riccardo/phd-thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

End of the theory introduction?

Browse Source 
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
Riccardo
2020-09-05 17:38:39 +02:00
parent 15d40c494b
commit 0cb4b091b0
4 changed files with 173 additions and 0 deletions
Download Patch File Download Diff File Expand all files Collapse all files

29
sec/part1/introduction.tex
View File

@@ -1320,6 +1320,35 @@ In string theory there are ways to deal with the requirement~\cite{Uranga:2003:C
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 $T^6 = T^2 \times T^2 \times T^2$).
This D-brane geometry supports chiral fermion at their intersection: while some of the modes of the stretched string become indeed massive, the spectrum of the fields is proportional to combinations of the angles and some of the modes can stay massless.
The light spectrum is thus composed of the desired matter content alongside with other particles arising from the string compactification.
\begin{figure}[tbp]
\centering
\def\svgwidth{0.7\linewidth}
\import{img}{smbranes.pdf_tex}
\caption{%
Example of \sm-like construction using intersecting D-branes with the indications of the hypercharge $Y$.
Perpendicular angles are only a matter of convenience: they are in principal arbitrary.
}
\label{fig:dbranes:smbranes}
\end{figure}
It is therefore possible to recover a \sm-like construction using multiple D-branes at angles as in~\Cref{fig:dbranes:smbranes}, where the angles have been drawn perpendicular but can in principle be arbitrary~\cite{Ibanez:2001:GettingJustStandard,Grimm:2005:EffectiveActionType,Sheikh-Jabbari:1998:ClassificationDifferentBranes,Berkooz:1996:BranesIntersectingAngles}.
For instance quarks are localised at the intersection of the \emph{baryonic} stack of D-branes, yielding the colour symmetry generators, with the \emph{left} and \emph{right} stacks, leading to the $( \vb{3}, \vb{2} )$ and $( \vb{3}, \vb{1})$ representations.
The same applies to leptons created by strings attached to the \emph{leptonic} stack.
Combinations of the additional \U{1} factors in the resulting gauge group finally lead to the definition of the hypercharge $Y$.
Physics in $4$ dimensions is eventually recovered by compactifying the extra-dimensions of the superstring.\footnotemark{}
\footnotetext{%
In general we reviewed particle physics interactions.
Gravitational interactions in general remain untouched by these constructions and still propagate in $10$-dimensional spacetime.
}
Fermions localised at the intersection of the D-branes are however naturally $4$-dimensional as they only propagate in the non compact Minkowski space $\ccM^{1,3}$.
The presence of compactified dimensions however leads to phenomena such as \emph{family replications} of the fermions.
With accurate calibration it is in fact possible to recover the quark and lepton families in the \sm.
Consider for example the simple \cy factorised manifold $T^6 = T^2 \times T^2 \times T^2$ and introduce stacks of D6-branes as lines in each of the bi-tori.
Even though the lines might never intersect on a plane, they can come across on a torus due to the identifications~\cite{Zwiebach::FirstCourseString}.
Since each intersections supports a different set of fermions with different spectrum, the angles of the intersecting branes can be calibrated to reproduce the separation in mass of the families of quarks and leptons in the \sm.
% vim ft=tex