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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{M-theory on G2-manifolds} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity} [[!include gravity contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{G2Manifolds}{Details}\dotfill \pageref*{G2Manifolds} \linebreak \noindent\hyperlink{VacuumSolutionsAndTorsion}{Vacuum solutions}\dotfill \pageref*{VacuumSolutionsAndTorsion} \linebreak \noindent\hyperlink{ComplexifiedModuli}{Complexified moduli space}\dotfill \pageref*{ComplexifiedModuli} \linebreak \noindent\hyperlink{EnhancedGaugeGroups}{Nonabelian gauge groups and chiral fermions at orbifold singularities}\dotfill \pageref*{EnhancedGaugeGroups} \linebreak \noindent\hyperlink{TheCField}{Solutions with non-vanishing $C$-field strength}\dotfill \pageref*{TheCField} \linebreak \noindent\hyperlink{Confinement}{Confinement?}\dotfill \pageref*{Confinement} \linebreak \noindent\hyperlink{relation_to_intersecting_dbrane_models}{Relation to intersecting D-brane models}\dotfill \pageref*{relation_to_intersecting_dbrane_models} \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{ReferencesPhenomenology}{Phenomenology}\dotfill \pageref*{ReferencesPhenomenology} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The [[Kaluza-Klein reduction]] of [[11-dimensional supergravity]] on [[G2 manifolds]] (notably [[Freund-Rubin compactifications]] and variants) yields an [[effective field theory|effective]] $N=1$ [[4-dimensional supergravity]] with [[gauge fields]] (arising from the KK-modes of the [[graviton]]) and charged [[fermions]] (arising from the KK-models of the [[gravitino]]). This construction is thought to lift to [[M-theory]] as the analog of the KK-compactification of [[heterotic string theory]] on [[Calabi-Yau manifolds]] (see at \emph{[[string phenomenology]]}), and of [[F-theory on CY4-manifolds]]. [[!include N=1 susy compactifications -- table]] In order for this to yield [[phenomenology|phenomenologically]] interesting effective physics the compactification space must be a [[G2-orbifold]] (hence a [[Riemannian orbifold]] of [[special holonomy]]), its [[stabilizer groups]] will encode the [[nonabelian group|nonabelian]] [[gauge group]] of the effective theory by ``[[geometric engineering of quantum field theory]]'' (\hyperlink{Acharya98}{Acharya 98}, \hyperlink{AtiyahWitten01}{Atiyah-Witten 01, section 6}), see \hyperlink{EnhancedGaugeGroups}{below}. Specifically for discussion of [[string phenomenology]] obtaining or approximating the [[standard model of particle physics]] by this procedure see at \emph{[[G2-MSSM]]}. \hypertarget{G2Manifolds}{}\subsection*{{Details}}\label{G2Manifolds} \hypertarget{VacuumSolutionsAndTorsion}{}\subsubsection*{{Vacuum solutions}}\label{VacuumSolutionsAndTorsion} Genuine [[G2-manifold]]/[[orbifold]] fibers, these having vanishing [[Ricci curvature]], correspond to [[vacuum]] solutions of the [[Einstein equations]] of [[11d supergravity]], i.e. with vanishing [[field strength]] of the [[gravitino]] and the [[supergravity C-field]] (see e.g. \hyperlink{Acharya02}{Acharya 02, p. 9}). (If one includes non-vanishing $C$-field strength one finds ``weak $G_2$-holonomy'' instead, see \hyperlink{TheCField}{below}). Notice that vanishing [[gravitino]] [[field strength]] (i.e. [[covariant derivative]]) means that the [[torsion of a Cartan connection|torsion]] of the super-[[vielbein]] is in each [[tangent space]] the canonical torsion of the [[super Minkowski spacetime]]. This [[supergravity torsion constraint|torsion constraint]] already just for the bosonic part $(E^a)$ of the super-vielbein $(E^a, E^\alpha)$ implies (together with the [[Bianchi identities]]) the [[equations of motion]] of supergravity, hence here the vacuum [[Einstein equations]] in the 11d [[spacetime]]. \hypertarget{ComplexifiedModuli}{}\subsubsection*{{Complexified moduli space}}\label{ComplexifiedModuli} For vanishing [[field strength]] of the [[supergravity C-field]], the formal linear combination \begin{displaymath} \tau \coloneqq C_3 + i \phi_3 \end{displaymath} of the (flat) supergravity $C$-field $C_3$ and the the 3-form $\phi_3$ of the $G_2$-structure is the natural [[holomorphic function|holomorphic coordinate]] on the [[moduli space]] of the [[KK-compactification]] of a $G_2$-manifold, in M-theoretic higher analogy of the complexified K\"a{}hler classes of CY compactifications of 10d string theory (\hyperlink{HarveyMoore99}{Harvey-Moore 99, (2.7)}, \hyperlink{Acharya02}{Acharya 02, (32) (59) (74)}, \hyperlink{GrigorianYau08}{Grigorian-Yau 08, (4.57)}, \hyperlink{AcharyaBobkov08}{Acharya-Bobkov 08, (4)}). Notice that restricted to [[associative submanifolds]] this combination becomes $C_3 + i vol$, which also governs the [[membrane instanton]]-contributions (``[[complex volume]]''). \hypertarget{EnhancedGaugeGroups}{}\subsubsection*{{Nonabelian gauge groups and chiral fermions at orbifold singularities}}\label{EnhancedGaugeGroups} The [[KK-compactification]] of vacuum [[11-dimensional supergravity]] on a \emph{[[smooth manifold|smooth]]} [[G2-manifold]] $Y$ results in a [[effective field theory|effective]] [[N=1 D=4 super Yang-Mills theory]] with [[abelian group|abelian]] [[gauge group]] $U(1)^{b_2(Y)}$ and with $b_3(Y)$ [[complex scalar fields]] which are neutral (not charged) under this gauge group (with $b_\bullet(Y)$ the [[Betti numbers]] of $Y$) (e.g. \hyperlink{Acharya02}{Acharya 02, section 2.3}). This is of course unsuitable for [[phenomenology]]. But when $Y$ is a $G_2$-[[orbifold]] then: \begin{enumerate}% \item at an [[ADE singularity]] there is \emph{[[enhanced gauge symmetry]] in that the [[gauge group]] (which a priori is copies of the [[abelian group]] $U(1)$ of the [[supergravity C-field]]) becomes [[nonabelian group|nonabelian]] (\hyperlink{Acharya98}{Acharya 98}, \hyperlink{Acharya00}{Acharya 00}, review includes \hyperlink{Acharya02}{Acharya 02, section 3}, \hyperlink{BBS07}{BBS 07, p. 422, 436}, \href{string+phenomenology#IbanezUranga12}{Ib\'a{}\~n{}ez-Uranga 12, section 6.3.3}, \href{string+phenomenology#Wijnholt14}{Wijnholt 14, part III} (from which the graphics below is grabbed));} \item at a (non-orbifold) [[conifold singularity]] [[chiral fermions]] appear (\hyperlink{Witten01}{Witten 01, p. 3}, \hyperlink{AtiyahWitten01}{Atiyah-Witten 01}, \hyperlink{AcharyaWitten01}{Acharya-Witten 01}, \hyperlink{BerglundBrandhuber02}{Berglund-Brandhuber 02}, \hyperlink{BourjailyEspahbodi08}{Bourjaily-Espahbodi 08}). \end{enumerate} The conifold singularities are supposed/assumed to be isolated (\hyperlink{Witten01}{Witten 01, section 2}), while the ADE singularities are supposed/assumed to be of codimension-4 in the 7-dimensional fibers (\hyperlink{Witten01}{Witten 01, section 3}, \hyperlink{Barrett06}{Barrett 06}). In the absence of a proper microscopic definition of M-theory, the first point is argued for indirectly in at least these ways: \begin{enumerate}% \item The fact that under [[KK-compactification]] to [[type IIA string theory]] the singularity becomes special points of intersecting [[D6-branes]] for which the gauge enhancement is known (\hyperlink{Sen97}{Sen 97}, \hyperlink{Witten01}{Witten 01, p. 1}, based on \hyperlink{CveticShiuUranga01}{Cvetic-Shiu-Uranga 01}). \item The [[duality in string