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\section*{GUT}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{fields_and_quanta}{}\paragraph*{{Fields and quanta}}\label{fields_and_quanta}

[[!include fields and quanta - table]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general_idea}{General idea}\dotfill \pageref*{general_idea} \linebreak
\noindent\hyperlink{IdeaofSU5GUT}{The $SU(5)$-GUT (Georgi-Glashow)}\dotfill \pageref*{IdeaofSU5GUT} \linebreak
\noindent\hyperlink{dseries_guts}{D-Series GUTs}\dotfill \pageref*{dseries_guts} \linebreak
\noindent\hyperlink{TheGUTKnownAsSO10}{The $Spin(10)$-GUT (known as ``$SO(10)$'')}\dotfill \pageref*{TheGUTKnownAsSO10} \linebreak
\noindent\hyperlink{TheGUTKnownAsSO11}{The $Spin(11)$-GUT (known as ``$SO(11)$'')}\dotfill \pageref*{TheGUTKnownAsSO11} \linebreak
\noindent\hyperlink{TheGUTKnownAsSO12}{The $Spin(12)$-GUT (known as ``$SO(12)$'')}\dotfill \pageref*{TheGUTKnownAsSO12} \linebreak
\noindent\hyperlink{TheGUTKnownAsSO16}{The $Spin(16), Spin(18)$-GUT (known as ``$SO(16), SO(18)$'')}\dotfill \pageref*{TheGUTKnownAsSO16} \linebreak
\noindent\hyperlink{eseries_guts}{E-series GUTs}\dotfill \pageref*{eseries_guts} \linebreak
\noindent\hyperlink{the_gut}{The $E_6$-GUT}\dotfill \pageref*{the_gut} \linebreak
\noindent\hyperlink{the_gut_}{The $E_7$-GUT (?)}\dotfill \pageref*{the_gut_} \linebreak
\noindent\hyperlink{the_gut__2}{The $E_8$-GUT (?)}\dotfill \pageref*{the_gut__2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToProtonDecay}{Relation to proton decay}\dotfill \pageref*{RelationToProtonDecay} \linebreak
\noindent\hyperlink{RelationToNeutrinoMasses}{Relation to neutrino masses}\dotfill \pageref*{RelationToNeutrinoMasses} \linebreak
\noindent\hyperlink{RelationToLeptoquarks}{Relation to leptoquarks}\dotfill \pageref*{RelationToLeptoquarks} \linebreak
\noindent\hyperlink{InStringPhenomenology}{Occurrence in string phenomenology}\dotfill \pageref*{InStringPhenomenology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesProtonNonDecay}{Proton (non-)decay}\dotfill \pageref*{ReferencesProtonNonDecay} \linebreak
\noindent\hyperlink{ReferencesRealisticModels}{Realistic models and phenomenology}\dotfill \pageref*{ReferencesRealisticModels} \linebreak
\noindent\hyperlink{gut}{$SO(10)$-GUT}\dotfill \pageref*{gut} \linebreak
\noindent\hyperlink{ReferencesSpin11}{$SO(11)$-GUT}\dotfill \pageref*{ReferencesSpin11} \linebreak
\noindent\hyperlink{ReferencesSpin12}{$SO(12)$-GUT}\dotfill \pageref*{ReferencesSpin12} \linebreak
\noindent\hyperlink{ReferencesSpin16}{$SO(16)$- and $Spin(18)$-GUT}\dotfill \pageref*{ReferencesSpin16} \linebreak
\noindent\hyperlink{in_string_theory}{In string theory}\dotfill \pageref*{in_string_theory} \linebreak
\noindent\hyperlink{in_conneslott_models}{In Connes-Lott models}\dotfill \pageref*{in_conneslott_models} \linebreak
\noindent\hyperlink{exotica_leptoquarks_bosons_etc}{Exotica: Leptoquarks, $Z'$-bosons, etc.}\dotfill \pageref*{exotica_leptoquarks_bosons_etc} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{general_idea}{}\subsubsection*{{General idea}}\label{general_idea}

The [[standard model of particle physics]] asserts that the fundamental quantum [[field (physics)|physical fields]] and [[particles]] are modeled as [[sections]] of and [[connection on a bundle|connections]] on a [[vector bundle]] that is [[associated bundle|associated]] to a $G$-[[principal bundle]], where the [[Lie group]] $G$ -- called the [[gauge group]] -- is the [[product]] of ([[special unitary group|special]]) [[unitary groups]] $G = SU(3) \times SU(2) \times U(1)$ (or rather a [[quotient]] of this by the [[cyclic group]] $Z/6$, see \href{standard+model+of+particle+physics#GaugeGroup}{there}) and where the [[representation]] of $G$ used to form the [[associated bundle|associated]] [[vector bundle]] looks fairly ad hoc on first sight.

A \textbf{grand unified theory} (``GUT'' for short) in this context is an attempt to realize the standard model as sitting inside a conceptually simpler [[model (in particle physics)|model]], in particular one for which the [[gauge group]] is a bigger but \emph{simpler} group $\hat{G}$, preferably a \emph{[[simple Lie group]]} in the technical sense, which contains $G$ as a [[subgroup]]. Such a grand unified theory would be [[phenomenology|phenomenologically]] viable if a process of [[spontaneous symmetry breaking]] at some high [[energy]] scale -- the ``GUT scale'' -- would reduce the model back to the [[standard model of particle physics]] without adding spurious extra effects that would not be in agreement with existing observations in [[experiment]].

