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Closed strings and CY

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
Riccardo
2020-09-03 18:26:02 +02:00
parent 7eac6f4191
commit cf13d2f918
5 changed files with 695 additions and 190 deletions
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311
sciencestuff.sty
View File

@@ -47,176 +47,179 @@
%---- remap greek letters
\renewcommand{\alpha}{\upalpha}
\renewcommand{\beta}{\upbeta}
\renewcommand{\gamma}{\upgamma}
\renewcommand{\delta}{\updelta}
\renewcommand{\epsilon}{\upepsilon}
\renewcommand{\zeta}{\upzeta}
\renewcommand{\eta}{\upeta}
\renewcommand{\theta}{\uptheta}
\renewcommand{\iota}{\upiota}
\renewcommand{\kappa}{\upkappa}
\renewcommand{\lambda}{\uplambda}
\renewcommand{\mu}{\upmu}
\renewcommand{\nu}{\upnu}
\renewcommand{\xi}{\upxi}
\renewcommand{\pi}{\uppi}
\renewcommand{\rho}{\uprho}
\renewcommand{\sigma}{\upsigma}
\renewcommand{\tau}{\uptau}
\renewcommand{\upsilon}{\upupsilon}
\renewcommand{\phi}{\upphi}
\renewcommand{\chi}{\upchi}
\renewcommand{\psi}{\uppsi}
\renewcommand{\omega}{\upomega}
\renewcommand{\varepsilon}{\upvarepsilon}
\renewcommand{\vartheta}{\upvartheta}
\renewcommand{\varpi}{\upvarpi}
\renewcommand{\varphi}{\upvarphi}
\renewcommand{\Gamma}{\Upgamma}
\renewcommand{\Delta}{\Updelta}
\renewcommand{\Theta}{\Uptheta}
\renewcommand{\Lambda}{\Uplambda}
\renewcommand{\Xi}{\Upxi}
\renewcommand{\Pi}{\Uppi}
\renewcommand{\Sigma}{\Upsigma}
\renewcommand{\Upsilon}{\Upupsilon}
\renewcommand{\Phi}{\Upphi}
\renewcommand{\Psi}{\Uppsi}
\renewcommand{\Omega}{\Upomega}
\renewcommand{\alpha}{\upalpha\xspace}
\renewcommand{\beta}{\upbeta\xspace}
\renewcommand{\gamma}{\upgamma\xspace}
\renewcommand{\delta}{\updelta\xspace}
\renewcommand{\epsilon}{\upepsilon\xspace}
\renewcommand{\zeta}{\upzeta\xspace}
\renewcommand{\eta}{\upeta\xspace}
\renewcommand{\theta}{\uptheta\xspace}
\renewcommand{\iota}{\upiota\xspace}
\renewcommand{\kappa}{\upkappa\xspace}
\renewcommand{\lambda}{\uplambda\xspace}
\renewcommand{\mu}{\upmu\xspace}
\renewcommand{\nu}{\upnu\xspace}
\renewcommand{\xi}{\upxi\xspace}
\renewcommand{\pi}{\uppi\xspace}
\renewcommand{\rho}{\uprho\xspace}
\renewcommand{\sigma}{\upsigma\xspace}
\renewcommand{\tau}{\uptau\xspace}
\renewcommand{\upsilon}{\upupsilon\xspace}
\renewcommand{\phi}{\upphi\xspace}
\renewcommand{\chi}{\upchi\xspace}
\renewcommand{\psi}{\uppsi\xspace}
\renewcommand{\omega}{\upomega\xspace}
\renewcommand{\varepsilon}{\upvarepsilon\xspace}
\renewcommand{\vartheta}{\upvartheta\xspace}
\renewcommand{\varpi}{\upvarpi\xspace}
\renewcommand{\varphi}{\upvarphi\xspace}
\renewcommand{\Gamma}{\Upgamma\xspace}
\renewcommand{\Delta}{\Updelta\xspace}
\renewcommand{\Theta}{\Uptheta\xspace}
\renewcommand{\Lambda}{\Uplambda\xspace}
\renewcommand{\Xi}{\Upxi\xspace}
\renewcommand{\Pi}{\Uppi\xspace}
\renewcommand{\Sigma}{\Upsigma\xspace}
\renewcommand{\Upsilon}{\Upupsilon\xspace}
\renewcommand{\Phi}{\Upphi\xspace}
\renewcommand{\Psi}{\Uppsi\xspace}
\renewcommand{\Omega}{\Upomega\xspace}
%---- numerical sets
\providecommand{\1}{\ensuremath{\mathds{1}}}
\providecommand{\N}{\ensuremath{\mathds{N}}}
\providecommand{\Q}{\ensuremath{\mathds{Q}}}
\providecommand{\R}{\ensuremath{\mathds{R}}}
\providecommand{\Z}{\ensuremath{\mathds{Z}}}
\providecommand{\C}{\ensuremath{\mathds{C}}}
\providecommand{\1}{\ensuremath{\mathds{1}}\xspace}
\providecommand{\N}{\ensuremath{\mathds{N}}\xspace}
\providecommand{\Q}{\ensuremath{\mathds{Q}}\xspace}
\providecommand{\R}{\ensuremath{\mathds{R}}\xspace}
\providecommand{\Z}{\ensuremath{\mathds{Z}}\xspace}
\providecommand{\C}{\ensuremath{\mathds{C}}\xspace}
%---- calligraphic letters
\providecommand{\cA}{\ensuremath{\mathcal{A}}}
\providecommand{\cB}{\ensuremath{\mathcal{B}}}
\providecommand{\cC}{\ensuremath{\mathcal{C}}}
\providecommand{\cD}{\ensuremath{\mathcal{D}}}
\providecommand{\cE}{\ensuremath{\mathcal{E}}}
\providecommand{\cF}{\ensuremath{\mathcal{F}}}
\providecommand{\cG}{\ensuremath{\mathcal{G}}}
\providecommand{\cH}{\ensuremath{\mathcal{H}}}
\providecommand{\cI}{\ensuremath{\mathcal{I}}}
\providecommand{\cJ}{\ensuremath{\mathcal{J}}}
\providecommand{\cK}{\ensuremath{\mathcal{K}}}
\providecommand{\cL}{\ensuremath{\mathcal{L}}}
\providecommand{\cM}{\ensuremath{\mathcal{M}}}
\providecommand{\cN}{\ensuremath{\mathcal{N}}}
\providecommand{\cO}{\ensuremath{\mathcal{O}}}
\providecommand{\cP}{\ensuremath{\mathcal{P}}}
\providecommand{\cQ}{\ensuremath{\mathcal{Q}}}
\providecommand{\cR}{\ensuremath{\mathcal{R}}}
\providecommand{\cS}{\ensuremath{\mathcal{S}}}
\providecommand{\cT}{\ensuremath{\mathcal{T}}}
\providecommand{\cU}{\ensuremath{\mathcal{U}}}
\providecommand{\cV}{\ensuremath{\mathcal{V}}}
\providecommand{\cW}{\ensuremath{\mathcal{W}}}
\providecommand{\cX}{\ensuremath{\mathcal{X}}}
\providecommand{\cY}{\ensuremath{\mathcal{Y}}}
\providecommand{\cZ}{\ensuremath{\mathcal{Z}}}
\providecommand{\cA}{\ensuremath{\mathcal{A}}\xspace}
\providecommand{\cB}{\ensuremath{\mathcal{B}}\xspace}
\providecommand{\cC}{\ensuremath{\mathcal{C}}\xspace}
