dp-brane}, where $p+1$ is the dimension of the surface (in this case $p = D - 2$):
\begin{equation}
\begin{split}
Y^{D-1}( \tau, \pi ) - Y^{D-1}( \tau, 0 )
@@ -1189,4 +1190,136 @@ They also present physical properties such as tension and charge~\cite{DiVecchia
However these aspects will not be discussed here as the following analysis will mainly focus on geometrical aspects of D-branes in spacetime.
\subsubsection{Gauge Groups from D-branes}
As previously stated, in order to recover $4$-dimensional physics we need to compactify the $6$ extra-dimensions of the superstring.
There are in general multiple ways to do such operation consistently~\cite{Brown:1988:NeutralizationCosmologicalConstant,Bousso:2000:QuantizationFourformFluxes,Susskind:2003:AnthropicLandscapeString,tHooft:2009:DimensionalReductionQuantum,Kachru:2003:SitterVacuaString}.
Reproducing the \sm or beyond \sm spectra are however strong constraints on the possible compactification procedures~\cite{Cleaver:2007:SearchMinimalSupersymmetric,Lust:2009:LHCStringHunter}.
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-branes breaks the original \SO{1, D-1} symmetry to $\SO{1, p} \otimes \SO{D - 1 - p}$.\footnotemark{}
\footnotetext{%
Notice that usually $D = 10$, but we keep a generic indication of the spacetime dimensions when possible.
}
The massless spectrum of the theory on the D-brane is easily computed in lightcone gauge~\cite{Goddard:1973:QuantumDynamicsMassless,Polchinski:1998:StringTheoryIntroduction,Green:1988:SuperstringTheoryIntroduction,Angelantonj:2002:OpenStrings}.
Using the residual symmetries of the two-dimensional diffeomorphism (i.e.\ armonic functions of $\tau$ and $\sigma$) we can set
\begin{equation}
X^+( \tau, \sigma ) = x_0^+ + 2 \ap\, p^+\, \tau,
\end{equation}
where $X^{\pm} = \frac{1}{\sqrt{2}} (X^0 \pm X^{D-1})$.
The vanishing of the stress-energy tensor fixes the oscillators in $X^-$ in terms of the physical transverse modes.
The mass shell condition for open strings then becomes:\footnotemark{}
\footnotetext{%
The constant $a$ in~\eqref{eq:dbranes:closedspectrum} takes here the value $-1$ from the imposition of the canonical commutation relations and a $\zeta$-regularisation.
}
\begin{equation}
M^2 = \frac{1}{\ap} \left( N - 1 \right).
\end{equation}
Consider for a moment bosonic string theory and define the usual vacuum as
\begin{equation}
\alpha_n^i \regvacuum = 0,
\qquad
n \ge 0,
\qquad
i = 1, 2, \dots, D - 2,
\end{equation}
we find that at the massless level we have a single \U{1} gauge field in the representation of the Little Group \SO{D-2}:
\begin{equation}
\cA^i
\qquad
\rightarrow
\qquad
\alpha_{-1}^i \regvacuum.
\end{equation}
The introduction of a Dp-brane however breaks the Lorentz invariance down to $\SO{1, p} \otimes \SO{D - 1 - p}$.
Thus the gauge field in the original theory is split into
\begin{equation}
\begin{split}
\cA^A
\qquad
& \rightarrow
\qquad
\alpha_{-1}^A \regvacuum,
\qquad
A = 1, \dots, p - 2,
\\
\cA^a
\qquad
& \rightarrow
\qquad
\alpha_{-1}^a \regvacuum,
\qquad
a = 1, 2, \dots, D - 1 -p.
\end{split}
\end{equation}
In the last expression $\cA^A$ forms a representation of the Little Group \SO{p-2} and as such it is a vector gauge field in $p$ dimensions.
The field $\cA^a$ form a vector representation of the group \SO{D-1-p} and from the point of view of the Lorentz group they are $D - 1 - p$ scalars in the light spectrum.
\begin{figure}[tbp]
\centering
\begin{subfigure}[b]{0.45\linewidth}
\centering
\def\svgwidth{0.8\linewidth}
\import{img}{chanpaton.pdf_tex}
\caption{Chan-Paton factors labelling strings.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\linewidth}
\centering
\def\svgwidth{0.7\linewidth}
\import{img}{quark.pdf_tex}
\caption{Naive model of a left handed massive quark.}
\end{subfigure}
\caption{Strings attached to different D-branes.}
\label{fig:dbranes:chanpaton}
\end{figure}
It is also possible to add non dynamical degrees of freedom to the open string endpoints.