The terminology ``grand unified'' here refers to the fact that such a single simple group $\hat{G}$ would unify the fundamental [[forces]] of [[electromagnetism]], the [[weak nuclear force]] and the [[strong nuclear force]] in a way that generalizes the way in which the [[electroweak field]] already unifies the [[weak nuclear force]] and [[electromagnetism]], and electromagnetism already unifies, as the word says, electricity and magnetism.

Since no GUT model has been fully validated yet (but see for instance \hyperlink{FongMeloni14}{Fong-Meloni 14}), GUTs are [[model (physics)|models]] ``beyond the [[standard model of particle physics|standard model]]''. Often quantum physics ``beyond the standard model'' is expected to also say something sensible about [[quantum gravity]] and hence unify not just the three gauge forces but also the fourth known fundamental force, which is [[gravity]]. Models that aim to do all of this would then ``unify'' the entire content of the [[standard model of particle physics]] plus the [[standard model of cosmology]], hence ``everything that is known about fundamental physics'' to date. Therefore such theories are then sometimes called a \emph{[[theory of everything]]}.

(Here it is important to remember the context, both ``grand unified'' and ``of everything'' refers to aspects of presently available models of fundamental physics, and not to deeper philosophical questions of [[ontology]].)

\hypertarget{IdeaofSU5GUT}{}\subsubsection*{{The $SU(5)$-GUT (Georgi-Glashow)}}\label{IdeaofSU5GUT}

The argument for the hypothesis of ``grand unification'' is fairly compelling if one asks for \emph{[[simple objects|simple algebraic structures]]} in the technical sense ([[simple Lie groups]] and their [[irreducible representations]]):

The \emph{exact} gauge group of the [[standard model of particle physics]] is really a [[quotient group]]

\begin{displaymath}
G_{SM}
  \;=\; 
  \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6
  \,,
\end{displaymath}
where the [[cyclic group]] $\mathbb{Z}_6$ [[free action|acts freely]], hence exhibiting a subtle global twist in the gauge structure. This seemingly ad hoc group turns out to be [[isomorphism|isomorphic]] to the [[subgroup]]

\begin{displaymath}
\underset{
    \simeq \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6  }{
  \underbrace{
    S \big( U(2) \times U(3)\big)
  }}
  \;\subset\;
  SU(5)
\end{displaymath}
of [[special unitary group|SU(5)]] (see \hyperlink{BaezHuerta09}{Baez-Huerta 09, p. 33-36}). The latter happens to be a [[simple Lie group]], thus exhibiting the exact standard model Lie group as being ``simply'' a ``(2+3)-breaking'' of a [[simple Lie group]].

Moreover, the [[gauge group]]-[[representation]] $V_{SM}$ that captures one [[generation of fundamental particles]] of the [[standard model of particle physics]], which looks fairly ad hoc as a representation of $G_{SM}$ (e.g. \hyperlink{BaezHuerta09}{Baez-Huerta 09, table 1}), but as a representation of $SU(5)$ it is simply

\begin{displaymath}
V_{SM} \simeq \Lambda \mathbb{C}^5
\end{displaymath}
(see \hyperlink{BaezHuerta09}{Baez-Huerta 09, p. 36-41}).

This leads to the $SU(5)$-GUT due to \hyperlink{GeorgiGlashow74}{Georgi-Glashow 74}

\hypertarget{dseries_guts}{}\subsubsection*{{D-Series GUTs}}\label{dseries_guts}

\hypertarget{TheGUTKnownAsSO10}{}\paragraph*{{The $Spin(10)$-GUT (known as ``$SO(10)$'')}}\label{TheGUTKnownAsSO10}

There is a further inclusion $SU(5) \hookrightarrow$ [[Spin(10)]] into the [[spin group]] in 10 (Euclidean) dimensions (still a [[simple Lie group]]), and one [[generation of fundamental particles]] regarded as an $SU(5)$-[[representation]] $\Lambda \mathbb{C}^5$ as \hyperlink{IdeaofSU5GUT}{above} extends to a [[spin representation]] (see \hyperlink{BaezHuerta09}{Baez-Huerta 09, theorem 2}). This has the aesthetically pleasing effect that under this identification the 1-generation rep $V_{SM}$ is now identified as

\begin{displaymath}
V_{SM} \;\simeq\; \mathbf{16} \oplus \overline{\mathbf{16}}
\end{displaymath}
namely as the [[direct sum]] of [[generalized the|the]] two (complex) [[irreducible representations]] of [[Spin(10)]], together being the [[Dirac representation]] of [[Spin(10)]].

Again, this means that under the embedding of the gauge group $G_{SM}$ all the way into the [[simple Lie group]] [[Spin(10)]], its ingredients become \emph{simpler}, not just in a naive sense, but in the technical mathematical sense of \emph{[[simple objects|simple algebraic objects]]}.

