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Kaehler manifolds

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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.gitignore vendored
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@@ -10,3 +10,4 @@
*.run.xml
*.toc
*.xdv
*.pdf

125
sec/part1/introduction.tex
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@@ -484,11 +484,11 @@ Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \bz )
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 $\overline{b}(z)$ and $\overline{c}(z)$.
The non vanishing components of their stress-energy tensor can be computed as:\footnotemark{}
\footnotetext{%
In general ghosts $b( z )$ and $c( z )$ with weight $(\lambda, 0)$ and $(1 - \lambda, 0)$ can be introduced as a standalone \cft \cite{Friedan:1986:ConformalInvarianceSupersymmetry,Polchinski:1998:StringTheorySuperstring} with action
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,Polchinski:1998:StringTheorySuperstring}
\begin{equation*}
S = \frac{1}{2 \pi} \iint \dd{z} \dd{\bz} b( z )\, \ipd{\bz} c( z )
S = \frac{1}{2 \pi} \iint \dd{z} \dd{\bz} b( z )\, \ipd{\bz} c( z ).
\end{equation*}
whose equations of motion are $\ipd{\bz} c( z ) = \ipd{\bz} b( z ) = 0$.
The equations of motion are $\ipd{\bz} c( z ) = \ipd{\bz} b( z ) = 0$.
The \ope is
\begin{equation*}
b( z )\, c( z ) = \frac{\varepsilon}{z - w} + \order{1},
@@ -509,7 +509,8 @@ The non vanishing components of their stress-energy tensor can be computed as:\f
T_{\text{ghost}}( z )\, j( w ) = \frac{Q}{( z - w )^3} + \order{(z - w)^{-2}}.
\end{equation*}
This is the case of the worldsheet fermions in~\eqref{eq:super:action} for which $\lambda = \frac{1}{2}$.
For instance the reparametrisation ghosts with $\lambda = 2$ have $Q = -3$.
For instance the reparametrisation ghosts with $\lambda = 2$ have $Q = -3$, while the superghosts with $\lambda = \frac{3}{2}$ present $Q = 2$.
\label{note:conf:ghosts}
}
\begin{equation}
\begin{split}
@@ -615,21 +616,21 @@ The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmet
\begin{equation}
\begin{split}
\sqrt{\frac{2}{\ap}}\,
\delta_{\epsilon, \bepsilon}\,
\delta_{\epsilon, \bepsilon}
X^{\mu}( z, \bz )
& =
\epsilon( z ) \psi^{\mu}( z ) + \bepsilon( \bz ) \bpsi^{\mu}( \bz ),
\epsilon( z )\, \psi^{\mu}( z ) + \bepsilon( \bz )\, \bpsi^{\mu}( \bz ),
\\
\sqrt{\frac{2}{\ap}} \delta_{\epsilon} \psi^{\mu}( z )
& =
- \epsilon( z ) \ipd{z} X^{\mu}( z ),
- \epsilon( z )\, \ipd{z} X^{\mu}( z ),
\\
\sqrt{\frac{2}{\ap}} \delta_{\bepsilon} \bpsi^{\mu}( \bz )
& =
- \bepsilon( \bz ) \ipd{\bz} \bX^{\mu}( \bz )
- \bepsilon( \bz )\, \ipd{\bz} \bX^{\mu}( \bz )
\end{split}
\end{equation}
generated by the currents $J( z ) = \epsilon( z ) T_F( z )$ and $\overline{J}( \bz ) = \bepsilon( \bz )\, \bT_F( \bz )$, where $\epsilon( z )$ and $\bepsilon( \bz ) = \left( \epsilon( z ) \right)^*$ are anti-commuting fermions and
generated by the currents $J( z ) = \epsilon( z )\, T_F( z )$ and $\overline{J}( \bz ) = \bepsilon( \bz )\, \bT_F( \bz )$, where $\epsilon( z )$ and $\bepsilon( \bz ) = \left( \epsilon( z ) \right)^*$ are anti-commuting fermions and
\begin{equation}
\begin{split}
T_F( z )
@@ -670,7 +671,7 @@ The central charge is therefore $c = \frac{3}{2} D$ for the \cft defined in~\eqr
As in the case of the bosonic string, in order to cancel the central charge we introduce the reparametrisation anti-commuting ghosts $b( z )$ and $c( z )$ and their anti-holomorphic components as well as the commuting \emph{superghosts} $\beta( z )$ and $\gamma( z )$ and their anti-holomorphic counterparts.