They are known as \emph{Chan-Paton factors}~\cite{Paton:1969:GeneralizedVenezianoModel}.
They have no dynamics and do not spoil Poincaré or conformal invariance in the action of the string.
Each state can then be labelled by $i$ and $j$ running from $1$ to $N$.
Matrices $\tensor{\lambda}{^a_{ij}}$ thus form a basis for expanding wave functions and states:
\begin{equation}
\ket{n;\, a} = \sum\limits_{i,\, j = 1}^N \ket{n;\, i, j}\, \lambda^a_{ij}.
\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 $\lambda^a_{ij}$ 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.
It is also possible to show that in the field theory limit the resulting gauge theory is a 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}.
These are the basic building blocks for a consistent string phenomenology involving both gauge bosons and matter.
\subsubsection{Standard Model Scenarios}
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}.
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.
It would however be theory of pure force, without matter content.
Moreover we should also worry about the extra \U{1} groups appearing: these need careful consideration but go beyond the necessary analysis for what follows.
Matter fields are notoriously fermions transforming in the bi-fundamental representation $(\vb{N}, \vb{M})$ of the \sm gauge group~\eqref{eq:intro:smgroup}.
For example left handed quarks in the \sm transform under the $(\vb{3}, \vb{2})$ representation of the group $\SU{3}_C \otimes \SU{2}_L$.
This is realised in string theory by a string stretched across two stacks of $3$ and $2$ D-branes as in the right of~\Cref{fig:dbranes:chanpaton}.
The fermion would then be characterised by the charge under the gauge bosons living on the D-branes.
The corresponding anti-particle would then simply be a string oriented in the opposite direction.
Things get complicated when introducing also left handed leptons transforming in the $(\vb{1}, \vb{2})$ representation: they cannot have endpoints on the same stack of D-branes as quarks since they do not have colour charge.
We therefore need to introduce more D-branes to account for all the possible combinations.
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,Uranga:2005:TASILecturesString,Zwiebach::FirstCourseString,Aldazabal:2000:DBranesSingularitiesBottomUp}.
These range from D-branes located at singular points of orbifolds to D-branes intersecting at angles~\cite{Finotello:2019:ClassicalSolutionBosonic}.
We focus in particular on the latter.
Specifically 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 $T^6 = T^2 \times T^2 \times T^2$).
% vim ft=tex

171
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@@ -1,4 +1,23 @@
@article{Aldazabal:2000:DBranesSingularitiesBottomUp,
title = {D-{{Branes}} at {{Singularities}} : {{A Bottom}}-{{Up Approach}} to the {{String Embedding}} of the {{Standard Model}}},
shorttitle = {D-{{Branes}} at {{Singularities}}},
author = {Aldazabal, G. and Ibanez, L. E. and Quevedo, F. and Uranga, A. M.},
date = {2000-08-01},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2000},
pages = {002--002},
issn = {1029-8479},
doi = {10.1088/1126-6708/2000/08/002},
abstract = {We propose a bottom-up approach to the building of particle physics models from string theory. Our building blocks are Type II D-branes which we combine appropriately to reproduce desirable features of a particle theory model: 1) Chirality ; 2) Standard Model group ; 3) N=1 or N=0 supersymmetry ; 4) Three quark-lepton generations. We start such a program by studying configurations of D=10, Type IIB D3-branes located at singularities. We study in detail the case of Z\_N, N=1,0 orbifold singularities leading to the SM group or some left-right symmetricextension. In general, tadpole cancellation conditions require the presence of additional branes, e.g. D7-branes. For the N=1 supersymmetric case the unique twist leading to three quark-lepton generations is Z\_3, predicting \$\textbackslash sin\^2\textbackslash theta\_W=3/14=0.21\$. The models obtained are the simplest semirealistic string models ever built. In the non-supersymmetric case there is a three-generation model for each Z\_N, N{$>$}4, but the Weinberg angle is in general too small. One can obtain a large class of D=4 compact models by considering the above structure embedded into a Calabi Yau compactification. We explicitly construct examples of such compact models using Z\_3 toroidal orbifolds and orientifolds, and discuss their properties. In these examples, global cancellation of RR charge may be achieved by adding anti-branes stuck at the fixed points, leading to models with hidden sector gravity-induced supersymmetry breaking. More general frameworks, like F-theory compactifications, allow completely \$\textbackslash NN=1\$ supersymmetric embeddings of our local structures, as we show in an explicit example.},