Discussion of [[SO(10)]] (i.e. [[Spin(10)]]) GUT-models with realistic [[phenomenology]] is in \hyperlink{BLM09}{BLM 09} \hyperlink{Malinsky09}{Malinský 09}, \hyperlink{LavouraWolfenstein10}{Lavoura-Wolfenstein 10} \hyperlink{AltarelliMeloni13}{Altarelli-Meloni 13} \hyperlink{DueckRodejohann13}{Dueck-Rodejohann 13} \hyperlink{OhlssonPernow19}{Ohlsson-Pernow 19} \hyperlink{CPS19}{CPS 19}.

$\backslash$begin\{imagefromfile\} ``file\_name'': ``MinimalNonSusySO10GUTMalinsky.jpg'', ``width'': 500 $\backslash$end\{imagefromfile\}

\begin{quote}%
slide grabbed from \hyperlink{Malinsky09}{Malinský 09}


\end{quote}
Discussion of [[leptoquarks]] in $SO(10)$-models possibly explaining the [[flavour anomalies]]: \hyperlink{AMM19}{AMM 19}

$\backslash$linebreak

\hypertarget{TheGUTKnownAsSO11}{}\paragraph*{{The $Spin(11)$-GUT (known as ``$SO(11)$'')}}\label{TheGUTKnownAsSO11}

Models with [[Spin(11)]] (``[[SO(11)]]'') GUT group.

Specifically with [[gauge-Higgs unification]] in a [[Randall-Sundrum model]]-like 6d spacetime: \hyperlink{HosotaniYamatsu15}{Hosotani-Yamatsu 15}, \hyperlink{FuruiHosotaniYamatsu16}{Furui-Hosotani-Yamatsu 16}, \hyperlink{Hosotani17}{Hosotani 17}, \hyperlink{HosotaniYamatsu17}{Hosotani-Yamatsu 17}

See the references \hyperlink{ReferencesSpin11}{below}.

$\backslash$linebreak

\hypertarget{TheGUTKnownAsSO12}{}\paragraph*{{The $Spin(12)$-GUT (known as ``$SO(12)$'')}}\label{TheGUTKnownAsSO12}

Models with [[Spin(12)]] (``[[SO(12)]]'') GUT group.

Specifically with [[gauge-Higgs unification]] in a [[Randall-Sundrum model]]-like 6d spacetime: \hyperlink{NomuraSato08}{Nomura-Sato 08}, \hyperlink{Nomura09}{Nomura 09}, \hyperlink{ChiangNomuraSato11}{Chiang-Nomura-Sato 11})

See the references \hyperlink{ReferencesSpin12}{below}.

$\backslash$linebreak

\hypertarget{TheGUTKnownAsSO16}{}\paragraph*{{The $Spin(16), Spin(18)$-GUT (known as ``$SO(16), SO(18)$'')}}\label{TheGUTKnownAsSO16}

Models with [[Spin(16)]] (``[[SO(16)]]'') GUT group.

\hyperlink{WilczekZee82}{Wilczek-Zee 82}, \hyperlink{SenjanovicWilczekZee84}{Senjanovic-Wilczek-Zee 84}, \hyperlink{MartínezMelfoNestiSenjanovic11}{Martínez-Melfo-Nesti-Senjanovic 11}

See also \hyperlink{diLucio11}{di Lucio 11, p. 44} and see the references \hyperlink{ReferencesSpin16}{below}.

Predicts [[fourth generation of fermions]]\ldots{}

\hypertarget{eseries_guts}{}\subsubsection*{{E-series GUTs}}\label{eseries_guts}

The most studied choices of GUT-groups $G$ are [[special unitary group|SU(5)]], [[spin group|Spin(10)]] (in the physics literature often referred to as [[special orthogonal group|SO(10)]]) and [[E6]] (review includes \hyperlink{Witten86}{Witten 86, sections 1 and 2}).

It so happens that, mathematically, the sequence [[special unitary groups|SU(5)]], [[spin group|Spin(10)]], [[E6]] naturally continues (each step by consecutively adding a node to the [[Dynkin diagrams]]) with the [[exceptional Lie groups]] [[E7]], [[E8]] that naturally appear in [[heterotic string theory|heterotic]] [[string phenomenology]] (exposition is in \hyperlink{Witten02a}{Witten 02a}) and conjecturally further via the [[U-duality]] [[Kac-Moody groups]] [[E9]], [[E10]], [[E11]] that are being argued to underly [[M-theory]]. In the context of [[F-theory]] model building, also properties of the observes [[Yukawa couplings]] may point to exceptional GUT groups (\hyperlink{Zoccarato14}{Zoccarato 14, slide 11}, \hyperlink{Vafa15}{Vafa 15, slide 11}).

\hypertarget{the_gut}{}\paragraph*{{The $E_6$-GUT}}\label{the_gut}

(\ldots{})

\hypertarget{the_gut_}{}\paragraph*{{The $E_7$-GUT (?)}}\label{the_gut_}

(\ldots{})

\hypertarget{the_gut__2}{}\paragraph*{{The $E_8$-GUT (?)}}\label{the_gut__2}

(\ldots{})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToProtonDecay}{}\subsubsection*{{Relation to proton decay}}\label{RelationToProtonDecay}

Many GUT models imply that the [[proton]] -- which in the [[standard model of particle physics]] is a stable [[bound state]] (of [[quarks]]) -- is in fact unstable, albeit with an extremely long mean liftetime, and hence may decay (e.g. \hyperlink{KM14}{KM 14}). [[experiment|Experimental]] searches for such \emph{[[proton decay]]} (see there for more) put strong bounds on this effect and hence heavily constrain or rule out many [[GUT]] [[model (physics)|models]].