theory|duality]] between M-theory compactified on [[K3]] and [[heterotic string theory]] on a 3-torus (\hyperlink{AcharyaWitten01}{Acharya-Witten 01}). Here it is fairly well understood that at the degeneration points of the K3-[[moduli space]] enhanced nonabelian gauge symmetry appears (e.g. \hyperlink{AcharyaGukov04}{Acharya-Gukov 04, section 5.1}). This comes down (\href{ALE+space#IntriligatorSeiberg96}{Intriligator-Seiberg 96}) to the fact that an [[ADE singularity]] $\mathbb{C}^2/\Gamma$ generically constitutes a point in the [[moduli space]] of [[vacua]] in the \href{N%3D2+D%3D4+super+Yang-Mills+theory#ModuliSpacesOfVacua}{Higgs branch} of a super Yang-Mills theory. \item The [[blow-up]] of an [[ADE-singularity]] happens to be a union of [[2-spheres]] touching pairwise in one point, such as to form the [[Dynkin diagram]] of the [[simple Lie group]] which under the [[ADE classification]] corresponds to the given [[orbifold]] isotropy group. (graphics grabbed from \href{http://ncatlab.org/schreiber/show/Equivariant+homotopy+and+super+M-branes}{HSS18}) [[M2-branes]] may [[wrapped brane|wrap]] these 2-cycles and since before blow-up they are of vanishing size, this corresponds to [[double dimensional reduction]] under which the M2-branes become [[strings]] stretching between coincident [[D-branes]]. These are well-understood to be the quanta of nonabelian gauge [[Chan-Paton gauge fields]] located on the D-branes, and hence these same nonabelian degrees of freedom have had to be present already at the level of the M2-branes. This is due to (\hyperlink{Sen97}{Sen 97}), for more see at \emph{[[M-theory lift of gauge enhancement on D6-branes]]}. \item In the [[F-theory]] description the ADE singularity maps to the locus where the F-theory [[elliptic fibration]] degenerates with 2-cycles in the elliptic fibers shrinking to 0. Via [[double dimensional reduction]] this manifestly takes the [[M2-brane]] [[wrapped brane|wrapping]] these elliptic fibers to an [[open string]] stretching between [[D7-branes]]. This yields at least $SU(N)$ gauge symmetry by the usual string theory argument about [[Chan-Paton gauge fields]]. \end{enumerate} Also notice that at least the $SU(N)$-enhancement of the [[effective field theory]] at $\mathbb{Z}_k$-singularities matches the $SU(N)$-enhancement of the [[worldvolume]] theory of $N$-coincident [[M2-branes]] sitting at the orbifold singularity: this is the statement of the [[ABJM model]]. \hypertarget{TheCField}{}\subsubsection*{{Solutions with non-vanishing $C$-field strength}}\label{TheCField} \textbf{Claim:} \emph{There is no warped [[KK-compactification]] of M-theory on $X_4 \times F_7$ which retains at least $N = 1$ [[supersymmetry]] in 4d while at the same time having non-vanishning $G_4$-[[flux]] ([[field strength]] of the [[supergravity C-field]]). In other words, non-vanishing flux always breaks the supersymmetry.} e.g. (\hyperlink{AcharyaSpence00}{Acharya-Spence 00}) see the Introduction of (\hyperlink{BeasleyWitten02}{Beasley-Witten 02}) $\,$ In compactifications with [[weak G2 holonomy]] it is the defining 4-form $\phi_4$ (the one which for strict [[G2 manifolds]] is the [[Hodge star operator|Hodge dual]] of the [[associative 3-form]]) which is the [[flux]]/[[field strength]] of the [[supergravity C-field]]. See for instance (\hyperlink{BilalDerendingerSfetos}{Bilal-Serendinger-Sfetos 02, section 6}): Consider a [[KK-compactification]]-Ansatz $X_{11} = (X_4,g_4) \times (X_7,g_7)$ and \begin{itemize}% \item $F_4 = f vol_{X_4}$; \item $F_7 = \tilde g e_7^\ast \phi_4$ \end{itemize} where $e_4$, $e_7$ are given [[vielbein]] fields on $X_4$ and $X_7$ and $\phi_4$ is