\providecommand{\cD}{\ensuremath{\mathcal{D}}\xspace}
\providecommand{\cE}{\ensuremath{\mathcal{E}}\xspace}
\providecommand{\cF}{\ensuremath{\mathcal{F}}\xspace}
\providecommand{\cG}{\ensuremath{\mathcal{G}}\xspace}
\providecommand{\cH}{\ensuremath{\mathcal{H}}\xspace}
\providecommand{\cI}{\ensuremath{\mathcal{I}}\xspace}
\providecommand{\cJ}{\ensuremath{\mathcal{J}}\xspace}
\providecommand{\cK}{\ensuremath{\mathcal{K}}\xspace}
\providecommand{\cL}{\ensuremath{\mathcal{L}}\xspace}
\providecommand{\cM}{\ensuremath{\mathcal{M}}\xspace}
\providecommand{\cN}{\ensuremath{\mathcal{N}}\xspace}
\providecommand{\cO}{\ensuremath{\mathcal{O}}\xspace}
\providecommand{\cP}{\ensuremath{\mathcal{P}}\xspace}
\providecommand{\cQ}{\ensuremath{\mathcal{Q}}\xspace}
\providecommand{\cR}{\ensuremath{\mathcal{R}}\xspace}
\providecommand{\cS}{\ensuremath{\mathcal{S}}\xspace}
\providecommand{\cT}{\ensuremath{\mathcal{T}}\xspace}
\providecommand{\cU}{\ensuremath{\mathcal{U}}\xspace}
\providecommand{\cV}{\ensuremath{\mathcal{V}}\xspace}
\providecommand{\cW}{\ensuremath{\mathcal{W}}\xspace}
\providecommand{\cX}{\ensuremath{\mathcal{X}}\xspace}
\providecommand{\cY}{\ensuremath{\mathcal{Y}}\xspace}
\providecommand{\cZ}{\ensuremath{\mathcal{Z}}\xspace}
\providecommand{\ccA}{\ensuremath{\mathscr{A}}}
\providecommand{\ccB}{\ensuremath{\mathscr{B}}}
\providecommand{\ccC}{\ensuremath{\mathscr{C}}}
\providecommand{\ccD}{\ensuremath{\mathscr{D}}}
\providecommand{\ccE}{\ensuremath{\mathscr{E}}}
\providecommand{\ccF}{\ensuremath{\mathscr{F}}}
\providecommand{\ccG}{\ensuremath{\mathscr{G}}}
\providecommand{\ccH}{\ensuremath{\mathscr{H}}}
\providecommand{\ccI}{\ensuremath{\mathscr{I}}}
\providecommand{\ccJ}{\ensuremath{\mathscr{J}}}
\providecommand{\ccK}{\ensuremath{\mathscr{K}}}
\providecommand{\ccL}{\ensuremath{\mathscr{L}}}
\providecommand{\ccM}{\ensuremath{\mathscr{M}}}
\providecommand{\ccN}{\ensuremath{\mathscr{N}}}
\providecommand{\ccO}{\ensuremath{\mathscr{O}}}
\providecommand{\ccP}{\ensuremath{\mathscr{P}}}
\providecommand{\ccQ}{\ensuremath{\mathscr{Q}}}
\providecommand{\ccR}{\ensuremath{\mathscr{R}}}
\providecommand{\ccS}{\ensuremath{\mathscr{S}}}
\providecommand{\ccT}{\ensuremath{\mathscr{T}}}
\providecommand{\ccU}{\ensuremath{\mathscr{U}}}
\providecommand{\ccV}{\ensuremath{\mathscr{V}}}
\providecommand{\ccW}{\ensuremath{\mathscr{W}}}
\providecommand{\ccX}{\ensuremath{\mathscr{X}}}
\providecommand{\ccY}{\ensuremath{\mathscr{Y}}}
\providecommand{\ccZ}{\ensuremath{\mathscr{Z}}}
\providecommand{\ccA}{\ensuremath{\mathscr{A}}\xspace}
\providecommand{\ccB}{\ensuremath{\mathscr{B}}\xspace}
\providecommand{\ccC}{\ensuremath{\mathscr{C}}\xspace}
\providecommand{\ccD}{\ensuremath{\mathscr{D}}\xspace}
\providecommand{\ccE}{\ensuremath{\mathscr{E}}\xspace}
\providecommand{\ccF}{\ensuremath{\mathscr{F}}\xspace}
\providecommand{\ccG}{\ensuremath{\mathscr{G}}\xspace}
\providecommand{\ccH}{\ensuremath{\mathscr{H}}\xspace}
\providecommand{\ccI}{\ensuremath{\mathscr{I}}\xspace}
\providecommand{\ccJ}{\ensuremath{\mathscr{J}}\xspace}
\providecommand{\ccK}{\ensuremath{\mathscr{K}}\xspace}
\providecommand{\ccL}{\ensuremath{\mathscr{L}}\xspace}
\providecommand{\ccM}{\ensuremath{\mathscr{M}}\xspace}
\providecommand{\ccN}{\ensuremath{\mathscr{N}}\xspace}
\providecommand{\ccO}{\ensuremath{\mathscr{O}}\xspace}
\providecommand{\ccP}{\ensuremath{\mathscr{P}}\xspace}
\providecommand{\ccQ}{\ensuremath{\mathscr{Q}}\xspace}
\providecommand{\ccR}{\ensuremath{\mathscr{R}}\xspace}
\providecommand{\ccS}{\ensuremath{\mathscr{S}}\xspace}
\providecommand{\ccT}{\ensuremath{\mathscr{T}}\xspace}
\providecommand{\ccU}{\ensuremath{\mathscr{U}}\xspace}
\providecommand{\ccV}{\ensuremath{\mathscr{V}}\xspace}
\providecommand{\ccW}{\ensuremath{\mathscr{W}}\xspace}
\providecommand{\ccX}{\ensuremath{\mathscr{X}}\xspace}
\providecommand{\ccY}{\ensuremath{\mathscr{Y}}\xspace}
\providecommand{\ccZ}{\ensuremath{\mathscr{Z}}\xspace}
%---- roman letters
\providecommand{\rA}{\ensuremath{\mathrm{A}}}
\providecommand{\rB}{\ensuremath{\mathrm{B}}}
\providecommand{\rC}{\ensuremath{\mathrm{C}}}
\providecommand{\rD}{\ensuremath{\mathrm{D}}}
\providecommand{\rE}{\ensuremath{\mathrm{E}}}
\providecommand{\rF}{\ensuremath{\mathrm{F}}}
\providecommand{\rG}{\ensuremath{\mathrm{G}}}
\providecommand{\rH}{\ensuremath{\mathrm{H}}}
\providecommand{\rI}{\ensuremath{\mathrm{I}}}
\providecommand{\rJ}{\ensuremath{\mathrm{J}}}
\providecommand{\rK}{\ensuremath{\mathrm{K}}}
\providecommand{\rL}{\ensuremath{\mathrm{L}}}
\providecommand{\rM}{\ensuremath{\mathrm{M}}}
\providecommand{\rN}{\ensuremath{\mathrm{N}}}
\providecommand{\rO}{\ensuremath{\mathrm{O}}}
\providecommand{\rP}{\ensuremath{\mathrm{P}}}
\providecommand{\rQ}{\ensuremath{\mathrm{Q}}}
\providecommand{\rR}{\ensuremath{\mathrm{R}}}
\providecommand{\rS}{\ensuremath{\mathrm{S}}}
\providecommand{\rT}{\ensuremath{\mathrm{T}}}
\providecommand{\rU}{\ensuremath{\mathrm{U}}}
\providecommand{\rV}{\ensuremath{\mathrm{V}}}
\providecommand{\rW}{\ensuremath{\mathrm{W}}}
\providecommand{\rX}{\ensuremath{\mathrm{X}}}