But in recent years it is claimed that there are in fact realistic $SU(5)$ [[GUT]] [[model (physics)|models]] that do not imply any [[proton decay]], quite generically so for [[MSSM]]-models (\hyperlink{MuetterRatzVaudrvange16}{Mütter-Ratz-Vaudrvange 16}), but also for non-supersymmetric models ( \hyperlink{FornalGrinstein17}{Fornal-Grinstein 17}, \hyperlink{FornalGrinstein18}{Fornal-Grinstein 18}, in particular in [[gauge-Higgs grand unification]] such as [[Spin(11)]]- (``[[SO(11)]]''-) and [[Spin(12)]]- (``[[SO(12)]]''-) models: (\hyperlink{HosotaniYamatsu15}{Hosotani-Yamatsu 15}, \hyperlink{FuruiHosotaniYamatsu16}{Furui-Hosotani-Yamatsu 16, Sec. 2.6} \hyperlink{Hosotani17}{Hosotani 17, Section 6}).

\hypertarget{RelationToNeutrinoMasses}{}\subsubsection*{{Relation to neutrino masses}}\label{RelationToNeutrinoMasses}

The high energy scale required by a [[seesaw mechanism]] to produce the experimentally observer [[neutrino]] masses happens to be about the conventional [[GUT scale]], for review see for instance (\hyperlink{Mohapatra06}{Mohapatra 06}).

\begin{quote}%
I also noted at the same time that interactions between a pair of lepton doublets and a pair of scalar doublets can generate a neutrino mass, which is suppressed only by a factor $M^{-1}$, and that therefore with a reasonable estimate of $M$ could produce observable neutrino oscillations. The subsequent confirmation of neutrino oscillations lends support to the view of the Standard Model as an effective field theory, with M somewhere in the neighborhood of $10^{16} GeV$. (\href{neutrino#Weinberg09}{Weinberg 09, p. 15})


\end{quote}
Detailed matching of parameters of non-supersymmetric $Spin(10)$-GUT to [[neutrino]] [[masses]] is discussed in \hyperlink{OhlssonPernow19}{Ohlsson-Pernow 19}

\hypertarget{RelationToLeptoquarks}{}\subsubsection*{{Relation to leptoquarks}}\label{RelationToLeptoquarks}

Generically, GUT-theories predict the existence of [[leptoquarks]] (\hyperlink{MurayamaYanagida92}{Murayama-Yanagida 92}), possibly related to the apparently observed

\begin{itemize}%
\item [[flavour anomalies]] (\hyperlink{BDFKFS18}{BDFKFS 18}, \hyperlink{AMM19}{AMM 19}, \hyperlink{HeekTeresi18}{Heek-Teresi 18}, \hyperlink{HeekTeresi19}{Heek-Teresi 19})


\item \href{anomalous+magnetic+moment#Anomalies}{anomalies} in [[anomalous magnetic moment]] of [[muon]] and [[electron]]



\end{itemize}
\hypertarget{InStringPhenomenology}{}\subsubsection*{{Occurrence in string phenomenology}}\label{InStringPhenomenology}

Discussion of [[string phenomenology]] of [[intersecting D-brane models]] [[KK-compactification|KK-compactified]] with non-geometric [[fibers]] such that the would-be string [[sigma-models]] with these [[target spaces]] are in fact [[Gepner models]] (in the sense of \emph{\href{https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/}{Spectral Standard Model and String Compactifications}}) is in (\hyperlink{DijkstraHuiszoonSchellekens04a}{Dijkstra-Huiszoon-Schellekens 04a}, \hyperlink{DijkstraHuiszoonSchellekens04b}{Dijkstra-Huiszoon-Schellekens 04b}):

\begin{quote}%
A plot of [[standard model of particle physics|standard model]]-like [[coupling constants]] in a computer scan of [[Gepner model]]-[[KK-compactification]] of [[intersecting D-brane models]] according to \hyperlink{DijkstraHuiszoonSchellekens04b}{Dijkstra-Huiszoon-Schellekens 04b}.

The blue dot indicates the couplings in $SU(5)$-[[GUT]] theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.


\end{quote}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[theory of everything]]


\item [[gauge coupling unification]]


\item [[supersymmetry]]


\item [[Connes-Lott-Chamseddine-Barrett model]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Original articles include

\begin{itemize}%
\item [[Jogesh Pati]], [[Abdus Salam]], \emph{Lepton number as the fourth ``color''}, Phys. Rev. D 10, 275 – Published 1 July 1974 (\href{https://doi.org/10.1103/PhysRevD.10.275}{doi:10.1103/PhysRevD.10.275})


\item [[Howard Georgi]], [[Sheldon Glashow]], \emph{Unity of All Elementary-Particle Forces}, Phys. Rev. Lett. 32, 438, 1974 (\href{https://doi.org/10.1103/PhysRevLett.32.438}{doi:10.1103/PhysRevLett.32.438})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Pati%E2%80%93Salam_model}{Pati-Salam model}}


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Grand_unification_energy}{Grand unification energy}}



\end{itemize}
Discussion with an eye towards [[supergravity]] unification:

\begin{itemize}%
\item [[Murray Gell-Mann]], introductory talk at \emph{\href{https://en.wikipedia.org/wiki/Shelter_Island_Conference}{Shelter Island II}}, 1983 ([[Gell-Mann\_ShelterIslandII\_1983.pdf:file]])

in: \emph{Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics}. MIT Press. pp. 301--343. ISBN 0-262-10031-2.