archivePrefix = {arXiv},
eprint = {hep-th/0005067},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/aldazabal_et_al_2000_d-branes_at_singularities.pdf;/home/riccardo/.local/share/zotero/storage/BTI6TASA/0005067.html},
number = {08}
}
@article{Anderson:2018:TASILecturesGeometric,
title = {{{TASI Lectures}} on {{Geometric Tools}} for {{String Compactifications}}},
author = {Anderson, Lara B. and Karkheiran, Mohsen},
@@ -66,6 +85,41 @@
series = {Theoretical and {{Mathematical Physics}}}
}
@article{Bousso:2000:QuantizationFourformFluxes,
title = {Quantization of {{Four}}-Form {{Fluxes}} and {{Dynamical Neutralization}} of the {{Cosmological Constant}}},
author = {Bousso, Raphael and Polchinski, Joseph},
date = {2000-06-04},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energy Phys.},
volume = {2000},
pages = {006--006},
issn = {1029-8479},
doi = {10.1088/1126-6708/2000/06/006},
abstract = {A four-form gauge flux makes a variable contribution to the cosmological constant. This has often been assumed to take continuous values, but we argue that it has a generalized Dirac quantization condition. For a single flux the steps are much larger than the observational limit, but we show that with multiple fluxes the allowed values can form a sufficiently dense `discretuum'. Multiple fluxes generally arise in M theory compactifications on manifolds with non-trivial three-cycles. In theories with large extra dimensions a few four-forms suffice; otherwise of order 100 are needed. Starting from generic initial conditions, the repeated nucleation of membranes dynamically generates regions with a cosmological constant in the observational range. Entropy and density perturbations can be produced.},
archivePrefix = {arXiv},
eprint = {hep-th/0004134},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/bousso_polchinski_2000_quantization_of_four-form_fluxes_and_dynamical_neutralization_of_the.pdf;/home/riccardo/.local/share/zotero/storage/2LKAEYI3/0004134.html},
number = {06}
}
@article{Brown:1988:NeutralizationCosmologicalConstant,
title = {Neutralization of the Cosmological Constant by Membrane Creation},
author = {Brown, J.David and Teitelboim, Claudio},
date = {1988-02},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {297},
pages = {787--836},
issn = {05503213},
doi = {10.1016/0550-3213(88)90559-7},
annotation = {http://web.archive.org/web/20200904101758/https://linkinghub.elsevier.com/retrieve/pii/0550321388905597},
file = {/home/riccardo/.local/share/zotero/files/brown_teitelboim_1988_neutralization_of_the_cosmological_constant_by_membrane_creation.pdf},
keywords = {archived},
langid = {english},
number = {4}
}
@inproceedings{Calabi:1957:KahlerManifoldsVanishing,
title = {On {{Kähler}} Manifolds with Vanishing Canonical Class},
booktitle = {Algebraic Geometry and Topology. {{A}} Symposium in Honor of {{S}}. {{Lefschetz}}},
@@ -92,6 +146,17 @@
langid = {english}
}
@article{Cleaver:2007:SearchMinimalSupersymmetric,
title = {In {{Search}} of the ({{Minimal Supersymmetric}}) {{Standard Model String}}},
author = {Cleaver, Gerald B.},
date = {2007-03},
abstract = {This paper summarizes several developments in string-derived (Minimal Supersymmetric) Standard Models. Part one reviews the first string model containing solely the three generations of the Minimal Supersymmetric Standard Model and a single pair of Higgs as the matter in the observable sector of the low energy effective field theory. This model was constructed by Cleaver, Faraggi, and Nanopoulos in the Z\_2 x Z\_2 free fermionic formulation of weak coupled heterotic strings. Part two examines a representative collection of string/brane-derived MSSMs that followed. These additional models were obtained from various construction methods, including weak coupled Z\_6 heterotic orbifolds, strong coupled heterotic on elliptically fibered Calabi-Yau's, Type IIB orientifolds with magnetic charged branes, and Type IIA orientifolds with intersecting branes (duals of the Type IIB). Phenomenology of the models is compared. To appear in String Theory Research Progress, Ferenc N. Balogh, editor., (ISBN 978-1-60456-075-6), Nova Science Publishers, Inc.\vphantom\{\}},