the [[Hodge star operator|Hodge dual]] of the [[associative 3-form]]. Then the [[Einstein equations]] of [[11-dimensional supergravity]] give \begin{displaymath} R_4 = - \frac{1}{3}\left(f^2 + \frac{7}{2} \tilde g^2\right) g_4 \end{displaymath} \begin{displaymath} R_7 = \frac{1}{6}\left(f^2 + 5 \tilde g^2\right) g_7 \end{displaymath} (where $g_4$, $g_7$ is the [[pseudo-Riemannian metric|metric tensor]]) saying that both spaces are [[Einstein manifolds]] (\hyperlink{BilalDerendingerSfetos}{BSS 02, (5.4)}). The [[equations of motion]] for the [[supergravity C-field]] is \begin{displaymath} \tilde g\left( d \phi - f \star\phi \right) = 0 \end{displaymath} for $\phi = e_7^\ast \phi_3$ the pullback of the [[associative 3-form]] (\hyperlink{BilalDerendingerSfetos}{BSS 02, (5.5)}), saying that $\phi \propto \star F_7$ exhibits \href{G2+manifold#WeakG2Holonomy}{weak G2-holonomy} with weakness parameter given by the component of the [[C-field]] on $X_4$. \hypertarget{Confinement}{}\subsubsection*{{Confinement?}}\label{Confinement} An idea for a strategy towards a proof of [[confinement]] in [[N=1 D=4 super Yang-Mills theory]] via two different but conjecturally equivalent realizations as [[M-theory on G2-manifolds]] has been given in \hyperlink{AtiyahWitten01}{Atiyah-Witten 01, section 6}, review is in \hyperlink{AcharyaGukov04}{Acharya-Gukov 04, section 5.3}. The idea here is to consider a [[KK-compactification]] of [[M-theory]] on [[fibers]] which are [[G2-manifolds]] that locally around a special point are of the form \begin{displaymath} X_{1,\Gamma} \;\coloneqq\; \big( S^3 / \Gamma \big) \times Cone\big(S^3\big) \phantom{AA} \text{or} \phantom{AA} X_{2,\Gamma} \;\coloneqq\; S^3 \times Cone\big(S^3/\Gamma\big) \end{displaymath} where \begin{itemize}% \item $\Gamma$ is a [[finite subgroup of SU(2)]] that [[action|acts]] canonically by left-multiplication on $S^3 \simeq$ [[SU(2)]]; \item $Cone(\cdots)$ denotes the [[metric cone]] construction. \end{itemize} This means that $X_{1,\Gamma}$ is a [[smooth manifold]], but $X_{2,\Gamma}$, as soon as $\Gamma$ is not the [[trivial group]], $\Gamma \neq 1$, is an [[orbifold]] with an [[ADE singularity]]. Now the lore of [[M-theory on G2-manifolds]] predicts that [[KK-compactification]] \begin{enumerate}% \item on $X_{1,\Gamma}$ yields a 4d theory without massless fields (since there are no massless modes on the [[covering space]] $S^3$ of $X_{1,\Gamma}$) \item on the [[ADE-singularity]] $X_{2,\Gamma}$ yields [[non-abelian group|non-abelian]] [[Yang-Mills theory]] in 4d coupled to [[chiral fermions]]. \end{enumerate} both of these [[duality in string theory|dual]] by thinking of them in two different ways as [[M-theory on 8-manifold|M-theory on the 8-manifold]] [[HP{\tt \symbol{94}}2]] (\hyperlink{AtiyahWitten01}{Atiyah-Witten 01, p. 75 onwards}). So in the first case a [[mass gap]] is manifest, while non-abelian gauge theory is not visible, while in the second case it is the other way around. But if there were an argument that [[M-theory on G2-manifolds]] is in fact equivalent for compactification both on $X_{1,\Gamma}$ and on $X_{2,\Gamma}$. To the extent that this is true, it looks like an argument that could demonstrate confinement in non-abelian 4d gauge theory. This approach is suggested in \hyperlink{AtiyahWitten01}{Atiyah-Witten 01, pages 84-85}. An argument that this equivalence is indeed the case is then provided in sections 6.1-6.4, based on an argument in \hyperlink{AtiyahMaldacenaVafa00}{Atiyah-Maldacena-Vafa 00}. \hypertarget{relation_to_intersecting_dbrane_models}{}\subsubsection*{{Relation to intersecting D-brane models}}\label{relation_to_intersecting_dbrane_models} relation