\item [[Murray Gell-Mann]], [[Pierre Ramond]], Richard Slansky, \emph{Complex Spinors and Unified Theories}, Supergravity, [[Peter van Nieuwenhuizen]] and D.Z. Freedman, eds, North Holland Publishing Co, 1979, (reprinted as \href{http://arxiv.org/abs/1306.4669}{arXiv:1306.4669})



\end{itemize}
A basic textbook account is in

\begin{itemize}%
\item [[Luis Ibáñez]], [[Angel Uranga]], section 1.2 of \emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]}, Cambridge University Press 2012

\end{itemize}
and a detailed one is in

\begin{itemize}%
\item A. Hebecker, J. Hisano, \emph{Grand Unified Theories}, chapter 114. in Particle Data Group's \emph{\href{http://pdg.lbl.gov/2018/}{The Review of Particle Physics}} 2018 (\href{http://pdg.lbl.gov/2018/reviews/rpp2018-rev-guts.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item Luca di Lucio, \emph{Aspects of Symmetry Breaking inGrand Unified Theories}, 2011 (\href{https://www.sissa.it/tpp/phdsection/AlumniThesis/Luca%20Di%20Luzio.pdf}{pdf})

\end{itemize}
Survey of arguments for the hypothesis of grand unification includes

\begin{itemize}%
\item [[Michael Peskin]], \emph{Beyond the Standard Model} (\href{http://arxiv.org/abs/hep-ph/9705479}{arXiv:hep-ph/9705479})


\item [[Jogesh Pati]], \emph{Discovery of Proton Decay: A Must for Theory, a Challenge for Experiment} (\href{http://arxiv.org/abs/hep-ph/0005095}{arXiv:hep-ph/0005095})


\item [[Edward Witten]], \emph{Quest For Unification}, Heinrich Hertz lecture at \href{http://www.desy.de/susy02/}{SUSY 2002} at DESY, Hamburg (\href{http://arxiv.org/abs/hep-ph/0207124}{arXiv:hep-ph/0207124})



\end{itemize}
Introduction to GUTs aimed more at mathematicians include

\begin{itemize}%
\item [[Edward Witten]], section 1 and 2 of \emph{Physics and geometry}, Proceedings of the international congress of mathematicians, 1986 (\href{http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0267.0306.ocr.pdf}{pdf})


\item [[John Baez]], [[John Huerta]], \emph{The Algebra of Grand Unified Theories}, Bull.Am.Math.Soc.47:483-552,2010 (\href{http://arxiv.org/abs/0904.1556}{arXiv:0904.1556})



\end{itemize}
\hypertarget{ReferencesProtonNonDecay}{}\subsubsection*{{Proton (non-)decay}}\label{ReferencesProtonNonDecay}

Discussion of experimental bounds on [[proton decay]] in GUTs includes

\begin{itemize}%
\item Helena Kole\v{s}ov\'a{}, [[Michal Malinský]], \emph{Proton lifetime in the minimal $SO(10)$ GUT and its implications for the LHC}, Phys. Rev. D 90, 115001 (2014) (\href{http://arxiv.org/abs/1409.4961}{arXiv:1409.4961})

\end{itemize}
Claim that [[proton decay]] may be entirely avoided:

\begin{itemize}%
\item Andreas Mütter, Michael Ratz, Patrick K.S. Vaudrevange, \emph{Grand Unification without Proton Decay} (\href{https://arxiv.org/abs/1606.02303}{arXiv:1606.02303})

(claims that many [[string theory]] and [[supergravity]] models have this property)


\item Bartosz Fornal, Benjamin Grinstein, \emph{$SU(5)$ Unification without Proton Decay}, Physics Review Letters (\href{https://arxiv.org/abs/1706.08535}{arXiv:1706.08535})


\item Bartosz Fornal, Benjamin Grinstein, \emph{Grand Unified Theory with a Stable Proton}, Int. J. Mod. Phys. A 33 (2018) 1844013 (\href{https://arxiv.org/abs/1808.00953}{arXiv:1808.00953})



\end{itemize}
Claim that [[threshold corrections]] can considerably alter (decrease) proton decay rate predictions in non-supersymmetric GUTs:

\begin{itemize}%
\item Joydeep Chakrabortty, Stephen F. King, Rinku Maji, \emph{Unification, Proton Decay and Topological Defects in non-SUSY GUTs with Thresholds} (\href{https://arxiv.org/abs/1901.05867}{arXiv:1901.05867})

\end{itemize}
\hypertarget{ReferencesRealisticModels}{}\subsubsection*{{Realistic models and phenomenology}}\label{ReferencesRealisticModels}

