End of the theory introduction?
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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@@ -1320,6 +1320,35 @@ In string theory there are ways to deal with the requirement~\cite{Uranga:2003:C
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These range from D-branes located at singular points of orbifolds to D-branes intersecting at angles~\cite{Finotello:2019:ClassicalSolutionBosonic}.
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We focus in particular on the latter.
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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 $T^6 = T^2 \times T^2 \times T^2$).
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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.
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The light spectrum is thus composed of the desired matter content alongside with other particles arising from the string compactification.
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\begin{figure}[tbp]
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\centering
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\def\svgwidth{0.7\linewidth}
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\import{img}{smbranes.pdf_tex}
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\caption{%
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Example of \sm-like construction using intersecting D-branes with the indications of the hypercharge $Y$.
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Perpendicular angles are only a matter of convenience: they are in principal arbitrary.
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}
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\label{fig:dbranes:smbranes}
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\end{figure}
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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}.
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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.
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The same applies to leptons created by strings attached to the \emph{leptonic} stack.
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Combinations of the additional \U{1} factors in the resulting gauge group finally lead to the definition of the hypercharge $Y$.
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Physics in $4$ dimensions is eventually recovered by compactifying the extra-dimensions of the superstring.\footnotemark{}
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\footnotetext{%
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In general we reviewed particle physics interactions.
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Gravitational interactions in general remain untouched by these constructions and still propagate in $10$-dimensional spacetime.
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}
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Fermions localised at the intersection of the D-branes are however naturally $4$-dimensional as they only propagate in the non compact Minkowski space $\ccM^{1,3}$.
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The presence of compactified dimensions however leads to phenomena such as \emph{family replications} of the fermions.
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With accurate calibration it is in fact possible to recover the quark and lepton families in the \sm.
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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.
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Even though the lines might never intersect on a plane, they can come across on a torus due to the identifications~\cite{Zwiebach::FirstCourseString}.
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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.
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% vim ft=tex
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71
thesis.bib
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thesis.bib
@@ -41,6 +41,24 @@
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file = {/home/riccardo/.local/share/zotero/files/angelantonj_sagnotti_2002_open_strings.pdf}
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}
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@article{Berkooz:1996:BranesIntersectingAngles,
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title = {Branes {{Intersecting}} at {{Angles}}},
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author = {Berkooz, Micha and Douglas, Michael R. and Leigh, Robert G.},
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date = {1996-11},
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journaltitle = {Nuclear Physics B},
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shortjournal = {Nuclear Physics B},
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volume = {480},
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pages = {265--278},
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issn = {05503213},
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doi = {10.1016/S0550-3213(96)00452-X},
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abstract = {We show that configurations of multiple D-branes related by SU(N) rotations will preserve unbroken supersymmetry. This includes cases in which two D-branes are related by a rotation of arbitrarily small angle, and we discuss some of the physics of this. In particular, we discuss a way of obtaining 4D chiral fermions on the intersection of D-branes. We also rephrase the condition for unbroken supersymmety as the condition that a `generalized holonomy group' associated with the brane configuration and manifold is reduced, and relate this condition (in Type IIA string theory) to a condition in eleven dimensions.},
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archivePrefix = {arXiv},
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eprint = {hep-th/9606139},
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eprinttype = {arxiv},
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file = {/home/riccardo/.local/share/zotero/files/berkooz_et_al_1996_branes_intersecting_at_angles5.pdf;/home/riccardo/.local/share/zotero/storage/A5B8IC2L/9606139.html},
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number = {1-2}
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}
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@article{Blumenhagen:2007:FourdimensionalStringCompactifications,
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title = {Four-Dimensional String Compactifications with {{D}}-Branes, Orientifolds and Fluxes},
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author = {Blumenhagen, Ralph and Körs, Boris and Lüst, Dieter and Stieberger, Stephan},
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@@ -347,6 +365,23 @@
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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}
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}
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@article{Grimm:2005:EffectiveActionType,
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title = {The Effective Action of Type {{IIA Calabi}}–{{Yau}} Orientifolds},
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author = {Grimm, Thomas W. and Louis, Jan},
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date = {2005-07},
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journaltitle = {Nuclear Physics B},
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shortjournal = {Nuclear Physics B},
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volume = {718},
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pages = {153--202},
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issn = {05503213},
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doi = {10.1016/j.nuclphysb.2005.04.007},
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annotation = {http://web.archive.org/web/20200905150924/https://linkinghub.elsevier.com/retrieve/pii/S0550321305002920},
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file = {/home/riccardo/.local/share/zotero/files/grimm_louis_2005_the_effective_action_of_type_iia_calabi–yau_orientifolds.pdf},
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keywords = {archived},
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langid = {english},
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number = {1-2}
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}
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@article{He:2020:CalabiyauSpacesString,
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title = {Calabi-Yau Spaces in the String Landscape},
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author = {He, Yang-Hui},
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@@ -384,6 +419,24 @@
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isbn = {978-981-02-1927-7}
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}
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@article{Ibanez:2001:GettingJustStandard,
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title = {Getting Just the Standard Model at Intersecting Branes},
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author = {Ibanez, Luis E. and Marchesano, Fernando and Rabadán, Raúl},
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date = {2001-11-02},
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journaltitle = {Journal of High Energy Physics},
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shortjournal = {J. High Energy Phys.},
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volume = {2001},
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pages = {002--002},
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issn = {1029-8479},
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doi = {10.1088/1126-6708/2001/11/002},
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archivePrefix = {arXiv},
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eprint = {hep-th/0105155},
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eprinttype = {arxiv},
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file = {/home/riccardo/.local/share/zotero/files/nez_et_al_2001_getting_just_the_standard_model_at_intersecting_branes.pdf},
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keywords = {archived},
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number = {11}
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}
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@article{Johnson:2000:DBranePrimer,
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title = {D-{{Brane Primer}}},
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author = {Johnson, Clifford V.},
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@@ -552,6 +605,24 @@
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number = {3}
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}
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@article{Sheikh-Jabbari:1998:ClassificationDifferentBranes,
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title = {Classification of {{Different Branes}} at {{Angles}}},
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author = {Sheikh-Jabbari, M. M.},
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date = {1998-02},
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journaltitle = {Physics Letters B},
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shortjournal = {Physics Letters B},
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volume = {420},
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pages = {279--284},
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issn = {03702693},
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doi = {10.1016/S0370-2693(97)01550-5},
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abstract = {In this paper, we consider two D-branes rotated with respect to each other, and argue that in this way one can find brane configurations preserving \$\{1 \textbackslash f 16\}\$ of SUSY. Also we classify different brane configurations preserving \$\{1 \textbackslash f 2\}\$, \$\{1 \textbackslash f 4\}\$, \$\{3 \textbackslash f 16\}\$,\$\{1 \textbackslash f 8\}\$, \$\{1 \textbackslash f 16\}\$ of SUSY.},
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archivePrefix = {arXiv},
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eprint = {hep-th/9710121},
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eprinttype = {arxiv},
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file = {/home/riccardo/.local/share/zotero/files/sheikh-jabbari_1998_classification_of_different_branes_at_angles.pdf;/home/riccardo/.local/share/zotero/storage/8IXE34EP/9710121.html},
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number = {3-4}
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}
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@article{Susskind:2003:AnthropicLandscapeString,
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title = {The {{Anthropic Landscape}} of {{String Theory}}},
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author = {Susskind, Leonard},
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