to [[intersecting D-brane models]]: see \href{intersecting+D-brane+model#ReferencesLiftToMTheory}{there} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[G2]] \item [[G2 manifold]], [[generalized G2-manifold]] \item [[G2-MSSM]] \item [[Freund-Rubin compactification]] \item [[exceptional generalized geometry]] \item [[topological M-theory]], [[Hitchin functional]] \item [[7d Chern-Simons theory]] \end{itemize} [[!include KK-compactifications of M-theory -- table]] \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The [[KK-compactification]] of [[11d supergravity]] of [[fibers]] of [[special holonomy]] was originally considered in \begin{itemize}% \item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapter V.6 of \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991) \item [[George Papadopoulos]], [[Paul Townsend]], \emph{Compactification of D=11 supergravity on spaces of exceptional holonomy}, Phys. Lett. B357:300-306,1995 (\href{http://arxiv.org/abs/hep-th/9506150}{arXiv:hep-th/9506150}) \end{itemize} Specifically [[string phenomenology]] for the case of compactification on [[G2-manifolds]] (or rather [[orbifolds]] ) goes back to \begin{itemize}% \item [[Bobby Acharya]], \emph{M theory, Joyce Orbifolds and Super Yang-Mills}, Adv. Theor. Math. Phys. 3 (1999) 227-248 (\href{http://arxiv.org/abs/hep-th/9812205}{arXiv:hep-th/9812205}) \item [[Bobby Acharya]], \emph{On Realising $N=1$ Super Yang-Mills in M theory} (\href{http://arxiv.org/abs/hep-th/0011089}{arXiv:hep-th/0011089}) \item [[Bobby Acharya]], B. Spence, \emph{Flux, Supersymmetry and M theory on 7-manifolds} (\href{http://arxiv.org/abs/hep-th/0007213}{arXiv:hep-th/0007213}) \item [[Bobby Acharya]], \emph{M Theory, $G_2$-manifolds and Four Dimensional Physics}, Classical and Quantum Gravity Volume 19 Number 22, 2002 (\href{http://users.ictp.it/~pub_off/lectures/lns013/Acharya/Acharya_Final.pdf}{pdf}) \item [[Michael Atiyah]], [[Juan Maldacena]], [[Cumrun Vafa]], \emph{An M-theory Flop as a Large N Duality}, J. Math. Phys.42:3209-3220, 2001 (\href{https://arxiv.org/abs/hep-th/0011256}{arXiv:hep-th/0011256}) \item [[Chris Beasley]], [[Edward Witten]], \emph{A Note on Fluxes and Superpotentials in M-theory Compactifications on Manifolds of $G_2$ Holonomy}, JHEP 0207:046,2002 (\href{http://arxiv.org/abs/hep-th/0203061}{arXiv:hep-th/0203061}) \item [[Michael Atiyah]], [[Edward Witten]] \emph{$M$-Theory dynamics on a manifold of $G_2$-holonomy}, Adv. Theor. Math. Phys. 6 (2001) (\href{http://arxiv.org/abs/hep-th/0107177}{arXiv:hep-th/0107177}) \item [[Edward Witten]], \emph{Anomaly Cancellation On Manifolds Of $G_2$ Holonomy} (\href{http://arxiv.org/abs/hep-th/0108165}{arXiv:hep-th/0108165}) \item [[Bobby Acharya]], [[Edward Witten]], \emph{Chiral Fermions from Manifolds of $G_2$ Holonomy} (\href{http://arxiv.org/abs/hep-th/0109152}{arXiv:hep-th/0109152}) \item Per Berglund, Andreas Brandhuber, \emph{Matter from $G_2$-manifolds}, Nucl. Phys. B641 (2002) 351-375 (\href{https://arxiv.org/abs/hep-th/0205184}{arXiv:hep-th/0205184}) \item Adam B. Barrett, \emph{M-Theory on Manifolds with $G_2$ Holonomy}, 2006 (\href{http://arxiv.org/abs/hep-th/0612096}{arXiv:hep-th/0612096}) \item Jacob L. Bourjaily, Sam Espahbodi, \emph{Geometrically Engineerable Chiral Matter in M-Theory} (\href{https://arxiv.org/abs/0804.1132}{arXiv:0804.1132}) \end{itemize} See also \begin{itemize}% \item [[Ashoke Sen]], \emph{A Note on Enhanced Gauge Symmetries in M- and String Theory}, JHEP 9709:001,1997 (\href{http://arxiv.org/abs/hep-th/9707123}{arXiv:hep-th/9707123}) \item [[Mirjam Cvetic]], [[Gary Gibbons]], H. L\"u{} and [[Christopher