\providecommand{\rY}{\ensuremath{\mathrm{Y}}}
\providecommand{\rZ}{\ensuremath{\mathrm{Z}}}
\providecommand{\rA}{\ensuremath{\mathrm{A}}\xspace}
\providecommand{\rB}{\ensuremath{\mathrm{B}}\xspace}
\providecommand{\rC}{\ensuremath{\mathrm{C}}\xspace}
\providecommand{\rD}{\ensuremath{\mathrm{D}}\xspace}
\providecommand{\rE}{\ensuremath{\mathrm{E}}\xspace}
\providecommand{\rF}{\ensuremath{\mathrm{F}}\xspace}
\providecommand{\rG}{\ensuremath{\mathrm{G}}\xspace}
\providecommand{\rH}{\ensuremath{\mathrm{H}}\xspace}
\providecommand{\rI}{\ensuremath{\mathrm{I}}\xspace}
\providecommand{\rJ}{\ensuremath{\mathrm{J}}\xspace}
\providecommand{\rK}{\ensuremath{\mathrm{K}}\xspace}
\providecommand{\rL}{\ensuremath{\mathrm{L}}\xspace}
\providecommand{\rM}{\ensuremath{\mathrm{M}}\xspace}
\providecommand{\rN}{\ensuremath{\mathrm{N}}\xspace}
\providecommand{\rO}{\ensuremath{\mathrm{O}}\xspace}
\providecommand{\rP}{\ensuremath{\mathrm{P}}\xspace}
\providecommand{\rQ}{\ensuremath{\mathrm{Q}}\xspace}
\providecommand{\rR}{\ensuremath{\mathrm{R}}\xspace}
\providecommand{\rS}{\ensuremath{\mathrm{S}}\xspace}
\providecommand{\rT}{\ensuremath{\mathrm{T}}\xspace}
\providecommand{\rU}{\ensuremath{\mathrm{U}}\xspace}
\providecommand{\rV}{\ensuremath{\mathrm{V}}\xspace}
\providecommand{\rW}{\ensuremath{\mathrm{W}}\xspace}
\providecommand{\rX}{\ensuremath{\mathrm{X}}\xspace}
\providecommand{\rY}{\ensuremath{\mathrm{Y}}\xspace}
\providecommand{\rZ}{\ensuremath{\mathrm{Z}}\xspace}
%---- frak letters
\providecommand{\fA}{\ensuremath{\mathfrak{A}}}
\providecommand{\fB}{\ensuremath{\mathfrak{B}}}
\providecommand{\fC}{\ensuremath{\mathfrak{C}}}
\providecommand{\fD}{\ensuremath{\mathfrak{D}}}
\providecommand{\fE}{\ensuremath{\mathfrak{E}}}
\providecommand{\fF}{\ensuremath{\mathfrak{F}}}
\providecommand{\fG}{\ensuremath{\mathfrak{G}}}
\providecommand{\fH}{\ensuremath{\mathfrak{H}}}
\providecommand{\fI}{\ensuremath{\mathfrak{I}}}
\providecommand{\fJ}{\ensuremath{\mathfrak{J}}}
\providecommand{\fK}{\ensuremath{\mathfrak{K}}}
\providecommand{\fL}{\ensuremath{\mathfrak{L}}}
\providecommand{\fM}{\ensuremath{\mathfrak{M}}}
\providecommand{\fN}{\ensuremath{\mathfrak{N}}}
\providecommand{\fO}{\ensuremath{\mathfrak{O}}}
\providecommand{\fP}{\ensuremath{\mathfrak{P}}}
\providecommand{\fQ}{\ensuremath{\mathfrak{Q}}}
\providecommand{\fR}{\ensuremath{\mathfrak{R}}}
\providecommand{\fS}{\ensuremath{\mathfrak{S}}}
\providecommand{\fT}{\ensuremath{\mathfrak{T}}}
\providecommand{\fU}{\ensuremath{\mathfrak{U}}}
\providecommand{\fV}{\ensuremath{\mathfrak{V}}}
\providecommand{\fW}{\ensuremath{\mathfrak{W}}}
\providecommand{\fX}{\ensuremath{\mathfrak{X}}}
\providecommand{\fY}{\ensuremath{\mathfrak{Y}}}
\providecommand{\fZ}{\ensuremath{\mathfrak{Z}}}
\providecommand{\fA}{\ensuremath{\mathfrak{A}}\xspace}
\providecommand{\fB}{\ensuremath{\mathfrak{B}}\xspace}
\providecommand{\fC}{\ensuremath{\mathfrak{C}}\xspace}
\providecommand{\fD}{\ensuremath{\mathfrak{D}}\xspace}
\providecommand{\fE}{\ensuremath{\mathfrak{E}}\xspace}
\providecommand{\fF}{\ensuremath{\mathfrak{F}}\xspace}
\providecommand{\fG}{\ensuremath{\mathfrak{G}}\xspace}
\providecommand{\fH}{\ensuremath{\mathfrak{H}}\xspace}
\providecommand{\fI}{\ensuremath{\mathfrak{I}}\xspace}
\providecommand{\fJ}{\ensuremath{\mathfrak{J}}\xspace}
\providecommand{\fK}{\ensuremath{\mathfrak{K}}\xspace}
\providecommand{\fL}{\ensuremath{\mathfrak{L}}\xspace}
\providecommand{\fM}{\ensuremath{\mathfrak{M}}\xspace}
\providecommand{\fN}{\ensuremath{\mathfrak{N}}\xspace}
\providecommand{\fO}{\ensuremath{\mathfrak{O}}\xspace}
\providecommand{\fP}{\ensuremath{\mathfrak{P}}\xspace}
\providecommand{\fQ}{\ensuremath{\mathfrak{Q}}\xspace}
\providecommand{\fR}{\ensuremath{\mathfrak{R}}\xspace}
\providecommand{\fS}{\ensuremath{\mathfrak{S}}\xspace}
\providecommand{\fT}{\ensuremath{\mathfrak{T}}\xspace}
\providecommand{\fU}{\ensuremath{\mathfrak{U}}\xspace}
\providecommand{\fV}{\ensuremath{\mathfrak{V}}\xspace}
\providecommand{\fW}{\ensuremath{\mathfrak{W}}\xspace}
\providecommand{\fX}{\ensuremath{\mathfrak{X}}\xspace}
\providecommand{\fY}{\ensuremath{\mathfrak{Y}}\xspace}
\providecommand{\fZ}{\ensuremath{\mathfrak{Z}}\xspace}
%---- groups
\providecommand{\O}[1]{\ensuremath{\mathrm{O}(#1)}}
\providecommand{\SO}[1]{\ensuremath{\mathrm{SO}(#1)}}
\providecommand{\U}[1]{\ensuremath{\mathrm{U}(#1)}}
\providecommand{\SU}[1]{\ensuremath{\mathrm{SU}(#1)}}
\providecommand{\SL}[2]{\ensuremath{\mathrm{SL}_{#1}(#2)}}
\providecommand{\GL}[2]{\ensuremath{\mathrm{GL}_{#1}(#2)}}
\providecommand{\OO}[1]{\ensuremath{\mathrm{O}(#1)}\xspace}
\providecommand{\SO}[1]{\ensuremath{\mathrm{SO}(#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}
\providecommand{\GL}[2]{\ensuremath{\mathrm{GL}_{#1}(#2)}\xspace}
%---- algebras
\providecommand{\liebraket}[2]{\ensuremath{\left[ #1,\, #2 \right]}}
%---- inline partial derivatives (use $\ipd{s}$ for $\partial_s$)

363
sec/part1/introduction.tex
View File

@@ -11,6 +11,9 @@ 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.
