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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*{G2 manifold} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry} [[!include Riemannian geometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{G2Structure}{$G_2$-structure}\dotfill \pageref*{G2Structure} \linebreak \noindent\hyperlink{ClosedG2Structure}{Closed $G_2$-structure}\dotfill \pageref*{ClosedG2Structure} \linebreak \noindent\hyperlink{G2Holonomy}{$G_2$-holonomy / $G_2$-manifold}\dotfill \pageref*{G2Holonomy} \linebreak \noindent\hyperlink{variants_and_weakenings}{Variants and weakenings}\dotfill \pageref*{variants_and_weakenings} \linebreak \noindent\hyperlink{with_skewsymmetric_torsion}{With skew-symmetric torsion}\dotfill \pageref*{with_skewsymmetric_torsion} \linebreak \noindent\hyperlink{WeakG2Holonomy}{Weak $G_2$-holonomy}\dotfill \pageref*{WeakG2Holonomy} \linebreak \noindent\hyperlink{WithADEOrbifoldStructure}{With ADE orbifold structure}\dotfill \pageref*{WithADEOrbifoldStructure} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{existence}{Existence}\dotfill \pageref*{existence} \linebreak \noindent\hyperlink{metric_structure}{Metric structure}\dotfill \pageref*{metric_structure} \linebreak \noindent\hyperlink{as_part_of_the_berger_classification}{As part of the Berger classification}\dotfill \pageref*{as_part_of_the_berger_classification} \linebreak \noindent\hyperlink{as_riemannian_manifolds}{As $\mathbb{O}$-Riemannian manifolds}\dotfill \pageref*{as_riemannian_manifolds} \linebreak \noindent\hyperlink{as_exceptional_geometry}{As exceptional geometry}\dotfill \pageref*{as_exceptional_geometry} \linebreak \noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak \noindent\hyperlink{resolution_of_joyce_orbifolds}{Resolution of Joyce orbifolds}\dotfill \pageref*{resolution_of_joyce_orbifolds} \linebreak \noindent\hyperlink{TwistedConnectedSumConstruction}{Twisted connected sum construction}\dotfill \pageref*{TwistedConnectedSumConstruction} \linebreak \noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak \noindent\hyperlink{in_supergravity}{In supergravity}\dotfill \pageref*{in_supergravity} \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{ReferencesG2Orbifolds}{$G_2$-Orbifolds}\dotfill \pageref*{ReferencesG2Orbifolds} \linebreak \noindent\hyperlink{moduli}{Moduli}\dotfill \pageref*{moduli} \linebreak \noindent\hyperlink{variants_and_generalizations}{Variants and generalizations}\dotfill \pageref*{variants_and_generalizations} \linebreak \noindent\hyperlink{relation_to_killing_spinors}{Relation to Killing spinors}\dotfill \pageref*{relation_to_killing_spinors} \linebreak \noindent\hyperlink{application_in_supergravity}{Application in supergravity}\dotfill \pageref*{application_in_supergravity} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{$G_2$-structure} on a [[manifold]] $X$ of [[dimension]] 7 is a choice of [[G-structure]] on $X$, for $G$ the [[exceptional Lie group]] [[G2]]. Hence it is a reduction of the [[structure group]] of the [[frame bundle]] of $X$ along the canonical (the defining) inclusion $G_2 \hookrightarrow GL(\mathbb{R}^7)$ into the [[general linear group]]. Given that $G_2$ is the [[subgroup]] of the [[general linear group]] on the [[Cartesian space]] $\mathbb{R}^7$ which preserves the [[associative 3-form]] on $\mathbb{R}^7$, a $G_2$ structure is a higher analog of an [[almost symplectic structure]] under lifting from [[symplectic geometry]] to [[2-plectic geometry]] (\hyperlink{Ibort}{Ibort}). A \emph{$G_2$-manifold} is a manifold equipped with $G_2$-structure that is [[integrability of G-structures|integrable to first order]], i.e. [[torsion of a