archivePrefix = {arXiv},
eprint = {hep-ph/0703027},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/cleaver_2007_in_search_of_the_(minimal_supersymmetric)_standard_model_string.pdf}
}
@book{DiFrancesco:1997:ConformalFieldTheory,
title = {Conformal {{Field Theory}}},
author = {Di Francesco, Philippe and Mathieu, Pierre and Sénéchal, David},
@@ -160,6 +225,25 @@
file = {/home/riccardo/.local/share/zotero/files/di_vecchia_et_al_2006_boundary_state_for_magnetized_d9_branes_and_one-loop_calculation.pdf}
}
@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}}},
author = {Finotello, Riccardo and Pesando, Igor},
date = {2019-04},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {941},
pages = {158--194},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2019.02.010},
abstract = {We consider the classical instantonic contribution to the open string configuration associated with three D-branes with relative rotation matrices in SO(4) which corresponds to the computation of the classical part of the correlator of three non Abelian twist fields. We write the classical solution as a sum of a product of two hypergeometric functions. Differently from all the previous cases with three D-branes, the solution is not holomorphic and suggests that the classical bosonic string knows when the configuration may be supersymmetric. We show how this configuration reduces to the standard Abelian twist field computation. From the phenomenological point of view, the Yukawa couplings between chiral matter at the intersection in this configuration are more suppressed with respect to the factorized case in the literature.},
annotation = {ZSCC: 0000000},
archivePrefix = {arXiv},
eprint = {1812.04643},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/finotello_pesando_2019_the_classical_solution_for_the_bosonic_string_in_the_presence_of_three_d-branes.pdf;/home/riccardo/.local/share/zotero/files/finotello_pesando_2019_the_classical_solution_for_the_bosonic_string_in_the_presence_of_three_d-branes2.pdf}
}
@article{Friedan:1986:ConformalInvarianceSupersymmetry,
title = {Conformal Invariance, Supersymmetry and String Theory},
author = {Friedan, Daniel and Martinec, Emil and Shenker, Stephen},
@@ -185,6 +269,22 @@
file = {/home/riccardo/.local/share/zotero/files/ginsparg_1988_applied_conformal_field_theory.pdf}
}
@article{Goddard:1973:QuantumDynamicsMassless,
title = {Quantum Dynamics of a Massless Relativistic String},
author = {Goddard, P. and Goldstone, J. and Rebbi, C. and Thorn, C.B.},
date = {1973-05},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {56},
pages = {109--135},
issn = {05503213},
doi = {10.1016/0550-3213(73)90223-X},
file = {/home/riccardo/.local/share/zotero/files/goddard_et_al_1973_quantum_dynamics_of_a_massless_relativistic_string.pdf},
keywords = {archived},
langid = {english},
number = {1}
}
@article{Grana:2005:FluxCompactificationsString,
title = {Flux Compactifications in String Theory: A Comprehensive Review},
author = {Graña, Mariana},
@@ -319,6 +419,24 @@
keywords = {⛔ No DOI found}
}
@article{Kachru:2003:SitterVacuaString,
title = {De {{Sitter Vacua}} in {{String Theory}}},
author = {Kachru, Shamit and Kallosh, Renata and Linde, Andrei and Trivedi, Sandip P.},
date = {2003-08-07},
journaltitle = {Physical Review D},
shortjournal = {Phys. Rev. D},
volume = {68},
pages = {046005},
issn = {0556-2821, 1089-4918},
doi = {10.1103/PhysRevD.68.046005},
abstract = {We outline the construction of metastable de Sitter vacua of type IIB string theory. Our starting point is highly warped IIB compactifications with nontrivial NS and RR three-form fluxes. By incorporating known corrections to the superpotential from Euclidean D-brane instantons or gaugino condensation, one can make models with all moduli fixed, yielding a supersymmetric AdS vacuum. Inclusion of a small number of anti-D3 branes in the resulting warped geometry allows one to uplift the AdS minimum and make it a metastable de Sitter ground state. The lifetime of our metastable de Sitter vacua is much greater than the cosmological timescale of 10\^10 years. We also prove, under certain conditions, that the lifetime of dS space in string theory will always be shorter than the recurrence time.},