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 string theory.
In this introduction we present instruments and preparatory frameworks used throughout the manuscript as many tools 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 physics.
\subsection{Properties of String Theory and Conformal Symmetry}
@@ -257,7 +260,7 @@ The transformation on a field $\phi_{\omega, \bomega}$ of weight $(\omega, \bome
\begin{split}
\delta_{\epsilon, \bepsilon} \phi_{\omega, \bomega}
& =
\left[ Q_{\epsilon, \bepsilon}, \phi_{\omega, \bomega}( w, \bw ) \right]
\liebraket{Q_{\epsilon, \bepsilon}}{\phi_{\omega, \bomega}( w, \bw )}
\\
& =
\cint{0} \ddz \epsilon(z) \left[ T(z), \phi_{\omega, \bomega}( w, \bw ) \right]
@@ -371,15 +374,15 @@ This is a reflection of the anomalous algebra of the operator modes $L_n$ and $\
This ultimately leads to the quantum algebra
\begin{equation}
\begin{split}
\left[ L_n, L_m \right]
\liebraket{L_n}{L_m}
& =
(n - m)\, L_{n + m} + \frac{c}{12}\, n\, (n^2 - 1)\, \delta_{n, -m},
\\
\left[ \bL_n, \bL_m \right]
\liebraket{\bL_n}{\bL_m}
& =
(n - m)\, \bL_{n + m} + \frac{\overline{c}}{12}\, n\, (n^2 - 1)\, \delta_{n, -m},
\\
\left[ L_n, \bL_m \right]
\liebraket{L_n}{\bL_m}
& =
0,
\end{split}
@@ -443,6 +446,7 @@ Finally the definitions of the primary operators~\eqref{eq:conf:primary} define
\quad
n \ge 1.
\end{split}
\label{eq:conf:physical}
\end{equation}
From this definition we can build an entire representation of \emph{descendant} states applying any operator $L_{-n}$ (or $\bL_{-n}$) with $n \ge 1$ to $\ket{\phi_{\omega, \bomega}}$.
@@ -479,6 +483,7 @@ and the components of the stress-energy tensor~\eqref{eq:conf:stringT} are
\\
\bT( \bz ) & = \ipd{\bz} \bX( \bz ) \cdot \ipd{\bz} \bX( \bz ).
\end{split}
\label{eq:conf:bosonicstringT}
\end{equation}
Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \bz ) X^{\nu}( w, \bw ) \right\rangle = - \frac{1}{2} \eta^{\mu\nu} \ln\left| z - w \right|$ we can show 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 $\overline{b}(z)$ and $\overline{c}(z)$.
@@ -604,11 +609,11 @@ In this case the components of the stress-energy tensor of the theory are:
\begin{split}
T( z )
& =
-\frac{1}{\ap} \ipd{z} X( z ) \cdot \ipd{z} X( z ) - \frac{1}{2} \psi( z ) \cdot \ipd{z} \psi( z ),
-\frac{1}{\ap}\, \ipd{z} X( z ) \cdot \ipd{z} X( z ) - \frac{1}{2}\, \psi( z ) \cdot \ipd{z} \psi( z ),
\\
\bT( \bz )
& =
-\frac{1}{\ap} \ipd{\bz} \bX( \bz ) \cdot \ipd{\bz} \bX( \bz ) - \frac{1}{2} \bpsi( \bz ) \cdot \ipd{\bz} \bpsi( \bz ).
-\frac{1}{\ap}\, \ipd{\bz} \bX( \bz ) \cdot \ipd{\bz} \bX( \bz ) - \frac{1}{2}\, \bpsi( \bz ) \cdot \ipd{\bz} \bpsi( \bz ).
\end{split}
\end{equation}
@@ -621,11 +626,13 @@ The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmet
& =
\epsilon( z )\, \psi^{\mu}( z ) + \bepsilon( \bz )\, \bpsi^{\mu}( \bz ),
\\
\sqrt{\frac{2}{\ap}} \delta_{\epsilon} \psi^{\mu}( z )
\sqrt{\frac{2}{\ap}}\,
\delta_{\epsilon} \psi^{\mu}( z )
& =
- \epsilon( z )\, \ipd{z} X^{\mu}( z ),
\\
\sqrt{\frac{2}{\ap}} \delta_{\bepsilon} \bpsi^{\mu}( \bz )
\sqrt{\frac{2}{\ap}}\,
\delta_{\bepsilon} \bpsi^{\mu}( \bz )
& =
- \bepsilon( \bz )\, \ipd{\bz} \bX^{\mu}( \bz )
\end{split}
@@ -715,8 +722,8 @@ In particular $\ccX_6$ should be a compact manifold to ``hide'' the 6 extra-dime
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}).
Their existence was later proved by Shing-Tung Yau~\cite{Yau:1977:CalabiConjectureNew}, hence the name Calabi-Yau (\cy) manifolds.
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,Greene:1997:StringTheoryCalabiYau}).
\subsubsection{Complex and Kähler Manifolds}
@@ -725,20 +732,20 @@ In general an \emph{almost complex structure} $J$ is a tensor such that $\tensor
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
\tensor{N}{^a_{bc}}\, v_p^b\, w_p^c
=
\left(
[ v_p, w_p ]
\liebraket{v_p}{w_p}
+
J
\left( [ J\, v_p, w_p ] + [ v_p, J\, w_p ] \right)
\left( \liebraket{J\, v_p}{w_p} + \liebraket{v_p}{J\, w_p} \right)
-
[ J\, v_p, J\, w_p ]
\liebraket{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.
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.
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.
@@ -768,14 +775,14 @@ The metric is \emph{Hermitian} if
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}}.
% \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}
@@ -784,31 +791,319 @@ In this case we can define a $(1, 1)$-form $\omega$ as
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}}.
% \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}:
$(M, J, g)$ is a \emph{Kähler} manifold if:
\begin{equation}
\dd{\omega}
=
\left( \ipd{z} + \ipd{\bz} \right)
\left( \pd + \bpd \right)
\omega(z, \bz)
=
0,
\label{eq:cy:kaehler}
\end{equation}
or equivalently $\nabla J = 0$ or $\nabla \omega = 0$, where $\nabla$ is the connection of $g$.
Notice that the operators $\pd$ and $\bpd$ are operators such that $\pd^2 = \bpd^2 = 0$: they replace the \emph{de Rham cohomology} operator $\mathrm{d}^2 = 0$ in complex space with the holomorphic and antiholomorphic \emph{Dolbeault cohomology} operators.
The covariant conservation of $J$ and $\omega$ implies that the holonomy group must preserve these objects in $\R^{2m}$.
Thus we have $\mathrm{Hol}(g) \subseteq \U{m} \subset \mathrm{O}(2m)$.
Thus we have $\mathrm{Hol}(g) \subseteq \U{m} \subset \OO{2m}$.
\subsubsection{Calabi-Yau Manifolds}
With the general definitions of the Kähler geometry we can now explicitly compute the conditions needed for a \cy manifold.
In local coordinates a Hermitian metric is such that
\begin{equation}
g
=
g_{a\overline{b}}\, \dd{z}^a \otimes \dd{\bz}^{\overline{b}}
+
g_{\overline{a}b}\, \dd{\bz}^{\overline{a}} \otimes \dd{z}^b,
\end{equation}
thus the Kähler form becomes $\omega = i g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}$.