Discussion of [[phenomenology|phenomenologically]] viable GUT-[[model (in theoretical physics)|models]] (compatible with [[experiment]] and the [[standard model of particle physics]]):

\hypertarget{gut}{}\paragraph*{{$SO(10)$-GUT}}\label{gut}

Discussion for [[Spin(10)]] GUT group (``[[SO(10)]]''):

\begin{itemize}%
\item Stefano Bertolini, Luca Di Luzio, [[Michal Malinský]], \emph{Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal}, Phys. Rev. D80:015013, 2009 (\href{https://arxiv.org/abs/0903.4049}{arXiv:0903.4049})

\end{itemize}
review:

\begin{itemize}%
\item [[Michal Malinský]], \emph{35 years of GUTs - where do we stand?}, 2009 (\href{https://www.mpi-hd.mpg.de/lin/seminar_theory/talks/Malinsky.pdf}{pdf})

\end{itemize}
for non-superymmetric [[model (physics)|models]]:

\begin{itemize}%
\item L. Lavoura and Lincoln Wolfenstein, \emph{Resuscitation of minimal $SO(10)$ grand unification}, Phys. Rev. D 48, 264 (\href{https://doi.org/10.1103/PhysRevD.48.264}{doi:10.1103/PhysRevD.48.264})


\item Guido Altarelli, Davide Meloni, \emph{A non Supersymmetric SO(10) Grand Unified Model for All the Physics below $M_{GUT}$} (\href{https://arxiv.org/abs/1305.1001}{arXiv:1305.1001})


\item Alexander Dueck, Werner Rodejohann, \emph{Fits to $SO(10)$ Grand Unified Models} (\href{http://arxiv.org/abs/1306.4468}{arXiv:1306.4468})


\item Chee Sheng Fong, Davide Meloni, Aurora Meroni, Enrico Nardi, \emph{Leptogenesis in $SO(10)$} (\href{http://arxiv.org/abs/1412.4776}{arXiv:1412.4776})

(in view of [[leptogenesis]])


\item Tommy Ohlsson, Marcus Pernow, \emph{Fits to Non-Supersymmetric SO(10) Models with Type I and II Seesaw Mechanisms Using Renormalization Group Evolution} (\href{https://arxiv.org/abs/1903.08241}{arXiv:1903.08241})


\item Mainak Chakraborty, M.K. Parida, Biswonath Sahoo, \emph{Triplet Leptogenesis, Type-II Seesaw Dominance, Intrinsic Dark Matter, Vacuum Stability and Proton Decay in Minimal SO(10) Breakings} (\href{https://arxiv.org/abs/1906.05601}{arXiv:1906.05601})

\begin{quote}%
Results indicating non-SUSY $SO(10)$ as self sufficient theory for neutrino masses, baryon asymmetry, dark matter, vacuum stability of SM scalar potential, origin of three gauge forces, and observed proton stability.


\end{quote}

\item Nobuchika Okada, Digesh Raut, Qaisar Shafi, \emph{Inflation, Proton Decay, and Higgs-Portal Dark Matter in $SO(10) \times U(1)_\pri$} (\href{https://arxiv.org/abs/1906.06869}{arXiv:1906.06869})



\end{itemize}
for [[supersymmetry|supersymmetric]] [[model (physics)|models]]:

\begin{itemize}%
\item Archana Anandakrishnan, B. Charles Bryant, Stuart Raby, \emph{LHC Phenomenology of $SO(10)$ Models with Yukawa Unification II} (\href{http://arxiv.org/abs/1404.5628}{arXiv:1404.5628})


\item Ila Garg, \emph{New minimal supersymmetric $SO(10)$ GUT phenomenology and its cosmological implications} (\href{http://arxiv.org/abs/1506.05204}{arXiv:1506.05204})



\end{itemize}
\hypertarget{ReferencesSpin11}{}\paragraph*{{$SO(11)$-GUT}}\label{ReferencesSpin11}

Discussion for [[Spin(11)]] GUT group (``[[SO(11)]]''):

\begin{itemize}%
\item [[Yutaka Hosotani]], Naoki Yamatsu, \emph{Gauge–Higgs grand unification}, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 11, November 2015 (\href{https://doi.org/10.1093/ptep/ptv153}{doi:10.1093/ptep/ptv153}, \href{https://doi.org/10.1093/ptep/ptw116}{doi:10.1093/ptep/ptw116})


\item Atsushi Furui, [[Yutaka Hosotani]], Naoki Yamatsu, \emph{Toward Realistic Gauge-Higgs Grand Unification}, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 9, September 2016, 093B01 (\href{https://arxiv.org/abs/1606.07222}{arXiv:1606.07222})


\item [[Yutaka Hosotani]], \emph{Gauge-Higgs EW and Grand Unification}, International Journal of Modern Physics AVol. 31, No. 20n21, 1630031 (2016) (\href{https://arxiv.org/abs/1606.08108}{arXiv:1606.08108})


\item [[Yutaka Hosotani]], \emph{New dimensions from gauge-Higgs unification} (\href{https://arxiv.org/abs/1702.08161}{arXiv:1702.08161})


\item Yutaka Hosotani, Naoki Yamatsu, \emph{Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification} (\href{https://arxiv.org/abs/1710.04811}{arXiv:1710.04811})