These are conformal fields with conformal weights $\left( \frac{3}{2}, 0 \right)$ and $\left( -\frac{1}{2}, 0 \right)$.
Their central charge becomes $c_{\text{ghost}} = c_{bc} + c_{\beta\gamma} = -26 + 11 = -15$.
Their central charge becomes $c_{\text{ghost}} = c_{bc} + c_{\beta\gamma} = -26 + 11 = -15$ (see \cref{note:conf:ghosts} for the general computation).
When considering the full theory $T_{\text{full}} = T + T_{\text{ghost}}$ and $\bT_{\text{full}} = \bT + \bT_{\text{ghost}}$ the central charge vanishes only in 10-dimensional spacetime:
\begin{equation}
\eval{T_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
@@ -686,6 +687,7 @@ When considering the full theory $T_{\text{full}} = T + T_{\text{ghost}}$ and $\
\Leftrightarrow
\quad
D = 10.
\label{eq:super:dimensions}
\end{equation}
@@ -700,6 +702,109 @@ In this section we briefly review for completeness the necessary tools to be abl
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,Blumenhagen:2013:BasicConceptsString,Grana:2005:FluxCompactificationsString,Grana:2017:StringTheoryCompactifications,Krippendorf:2010:CambridgeLecturesSupersymmetry,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
\begin{equation}
\ccM^{1,9}
=
\ccM^{1,3} \otimes \ccX_6,
\end{equation}
where $\ccX_6$ is a generic $6$-dimensional manifold at this stage.
This \emph{internal} manifold $\ccX_6$ is however subject to very stringent restrictions due to mathemtical consistency conditions and physical requests.
In particular $\ccX_6$ should be a compact manifold to ``hide'' the 6 extra-dimensions computed in~\eqref{eq:super:dimensions}.
Moreover the geometry of $\ccM^{1,3}$ should be a maximally symmetric space and there should be a $N = 1$ unbroken supersymmetry in $4$ dimensions.
Finally the gauge group of and the spectrum of fermions should be realistic (e.g.\ it should be possible to define chiral fermion states) \cite{Candelas:1985:VacuumConfigurationsSuperstrings}.
These manifolds were first conjectured to exist by Eugenio Calabi~\cite{Calabi:1957:KahlerManifoldsVanishing}.
Their existence was later proved by Shing-Tung Yau~\cite{Yau:1977:CalabiConjectureNew}.
They are defined as complex Ricci-flat Kähler manifolds $M$ of dimensions $2m$ and with holonomy \SU{3} (see for instance \cite{Joyce:2000:CompactManifoldsSpecial,Joyce:2002:LecturesCalabiYauSpecial}).
\subsubsection{Complex and Kähler Manifolds}
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
\begin{equation}
\tensor{N}{^a_{bc}} v_p^b w_p^c
=
\left(
[ v_p, w_p ]
+
J
\left( [ J\, v_p, w_p ] + [ v_p, J\, w_p ] \right)
-
[ J\, v_p, J\, w_p ]
\right)^a
=
0
\end{equation}
for any $v_p,\, w_p \in \rT_p M$, where $[ \cdot, \cdot ]$ is the 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 $f_{*\, p}\colon\, \rT_p M \to \rT_{f(p)} \C$ is called holomorphic if $(J\, f_{*\, p}( v_p ))^a = i ( f_{*\, p}( v_p ) )^a$ as such expression encodes the Cauchy-Riemann equations.