The relation~\eqref{eq:cy:kaehler} then translates into:
\begin{equation}
\dd{\omega} = i\, \left( \pd + \bpd \right)\, g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}
=
0
\quad
\Leftrightarrow
\quad
\begin{cases}
\ipd{z^c} g_{a\overline{b}} & = \ipd{z^a} g_{c\overline{b}}
\\
\ipd{\bz^c} g_{\overline{a}b} & = \ipd{\bz^a} g_{\overline{c}b}
\end{cases}.
\end{equation}
The $(1,1)$-form $\omega$ can locally be written as $\omega = i\, \pd \bpd\, \phi( z, \bz )$ up to a constant.
This ultimately leads to
\begin{equation}
g_{a\overline{b}}
=
\pdv{\phi( z, \bz )}{z^a}{\bz^{\overline{b}}}
=
\ipd{a} \ipd{\overline{b}}\, \phi( z, \bz ),
\end{equation}
Since $\omega$ is the Kähler form then the Levi-Civita connection has only fully holomorphic and anti-holomorphic components:
\begin{equation}
\tensor{\Gamma}{^a_{bc}}
=
\tensor{g}{^{a\overline{d}}}\,
\ipd{b}\,
\tensor{g}{_{\overline{d}c}},
\qquad
\tensor{\Gamma}{^{\overline{a}}_{\overline{b}\overline{c}}}
=
\tensor{g}{^{\overline{a}d}}\,
\ipd{\overline{b}}\,
\tensor{g}{_{d\overline{c}}}.
\end{equation}
As a consequence the Ricci tensor becomes
\begin{equation}
\tensor{R}{_{\overline{a}b}}
=
-
\pdv{\tensor{\Gamma}{^{\overline{c}}_{\overline{a}\overline{c}}}}{z^b}.
\end{equation}
Since \cy manifolds have \SU{m} holonomy, the trace part of the coefficients of the connection vanishes.
\cy manifolds thus have $\tensor{R}{_{\overline{a}b}} = 0$, that is they are complex Ricci-flat Kähler manifolds.
\subsubsection{Cohomology and Hodge Numbers}
\cy manifolds $M$ of complex dimension $m$ present geometric characteristics of general interest both in pure mathematics and string theory.
They can be characterised in different ways.
For instance the study of the cohomology groups of the manifold has a direct connection with the analysis of topological invariants.
For real manifolds $M$ of dimension $2m$, closed $p$-forms $\omega$ are always defined up to an \emph{exact} term.
In fact:
\begin{equation}
\dd{\omega'_{(p)}} = \dd{(\omega_{(p)} + \dd{\eta_{(p-1)}})} = 0
\label{eq:cy:closedform}
\end{equation}
implies an equivalence relation $\omega'_{(p)} \sim \omega_{(p)} + \dd{\eta_{(p-1)}}$.
This translates in the fact that elements of the de Rham cohomology group $H^{(p)}_{\mathrm{d}}(M, \R)$ are equivalence classes $[ \omega ]$ computed through the operator $\mathrm{d}$.
The term $b^{p} = \dim{H^{(p)}_{\mathrm{d}}( M, \R )}$ counts the total number of possible $p$-forms we can build on $X$, up to \emph{gauge transformations}.
These are known as \emph{Betti numbers}.
The extension to the Dolbeault cohomology in complex space is possible through the operators $\pd$ and $\bpd$ over $(r, s)$-forms on manifolds $M$ of real dimension $2m$.
The equivalence relation~\eqref{eq:cy:closedform} has a similar expression in complex space as
\begin{equation}
\omega'_{(r,s)} \sim \omega_{(r,s)} + \bpd \omega_{(r,s-1)},
\end{equation}
or an equivalent formulation using $\pd$.
The cohomology group in this case is $H^{(r,s)}_{\bpd}( M, \C )$ and the relation with the real counterpart is
\begin{equation}
H^{(p)}_{\mathrm{d}}( M, \R )
=
\bigoplus\limits_{p = r + s}\,
H^{(r,s)}_{\bpd}( M, \C ).
\end{equation}
As in the case of Betti numbers, we can define the complex equivalents, the \emph{Hodge numbers}, $h^{r,s} = \dim\limits_{\C} H^{(r,s)}_{\bpd}( M, \C )$ which count the number of harmonic $(r, s)$-forms on $M$.
Notice that in this case $h^{r,s}$ is the complex dimension $\dim\limits_{\C}$ of the cohomology group.
For \cy manifolds it is possible to show that the \SU{m} holonomy of $g$ implies that the vector space of $(r, 0)$-forms is \C if $r = 0$ or $r = m$.
Therefore $h^{0,0} = h^{m,0} = 1$, while $h^{r,0} = 0$ if $r \neq 0,\, m$.
Exploiting symmetries of the cohomology groups, Hodge numbers are usually collected in \emph{Hodge diamonds}.
In string theory we are ultimately interested in \cy manifolds of real dimensions $6$, thus we focus mainly on \cy $3$-folds (i.e.\ having $m = 3$).
The diamond in this case is
\begin{equation}
\mqty{%
& & & h^{0,0} & & &
\\
& & h^{1,0} & & h^{0,1} & &
\\
& h^{2,0} & & h^{1,1} & & h^{0,2} &
\\
h^{3,0} & & h^{2,1} & & h^{1,2} & & h^{0,3}
\\
& h^{3,1} & & h^{2,2} & & h^{1,3} &
\\
& & h^{3,2} & & h^{2,3} & &
\\
& & & h^{3,3} & & &
}
\quad
=
\quad
\mqty{%
& & & 1 & & &
\\
& & 0 & & 0 & &
\\
& 0 & & h^{1,1} & & 0 &
\\
1 & & h^{2,1} & & h^{2,1} & & 1
\\
& 0 & & h^{1,1} & & 0 &
\\
& & 0 & & 0 & &
\\
& & & 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 $m = 3$.
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.
\subsection{D-branes and Open Strings}
Dirichlet branes, or \emph{D-branes}, are another key mathematical object in string theory.
They are naturally included as extended object as hypersurfaces supporting strings with open topology and as physical objects with charge and tension~\cite{Polchinski:1995:DirichletBranesRamondRamond,Polchinski:1996:TASILecturesDBranes,DiVecchia:1999:DbranesStringTheory,DiVecchia:2000:BranesStringTheory,DiVecchia:1997:ClassicalPbranesBoundary,Taylor:2003:LecturesDbranesTachyon,Taylor:2004:DBranesTachyonsString,Johnson:2000:DBranePrimer}.
They are relevant in the definition of phenomenological models in string theory as they can be arranged to support chiral fermions and bosons in \sm-like scenarios as well as beyond~\cite{Honecker:2012:FieldTheoryStandard,Lust:2009:LHCStringHunter,Zwiebach::FirstCourseString}.
We are ultimately interested in their study to construct Yukawa couplings in string theory.
\subsubsection{Compactification of Closed Strings}
As a first approach to the definition of D-branes, consider the action~\eqref{eq:conf:polyakov}.
The variation of such action with respect to $\delta X$ leads to the equation of motion
\begin{equation}
\partial_{\alpha} \partial^{\alpha}\, X^{\mu}( \tau, \sigma ) = 0
\qquad
\mu = 0, 1, \dots, D - 1,
\end{equation}
and naturally to the \emph{Neumann} boundary conditions:\footnotemark{}
\footnotetext{%
As~\cite{Polchinski:1996:TASILecturesDBranes} shows, \emph{Dirichlet} conditions can be shown to descend from T-duality.