\end{itemize}
\hypertarget{ReferencesSpin12}{}\paragraph*{{$SO(12)$-GUT}}\label{ReferencesSpin12}

Discussion for [[Spin(12)]] GUT group (``[[SO(12)]]''):

\begin{itemize}%
\item S. Rajpoot and P. Sithikong, \emph{Implications of the $SO(12)$ gauge symmetry for grand unification}, Phys. Rev. D 23, 1649 (1981) (\href{https://doi.org/10.1103/PhysRevD.23.1649}{doi:10.1103/PhysRevD.23.1649})


\item Takaaki Nomura, Joe Sato, \emph{Standard(-like) Model from an $SO(12)$ Grand Unified Theory in six-dimensions with $S^2$ extra-space}, Nucl.Phys.B811:109-122, 2009 (\href{https://arxiv.org/abs/0810.0898}{arXiv:0810.0898})


\item Takaaki Nomura, \emph{Physics beyond the standard model with $S^2$ extra-space}, 2009 (\href{http://krishna.th.phy.saitama-u.ac.jp/joe/doctor/nomura-doctor.pdf}{pdf}, [[NomuraSO12GUT.pdf:file]])


\item Cheng-Wei Chiang, Takaaki Nomura, Joe Sato, \emph{Gauge-Higgs unification models in six dimensions with $S^2/\mathbb{Z}_2$ extra space and GUT gauge symmetry} (\href{https://arxiv.org/abs/1109.5835}{arXiv:1109.5835})



\end{itemize}
\hypertarget{ReferencesSpin16}{}\paragraph*{{$SO(16)$- and $Spin(18)$-GUT}}\label{ReferencesSpin16}

Discussion for [[Spin(16)]] and [[Spin(18)]] GUT group (``[[SO(16)]]'' and ``[[SO(18)]]''):

\begin{itemize}%
\item [[Frank Wilczek]], [[Anthony Zee]], \emph{Families from spinors}, Phys. Rev. D 25, 553 (1982) (\href{https://doi.org/10.1103/PhysRevD.25.553}{doi:10.1103/PhysRevD.25.55310.1103/PhysRevD.25.553})


\item Goran Senjanović, [[Frank Wilczek]], [[Anthony Zee]], \emph{Reflections on mirror fermions}, Physics Letters B Volume 141, Issues 5–6, 5 July 1984, Pages 389-394 Physics Letters B ()


\item Homero Martínez, Alejandra Melfo, Fabrizio Nesti, Goran Senjanović, \emph{Three Extra Mirror or Sequential Families: a Case for Heavy Higgs and Inert Doublet}, Phys. Rev. Lett.106:191802, 2011 (\href{https://arxiv.org/abs/1101.3796}{arXiv:1101.3796})


\item Michael McGuigan, \emph{Dark Horse, Dark Matter: Revisiting the $SO(16) \times SO(16)'$ Nonsupersymmetric Model in the LHC and Dark Energy Era} (\href{https://arxiv.org/abs/1907.01944}{arXiv:1907.01944})



\end{itemize}
\hypertarget{in_string_theory}{}\subsubsection*{{In string theory}}\label{in_string_theory}

Introductory overview to GUTs in [[string theory]] is in

\begin{itemize}%
\item [[Hans-Peter Nilles]], \emph{Strings, Exceptional Groups and Grand Unification}, talk at \emph{\href{https://indico.cern.ch/event/112851/}{Planck 2011}} (\href{http://www.th.physik.uni-bonn.de/nilles/db/HPtalks/114planck.pdf}{pdf}, [[Nilles11GUT.pdf:file]])

\end{itemize}
Amplifcation that $SO(32)$-GUT (as in [[heterotic string theory]] and [[type I string theory]]) is viable via special subgroup-breaking:

\begin{itemize}%
\item Naoki Yamatsu, \emph{String-Inspired Special Grand Unification}, Progress of Theoretical and Experimental Physics, Volume 2017, Issue 10, 1 (\href{https://arxiv.org/abs/1708.02078}{arXiv:1708.02078}, \href{https://doi.org/10.1093/ptep/ptx135}{doi:10.1093/ptep/ptx135})

\end{itemize}
Computer scan of [[Gepner model]]-compactifications in relation to GUT-models is in

\begin{itemize}%
\item T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], \emph{Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models}, Phys.Lett. B609 (2005) 408-417 (\href{https://arxiv.org/abs/hep-th/0403196}{arXiv:hep-th/0403196})


\item T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], \emph{Supersymmetric Standard Model Spectra from RCFT orientifolds}, Nucl.Phys.B710:3-57,2005 (\href{https://arxiv.org/abs/hep-th/0411129}{arXiv:hep-th/0411129})



\end{itemize}
Realization of GUTs in the context of [[M-theory on G2-manifolds]] and possible resolution of the [[doublet-triplet splitting problem]] is discussed in

\begin{itemize}%
\item [[Edward Witten]], \emph{Deconstruction, $G_2$ Holonomy, and Doublet-Triplet Splitting}, (\href{http://arxiv.org/abs/hep-ph/0201018}{arXiv:hep-ph/0201018})