G-structure|torsion-free]] (prop. \ref{CovariantlyConstantDefinite3FormMeansTorsionVanishes} below). This is equivalently a [[Riemannian manifold]] of [[dimension]] 7 with [[special holonomy]] group being the [[exceptional Lie group]] [[G2]]. $G_2$-manifolds may be understood as 7-dimensional [[analogs]] of real 6-dimensional [[Calabi-Yau manifolds]]. Accordingly the relation between [[Calabi-Yau manifolds and supersymmetry]] lifts from [[string theory]] to [[M-theory on G2-manifolds]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{G2Structure}{}\subsubsection*{{$G_2$-structure}}\label{G2Structure} \begin{defn} \label{G2Structure}\hypertarget{G2Structure}{} For $X$ a [[smooth manifold]] of [[dimension]] $7$ a \textbf{$G_2$-structure} on $X$ is a [[G-structure]] for $G =$ [[G2]] $\hookrightarrow GL(7)$. \end{defn} \begin{remark} \label{CanonicalRiemannianMetric}\hypertarget{CanonicalRiemannianMetric}{} A $G_2$-structure in particular implies an [[orthogonal structure]], hence a [[Riemannian metric]]. \end{remark} Given the definition of [[G2]] as the [[stabilizer group]] of the [[associative 3-form]] on $\mathbb{R}^7$, there is accordingly an equivalent formulation of def. \ref{G2Structure} in terms of [[differential forms]]: \begin{defn} \label{Definite3Forms}\hypertarget{Definite3Forms}{} Write $\Lambda^3_+(\mathbb{R}^7)^\ast \hookrightarrow \Lambda^3(\mathbb{R}^7)^\ast$ for the [[orbit]] of the [[associative 3-form]] $\phi$ under the canonical $GL(7)$-[[action]]. Similarly for $X$ a [[smooth manifold]] of [[dimension]] 7, write \begin{displaymath} \Omega^3_+(X) \hookrightarrow \Omega^3(X) \end{displaymath} for the subset of the set of [[differential 3-forms]] on those that, as [[sections]] to the exterior power of the [[cotangent bundle]], are pointwise in $\Lambda^3_+(\mathbb{R}^7)^\ast$. These are also called the \emph{positive forms} (\href{Joyce00}{Joyce 00, p. 243}) or the \emph{[[definite differential forms]]} (\hyperlink{Bryant05}{Bryant 05, section 3.1.1}) on $X$. \end{defn} (e.g. \hyperlink{Bryant05}{Bryant 05, definition 2}) \begin{prop} \label{G2StructureViaDefinite3Form}\hypertarget{G2StructureViaDefinite3Form}{} A $G_2$-structure on $X$, def. \ref{G2Structure}, is equivalently a choice of [[definite 3-form]] $\sigma$ on $X$, def. \ref{Definite3Forms}. \end{prop} (e.g. \href{Joyce00}{Joyce 00, p. 243}, \hyperlink{Bryant05}{Bryant 05, section 3.1.1}) Often it is useful to exhibit prop. \ref{G2StructureViaDefinite3Form} in the following way. \begin{example} \label{DefiniteFormsInTermsOfVielbeinFields}\hypertarget{DefiniteFormsInTermsOfVielbeinFields}{} For $X$ a [[smooth manifold]] of [[dimension]] 7, write $Fr(X) \to X$ for its [[frame bundle]]. By the discussion at \emph{\href{vielbein#InTermsOfBasicFormsOnFrameBundle}{vielbein -- in terms of basic forms on the frame bundle}} there is a universal $\mathbb{R}^7$-valued differential form on the total space of the frame bundle \begin{displaymath} E_u \in \Omega^1(Fr(X), \mathbb{R}^7) \end{displaymath} (whose components we write $(E_u^a)_{a = 1}^7$) such that given an [[orthogonal structure]] $i \colon Fr_O(X)\hookrightarrow Fr(X)$ and a local section $\sigma_i \colon (U_i \subset X) \to Fr_O(X)$ of orthogonal frames, then the [[pullback of differential forms]] \begin{displaymath} E_i \coloneqq \sigma_i^\ast i^\ast E_u \end{displaymath} is the corresponding local [[vielbein]] field. Hence one obtains a universal 3-form $\phi_u \in \Omega^3(Fr(x))$ on the frame bundle by setting \begin{displaymath} \phi_u \coloneqq \phi_{a b c} E_u^a \wedge E_u^b \wedge E_u^c \end{displaymath} with $(\phi_{a b c})$ the canonical components of the [[associative 3-form]] and with summation over repeated indices understood. By construction this is such