Let in fact $f( x, y ) = f_1( x, y ) + i\, f_2( x, y )$, then the expression implies
\begin{equation*}
\begin{cases}
\ipd{x} f_1( x, y )
& =
\ipd{y} f_2( x, y )
\\
\ipd{x} f_2( x, y )
& =
-\ipd{y} f_1( x, y )
\end{cases}
\Rightarrow
\ipd{x} f( x, y ) = -i \ipd{y} f( x, y )
\Rightarrow
\ipd{\bz} f( z, \bz ) = 0
\Rightarrow
f( z, \bz ) = f( z ).
\end{equation*}
}
Let then $(M, J, g)$ be a complex manifold with a Riemannian metric $g$.
The metric is \emph{Hermitian} if
\begin{equation}
g( v_p, w_p ) = g( J\, v_p, J\, w_p )
\quad
\forall v_p,\, w_p \in \rT_p M
\quad
\Leftrightarrow
\quad
\tensor{g}{_{ab}}
=
\tensor{J}{_a^c}\,
\tensor{J}{_b^d}\,
\tensor{g}{_{cd}}.
\end{equation}
In this case we can define a $(1, 1)$-form $\omega$ as
\begin{equation}
\omega( v_p, w_p )
=
g( J\, v_p, w_p )
\quad
\forall v_p,\, w_p \in \rT_p M.
\quad
\Leftrightarrow
\quad
\tensor{\omega}{_{ab}}
=
\tensor{J}{_a^c}\,
\tensor{g}{_{cb}}.
\end{equation}
$(M, J, g)$ is a \emph{Kähler} manifold if ~\cite{Joyce:2002:LecturesCalabiYauSpecial}:
\begin{equation}
\dd{\omega}
=
\left( \ipd{z} + \ipd{\bz} \right)
\omega(z, \bz)
=
0,
\end{equation}
or equivalently $\nabla J = 0$ or $\nabla \omega = 0$, where $\nabla$ is the connection of $g$.
The covariant conservation of $J$ and $\omega$ implies that the holonomy group must preserve these objects in $\R^{2m}$.
Thus we have $\mathrm{Hol}(g) \subseteq \U{m} \subset \mathrm{O}(2m)$.
\subsection{D-branes and Open Strings}

139
thesis.bib
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@@ -3,9 +3,10 @@
title = {{{TASI Lectures}} on {{Geometric Tools}} for {{String Compactifications}}},
author = {Anderson, Lara B. and Karkheiran, Mohsen},
date = {2018-04},
url = {http://arxiv.org/abs/1804.08792},
abstract = {In this work we provide a self-contained and modern introduction to some of the tools, obstacles and open questions arising in string compactifications. Techniques and current progress are illustrated in the context of smooth heterotic string compactifications to 4-dimensions. Progress is described on bounding and enumerating possible string backgrounds and their properties. We provide an overview of constructions, partial classifications, and moduli problems associated to Calabi-Yau manifolds and holomorphic bundles over them.},
annotation = {ZSCC: 0000002},
archivePrefix = {arXiv},
eprint = {1804.08792},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/anderson_karkheiran_2018_tasi_lectures_on_geometric_tools_for_string_compactifications.pdf}
}
@@ -18,9 +19,10 @@
pages = {1--193},
issn = {03701573},
doi = {10.1016/j.physrep.2007.04.003},
url = {http://arxiv.org/abs/hep-th/0610327 http://dx.doi.org/10.1016/j.physrep.2007.04.003 http://linkinghub.elsevier.com/retrieve/pii/S037015730700172X},
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.},
annotation = {00000},
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}
}
@@ -33,8 +35,6 @@
publisher = {{Springer Berlin Heidelberg}},
location = {{Berlin, Heidelberg}},
doi = {10.1007/978-3-642-00450-6},
url = {http://link.springer.com/10.1007/978-3-642-00450-6},
urldate = {2020-05-31},
file = {/home/riccardo/.local/share/zotero/files/blumenhagen_plauschinn_2009_introduction_to_conformal_field_theory.pdf},
isbn = {978-3-642-00449-0 978-3-642-00450-6},
langid = {english},
@@ -48,27 +48,33 @@
publisher = {{Springer Berlin Heidelberg}},
location = {{Berlin, Heidelberg}},
doi = {10.1007/978-3-642-29497-6},
url = {http://link.springer.com/10.1007/978-3-642-29497-6},
urldate = {2020-05-31},