Pope]], \emph{Supersymmetric M3-branes and G2 Manifolds} (\href{http://cdsweb.cern.ch/record/503160/files/0106026.pdf}{pdf}) \item [[Bobby Acharya]], F. Denef, C. Hofman, [[Neil Lambert]], \emph{Freund-Rubin Revisited} (\href{http://arxiv.org/abs/hep-th/0308046}{arXiv:hep-th/0308046}) \end{itemize} More discussion of the non-abelian gauge group enhancement at [[orbifold]] singularities includes \begin{itemize}% \item [[Mirjam Cvetic]], [[Gary Shiu]], [[Angel Uranga]], \emph{Chiral Four-Dimensional $N=1$ Supersymmetric Type IIA Orientifolds from Intersecting D6-Branes}, Nucl. Phys. B615:3-32,2001 (\href{http://arxiv.org/abs/hep-th/0107166}{arXiv:hep-th/0107166}) \item [[James Halverson]], [[David Morrison]], \emph{On Gauge Enhancement and Singular Limits in $G_2$ Compactifications of M-theory} (\href{http://arxiv.org/abs/1507.05965}{arXiv:1507.05965}) \item [[Antonella Grassi]], [[James Halverson]], Julius L. Shaneson, \emph{Matter From Geometry Without Resolution}, Journal of High Energy Physics October 2013, 2013:205 (\href{http://arxiv.org/abs/1306.1832}{arXiv:1306.1832}) \item [[Neil Lambert]], Miles Owen, \emph{Charged Chiral Fermions from M5-Branes} (\href{https://arxiv.org/abs/1802.07766}{arXiv:1802.07766}) \item [[Andreas Braun]], Sebastjan Cizel, Max Hubner, Sakura Schafer-Nameki, \emph{Higgs Bundles for M-theory on G2-Manifolds} (\href{https://arxiv.org/abs/1812.06072}{arXiv:1812.06072}) \end{itemize} Discussion of [[Freund-Rubin compactification]] on $\mathbb{R}^4 \times X_7$ ``with flux'', hence non-vanishing [[supergravity C-field]] and how they preserve one supersymmetry if $X_7$ is of [[weak G2 holonomy]] with $\lambda$ = [[cosmological constant]] = C-[[field strength]] on $\mathbb{R}^4$ is in \begin{itemize}% \item [[Adel Bilal]], J.-P. Derendinger, K. Sfetsos, \emph{(Weak) $G_2$ Holonomy from Self-duality, Flux and Supersymmetry}, Nucl.Phys. B628 (2002) 112-132 (\href{http://arxiv.org/abs/hep-th/0111274}{arXiv:hep-th/0111274}) \item Thomas House, Andrei Micu, \emph{M-theory Compactifications on Manifolds with $G_2$ Structure} (\href{http://arxiv.org/abs/hep-th/0412006}{arXiv:hep-th/0412006}) \end{itemize} Further discussion of [[membrane instantons]] in this context includes \begin{itemize}% \item [[Gottfried Curio]], \emph{Superpotentials for M-theory on a G}2 holonomy manifold and Triality symmetry\_, JHEP 0303:024,2003 (\href{http://arxiv.org/abs/hep-th/0212211}{arXiv:hep-th/0212211}) \end{itemize} Survey and further discussion includes \begin{itemize}% \item [[Michael Duff]], \emph{M-theory on manifolds of G2 holonomy: the first twenty years} (\href{http://arxiv.org/abs/hep-th/0201062}{arXiv:hep-th/0201062}) \item [[Sergei Gukov]], \emph{M-theory on manifolds with exceptional holonomy}, Fortschr. Phys. 51 (2003), 719--731 (\href{http://research.physics.unc.edu/string/gukov.pdf}{pdf}) \item [[Bobby Acharya]], [[Sergei Gukov]], \emph{M theory and Singularities of Exceptional Holonomy Manifolds}, Phys.Rept.392:121-189,2004 (\href{http://arxiv.org/abs/hep-th/0409191}{arXiv:hep-th/0409191}) \item Adil Belhaj, \emph{M-theory on G2 manifolds and the method of (p, q) brane webs} (2004) (\href{http://iopscience.iop.org/0305-4470/37/18/011}{web}) \item Adam B. Barrett, \emph{M-Theory on Manifolds with $G_2$ Holonomy} (\href{http://arxiv.org/abs/hep-th/0612096}{arXiv:hep-th/0612096}) \item [[James Halverson]], [[David Morrison]], \emph{The Landscape of M-theory Compactifications on Seven-Manifolds with $G_2$ Holonomy} (\href{http://arxiv.org/abs/1412.4123}{arXiv:1412.4123}) \item Aaron Kennon, \emph{$G_2$-Manifolds and M-Theory Compactifications} (\href{https://arxiv.org/abs/1810.12659}{arXiv:1810.12659}) \end{itemize} The