that on a [[chart]] $(U_i \subset X)$ any [[definite 3-form]], def. \ref{Definite3Forms}, restricts to the [[pullback of differential forms|pullback]] of $\phi_u$ via a [[section]] $\sigma_i \colon U_i \to Fr(X)$ and hence is of the form \begin{displaymath} \phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c \,. \end{displaymath} Conversely, given a 3-form $\sigma \in \Omega^3(X)$ such that on an [[atlas]] $(U_i \to X)$ over which the frame bundle trivializes it is of this form \begin{displaymath} \sigma|_{U_i} = \phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c \end{displaymath} then the $GL(7)$-valued transition functions $g_{i j}$ of the given local trivialization must factor through $G_2\hookrightarrow SO(7) \hookrightarrow GL(7)$ and hence exhibit a $G_2$-structure: because we have $\sigma|_{U_i} = \sigma|_{U_j} \;\;\; on \;U_i \cap U_j$ and hence \begin{equation} \phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c = \phi_{a b c} E_j^a \wedge E_j^b \wedge E_j^c \;\;\; on \; U_i \cap U_j \,. \label{Definite3FormsOnTwoPatches}\end{equation} But by the nature of the \href{vielbein#InTermsOfBasicFormsOnFrameBundle}{universal vielbein}, its local pullbacks are related by \begin{displaymath} E_j = g_{i j} E_i \end{displaymath} i.e. \begin{displaymath} E_j^a = (g_{i j})^a{}_b E_i^b \end{displaymath} and hence \eqref{Definite3FormsOnTwoPatches} says that \begin{displaymath} \phi_{a b c} = \phi_{a' b' c'} (g_{i j})^{a'}{}_a (g_{i j})^{b'}{}_b (g_{i j})^{c'}{}_c \;\;\; on \; U_i \cap U_j \end{displaymath} which is precisely the defining condition for $g_{i j}$ to take values in $G_2$. Viewed this way, the [[definite 3-forms]] characterizing $G_2$-structures are an example of a more general kind of differential forms obtained from a constant form on some linear model space $V$ by locally contracting with a [[vielbein]] field. For instance on a [[super-spacetime]] solving the [[equations of motion]] of [[11-dimensional supergravity]] there is a super-4-form part of the [[field strength]] of the [[supergravity C-field]] which is constrained to be locally of the form \begin{displaymath} \Gamma_{a b \alpha \beta} E_i^a \wedge E_i^b \wedge E_i^\alpha \wedge E_i^\beta \end{displaymath} for $(E^A)= (E^a, E^\alpha) = (E^a, \Psi^\alpha)$ the [[super-vielbein]]. See at \emph{\href{Green-Schwarz+action+functional#MembraneIn11dSuGraBackground}{Green-Schwarz action functional -- Membrane in 11d SuGra Background}}. Indeed, by the discussion there this 4-form is required to be [[covariant derivative|covariantly constant]], which is precisely the analog of $G_2$-manifold structure as in def. \ref{G2manifold}. \end{example} References that write [[definite 3-forms]] in this form locally as $\phi_{a b c}E^a \wedge E^b \wedge E^c$ include (\hyperlink{BGGG01}{BGGG 01 (2.9)}, \ldots{}). The following is important for the analysis: \begin{remark} \label{Definition3FormsGiveOpenSubset}\hypertarget{Definition3FormsGiveOpenSubset}{} The subset $\Lambda^3_+(\mathbb{R}^7)^\ast \hookrightarrow \Lambda^3(\mathbb{R}^7)^\ast$ in def. \ref{Definite3Forms} is an [[open subset]], hence $\phi$ is a \emph{[[stable form]]} (e.g. \hyperlink{Hitchin}{Hitchin, def. 1.1}). \end{remark} (e.g. \hyperlink{Joyce00}{Joyce 00, p. 243}, \hyperlink{Bryant05}{Bryant 05, 2.8}) \begin{proof} By definition of $G_2$ as the [[stabilizer group]] of the [[associative 3-form]], the [[orbit]] it generates under the $GL_+(7)$-action is the [[coset]] $GL_+(7)/G_2$. The [[dimension]] of this as a [[smooth manifold]] is 49-14 = 35. This is however already the full dimension $\left(7 \atop 3\right) = 35$ of the space of 3-forms in 7d that the orbit sits in. Therefore (since $G_+(7)/G_2$ does not have a [[manifold