file = {/home/riccardo/.local/share/zotero/files/blumenhagen_et_al_2013_basic_concepts_of_string_theory.pdf},
isbn = {978-3-642-29496-9 978-3-642-29497-6},
langid = {english},
series = {Theoretical and {{Mathematical Physics}}}
}
@inproceedings{Calabi:1957:KahlerManifoldsVanishing,
title = {On {{Kähler}} Manifolds with Vanishing Canonical Class},
booktitle = {Algebraic Geometry and Topology. {{A}} Symposium in Honor of {{S}}. {{Lefschetz}}},
author = {Calabi, Eugenio},
date = {1957},
volume = {12},
pages = {78--89},
doi = {10.1515/9781400879915-006}
}
@article{Candelas:1985:VacuumConfigurationsSuperstrings,
title = {Vacuum Configurations for Superstrings},
author = {Candelas, P. and Horowitz, Gary T. and Strominger, Andrew and Witten, Edward},
date = {1985-01-01},
author = {Candelas, Philip and Horowitz, Gary T. and Strominger, Andrew and Witten, Edward},
date = {1985-01},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {258},
pages = {46--74},
issn = {05503213},
doi = {10.1016/0550-3213(85)90602-9},
url = {http://www.sciencedirect.com/science/article/pii/0550321385906029},
urldate = {2020-08-31},
abstract = {We study candidate vacuum configurations in ten-dimensional O(32) and E8 × E8 supergravity and superstring theory that have unbroken N = 1 supersymmetry in four dimensions. This condition permits only a few possibilities, all of which have vanishing cosmological constant. In the E8 × E8 case, one of these possibilities leads to a model that in four dimensions has an E6 gauge group with four standard generations of fermions.},
annotation = {http://web.archive.org/web/20200831103147/https://linkinghub.elsevier.com/retrieve/pii/0550321385906029},
file = {/home/riccardo/.local/share/zotero/files/candelas_et_al_1985_vacuum_configurations_for_superstrings.pdf},
keywords = {archived},
langid = {english}
@@ -81,8 +87,6 @@
publisher = {{Springer New York}},
location = {{New York, NY}},
doi = {10.1007/978-1-4612-2256-9},
url = {http://link.springer.com/10.1007/978-1-4612-2256-9},
urldate = {2020-05-31},
file = {/home/riccardo/.local/share/zotero/files/di_francesco_et_al_1997_conformal_field_theory.pdf},
isbn = {978-1-4612-7475-9 978-1-4612-2256-9},
langid = {english},
@@ -90,7 +94,6 @@
}
@article{Friedan:1986:ConformalInvarianceSupersymmetry,
ids = {Friedan:1986:ConformalInvarianceSupersymmetrya},
title = {Conformal Invariance, Supersymmetry and String Theory},
author = {Friedan, Daniel and Martinec, Emil and Shenker, Stephen},
date = {1986-01},
@@ -99,21 +102,19 @@
pages = {93--165},
issn = {05503213},
doi = {10.1016/s0550-3213(86)80006-2},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0550321386800062},
abstract = {Covariant quantization of string theories is developed in the context of conformal field theory and the BRST quantization procedure. The BRST method is used to covariantly quantize superstrings, and in particular to construct the vertex operators for string emission as well as the supersymmetry charge. The calculation of string loop diagrams is sketched. We discuss how conformal methods can be used to study string compactification and dynamics. © 1986, Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division). All rights reserved. All rights reserved.},
annotation = {ZSCC: 0002220},
file = {/home/riccardo/.local/share/zotero/files/friedan_et_al_1986_conformal_invariance,_supersymmetry_and_string_theory.pdf},
issue = {3-4},
number = {PRINT-86-0024 (CHICAGO), EFI-85-89-CHICAGO}
number = {3-4}
}
@article{Ginsparg:1988:AppliedConformalField,
title = {Applied {{Conformal Field Theory}}},
author = {Ginsparg, Paul},
date = {1988-11},
