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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 8-manifolds} [[!redirects M-theory on Spin(8)-manifolds]] \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{cfield_tadpole_cancellation_condition}{C-field tadpole cancellation condition}\dotfill \pageref*{cfield_tadpole_cancellation_condition} \linebreak \noindent\hyperlink{RelationToFTheory}{Relation to F-theory}\dotfill \pageref*{RelationToFTheory} \linebreak \noindent\hyperlink{BlackM2BranesAndExotic7Spheres}{Black M2-branes and Exotic 7-spheres}\dotfill \pageref*{BlackM2BranesAndExotic7Spheres} \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{ReferencesWithSpin7Structure}{With $Spin(7)$-structure}\dotfill \pageref*{ReferencesWithSpin7Structure} \linebreak \noindent\hyperlink{with_structure_2}{With $Sp(2)\cdot Sp(1)$-structure}\dotfill \pageref*{with_structure_2} \linebreak \noindent\hyperlink{phenomenology_and_cosmological_constant}{Phenomenology and cosmological constant}\dotfill \pageref*{phenomenology_and_cosmological_constant} \linebreak \noindent\hyperlink{m2brane_spacetimes}{M2-brane spacetimes}\dotfill \pageref*{m2brane_spacetimes} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The [[KK-compactification]] of [[M-theory]] on [[fibers]] which are [[8-manifolds]]. In the low-energy limiting [[11-dimensional supergravity]] this is KK-compactification to [[3d supergravity]]. Typically this is considered with a [[reduction of the structure group]] on the compactification fiber from [[Spin(8)]] to [[Spin(7)]], in which case one speaks of \emph{M-theory on [[Spin(7)-manifolds]]} (see the references \hyperlink{ReferencesWithSpin7Structure}{below}). Further reduction to [[G2-structure]] yields [[M-theory on G2-manifolds]]. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{cfield_tadpole_cancellation_condition}{}\subsubsection*{{C-field tadpole cancellation condition}}\label{cfield_tadpole_cancellation_condition} In [[M-theory]] [[KK-compactification|compactified]] [[compact topological space|compact]] [[8-manifold]] [[fibers]], [[tadpole cancellation]] for the [[supergravity C-field]] (see also at \emph{[[C-field tadpole cancellation]]}) is equivalently the condition \begin{displaymath} N_{M2} \;+\; \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;=\; \underset{ I_8(X^8) }{ \underbrace{ \tfrac{1}{48}\big( p_2 - \tfrac{1}{2}p_1^2 \big)[X^{8}] } } \;\;\;\; \in \mathbb{Z} \,, \end{displaymath} where \begin{enumerate}% \item $N_{M2}$ is the net number of [[M2-branes]] in the spacetime (whose [[worldvolume]] appears as points in $X^{(8)}$); \item $G_4$ is the [[field strength]]/flux of the [[supergravity C-field]] \item $p_1$ is the [[first Pontryagin class]] and $p_2$ the [[second Pontryagin class]] combining to [[I8]], all regarded here in [[rational homotopy theory]]. \end{enumerate} If $X^{8}$ has \begin{itemize}% \item [[Spin(7)-structure]] (hence in particular if it is a [[Calabi-Yau manifold]], which has $SU(4) =$ [[Spin(6)]]-structure) \end{itemize} or \begin{itemize}% \item [[Sp(2).Sp(1)-structure]] \end{itemize} then \begin{displaymath} \tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2 \big) \;=\; \chi \end{displaymath} is the [[Euler class]] (see \href{Spin7-manifold#CharacteristicClassesForSpinStructure}{this Prop.} and \href{quaternion-Kähler+manifold#CharacteristicClassesForSpin5Spin3Structure}{this Prop.