with boundary|boundary]]) the orbit must be an open subset. \end{proof} \hypertarget{ClosedG2Structure}{}\subsubsection*{{Closed $G_2$-structure}}\label{ClosedG2Structure} \begin{defn} \label{ClosedG2Structure}\hypertarget{ClosedG2Structure}{} A $G_2$-structure, def. \ref{G2Structure}, is called \emph{closed} if the definite 3-form $\sigma$ corresponding to it via prop. \ref{G2StructureViaDefinite3Form} is a closed differential form, $\mathbf{d}\sigma = 0$. \end{defn} (e.g. \hyperlink{Bryant05}{Bryant 05, (4.31)}) \begin{prop} \label{ClosedG2StructureByAtlas}\hypertarget{ClosedG2StructureByAtlas}{} For a closed $G_2$-structure, def. \ref{ClosedG2Structure}, on a manifold $X$ there exists an [[atlas]] by [[open subsets]] \begin{displaymath} \mathbb{R}^7 \underoverset{et}{f}{\leftarrow} U \underset{et}{\rightarrow} X \end{displaymath} such that the globally defined 3-form $\sigma \in \Omega^3_+(X)$ is locally gauge equivalent to the canonical [[associative 3-form]] $\phi$ \begin{displaymath} \sigma|_U = f^\ast \phi + \mathbf{d}\beta \end{displaymath} via a 2-form $\beta$ on $U$. \end{prop} (e.g. \hyperlink{Bryant05}{Bryant 05, p. 21}) This follows from the fact, remark \ref{Definition3FormsGiveOpenSubset}, that the definite 3-forms are an [[open subset]] inside all 3-forms: given a chart centered around any point then there is $\beta$ with $\mathbf{d}\beta$ vanishing at that point such that $\sigma|_U \simeq f^\ast \phi + \mathbf{d}\beta$ at that point. But since the $GL(7)$-action on $\phi$ is open, there is an open neighbourhood around that point where this is still the case. \begin{remark} \label{AtlasForClosedG2tructureInTermsOfHigherGeometry}\hypertarget{AtlasForClosedG2tructureInTermsOfHigherGeometry}{} When regarding [[smooth manifolds]] in the wider context of [[higher differential geometry]], then the situation of prop. \ref{ClosedG2StructureByAtlas} corresponds to a diagram of [[formal smooth infinity-groupoids]] of the following form: \begin{displaymath} \itexarray{ && U \\ & {}^{\mathllap{f}}\swarrow && \searrow \\ \mathbb{R}^7 && \swArrow_{\mathrlap{\beta}} && X \\ & {}_{\mathllap{\phi}}\searrow && \swarrow_{\mathrlap{\sigma}} \\ && \flat_{dR}\mathbf{B}^3\mathbb{R} } \,, \end{displaymath} where $\flat_{dR}\mathbf{B}^3\mathbb{R}$ is the [[moduli infinity-stack|higher moduli stack]] of flat 3-forms with 2-form gauge transformations between them (and 1-form gauge transformation between these). The diagram expresses the 3-form $\sigma$ as a map to this moduli stack, which when restricted to the cover $U$ becomes gauge equivalent to the pullback of the [[associative 3-form]] $\phi$, similarly regarded as a map, to the cover, where the gauge equivalence is exhibited by a [[homotopy]] (of maps of formal smooth $\infty$-groupoids) which is the 2-form $\beta$ on $U$. \end{remark} \hypertarget{G2Holonomy}{}\subsubsection*{{$G_2$-holonomy / $G_2$-manifold}}\label{G2Holonomy} \begin{defn} \label{G2manifold}\hypertarget{G2manifold}{} A manifold $X$ equipped with a $G_2$-structure, def. \ref{G2Structure}, is called a \textbf{$G_2$-manifold} if the following equivalent conditions hold \begin{enumerate}% \item we have \begin{enumerate}% \item $\mathbf{d} \omega = 0$ (\hyperlink{ClosedG2Structure}{closed}) \item $\mathbf{d} \star_g \omega = 0$ (co-closed); \end{enumerate} \item $\nabla^g \omega = 0$; \item $(X,g)$ has [[special holonomy]] $Hol(g) \subset G_2$; \item $Ric(g) = 0$ (vanishing [[Ricci curvature]]); \item $R(g) = 0$ (vanishing [[scalar curvature]]); \item $\tau = 0$ (vanishing [[torsion of a G-structure|torsion of the G2-structure]]). \end{enumerate} Here \begin{itemize}% \item $d$ is the [[de Rham differential]]; \item $\omega$ is the 3-form $\omega$ corresponding to the given $G_2$-structure via prop. \ref{G2StructureViaDefinite3Form}; \item $g$ is the induced [[Riemannian metric]] of remark \ref{CanonicalRiemannianMetric}); \item $\star_g$ is the [[Hodge star operator]] of this metric; \item $\nabla^g$ is the [[covariant derivative]] of this metric; \end{itemize} \end{defn} For the equivalence of the first items see for instance (\hyperlink{Joyce}{Joyce, p. 4}, \hyperlink{Joyce00}{Joyce 00, prop. 10.1.3}). For the equivalence to the vanishing curvature invariant see also (\hyperlink{Bryant05}{Bryant 05, corollary 1}), and for the equivalence to the vanishing [[torsion of a G-structure]] see (\hyperlink{Bryant05}{Bryant 05, prop. 2}). \begin{remark} \label{}\hypertarget{}{} The higher [[torsion of a G-structure|torsion invariants]] of $G_2$-structures do not necessarily vanish (contrary to the case for instance of [[symplectic structure]] and [[complex structure]], see at \href{integrability+of+G-structures#Examples}{integrability of G-structures -- Examples}). Therefore, even in view of prop. \ref{CovariantlyConstantDefinite3FormMeansTorsionVanishes}, a $G_2$-manifold, def. \ref{G2manifold}, does not, in general admit an [[atlas]] be adapted [[coordinate charts]] equal to $(\mathbb{R}^7, \phi)$. The space of second order torsion invariants of $G_2$-structures is for instance in (\hyperlink{Bryant05}{Bryant 05 (4.7)}). \end{remark} \hypertarget{variants_and_weakenings}{}\subsubsection*{{Variants and weakenings}}\label{variants_and_weakenings} There are several variants of the definition of $G_2$-manifolds, def.\ref{G2manifold}, given by imposing other constraints on the [[torsion of a Cartan connection|torsion]]. \hypertarget{with_skewsymmetric_torsion}{}\paragraph*{{With skew-symmetric torsion}}\label{with_skewsymmetric_torsion} Discussion for totally skew symmetric [[torsion of a Cartan connection]] includes (\hyperlink{FriedrichIvanov01}{Friedrich-Ivanov 01, theorem 4.7, theorem 4.8}) \hypertarget{WeakG2Holonomy}{}\paragraph*{{Weak $G_2$-holonomy}}\label{WeakG2Holonomy} \begin{defn} \label{WeakG2Holonomy}\hypertarget{WeakG2Holonomy}{} A 7-dimensional manifold is said to be of \emph{weak $G_2$-holonomy} if it carries a 3-form $\omega$ with the relation of def. \ref{G2manifold} generalized to \begin{displaymath} \mathbf{d} \omega = \lambda \star \omega \end{displaymath} and hence \begin{displaymath} \mathbf{d} \star \omega = 0 \end{displaymath} for $\lambda \in \mathbb{R}$. For $\lambda = 0$ this reduces to strict $G_2$-holonomy, by \ref{G2manifold}. \end{defn} (See for instance (\hyperlink{BilalDerendingerSfetsos}{Bilal-Derendinger-Sfetsos 02}, \hyperlink{BilalMetzger03}{Bilal-Metzger 03}).) \hypertarget{WithADEOrbifoldStructure}{}\paragraph*{{With ADE orbifold structure}}\label{WithADEOrbifoldStructure} When used as [[KK-compactification]]-fibers for [[M-theory on G2-manifolds]], then for realistic [[string phenomenology|phenomenology]] one needs to consider [[ADE orbifolds]] with ``$G_2$-manifold'' structure, i.e. [[G2-orbifolds]], also called \emph{Joyce orbifolds}. Moreover, for [[F-theory]] purposes this $G_2$-orbifold is to be a fibration by a [[K3 surface]] $X_{K3}$. For instance the [[Cartesian product]] $X_{K3} \times T^3$ admits a $G_2$-manifold structure. There is a canonical [[special orthogonal group|SO(3)]]-[[action]] on the tangent spaces of $X_{K3} \times T^3$, given on $X_{K3}$ by rotation of the [[hyper-Kähler manifold]]-structure of $X_{K_3}$ and on $T^3$ by the standard rotation. For $K_{ADE}$ a [[finite group|finite]] [[subgroup]] of $SO(3)$, hence a finite group in the [[ADE classification]], then $(X_{K3}\times T^3)/K_{ADE}$ is a [[G2-orbifold]]. (\hyperlink{Acharya98}{Acharya 