url = {http://arxiv.org/abs/hep-th/9108028},
abstract = {These lectures consisted of an elementary introduction to conformal field theory, with some applications to statistical mechanical systems, and fewer to string theory. Contents: 1. Conformal theories in d dimensions 2. Conformal theories in 2 dimensions 3. The central charge and the Virasoro algebra 4. Kac determinant and unitarity 5. Identication of m = 3 with the critical Ising model 6. Free bosons and fermions 7. Free fermions on a torus 8. Free bosons on a torus 9. Affine Kac-Moody algebras and coset constructions 10. Advanced applications},
annotation = {ZSCC: 0000776},
archivePrefix = {arXiv},
eprint = {hep-th/9108028},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/ginsparg_1988_applied_conformal_field_theory.pdf}
}
@@ -126,9 +127,7 @@
pages = {91--158},
issn = {03701573},
doi = {10.1016/j.physrep.2005.10.008},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0370157305004618 http://arxiv.org/abs/hep-th/0509003 http://dx.doi.org/10.1016/j.physrep.2005.10.008},
abstract = {We present a pedagogical overview of flux compactifications in string theory, from the basic ideas to the most recent developments. We concentrate on closed string fluxes in type II theories. We start by reviewing the supersymmetric flux configurations with maximally symmetric four-dimensional spaces. We then discuss the no-go theorems (and their evasion) for compactifications with fluxes. We analyze the resulting four-dimensional effective theories, as well as some of its perturbative and non-perturbative corrections, focusing on moduli stabilization. Finally, we briefly review statistical studies of flux backgrounds.},
annotation = {ZSCC: 0000000[s0]},
file = {/home/riccardo/.local/share/zotero/files/graña_2005_flux_compactifications_in_string_theory.pdf},
number = {3}
}
@@ -140,94 +139,100 @@
publisher = {{Springer International Publishing}},
location = {{Cham}},
doi = {10.1007/978-3-319-54316-1},
url = {http://link.springer.com/10.1007/978-3-319-54316-1},
urldate = {2020-05-31},
file = {/home/riccardo/.local/share/zotero/files/graña_triendl_2017_string_theory_compactifications.pdf},
isbn = {978-3-319-54315-4 978-3-319-54316-1},
langid = {english},
series = {{{SpringerBriefs}} in {{Physics}}}
}
@mvbook{Green:1988:SuperstringTheoryIntroduction,
@book{Green:1988:SuperstringTheoryIntroduction,
title = {Superstring {{Theory}}. {{Introduction}}.},
author = {Green, Michael B. and Schwarz, J.H. and Witten, Edward},
date = {1988-07},
author = {Green, Michael B. and Schwarz, John H. and Witten, Edward},
date = {1988},
volume = {1},
doi = {10.1017/CBO9781139248563},
file = {/home/riccardo/.local/share/zotero/files/green_et_al_1988_superstring_theory.pdf},
isbn = {978-0-521-35752-4},
series = {Cambridge Monographs on Mathematical Physics},
volumes = {2}
series = {Cambridge Monographs on Mathematical Physics}
}
@mvbook{Green:1988:SuperstringTheoryLoop,
@book{Green:1988:SuperstringTheoryLoop,
title = {Superstring {{Theory}}. {{Loop Amplitudes}}, {{Anomalies}} and {{Phenomenology}}.},
author = {Green, Michael B. and Schwarz, J.H. and Witten, Edward},
date = {1988-07},
author = {Green, Michael B. and Schwarz, John H. and Witten, Edward},
date = {1988},
volume = {2},
doi = {10.1017/CBO9781139248570},
file = {/home/riccardo/.local/share/zotero/files/green_et_al_1988_superstring_theory2.pdf},
isbn = {978-0-521-35753-1},
volumes = {2}
series = {Cambridge Monographs on Mathematical Physics}
}
@book{Joyce:2000:CompactManifoldsSpecial,
title = {Compact Manifolds with Special Holonomy},