archivePrefix = {arXiv},
eprint = {hep-th/0301240},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/kachru_et_al_2003_de_sitter_vacua_in_string_theory.pdf;/home/riccardo/.local/share/zotero/storage/AABMA8ED/0301240.html},
number = {4}
}
@article{Krippendorf:2010:CambridgeLecturesSupersymmetry,
title = {Cambridge {{Lectures}} on {{Supersymmetry}} and {{Extra Dimensions}}},
author = {Krippendorf, Sven and Quevedo, Fernando and Schlotterer, Oliver},
@@ -349,6 +467,22 @@
number = {1-2}
}
@article{Paton:1969:GeneralizedVenezianoModel,
title = {Generalized {{Veneziano}} Model with Isospin},
author = {Paton, J.E. and {Chan Hong-Mo}},
date = {1969-05},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {10},
pages = {516--520},
issn = {05503213},
doi = {10.1016/0550-3213(69)90038-8},
file = {/home/riccardo/.local/share/zotero/files/paton_chan_hong-mo_1969_generalized_veneziano_model_with_isospin.pdf},
keywords = {archived},
langid = {english},
number = {3}
}
@article{Polchinski:1995:DirichletBranesRamondRamond,
title = {Dirichlet Branes and {{Ramond}}-{{Ramond}} Charges},
author = {Polchinski, Joseph},
@@ -418,6 +552,17 @@
number = {3}
}
@article{Susskind:2003:AnthropicLandscapeString,
title = {The {{Anthropic Landscape}} of {{String Theory}}},
author = {Susskind, Leonard},
date = {2003-02},
abstract = {In this lecture I make some educated guesses, about the landscape of string theory vacua. Based on the recent work of a number of authors, it seems plausible that the lanscape is unimaginably large and diverse. Whether we like it or not, this is the kind of behavior that gives credence to the Anthropic Principle. I discuss the theoretical and conceptual issues that arise in developing a cosmology based on the diversity of environments implicit in string theory.},
archivePrefix = {arXiv},
eprint = {hep-th/0302219},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/susskind_2003_the_anthropic_landscape_of_string_theory.pdf}
}
@article{Taylor:2003:LecturesDbranesTachyon,
title = {Lectures on {{D}}-Branes, Tachyon Condensation, and String Field Theory},
author = {Taylor, Washington},
@@ -443,6 +588,32 @@
file = {/home/riccardo/.local/share/zotero/files/taylor_zwiebach_2004_d-branes,_tachyons,_and_string_field_theory.pdf}
}
@article{tHooft:2009:DimensionalReductionQuantum,
title = {Dimensional {{Reduction}} in {{Quantum Gravity}}},
author = {'t Hooft, Gerard},
date = {2009-03-20},
abstract = {The requirement that physical phenomena associated with gravitational collapse should be duly reconciled with the postulates of quantum mechanics implies that at a Planckian scale our world is not 3+1 dimensional. Rather, the observable degrees of freedom can best be described as if they were Boolean variables defined on a two-dimensional lattice, evolving with time. This observation, deduced from not much more than unitarity, entropy and counting arguments, implies severe restrictions on possible models of quantum gravity. Using cellular automata as an example it is argued that this dimensional reduction implies more constraints than the freedom we have in constructing models. This is the main reason why so-far no completely consistent mathematical models of quantum black holes have been found. Essay dedicated to Abdus Salam.},
archivePrefix = {arXiv},
eprint = {gr-qc/9310026},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/hooft_2009_dimensional_reduction_in_quantum_gravity.pdf;/home/riccardo/.local/share/zotero/storage/XXKQV4D9/9310026.html},
options = {useprefix=true}
}
@article{Uranga:2003:ChiralFourdimensionalString,
title = {Chiral Four-Dimensional String Compactifications with Intersecting {{D}}-Branes},
author = {Uranga, Angel M},
date = {2003-06-21},
journaltitle = {Classical and Quantum Gravity},
shortjournal = {Class. Quantum Grav.},
volume = {20},
pages = {S373-S393},
issn = {0264-9381, 1361-6382},
doi = {10.1088/0264-9381/20/12/303},
file = {/home/riccardo/.local/share/zotero/files/uranga_2003_chiral_four-dimensional_string_compactifications_with_intersecting_d-branes.pdf},
number = {12}
}
@article{Uranga:2005:TASILecturesString,
title = {{{TASI}} Lectures on {{String Compactification}}, {{Model Building}}, and {{Fluxes}}},
author = {Uranga, Angel M.},