}
\begin{equation}
\eval{\ipd{\sigma} X^{\mu}( \tau, \sigma )}_{\sigma = 0}^{\sigma = \ell} = 0,
\qquad
\mu = 0, 1, \dots, D - 1.
\end{equation}
Closed strings are such that $X^{\mu}( \tau, \sigma + \ell ) = X^{\mu}( \tau, \sigma )$.
The usual mode expansion in conformal coordinates $X^{\mu}( z, \bz ) = X( z ) + \bX( \bz )$ leads to
\begin{equation}
\begin{split}
X^{\mu}( z )
& =
x_0^{\mu}
+
i\, \sqrt{\frac{\ap}{2}}\,
\left(
- \alpha_0^{\mu}\, \ln{z}
+ \sum\limits_{n \in \Z \setminus \{0\}} \frac{\alpha_n^{\mu}}{n} z^{-n}
\right),
\\
\bX^{\mu}( \bz )
& =
\overline{x}_0^{\mu}
+
i\, \sqrt{\frac{\ap}{2}}\,
\left(
- \balpha_0^{\mu}\, \ln{\bz}
+ \sum\limits_{n \in \Z \setminus \{0\}} \frac{\balpha_n^{\mu}}{n} \bz^{-n}
\right),
\end{split}
\end{equation}
where $\alpha_0^{\mu} = \balpha_0^{\mu}$ and $\ell = 2 \pi$.
When the string is free to move in the entire $D$-dimensional space, then the momentum of the center of mass is $p^{\mu} = \frac{1}{\sqrt{2 \ap}} ( \alpha_0^{\mu} + \balpha_0^{\mu} )$.
Now let
\begin{equation}
\ccM^{1, D - 1} = \ccM^{1, D - 2} \otimes S^1( R ),
\end{equation}
where $S^1( R )$ is a compact $1$-dimensional circle of radius $R$ such that the boundary conditions for the compact coordinate are
\begin{equation}
X^{D - 1}( z\, e^{2\pi i}, \bz\, e^{-2\pi i} )
=
X^{D - 1}( z, \bz ) + 2 \pi\, m\, R,
\qquad
m \in \Z.
\label{eq:dbranes:winding}
\end{equation}
This is cast into
\begin{equation}
\begin{split}
\alpha_0^{D-1} + \balpha_0^{D-1} & = \sqrt{\frac{2}{\ap}}\, n\, \frac{\ap}{R},
\qquad
n \in \Z,
\\
\alpha_0^{D-1} - \balpha_0^{D-1} & = \sqrt{\frac{2}{\ap}}\, m\, R,
\qquad
m \in \Z,
\end{split}
\end{equation}
respectively encoding the quantisation of the momentum for a compact coordinate and the \emph{winding} in the compact direction~\eqref{eq:dbranes:winding}.
We finally have
\begin{equation}
\begin{split}
\alpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \left( n\, \frac{\ap}{R} + m\, R \right),
\\
\balpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \left( n\, \frac{\ap}{R} - m\, R \right),
\end{split}
\end{equation}
An interesting phenomenon involving these quantities appears when computing the mass spectrum of the theory.
From~\eqref{eq:conf:Texpansion} and \eqref{eq:conf:bosonicstringT} we find
\begin{equation}
\begin{split}
L_0
&=
\frac{\ap}{2}\,
\left(
\left( \alpha_0^{D-1} \right)^2
+
\sum\limits_{i = 0}^{D-2}\, \left( \alpha_0^i \right)^2
+
\sum\limits_{n = 1}^{+\infty}\, \left( 2 \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a \right)
\right),
\\
\bL_0
&=
\frac{\ap}{2}\,
\left(
\left( \balpha_0^{D-1} \right)^2
+
\sum\limits_{i = 0}^{D-2}\, \left( \balpha_0^i \right)^2
+
\sum\limits_{n = 1}^{+\infty}\, \left( 2 \balpha_{-n}^{\mu} \balpha_n^{\nu}\, \eta_{\mu\nu} + a \right)
\right),
\end{split}
\end{equation}
where $a$ is constant given by normal ordering, representing the zero point energy of the theory.
Imposing physical conditions~\eqref{eq:conf:physical} and the \emph{level matching} $(L_0 - \bL_0) \ket{\phi} = 0$ for closed strings, we find
\begin{equation}
\begin{split}
M^2
& =
\frac{1}{\ap^2}\, \left( n\, \frac{\ap}{R} + m\, R \right)^2
+
\frac{4}{\ap}\, \left( \rN + a \right)
\\
& =
\frac{1}{\ap^2}\, \left( n\, \frac{\ap}{R} - m\, R \right)^2
+
\frac{4}{\ap}\, \left( \overline{\rN} + a \right),
\end{split}
\label{eq:dbranes:closedspectrum}
\end{equation}
where $\rN = \sum\limits_{n = 1}^{+\infty}\, \alpha_{-n} \cdot \alpha_n$ and $\overline{\rN} = \sum\limits_{n = 1}^{+\infty}\, \balpha_{-n} \cdot \balpha_n$.
We then notice that as $R \to \infty$ all states with $m \neq 0$ become infinitely massive while the states for $m = 0$ and all values of $n$ become a continuum.
Conversely, as $R \to 0$ all states with $n \neq 0$ become infinitely heavy.
In field theory this would translate into a reduction of the number of dimensions since the remaining fields would be independent of the compact coordinate~\cite{Polchinski:1996:TASILecturesDBranes,Zwiebach::FirstCourseString}.
However in closed string theory as $R \to 0$ the compactified dimension is again present.
As seen in~\eqref{eq:dbranes:closedspectrum} the mass spectra of the theories compactified at radius $R$ or $\ap\, R^{-1}$ are the same under the exchange of $n$ and $m$.
At the level of the modes this \emph{T-duality} acts by swapping the sign of the right zero-modes in the compact direction
\begin{equation}
\alpha_0^{D-1} \stackrel{T}{\longmapsto} \alpha_0^{D-1},
\qquad
\balpha_0^{D-1} \stackrel{T}{\longmapsto} - \balpha_0^{D-1}.