author = {Joyce, Dominic},
date = {2000},
publisher = {{Oxford University Press on Demand}},
isbn = {978-0-19-850601-0}
}
@article{Joyce:2002:LecturesCalabiYauSpecial,
title = {Lectures on {{Calabi}}-{{Yau}} and Special {{Lagrangian}} Geometry},
author = {Joyce, Dominic},
date = {2002-06},
abstract = {This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject, and is designed to be fairly self-contained. It is based on lecture courses given at Nordfjordeid, Norway and MSRI, Berkeley in June and July 2001. We introduce Calabi-Yau m-folds via holonomy groups, Kahler geometry and the Calabi Conjecture, and special Lagrangian m-folds via calibrated geometry. `Almost Calabi-Yau m-folds' (a generalization of Calabi-Yau m-folds useful in special Lagrangian geometry) are explained and the deformation theory and moduli spaces of compact special Lagrangian submanifolds in (almost) Calabi-Yau m-folds is described. In the final part we consider isolated singularities of special Lagrangian m-folds, focussing mainly on singularities locally modelled on cones, and the expected behaviour of singularities of compact special Lagrangian m-folds in generic (almost) Calabi-Yau m-folds. String Theory, Mirror Symmetry and the SYZ Conjecture are briefly discussed, and some results of the author on singularities of special Lagrangian fibrations of Calabi-Yau 3-folds are described.},
archivePrefix = {arXiv},
eprint = {math/0108088},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/joyce_2002_lectures_on_calabi-yau_and_special_lagrangian_geometry2.pdf},
keywords = {⛔ No DOI found}
}
@article{Krippendorf:2010:CambridgeLecturesSupersymmetry,
ids = {krippendorfCambridgeLecturesSupersymmetry2010a},
title = {Cambridge {{Lectures}} on {{Supersymmetry}} and {{Extra Dimensions}}},
author = {Krippendorf, Sven and Quevedo, Fernando and Schlotterer, Oliver},
date = {2010-11-05},
url = {http://arxiv.org/abs/1011.1491},
abstract = {These lectures on supersymmetry and extra dimensions are aimed at finishing undergraduate and beginning postgraduate students with a background in quantum field theory and group theory. Basic knowledge in general relativity might be advantageous for the discussion of extra dimensions. This course was taught as a 24+1 lecture course in Part III of the Mathematical Tripos in recent years. The first six chapters give an introduction to supersymmetry in four spacetime dimensions, they fill about two thirds of the lecture notes and are in principle self-contained. The remaining two chapters are devoted to extra spacetime dimensions which are in the end combined with the concept of supersymmetry. Videos from the course lectured in 2006 can be found online at http://www.sms.cam.ac.uk/collection/659537 .},
archivePrefix = {arXiv},
eprint = {1011.1491},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/krippendorf_et_al_2010_cambridge_lectures_on_supersymmetry_and_extra_dimensions.pdf},
langid = {english},
primaryClass = {hep-ph, physics:hep-th}
langid = {english}
}
@mvbook{Polchinski:1998:StringTheoryIntroduction,
@book{Polchinski:1998:StringTheoryIntroduction,
title = {String {{Theory}}. {{An}} Introduction to the Bosonic String.},
author = {Polchinski, Joseph},
date = {1998-10-13},
edition = {1},
date = {1998},
volume = {1},
publisher = {{Cambridge University Press}},
doi = {10.1017/CBO9780511816079; http://web.archive.org/web/20200831133025/https://www.cambridge.org/core/books/string-theory/30409AF2BDE27D53E275FDA395AB667A},
url = {https://doi.org/10.1017/CBO9780511816079},
urldate = {2020-08-31},
doi = {10.1017/CBO9780511816079},