corresponding [[membrane]] [[instanton]] corrections to the [[superpotential]] are discussed in \begin{itemize}% \item [[Jeffrey Harvey]], [[Greg Moore]], \emph{Superpotentials and Membrane Instantons} (\href{http://arxiv.org/abs/hep-th/9907026}{arXiv:hep-th/9907026}) \item [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], p. 333 of \emph{String Theory and M-Theory: A Modern Introduction}, 2007 \end{itemize} Discussion of [[duality in string theory|duality]] with [[F-theory on CY4-manifolds]] includes \begin{itemize}% \item [[Sergei Gukov]], [[Shing-Tung Yau]], [[Eric Zaslow]], \emph{Duality and Fibrations on $G_2$ Manifolds} (\href{http://arxiv.org/abs/hep-th/0203217}{arXiv:hep-th/0203217}) \item Adil Belhaj, \emph{F-theory Duals of M-theory on $G_2$ Manifolds from Mirror Symmetry} (\href{http://arxiv.org/abs/hep-th/0207208}{arXiv:hep-th/0207208}) \item [[Mariana Graña]], C. S. Shahbazi, [[Marco Zambon]], \emph{$Spin(7)$-manifolds in compactifications to four dimensions}, JHEP11(2014)046 (\href{http://arxiv.org/abs/1405.3698}{arXiv:1405.3698}) \end{itemize} Discussion of [[duality in string theory|duality]] with [[heterotic string theory on CY3-manifolds]]: \begin{itemize}% \item [[Andreas Braun]], Sakura Schaefer-Nameki, \emph{Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality}, JHEP04(2018)126 (\href{https://arxiv.org/abs/1708.07215}{arXiv:1708.07215}) \end{itemize} The [[moduli space]] is discussed in \begin{itemize}% \item [[Sergey Grigorian]], [[Shing-Tung Yau]], \emph{Local geometry of the $G_2$ moduli space}, Commun.Math.Phys.287:459-488,2009 (\href{http://arxiv.org/abs/0802.0723}{arXiv:0802.0723}) \item [[Bobby Acharya]], Konstantin Bobkov, \emph{K\"a{}hler Independence of the G2-MSSM}, HEP 1009:001,2010 (\href{http://arxiv.org/abs/0810.3285}{arXiv:0810.3285}) \item [[Spiro Karigiannis]], [[Naichung Conan Leung]]\_, \emph{Hodge Theory for $G_2$-manifolds: Intermediate Jacobians and Abel-Jacobi maps}, Proceedings of the London Mathematical Society (3) 99, 297-325 (2009) (\href{http://arxiv.org/abs/0709.2987}{arXiv:0709.2987} \href{http://www.math.uwaterloo.ca/~karigian/talks/g2modulispace.pdf}{talk slides pdf} \end{itemize} \hypertarget{ReferencesPhenomenology}{}\subsubsection*{{Phenomenology}}\label{ReferencesPhenomenology} Popular exposition of the [[G2-MSSM]] [[phenomenology]] is in \begin{itemize}% \item [[Gordon Kane]], \emph{String theory and the real world}, Morgan \& Claypool, 2017 () \end{itemize} Further discussion of [[string phenomenology]] in terms of $M$-theory on $G_2$-manifolds, beyond the original (\hyperlink{Acharya98}{Acharya 98}, \hyperlink{AtiyahWitten01}{Atiyah-Witten 01}, \hyperlink{AcharyaWitten01}{Acharya-Witten 01}), includes \begin{itemize}% \item [[Bobby Acharya]], [[Gordon Kane]], [[Piyush Kumar]], \emph{Compactified String Theories -- Generic Predictions for Particle Physics}, Int. J. Mod. Phys. A, Volume 27 (2012) 1230012 (\href{http://arxiv.org/abs/1204.2795}{arXiv:1204.2795}) \item [[Bobby Acharya]], \emph{$G_2$-manifolds at the CERN Large Hadron collider and in the Galaxy}, talk at \emph{$G_2$-days} (2012) (\href{http://www.mth.kcl.ac.uk/~tbmadsen/acharya.pdf}{pdf}) \item [[Gordon Kane]], \emph{String/M-theories About Our World Are Testable in the traditional Physics Way} (\href{http://arxiv.org/abs/1601.07511}{arXiv:1601.07511}, \href{https://videoonline.edu.lmu.de/en/node/7485}{video recording}) \end{itemize} Discussion of [[moduli stabilization]] for stabilization via ``[[flux]]'' (non-vanishing bosonic [[field strength]] of the [[supergravity C-field]]) is in \begin{itemize}% \item [[Bobby