\item [[Bobby Acharya]], Krzysztof Bozek, Miguel Crispim Romao, Stephen F. King, Chakrit Pongkitivanichkul, \emph{$SO(10)$ Grand Unification in M theory on a $G_2$ manifold} (\href{http://arxiv.org/abs/1502.01727}{arXiv:1502.01727})



\end{itemize}
Discussion of GUTs in [[F-theory]] includes

\begin{itemize}%
\item [[Chris Beasley]], [[Jonathan Heckman]], [[Cumrun Vafa]], \emph{GUTs and Exceptional Branes in F-theory - I} (\href{http://arxiv.org/abs/0802.3391}{arxiv:0802.3391}), \emph{II: Experimental Predictions} (\href{http://arxiv.org/abs/0806.0102}{arxiv:0806.0102})


\item [[Chris Beasley]], [[Jonathan Heckman]], [[Cumrun Vafa]], \emph{GUTs and Exceptional Branes in F-theory - I}, JHEP 0901:058,2009 (\href{http://arxiv.org/abs/0802.3391}{arXiv:0802.3391})


\item Gianluca Zoccarato, \emph{Yukawa couplings at the point of $E_8$ in F-theory}, 2014 (\href{http://stringpheno2014.ictp.it/parallels/tuesday/F-theory(B}{pdf}/zoccarato.pdf))


\item [[Cumrun Vafa]], \emph{Reflections on F-theory}, 2015 ()



\end{itemize}
\hypertarget{in_conneslott_models}{}\subsubsection*{{In Connes-Lott models}}\label{in_conneslott_models}

Discussion of GUTs within [[Connes-Lott models]]:

\begin{itemize}%
\item [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, \emph{Quanta of Geometry: Noncommutative Aspects}, Phys. Rev. Lett. 114 (2015) 9, 091302 (\href{https://arxiv.org/abs/1409.2471}{arXiv:1409.2471})


\item [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, \emph{Geometry and the Quantum: Basics}, JHEP 12 (2014) 098 (\href{https://arxiv.org/abs/1411.0977}{arXiv:1411.0977})


\item [[Alain Connes]], section 4 of \emph{Geometry and the Quantum}, in \emph{Foundations of Mathematics and Physics One Century After Hilbert}, Springer 2018. 159-196 (\href{https://arxiv.org/abs/1703.02470}{arXiv:1703.02470}, \href{https://www.springer.com/gp/book/9783319648125}{doi:10.1007/978-3-319-64813-2})



\end{itemize}
\hypertarget{exotica_leptoquarks_bosons_etc}{}\subsubsection*{{Exotica: Leptoquarks, $Z'$-bosons, etc.}}\label{exotica_leptoquarks_bosons_etc}

Topological defects can play considerable role to constrain the non-SUSY and SUSY GUTs:

\begin{itemize}%
\item Joydeep Chakrabortty, Rinku Maji, Sunando Kumar Patra, Tripurari Srivastava, Subhendra Mohanty, \emph{Roadmap of left-right models based on GUTs} (\href{https://arxiv.org/abs/1711.11391}{arXiv:1711.11391}, Phys.Rev. D97 (2018) no.9, 095010.)

\end{itemize}
Relation to [[Z'-bosons]]:

\begin{itemize}%
\item S. Sahoo, \emph{The prediction of mass of $Z'$-boson in an $SO(10)$-based model}, Indian J. Phys. 80 (2), 191-194 (2006) (\href{http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/6019/1/The%20Prediction%20of%20Mass%20of%20Z%27-Boson_By%20S%20Sahoo.pdf}{pdf})

\end{itemize}
Relation to [[leptoquarks]] and [[flavour anomalies]]:

\begin{itemize}%
\item H. Murayama, T. Yanagida, \emph{A viable $SU(5)$ GUT with light leptoquark bosons}, Mod.Phys.Lett. A7 (1992) 147-152 (\href{inspirehep.net/record/315898}{arXiv:315898}, \href{https://doi.org/10.1142/S0217732392000070}{doi:10.1142/S0217732392000070})


\item Damir Bečirević, Ilja Doršner, Svjetlana Fajfer, Nejc Košnik, Darius A. Faroughy, Olcyr Sumensari, \emph{Scalar leptoquarks from GUT to accommodate the $B$-physics anomalies}, Phys. Rev. D 98, 055003 (2018) (\href{https://arxiv.org/abs/1806.05689}{arXiv:1806.05689})


\item Ufuk Aydemir, Tanumoy Mandal, Subhadip Mitra, \emph{A single TeV-scale scalar leptoquark in SO(10) grand unification and B-decay anomalies} (\href{https://arxiv.org/abs/1902.08108}{arXiv:1902.08108})


\item Julian Heeck, Daniele Teresi, \emph{Pati-Salam explanations of the B-meson anomalies}, JHEP 12 (2018) 103 (\href{https://arxiv.org/abs/1808.07492}{arXiv:1808.07492})


\item Julian Heeck, Daniele Teresi, \emph{Pati-Salam and lepton universality in B decays} (\href{https://arxiv.org/abs/1905.05211}{arXiv:1905.05211})


\item [[Michal Malinský]], \emph{Lepton non-universality in $B$-decays in the minimal leptoquark gauge model} (\href{https://arxiv.org/abs/1906.09174}{arXiv:1906.09174})



\end{itemize}
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