file = {/home/riccardo/.local/share/zotero/files/polchinski_1998_string_theory.pdf},
isbn = {978-0-521-67227-6 978-0-521-63303-1 978-0-511-81607-9},
keywords = {archived},
volumes = {2}
keywords = {archived}
}
@mvbook{Polchinski:1998:StringTheorySuperstring,
@book{Polchinski:1998:StringTheorySuperstring,
title = {String {{Theory}}. {{Superstring}} Theory and Beyond.},
author = {Polchinski, Joseph},
date = {1998-10-13},
edition = {1},
date = {1998},
volume = {2},
publisher = {{Cambridge University Press}},
doi = {10.1017/CBO9780511618123; http://web.archive.org/web/20200831133053/https://www.cambridge.org/core/books/string-theory/2D456468D20AA8A9CE10CEB08B95B9DC},
url = {https://doi.org/10.1017/CBO9780511618123},
urldate = {2020-08-31},
doi = {10.1017/CBO9780511618123},
file = {/home/riccardo/.local/share/zotero/files/polchinski_1998_string_theory2.pdf},
isbn = {978-0-521-63304-8 978-0-521-67228-3 978-0-511-61812-3},
keywords = {archived},
volumes = {2}
keywords = {archived}
}
@article{Polyakov:1981:QuantumGeometryBosonic,
title = {Quantum Geometry of Bosonic Strings},
author = {Polyakov, A.M.},
author = {Polyakov, Alexander M.},
date = {1981-07},
journaltitle = {Physics Letters B},
shortjournal = {Physics Letters B},
volume = {103},
pages = {207--210},
issn = {03702693},
doi = {10.1016/0370-2693(81)90743-7},
url = {10.1016/0370-2693(81)90743-7},
urldate = {2020-09-01},
annotation = {http://web.archive.org/web/20200901072439/https://linkinghub.elsevier.com/retrieve/pii/0370269381907437},
keywords = {archived},
langid = {english},
number = {3}
@@ -239,8 +244,22 @@
date = {2005},
url = {http://cds.cern.ch/record/933469/files/cer-002601054.pdf},
abstract = {We review the construction of chiral four-dimensional compactifications of string theory with different systems of D-branes, including type IIA intersecting D6-branes and type IIB magnetised D-branes. Such models lead to four-dimensional theories with non-abelian gauge interactions and charged chiral fermions. We discuss the application of these techniques to building of models with spectrum as close as possible to the Standard Model, and review their main phenomenological properties. We finally describe how to implement the tecniques to construct these models in flux compactifications, leading to models with realistic gauge sectors, moduli stabilization and supersymmetry breaking soft terms.},
annotation = {ZSCC: 0000000},
file = {/home/riccardo/.local/share/zotero/files/uranga_2005_tasi_lectures_on_string_compactification,_model_building,_and_fluxes.pdf}
}
@article{Yau:1977:CalabiConjectureNew,
title = {Calabi's Conjecture and Some New Results in Algebraic Geometry},
author = {Yau, Shing-Tung},
date = {1977-05-01},
journaltitle = {Proceedings of the National Academy of Sciences},
shortjournal = {Proceedings of the National Academy of Sciences},
volume = {74},
pages = {1798--1799},
issn = {0027-8424, 1091-6490},
doi = {10.1073/pnas.74.5.1798},
file = {/home/riccardo/.local/share/zotero/files/yau_1977_calabi's_conjecture_and_some_new_results_in_algebraic_geometry2.pdf},
langid = {english},
number = {5}
}

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@@ -44,7 +44,7 @@
citestyle=numeric-comp,
sorting=none,
sortcites=true,
%style=ieee,
style=ieee,
maxnames=3]{biblatex} %--------- bibliography backend
\RequirePackage{bookmark} %--------------------- hyperref and links

1
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@@ -108,6 +108,7 @@
\appendix
%---- BIBLIOGRAPHY
\cleardoubleplainpage{}
\printbibliography[heading=bibintoc]
\end{document}