98, p.3}). (For $K_{ADE}$ \emph{not} a [[cyclic group]] then this has precisely one [[parallel spinor]].) In a local [[coordinate chart]] of $X_{K3}$ by $\mathbb{C}^2$ the orbifold $X_{K3}/K_{ADE}$ locally looks like $\mathbb{C}^2/{G_{ADE}}$, where now $G_{ADE}$ is a [[finite group|finite]] [[subgroup]] of [[special unitary group|SU(2)]]. Such local [[G2-orbifolds]] are discussed in some detail in (\hyperlink{AtiyahWitten01}{Atiyah-Witten 01}). Families of examples are constructed in \hyperlink{Reidegeld15}{Reidegeld 15}. Codimension-4 ADE singularities in $G_2$-manifolds are discussed in (\href{AcharyaGukov04}{Acharya-Gukov 04, section 5.1}, \hyperlink{Barrett06}{Barrett 06}). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{existence}{}\subsubsection*{{Existence}}\label{existence} \begin{prop} \label{}\hypertarget{}{} A 7-manifold admits a $G_2$-structure, def. \ref{G2Structure}, precisely if it admits an [[orientation]] and a [[spin structure]]. \end{prop} That orientability and spinnability is necessary follows directly from the fact that $G_2 \hookrightarrow GL(7)$ is connected and simply connected. That these conditions are already sufficient is due to (\hyperlink{Gray69}{Gray 69}), see also (\hyperlink{Bryant05}{Bryant 05, remark 3}). \hypertarget{metric_structure}{}\subsubsection*{{Metric structure}}\label{metric_structure} The canonical [[Riemannian metric]] $G_2$ manifold is [[Ricci tensor|Ricci flat]]. More generally a manifold of weak $G_2$-holonomy, def. \ref{WeakG2Holonomy}, with weakness parameter $\lambda$ is an [[Einstein manifold]] with [[cosmological constant]] $\lambda$. \hypertarget{as_part_of_the_berger_classification}{}\subsubsection*{{As part of the Berger classification}}\label{as_part_of_the_berger_classification} [[!include special holonomy table]] \hypertarget{as_riemannian_manifolds}{}\subsubsection*{{As $\mathbb{O}$-Riemannian manifolds}}\label{as_riemannian_manifolds} [[!include normed division algebra Riemannian geometry -- table]] \hypertarget{as_exceptional_geometry}{}\subsubsection*{{As exceptional geometry}}\label{as_exceptional_geometry} [[!include Spin(8)-subgroups and reductions -- table]] \hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples} \hypertarget{resolution_of_joyce_orbifolds}{}\subsubsection*{{Resolution of Joyce orbifolds}}\label{resolution_of_joyce_orbifolds} [[compact topological space|compact]] [[G2-manifolds]] by [[resolution of singularities]] in compact [[flat orbifolds]] (\hyperlink{Joyce96}{Joyce 96}, \hyperlink{Joyce00}{Joyce 00}) (\ldots{}) \hypertarget{TwistedConnectedSumConstruction}{}\subsubsection*{{Twisted connected sum construction}}\label{TwistedConnectedSumConstruction} [[compact topological space|compact]] [[G2-manifolds]] by twisted [[connected sum]]-constructions (\hyperlink{Kovalev00}{Kovalev 00}) $\backslash$begin\{center\} $\backslash$begin\{imagefromfile\} ``file\_name'': ``TwistedConnectedSumG2Manifold.jpg'', ``width'': 680 $\backslash$end\{imagefromfile\} $\backslash$end\{center\} \begin{quote}% graphics grabbed from \hyperlink{Klemm17}{Klemm 16} \end{quote} \hypertarget{applications}{}\subsection*{{Applications}}\label{applications} \hypertarget{in_supergravity}{}\subsubsection*{{In supergravity}}\label{in_supergravity} In [[string phenomenology]] [[model (in particle phyiscs)|models]] obtained from [[Kaluza-Klein mechanism|compactification]] of [[11-dimensional supergravity]]/[[M-theory on G2-manifolds]] (see for instance \hyperlink{Duff}{Duff}) can have attractive [[phenomenology|phenomenological]] properties, see for instance the \emph{[[G2-MSSM]]}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[associative submanifold]] \item [[Hitchin functional]] \item [[M-theory