}, respectively), hence in these cases the condition is equivalently \begin{displaymath} N_{M2} \;+\; \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;=\; \tfrac{1}{24}\chi[X^8] \;\;\;\; \in \mathbb{Z} \,, \end{displaymath} where $\chi[X]$ is the [[Euler characteristic]] of $X$. For references see \href{C-field+tadpole+cancellation#References}{there}. $\backslash$linebreak \hypertarget{RelationToFTheory}{}\subsubsection*{{Relation to F-theory}}\label{RelationToFTheory} If the 8-dimensional [[fibers]] themselves are [[elliptic fibrations]], then [[M-theory]] on these 8-manifolds is supposedly [[T-duality|T-dual]] to [[F-theory]] [[KK-compactification|KK-compactified]] to $3+1$ [[spacetime]]-[[dimensions]]. In particular, if there is an [[M2-brane]] [[wrapped brane|filling]] the base 2+1-dimensional spacetime, this is supposedly [[T-duality|T-dual]] to a 3+1-dimensional spacetime filling [[D3-brane]] in [[F-theory]] (e.g. \hyperlink{CondeescuMicuPalti14}{Condeescu-Micu-Palti 14, p. 2}) For more on this see at \emph{[[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]}. $\backslash$linebreak \hypertarget{BlackM2BranesAndExotic7Spheres}{}\subsubsection*{{Black M2-branes and Exotic 7-spheres}}\label{BlackM2BranesAndExotic7Spheres} The discovery of [[exotic 7-spheres]] proceeded via [[8-manifolds]] $X$ [[manifold with boundary|with boundary]] [[homeomorphism|homeomorphic]] to the [[7-sphere]] $\partial X \simeq_{homeo} S^7$, but not necessarily [[diffeomorphism|diffeomorphic]] to $S^7$ with its canonical [[smooth structure]] (for more see \href{8-manifold#ExoticBoundary7Spheres}{there}). Hence when regarded from the point of view of [[M-theory on 8-manifolds]], exotic 7-spheres arise as [[near horizon limits]] of peculiar [[black brane|black]] [[M2-brane]] [[spacetimes]] $\mathbb{R}^{2,1} \times X$. See also \hyperlink{MorrisonRPlesser99}{Morrison-Plesser 99, section 3.2}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} [[!include KK-compactifications of M-theory -- table]] \begin{itemize}% \item [[M2-M5 brane bound state]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item [[Edward Witten]], \emph{Strong Coupling and the Cosmological Constant}, Mod.Phys.Lett.A10:2153-2156, 1995 (\href{https://arxiv.org/abs/hep-th/9506101}{arXiv:hep-th/9506101}) (on possible relation to the [[cosmological constant]]) \item [[Katrin Becker]], [[Melanie Becker]], \emph{M-Theory on Eight-Manifolds}, Nucl.Phys. B477 (1996) 155-167 (\href{https://arxiv.org/abs/hep-th/9605053}{arXiv:hep-th/9605053}) \item [[Savdeep Sethi]], [[Cumrun Vafa]], [[Edward Witten]], \emph{Constraints on Low-Dimensional String Compactifications}, Nucl.Phys.B480:213-224, 1996 (\href{https://arxiv.org/abs/hep-th/9606122}{arXiv:hep-th/9606122}) \item [[Sergei Gukov]], [[James Sparks]], \emph{M-Theory on $Spin(7)$ Manifolds}, Nucl.Phys. B625 (2002) 3-69 (\href{https://arxiv.org/abs/hep-th/0109025}{arXiv:hep-th/0109025}) \item Cezar Condeescu, Andrei Micu, Eran Palti, \emph{M-theory Compactifications to Three Dimensions with M2-brane Potentials}, JHEP04(2014)026 (\href{https://arxiv.org/abs/1311.5901}{arXiv:1311.5901}) \item Daniël Prins, [[Dimitrios Tsimpis]], \emph{IIA supergravity and M-theory on manifolds with SU(4) structure}, PhysRevD.89.064030 (\href{https://arxiv.org/abs/1312.1692}{arXiv:1312.1692}) \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}) \item Elena Mirela Babalic, [[Calin Lazaroiu]], \emph{Foliated eight-manifolds for M-theory compactification}, JHEP01(2015)140 (\href{https://arxiv.org/abs/1411.3148}{arXiv:1411.3148}) \item C. S. Shahbazi, \emph{M-theory on non-Kähler manifolds}, JHEP09(2015)178 (\href{https://arxiv.org/abs/1503.00733}{arXiv:1503.00733}) \item Elena Mirela Babalic, [[Calin Lazaroiu]], \emph{The landscape of G-structures in eight-manifold compactifications of M-theory}, JHEP11 (2015) 007 (\href{https://arxiv.org/abs/1505.02270}{arXiv:1505.02270}) \item Elena Mirela Babalic, [[Calin Lazaroiu]], \emph{Internal circle uplifts, transversality and stratified G-structures}, JHEP11(2015)174 (\href{https://arxiv.org/abs/1505.05238}{arXiv:1505.05238}) \end{itemize} \hypertarget{ReferencesWithSpin7Structure}{}\subsubsection*{{With $Spin(7)$-structure}}\label{ReferencesWithSpin7Structure} Discussion of [[KK-compactification]] on 8-dimensional [[Spin(7)-manifolds]] (see also at \emph{[[M-theory on G2-manifolds]]}): \begin{itemize}% \item [[Mirjam Cvetic]], [[Gary Gibbons]], H. Lu, [[Christopher Pope]], \emph{New Complete Non-compact Spin(7) Manifolds}, Nucl.Phys.B620:29-54, 2002 (\href{https://arxiv.org/abs/hep-th/0103155}{arXiv:hep-th/0103155}) \item Jaydeep Majumder, \emph{Type IIA Orientifold Limit of M-Theory on Compact Joyce 8-Manifold of Spin(7)-Holonomy}, JHEP 0201 (2002) 048 (\href{https://arxiv.org/abs/hep-th/0109076}{arXiv:hep-th/0109076}) \item [[Ralph Blumenhagen]], [[Volker Braun]], \emph{Superconformal Field Theories for Compact Manifolds with Spin(7) Holonomy}, JHEP 0112:013, 2001 (\href{https://arxiv.org/abs/hep-th/0111048}{arXiv:hep-th/0111048}) \item [[Sergei Gukov]], James Sparks, [[David Tong]], \emph{Conifold Transitions and Five-Brane Condensation in M-Theory on $Spin(7)$ Manifolds}, Class.Quant.Grav.20:665-706, 2003 (\href{https://arxiv.org/abs/hep-th/0207244}{arXiv:hep-th/0207244}) \item [[Melanie Becker]], Dragos Constantin, [[Sylvester James Gates Jr.]], William D. Linch III, Willie Merrell, J. Phillips, \emph{M-theory on Spin(7) Manifolds, Fluxes and 3D, N=1 Supergravity}, Nucl.Phys. B683 (2004) 67-104 (\href{https://arxiv.org/abs/hep-th/0312040}{arXiv:hep-th/0312040}) \item Dragos Constantin, \emph{M-Theory Vacua from Warped Compactifications on Spin(7) Manifolds}, Nucl.Phys.B706:221-244, 2005 (\href{https://arxiv.org/abs/hep-th/0410157}{arXiv:hep-th/0410157}) \item Dragos Constantin, \emph{Flux Compactification of M-theory on Compact Manifolds with Spin(7) Holonomy}, Fortsch.Phys. 53 (2005) 1272-1329 (\href{https://arxiv.org/abs/hep-th/0507104}{arXiv:hep-th/0507104}) \item [[Dimitrios Tsimpis]], \emph{M-theory on eight-manifolds revisited: N=1 supersymmetry and generalized $Spin(7)$ structures}, JHEP 0604 (2006) 027 (\href{https://arxiv.org/abs/hep-th/0511047}{arXiv:hep-th/0511047}) \item S. Salur, O. Santillan, \emph{New Spin(7) holonomy metrics admiting G2 holonomy reductions and M-theory/IIA dualities}, Phys.Rev.D79:086009, 2009 (\href{https://arxiv.org/abs/0811.4422}{arXiv:0811.4422}) \item Adil Belhaj, Luis J. Boya, Antonio Segui, \emph{Holonomy Groups Coming From F-Theory Compactification}, Int J Theor Phys (2010) 49: 681. (\href{https://arxiv.org/abs/0911.2125}{arXiv:0911.2125}) \item Thomas Bruun Madsen, \emph{Spin(7)-manifolds with three-torus symmetry}, J.Geom.Phys.61:2285-2292, 2011 (\href{https://arxiv.org/abs/1104.3089}{arXiv:1104.3089}) \item Federico Bonetti, [[Thomas Grimm]], Tom G. Pugh, \emph{Non-Supersymmetric F-Theory Compactifications on Spin(7) Manifolds}, JHEP01(2014)112 (\href{https://arxiv.org/abs/1307.5858}{arXiv:1307.5858}) \item Federico Bonetti, [[Thomas Grimm]], Eran Palti, Tom G. Pugh, \emph{F-Theory on Spin(7) Manifolds: Weak-Coupling Limit}, JHEP02(2014)076 (\href{https://arxiv.org/abs/1309.2287}{arXiv:1309.2287}) \item