Acharya]], \emph{A Moduli Fixing Mechanism in M theory} (\href{http://arxiv.org/abs/hep-th/0212294}{arXiv:hep-th/0212294}) \end{itemize} and [[moduli stabilization]] for fluxless compactifications via [[nonperturbative effects]], claimed to be sufficient and necessary to solve the [[hierarchy problem]], is discussed in \begin{itemize}% \item [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Diana Vaman, \emph{An M theory Solution to the Hierarchy Problem}, Phys.Rev.Lett.97:191601,2006 (\href{http://arxiv.org/abs/hep-th/0606262}{arXiv:hep-th/0606262}) \item [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, \emph{Explaining the Electroweak Scale and Stabilizing Moduli in M Theory}, Phys.Rev.D76:126010,2007 (\href{http://arxiv.org/abs/hep-th/0701034}{arXiv:hep-th/0701034}) \item [[Bobby Acharya]], [[Piyush Kumar]], Konstantin Bobkov, [[Gordon Kane]], Jing Shao, Scott Watson, \emph{Non-thermal Dark Matter and the Moduli Problem in String Frameworks},JHEP 0806:064,2008 (\href{http://arxiv.org/abs/0804.0863}{arXiv:0804.0863}) \end{itemize} and specifically for the [[G2-MSSM]] in \begin{itemize}% \item [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, \emph{The $G_2$-MSSM - An $M$ Theory motivated model of Particle Physics} (\href{http://arxiv.org/abs/0801.0478}{arXiv:0801.0478}) \end{itemize} the [[strong CP problem]] is discussed in \begin{itemize}% \item Peter Svrcek, [[Edward Witten]], section 6 of \emph{Axions In String Theory}, JHEP 0606:051,2006 (\href{http://arxiv.org/abs/hep-th/0605206}{arXiv:hep-th/0605206}) \item [[Bobby Acharya]], Konstantin Bobkov, [[Piyush Kumar]], \emph{An M Theory Solution to the Strong CP Problem and Constraints on the Axiverse}, JHEP 1011:105,2010 (\href{http://arxiv.org/abs/1004.5138}{arXiv:1004.5138}) \end{itemize} and realization of [[GUTs]] 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} The phenomenology of compactifications on [[compact twisted connected sum G2-manifolds]] (\href{G2+manifold#Kovalev00}{Kovalev 00}) is in \begin{itemize}% \item Thaisa C. da C. Guio, [[Hans Jockers]], [[Albrecht Klemm]], Hung-Yu Yeh, \emph{Effective action from M-theory on twisted connected sum $G_2$-manifolds} (\href{https://arxiv.org/abs/1702.05435}{arXiv:1702.05435}, \href{https://lecture2go.uni-hamburg.de/l2go/-/get/v/21906}{talk video}) \end{itemize} Discussion of the [[cosmological constant]] in these models includes \begin{itemize}% \item Beatriz de Carlos, Andre Lukas, Stephen Morris, \emph{Non-perturbative vacua for M-theory on G2 manifolds}, JHEP 0412:018, 2004 (\href{https://arxiv.org/abs/hep-th/0409255}{arxiv:hep-th/0409255}) \end{itemize} which concludes that with taking [[non-perturbative effects]] from [[membrane instantons]] into account one gets 4d vacua with vanishing and negative [[cosmological constant]] ([[Minkowski spacetime]] and [[anti-de Sitter spacetime]]) but not with positive [[cosmological constant]] ([[de Sitter spacetime]]). They close by speculating that [[M5-brane]] instantons might yield [[de Sitter spacetime]]. Suggestion that [[higher curvature corrections]] allow [[de Sitter spacetime]] vacua: \begin{itemize}% \item Johan Blåbäck, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, \emph{Constructing stable de Sitter in M-theory from higher curvature corrections} (\href{https://arxiv.org/abs/1902.04053}{arXiv:1902.04053}) \end{itemize} See also \begin{itemize}% \item [[Andreas Braun]], Sebastjan Cizel, Max Hubner, Sakura Schafer-Nameki, \emph{Higgs Bundles for M-theory on $G_2$-Manifolds} (\href{https://arxiv.org/abs/1812.06072}{arXiv:1812.06072}) \end{itemize} [[!redirects G2 compactifications of M-theory]] \end{document}