on G2-manifolds]], [[G2-MSSM]] \item [[topological M-theory]], [[topological membrane]] \item [[generalized G2-manifold]] \item [[Calabi-Yau manifold]] \item [[exceptional generalized geometry]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The concept goes back to \begin{itemize}% \item E. Bonan, (1966), \emph{Sur les vari\'e{}t\'e{}s riemanniennes \`a{} groupe d'holonomie G2 ou Spin(7)}, C. R. Acad. Sci. Paris 262: 127--129. \end{itemize} Non-compact $G_2$-manifolds were first constructed in \begin{itemize}% \item [[Robert Bryant]], ; S.M. Salamon, (1989), \emph{On the construction of some complete metrics with exceptional holonomy}, Duke Mathematical Journal 58: 829--850. \end{itemize} [[compact topological space|Compact]] $G_2$-manifolds were first found in \begin{itemize}% \item [[Dominic Joyce]], \emph{Compact Riemannian 7-manifolds with holonomy $G_2$}, Journal of Differential Geometry vol 43, no 2, 1996 (\href{https://projecteuclid.org/euclid.jdg/1214458109}{Euclid}) \item [[Dominic Joyce]], \emph{Compact Manifolds with Special Holonomy}, Oxford Mathematical Monographs, Oxford University Press (2000) \end{itemize} The sufficiency of spin structure for $G_2$-structure is due to \begin{itemize}% \item A. Gray, \emph{Vector cross products on manifolds}, Trans. Amer. Math. Soc. 141 (1969), 465--504. \end{itemize} and the [[compact twisted connected sum G2-manifolds]] due to \begin{itemize}% \item [[Alexei Kovalev]], \emph{Twisted connected sums and special Riemannian holonomy}, J. Reine Angew. Math. 565 (2003) (\href{https://arxiv.org/abs/math/0012189}{arXiv:math/0012189}) \end{itemize} Review includes \begin{itemize}% \item [[Albrecht Klemm]], \emph{Effective Action from M-theory ontwisted connected sums}, talk at Ascona Monte Verita, 6 July 2017 (\href{http://conf.itp.phys.ethz.ch/string17/talks/Klemm.pdf}{pdf}) \end{itemize} More compact examples are constructed in \begin{itemize}% \item [[Dominic Joyce]], [[Spiro Karigiannis]], \emph{A new construction of compact $G_2$-manifolds by gluing families of Eguchi-Hanson spaces}, arXiv:\href{https://arxiv.org/abs/1707.09325}{1707.09325} \end{itemize} Surveys include \begin{itemize}% \item [[Spiro Karigiannis]], \emph{What is\ldots{} a $G_2$-manifold} (\href{http://www.ams.org/notices/201104/rtx110400580p.pdf}{pdf}) \item [[Spiro Karigiannis]], \emph{$G_2$-manifolds -- Exceptional structures in geometry arising from exceptional algebra} (\href{http://www.math.uwaterloo.ca/~karigian/talks/waterloo.pdf}{pdf}) \item [[Spiro Karigiannis]], \emph{$G_2$-Conifolds: A survey}, 2014 (\href{http://www.math.uni-hamburg.de/sgstructures/hamburg-talk-2.pdf}{pdf}) \item [[Nigel Hitchin]], \emph{Special holonomy and beyond}, Clay Mathematics Proceedings (\href{https://people.maths.ox.ac.uk/hitchin/hitchinlist/clay.pdf}{pdf}) \item [[Robert Bryant]], \emph{Some remarks on $G_2$-structures}, Proceedings of the 12th G\"o{}kova Geometry-Topology Conference 2005, pp. 75-109 \href{http://gokovagt.org/proceedings/2005/ggt05-bryant.pdf}{pdf} \end{itemize} The relation to [[multisymplectic geometry]]/[[2-plectic geometry]] is mentioned explicitly in \begin{itemize}% \item Alberto Ibort, \emph{Multisymplectic geometry: generic and exceptional}, \emph{\href{http://rsme.es/public/publi3.htm}{Proceedings of the IX Fall workshop on geometry and physics}} ([[IbortMultisymplectic.pdf:file]]) \end{itemize} (but beware of some mistakes in that article\ldots{}) For more see the references at \emph{[[exceptional geometry]]}. \hypertarget{ReferencesG2Orbifolds}{}\subsubsection*{{$G_2$-Orbifolds}}\label{ReferencesG2Orbifolds} Discussion of [[G2-orbifolds]] includes \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 [[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 [[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 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 [[Frank Reidegeld]], \emph{$G_2$-orbifolds from K3 surfaces with ADE-singularities} (\href{http://arxiv.org/abs/1512.05114}{arXiv:1512.05114}) \item [[Frank Reidegeld]], \emph{K3 surfaces with a pair of commuting non-symplectic involutions} (\href{https://arxiv.org/abs/1809.07501}{arXiv:1809.07501}) \item [[Bobby Acharya]], [[Andreas Braun]], Eirik Eik Svanes, Roberto Valandro, \emph{Counting Associatives in Compact $G_2$ Orbifolds} (\href{https://arxiv.org/abs/1812.04008}{arXiv:1812.04008}) \end{itemize} \hypertarget{moduli}{}\subsubsection*{{Moduli}}\label{moduli} Discussion of the [[moduli space]] of $G_2$-structures includes \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 [[Spiro Karigiannis]], [[Naichung Conan Leung]], \emph{Hodge Theory for G2-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{variants_and_generalizations}{}\subsubsection*{{Variants and generalizations}}\label{variants_and_generalizations} Discussion of the more general concept of Riemannian manifolds equipped with [[covariant derivative|covariantly constant]] 3-forms is in \begin{itemize}% \item Hong Van Le , \emph{Geometric structures associated with a simple Cartan 3-form}, Journal of Geometry and Physics (2013) (\href{http://arxiv.org/abs/1103.1201}{arXiv:1103.1201}) \end{itemize} \hypertarget{relation_to_killing_spinors}{}\subsubsection*{{Relation to Killing spinors}}\label{relation_to_killing_spinors} Discussion of $G_2$-structures in view of the existence of [[Killing spinors]] includes \begin{itemize}% \item [[Thomas Friedrich]], Stefan Ivanov, \emph{Parallel spinors and connections with skew-symmetric torsion in string theory}, AsianJ.Math.6:303-336,2002 (\href{http://arxiv.org/abs/math/0102142}{arXiv:math/0102142}) \end{itemize} \hypertarget{application_in_supergravity}{}\subsubsection*{{Application in supergravity}}\label{application_in_supergravity} The following references discuss the role of $G_2$-manifolds in [[M-theory on G2-manifolds]]: \begin{itemize}% \item [[Mike 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}) \end{itemize} A survey of the corresponding [[string phenomenology]] for [[M-theory on G2-manifolds]] (see there for more) is in \begin{itemize}% \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}) \end{itemize} See also \begin{itemize}% \item [[Andreas Brandhuber]], [[Jaume Gomis]], [[Steven Gubser]], [[Sergei Gukov]], \emph{Gauge Theory at Large N and New $G_2$ Holonomy Metrics}, Nucl.Phys. B611 (2001) 179-204 (\href{http://arxiv.org/abs/hep-th/0106034}{arXiv:hep-th/0106034}) \end{itemize} Weak $G_2$-holonomy is discussed 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 [[Adel Bilal]], Steffen Metzger, \emph{Compact weak $G_2$-manifolds with conical singularities} (\href{http://arxiv.org/abs/hep-th/0302021}{arXiv:hep-th/0302021}) \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} For more on this see at \emph{[[M-theory on G2-manifolds]]} [[!redirects G2 manifolds]] [[!redirects G2-manifold]] [[!redirects G2-manifolds]] [[!redirects G2 structure]] [[!redirects G2-structure]] [[!redirects G2 structures]] [[!redirects G2-structures]] [[!redirects G2-holonomy]] [[!redirects G2-holonomies]] [[!redirects weak G2-holonomy]] [[!redirects weak G2 holonomy]] [[!redirects twisted connected sum G2-manifold]] [[!redirects twisted connected sum G2-manifolds]] [[!redirects compact twisted connected sum G2-manifold]] [[!redirects compact twisted connected sum G2-manifolds]] [[!redirects compact twisted connected sum G2-manifold]] [[!redirects compact twisted connected sum G2-manifolds]] \end{document}