Tom Pugh, \emph{M-theory on $Spin(7)$-manifold duals and their F-theory duals} () \item [[Andreas Braun]], Sakura Schaefer-Nameki, \emph{Spin(7)-Manifolds as Generalized Connected Sums and 3d $N=1$ Theories}, JHEP06(2018)103 (\href{https://arxiv.org/abs/1803.10755}{arXiv:1803.10755}) (generalization of [[compact twisted connected sum G2-manifolds]]) \end{itemize} \hypertarget{with_structure_2}{}\subsubsection*{{With $Sp(2)\cdot Sp(1)$-structure}}\label{with_structure_2} M-theory on [[HP{\tt \symbol{94}}2]], hence on a [[quaternion-Kähler manifold]] of dimension 8 with [[holonomy]] [[Sp(2).Sp(1)]], is considered in \begin{itemize}% \item [[Michael Atiyah]], [[Edward Witten]], p. 75 onwards in \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}) \end{itemize} and argued to be [[duality in string theory|dual]] to [[M-theory on G2-manifolds]] in three different ways, which in turn is argued to lead to a a possible [[proof]] of [[confinement]] in the resulting 4d [[effective field theory]] (see \href{M-theory+on+G2-manifolds#Confinement}{there} for more). See also \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of \emph{[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]} (\href{https://arxiv.org/abs/1904.10207}{arXiv:1904.10207}) \end{itemize} \hypertarget{phenomenology_and_cosmological_constant}{}\subsubsection*{{Phenomenology and cosmological constant}}\label{phenomenology_and_cosmological_constant} An argument for [[non-perturbative effect|non-perturbative]] non-[[supersymmetry|supersymmetric]] 4d [[string phenomenology]] with fundamentally vanishing [[cosmological constant]], based on 3d [[M-theory on 8-manifolds]] decompactified at strong coupling to 4d via [[duality between M-theory and type IIA string theory]] (recall the [[super 2-brane in 4d]]): \begin{itemize}% \item [[Edward Witten]], \emph{The Cosmological Constant From The Viewpoint Of String Theory}, lecture at \href{http://inspirehep.net/record/972507}{DM2000} (\href{https://arxiv.org/abs/hep-ph/0002297}{arXiv:hep-ph/0002297}) (see p. 7) \item [[Edward Witten]], \emph{Strong coupling and the cosmological constant}, Mod. Phys. Lett. A 10:2153-2156, 1995 (\href{https://arxiv.org/abs/hep-th/9506101}{arXiv:hep-th/9506101}) \item [[Edward Witten]], Section 3 of \emph{Some Comments On String Dynamics}, talk at \href{https://cds.cern.ch/record/305869}{Strings95} (\href{http://arxiv.org/abs/hep-th/9507121}{arXiv:hep-th/9507121}) \end{itemize} The realization of this scenario in [[F-theory]]: \begin{itemize}% \item [[Cumrun Vafa]], Section 4.3 of: \emph{Evidence for F-Theory}, Nucl. Phys. B469:403-418, 1996 (\href{https://arxiv.org/abs/hep-th/9602022}{arxiv:hep-th/9602022}) \item [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Gianluca Zoccarato, \emph{F-theory and Dark Energy}, Fortschritte der Physik (\href{https://arxiv.org/abs/1811.01959}{arXiv:1811.01959}, \href{https://doi.org/10.1002/prop.201900057}{doi:10.1002/prop.201900057}) \item [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, Gianluca Zoccarato, \emph{Pixelated Dark Energy} (\href{https://arxiv.org/abs/1901.10489}{arXiv:1901.10489}) \end{itemize} \hypertarget{m2brane_spacetimes}{}\subsubsection*{{M2-brane spacetimes}}\label{m2brane_spacetimes} \begin{itemize}% \item [[David Morrison]], [[M. Ronen Plesser]], section 3.2 of \emph{Non-Spherical Horizons, I}, Adv. Theor. Math. Phys.3:1-81, 1999 (\href{https://arxiv.org/abs/hep-th/9810201}{arXiv:hep-th/9810201}) \end{itemize} [[!redirects M-theory on Spin(7)-manifolds]] \end{document}