\end{equation}
\subsubsection{D-branes from T-duality}
\subsection{Twist Fields and Spin Fields}
% vim ft=tex

205
thesis.bib
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@@ -10,6 +10,18 @@
file = {/home/riccardo/.local/share/zotero/files/anderson_karkheiran_2018_tasi_lectures_on_geometric_tools_for_string_compactifications.pdf}
}
@article{Angelantonj:2002:OpenStrings,
title = {Open {{Strings}}},
author = {Angelantonj, Carlo and Sagnotti, Augusto},
date = {2002-04},
doi = {10.1016/s0370-1573(02)00273-9},
abstract = {This review is devoted to open strings, and in particular to the often surprising features of their spectra. It follows and summarizes developments that took place mainly at the University of Rome “Tor Vergata” over the last decade, and centred on world-sheet aspects of the constructions now commonly referred to as “orientifolds”. Our presentation aims to bridge the gap between the world-sheet analysis, that first exhibited many of the novel features of these systems, and their geometric description in terms of extended objects, D-branes and O-planes, contributed by many other colleagues, and most notably by J. Polchinski. We therefore proceed through a number of prototype examples, starting from the bosonic string and moving on to ten-dimensional fermionic strings and their toroidal and orbifold compactifications, in an attempt to guide the reader in a self-contained journey to the more recent developments related to the breaking of supersymmetry.},
archivePrefix = {arXiv},
eprint = {hep-th/0204089},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/angelantonj_sagnotti_2002_open_strings.pdf}
}
@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},
@@ -85,7 +97,7 @@
author = {Di Francesco, Philippe and Mathieu, Pierre and Sénéchal, David},
date = {1997},
publisher = {{Springer New York}},
location = {{New York, NY}},
location = {{New York}},
doi = {10.1007/978-1-4612-2256-9},
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},
@@ -93,6 +105,51 @@
series = {Graduate {{Texts}} in {{Contemporary Physics}}}
}
@article{DiVecchia:1997:ClassicalPbranesBoundary,
title = {Classical P-Branes from Boundary State},
author = {Di Vecchia, Paolo and Frau, Marialuisa and Pesando, Igor and Sciuto, Stefano and Lerda, Alberto and Russo, Rodolfo},
date = {1997-07},
journaltitle = {Nuclear Physics B},
volume = {507},
pages = {259--276},
issn = {05503213},
doi = {10.1016/s0550-3213(97)00576-2},
abstract = {We show that the boundary state description of a Dp-brane is strictly related to the corresponding classical solution of the low-energy string effective action. By projecting the boundary state on the massless states of the closed string we obtain the tension, the R-R charge and the large distance behavior of the classical solution. We discuss both the case of a single D-brane and that of bound states of two D-branes. We also show that in the R-R sector the boundary state, written in a picture which treats asymmetrically the left and right components, directly yields the R-R gauge potentials.},
archivePrefix = {arXiv},
eprint = {hep-th/9707068},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/di_vecchia_et_al_1997_classical_p-branes_from_boundary_state.pdf},
number = {1-2}
}
@inproceedings{DiVecchia:1999:DbranesStringTheory,
title = {D-Branes in String Theory {{II}}},
booktitle = {{{YITP}} Workshop on Developments in Superstring and {{M}} Theory},
author = {Di Vecchia, P. and Liccardo, Antonella},
date = {1999-12},
pages = {7--48},
archivePrefix = {arXiv},
eprint = {hep-th/9912275},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/di_vecchia_liccardo_1999_d-branes_in_string_theory.pdf}
}
@article{DiVecchia:2000:BranesStringTheory,
title = {D Branes in String Theory {{I}}},
author = {Di Vecchia, Paolo and Liccardo, Antonella},
editor = {Thorlacius, Lárus and Jonsson, Thordur},
date = {2000},
journaltitle = {NATO Sci. Ser. C},
volume = {556},
pages = {1--60},
doi = {10.1007/978-94-011-4303-5_1},
archivePrefix = {arXiv},
eprint = {hep-th/9912161},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/di_vecchia_liccardo_2000_d_branes_in_string_theory,_i.pdf},
number = {NORDITA-1999-77-HE}
}
@article{Friedan:1986:ConformalInvarianceSupersymmetry,
title = {Conformal Invariance, Supersymmetry and String Theory},
author = {Friedan, Daniel and Martinec, Emil and Shenker, Stephen},
@@ -167,11 +224,76 @@
series = {Cambridge Monographs on Mathematical Physics}
}
@article{Greene:1997:StringTheoryCalabiYau,
title = {String {{Theory}} on {{Calabi}}-{{Yau Manifolds}}},
author = {Greene, Brian},
date = {1997-02},
url = {http://arxiv.org/abs/hep-th/9702155},
urldate = {2020-09-03},
abstract = {These lectures are devoted to introducing some of the basic features of quantum geometry that have been emerging from compactified string theory over the last couple of years. The developments discussed include new geometric features of string theory which occur even at the classical level as well as those which require non-perturbative effects. These lecture notes are based on an evolving set of lectures presented at a number of schools but most closely follow a series of seven lectures given at the TASI-96 summer school on Strings, Fields and Duality.},
archivePrefix = {arXiv},
eprint = {hep-th/9702155},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/greene_1997_string_theory_on_calabi-yau_manifolds.pdf;/home/riccardo/.local/share/zotero/storage/R7F26ND6/9702155.html}
}
@article{He:2020:CalabiyauSpacesString,
title = {Calabi-Yau Spaces in the String Landscape},
author = {He, Yang-Hui},
date = {2020-06},
archivePrefix = {arXiv},
eprint = {2006.16623},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/he_2020_calabi-yau_spaces_in_the_string_landscape.pdf},
keywords = {⛔ No DOI found}
}
@article{Honecker:2012:FieldTheoryStandard,
title = {Towards the Field Theory of the {{Standard Model}} on Fractional {{D6}}-Branes on {{T6}}/{{Z6}}: {{Yukawa}} Couplings and Masses},
author = {Honecker, Gabriele and Vanhoof, Joris},
date = {2012-01},
journaltitle = {Fortschritte der Physik},
volume = {60},
pages = {1050--1056},
issn = {00158208},
doi = {10.1002/prop.201200016},
abstract = {We present the perturbative Yukawa couplings of the Standard Model on fractional intersecting D6-branes on T6/Z6' and discuss two mechanisms of creating mass terms for the vector-like particles in the matter spectrum, through perturbative three-point couplings and through continuous D6-brane displacements.},
archivePrefix = {arXiv},
eprint = {1201.5872},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/honecker_vanhoof_2012_towards_the_field_theory_of_the_standard_model_on_fractional_d6-branes_on_t6-ℤ6′.pdf},
number = {9-10}
}
@book{Hubsch:1992:CalabiyauManifoldsBestiary,
title = {Calabi-Yau Manifolds: {{A}} Bestiary for Physicists},
author = {Hubsch, Tristan},
date = {1992},
publisher = {{World Scientific}},
file = {/home/riccardo/.local/share/zotero/files/hubsch_1992_calabi-yau_manifolds.pdf},
isbn = {978-981-02-1927-7}
}
@article{Johnson:2000:DBranePrimer,
title = {D-{{Brane Primer}}},
author = {Johnson, Clifford V.},
date = {2000-07},
journaltitle = {Strings, Branes and Gravity},
pages = {129--350},
doi = {10.1142/9789812799630_0002},
abstract = {Following is a collection of lecture notes on D-branes, which may be used by the reader as preparation for applications to modern research applications such as: the AdS/CFT and other gauge theory/geometry correspondences, Matrix Theory and stringy non-commutative geometry, etc. In attempting to be reasonably self-contained, the notes start from classical point-particles and develop the subject logically (but selectively) through classical strings, quantisation, D-branes, supergravity, superstrings, string duality, including many detailed applications. Selected focus topics feature D-branes as probes of both spacetime and gauge geometry, highlighting the role of world-volume curvature and gauge couplings, with some non-Abelian cases. Other advanced topics which are discussed are the (presently) novel tools of research such as fractional branes, the enhancon mechanism, D(ielectric)-branes and the emergence of the fuzzy/non-commutative sphere.},
archivePrefix = {arXiv},
eprint = {hep-th/0007170},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/johnson_2000_d-brane_primer.pdf}
}
@book{Joyce:2000:CompactManifoldsSpecial,
title = {Compact Manifolds with Special Holonomy},
author = {Joyce, Dominic},
date = {2000},
publisher = {{Oxford University Press on Demand}},
file = {/home/riccardo/.local/share/zotero/files/joyce_2000_compact_manifolds_with_special_holonomy.pdf},
isbn = {978-0-19-850601-0}
}
@@ -199,6 +321,54 @@
langid = {english}
}
@article{Lust:2009:LHCStringHunter,
title = {The {{LHC String Hunter}}'s {{Companion}}},
author = {Lust, Dieter and Stieberger, Stephan and Taylor, Tomasz R.},
date = {2009-02},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {808},
pages = {1--52},
issn = {05503213},
doi = {10.1016/j.nuclphysb.2008.09.012},
abstract = {The mass scale of fundamental strings can be as low as few TeV/c\^2 provided that spacetime extends into large extra dimensions. We discuss the phenomenological aspects of weakly coupled low mass string theory related to experimental searches for physics beyond the Standard Model at the Large Hadron Collider (LHC). We consider the extensions of the Standard Model based on open strings ending on D-branes, with gauge bosons due to strings attached to stacks of D-branes and chiral matter due to strings stretching between intersecting D-branes. We focus on the model-independent, universal features of low mass string theory. We compute, collect and tabulate the full-fledged string amplitudes describing all 2-{$>$}2 parton scattering subprocesses at the leading order of string perturbation theory. We cast our results in a form suitable for the implementation of stringy partonic cross sections in the LHC data analysis. The amplitudes involving four gluons as well as those with two gluons plus two quarks do not depend on the compactification details and are completely model-independent. They exhibit resonant behavior at the parton center of mass energies equal to the masses of Regge resonances. The existence of these resonances is the primary signal of string physics and should be easy to detect. On the other hand, the four-fermion processes like quark-antiquark scattering include also the exchanges of heavy Kaluza-Klein and winding states, whose details depend on the form of internal geometry. They could be used as ``precision tests'' in order to distinguish between various compactification scenarios.},
archivePrefix = {arXiv},
eprint = {0807.3333},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/lust_et_al_2009_the_lhc_string_hunter's_companion.pdf},
number = {1-2}
}
@article{Polchinski:1995:DirichletBranesRamondRamond,
title = {Dirichlet Branes and {{Ramond}}-{{Ramond}} Charges},
author = {Polchinski, Joseph},
date = {1995-12},
journaltitle = {Physical Review Letters},
volume = {75},
pages = {4724--4727},
issn = {00319007},
doi = {10.1103/physrevlett.75.4724},
abstract = {We show that Dirichlet-branes, extended objects defined by mixed Dirichlet-Neumann boundary conditions in string theory, break half of the supersymmetries of the type∼II superstring and carry a complete set of electric and magnetic Ramond-Ramond charges. We also find that the product of the electric and magnetic charges is a single Dirac unit, and that the quantum of charge takes the value required by string duality. This is strong evidence that the Dirchlet-branes are intrinsic to type II string theory and are the Ramond-Ramond sources required by string duality. We also note the existence of a previously overlooked 9-form potential in the IIa string, which gives rise to an effective cosmological constant of undetermined magnitude.},
archivePrefix = {arXiv},
eprint = {hep-th/9510017},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/polchinski_1995_dirichlet_branes_and_ramond-ramond_charges.pdf},
number = {26}
}
@article{Polchinski:1996:TASILecturesDBranes,
title = {{{TASI Lectures}} on {{D}}-{{Branes}}},
author = {Polchinski, Joseph},
date = {1996-11},
journaltitle = {New Frontiers in Fields and Strings},
pages = {75--136},
abstract = {This is an introduction to the properties of D-branes, topological defects in string theory on which string endpoints can live. D-branes provide a simple description of various nonperturbative objects required by string duality, and give new insight into the quantum mechanics of black holes and the nature of spacetime at the shortest distances. The first two thirds of these lectures closely follow the earlier ITP lectures hep-th/9602052, written with S. Chaudhuri and C. Johnson. The final third includes more extensive applications to string duality.},
archivePrefix = {arXiv},
eprint = {hep-th/9611050},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/polchinski_1996_tasi_lectures_on_d-branes.pdf}
}
@book{Polchinski:1998:StringTheoryIntroduction,
title = {String {{Theory}}. {{An}} Introduction to the Bosonic String.},
author = {Polchinski, Joseph},
@@ -238,6 +408,31 @@
number = {3}
}
@article{Taylor:2003:LecturesDbranesTachyon,
title = {Lectures on {{D}}-Branes, Tachyon Condensation, and String Field Theory},
author = {Taylor, Washington},
date = {2003-01-15},
abstract = {These lectures provide an introduction to the subject of tachyon condensation in the open bosonic string. The problem of tachyon condensation is first described in the context of the low-energy Yang-Mills description of a system of multiple D-branes, and then using the language of string field theory. An introduction is given to Witten's cubic open bosonic string field theory. The Sen conjectures on tachyon condensation in open bosonic string field theory are introduced, and evidence confirming these conjectures is reviewed.},
archivePrefix = {arXiv},
eprint = {hep-th/0301094},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/taylor_2003_lectures_on_d-branes,_tachyon_condensation,_and_string_field_theory.pdf}
}
@article{Taylor:2004:DBranesTachyonsString,
title = {D-{{Branes}}, {{Tachyons}}, and {{String Field Theory}}},
author = {Taylor, Washington and Zwiebach, Barton},
date = {2004-03},
journaltitle = {Strings, Branes and Extra Dimensions},
pages = {641--760},
doi = {10.1142/9789812702821_0012},
abstract = {In these notes we provide a pedagogical introduction to the subject of tachyon condensation in Witten's cubic bosonic open string field theory. We use both the low-energy Yang-Mills description and the language of string field theory to explain the problem of tachyon condensation on unstable D-branes. We give a self-contained introduction to open string field theory using both conformal field theory and overlap integrals. Our main subjects are the Sen conjectures on tachyon condensation in open string field theory and the evidence that supports these conjectures. We conclude with a discussion of vacuum string field theory and projectors of the star-algebra of open string fields. We comment on the possible role of string field theory in the construction of a nonperturbative formulation of string theory that captures all possible string backgrounds.},
archivePrefix = {arXiv},
eprint = {hep-th/0311017},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/taylor_zwiebach_2004_d-branes,_tachyons,_and_string_field_theory.pdf}
}
@article{Uranga:2005:TASILecturesString,
title = {{{TASI}} Lectures on {{String Compactification}}, {{Model Building}}, and {{Fluxes}}},
author = {Uranga, Angel M.},
@@ -262,4 +457,12 @@
number = {5}
}
@book{Zwiebach::FirstCourseString,
title = {A {{First Course}} in {{String Theory}}},
author = {Zwiebach, Barton},
file = {/home/riccardo/.local/share/zotero/files/zwiebach_a_first_course_in_string_theory.pdf},
isbn = {978-0-521-88032-9},
langid = {english}
}

3
thesis.cls
View File

@@ -181,10 +181,11 @@
\newcommand{\plaintoc}
{%
\thispagestyle{plain}
\pagestyle{plain}
\renewcommand*{\contentsname}{\Huge Table of Contents}
\tableofcontents
\cleardoubleplainpage{}
\pagestyle{fancy}
}
%---- abstract

3
thesis.tex
View File

@@ -26,6 +26,7 @@
\newcommand{\cft}{\textsc{cft}\xspace}
\newcommand{\ope}{\textsc{ope}\xspace}
\newcommand{\ap}{\ensuremath{\alpha'}}
\newcommand{\cy}{\textsc{CY}\xspace}
%---- derivatives
\newcommand{\pd}{\ensuremath{\partial}}
@@ -54,6 +55,7 @@
\newcommand{\bw}{\ensuremath{\overline{w}}}
\newcommand{\bomega}{\ensuremath{\overline{\omega}}}
\newcommand{\bepsilon}{\ensuremath{\overline{\epsilon}}}
\newcommand{\balpha}{\ensuremath{\overline{\alpha}}}
%---- BEGIN DOCUMENT
@@ -101,6 +103,7 @@
%---- DEEP LEARNING
\thesispart{Deep Learning the Geometry of String Theory}
\label{part:deeplearning}
\section{Introduction}
\input{sec/part3/introduction.tex}