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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*{M2-brane} \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{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{as_a_greenschwarz_sigma_model}{As a Green-Schwarz sigma model}\dotfill \pageref*{as_a_greenschwarz_sigma_model} \linebreak \noindent\hyperlink{AsABlackBrane}{As a black brane}\dotfill \pageref*{AsABlackBrane} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{QuantizationViaBFSSMatrixModel}{Quantization via BFSS matrix model}\dotfill \pageref*{QuantizationViaBFSSMatrixModel} \linebreak \noindent\hyperlink{WorldvolumeTheory}{Worldvolume theory -- BLG and ABJM}\dotfill \pageref*{WorldvolumeTheory} \linebreak \noindent\hyperlink{ads4cft3_duality}{AdS4-CFT3 duality}\dotfill \pageref*{ads4cft3_duality} \linebreak \noindent\hyperlink{M2M5BoundStates}{M2/M5 bound states}\dotfill \pageref*{M2M5BoundStates} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{as_a_fundamental_brane_gstype_model}{As a fundamental brane (GS-type $\sigma$-model)}\dotfill \pageref*{as_a_fundamental_brane_gstype_model} \linebreak \noindent\hyperlink{RegularizationReferences}{Regularization and relation to BFSS}\dotfill \pageref*{RegularizationReferences} \linebreak \noindent\hyperlink{as_a_black_brane_2}{As a black brane}\dotfill \pageref*{as_a_black_brane_2} \linebreak \noindent\hyperlink{dualities}{Dualities}\dotfill \pageref*{dualities} \linebreak \noindent\hyperlink{ReferencesDyonic}{M2-M5 bound states}\dotfill \pageref*{ReferencesDyonic} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The theory of [[11-dimensional supergravity]] contains a higher [[gauge field]] -- the [[supergravity C-field]] -- that naturally couples to higher electrically charged 2-[[branes]], [[membranes]] (\hyperlink{BergshoeffSezginTownsend87}{Bergshoeff-Sezgin-Townsend 87}). By \emph{[[double dimensional reduction]]}, these turn into the [[superstrings]] of [[type IIA string theory]] (\hyperlink{DuffHoweInamiStelle87}{Duff-Howe-Inami-Stelle 87}). (See at \emph{[[duality between M-theory and type IIA string theory]]}.) When in (\hyperlink{Witten95}{Witten95}) it was argued that the 10-dimensional [[target space]] theories of the five types of [[string theory|superstring theories]] are all limiting cases of one single 11-dimensional target space theory that extends [[11-dimensional supergravity]] ([[M-theory]]), it was natural to guess that this supergravity membrane accordingly yields a 3-dimensional [[sigma-model]] that reduces in limiting cases to the [[string]] [[sigma-models]]. But there were two aspects that make this idea a little subtle, even at this vague level: first, there is no good theory of the [[quantization]] of the membrane [[sigma-model]], as opposed to the well understood quantum [[string]]. Secondly, that hypothetical ``theory extending 11-dimensional supergravity'' (``M-theory'') has remained elusive enough that it is not clear in which sense the membrane would relate to it in a way analogous to how the string relates to its [[target space]] theories (which is fairly well understood). Later, with the [[BFSS matrix model]] some people gained more confidence in the idea, by identifying the corresponding degrees of freedom in a special case (\hyperlink{NicolaiHelling98}{Nicolai-Helling 98}, \hyperlink{DasguptaNicolaiPlefka02}{Dasgupta-Nicolai-Plefka 02}). See also at \emph{[[membrane matrix model]]}. In a more modern perspective, the M2-brane [[worldvolume]] theory appears under \emph{\href{AdS-CFT#AdS4CFT3}{AdS4-CFT3 duality}} as a [[holographic principle|holographic dual]] of a [[4-dimensional Chern-Simons theory]]. Indeed, its [[Green-Schwarz action functional]] is entirely controled by the [[super Lie algebra|super]]-[[Lie algebra cocycle|Lie algebra 4-cocycle]] of [[super Minkowski spacetime]] given by the [[brane scan]]. This exhibits the M2-brane worldvolume theory as a 3-dimensional [[higher dimensional WZW model]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} There are two different incarnations of the M2-brane. On the one hand it is defined as a [[Green-Schwarz sigma model]] with [[target space]] a [[spacetime]] that is a solution to the [[equations of motion]] of [[11-dimensional supergravity]]. One would call this the ``fundamental'' M2 in analogy with the ``fundamental string'', if only there were an ``M2-perturbation series'' which however is essentially ruled out. On the other hand the M2 also appears as a [[black brane]], hence as a solution to the [[equations of motion]] of [[11-dimensional supergravity]] with [[singularity]] that looks from outside like a charged 2 dimensional object. \hypertarget{as_a_greenschwarz_sigma_model}{}\subsubsection*{{As a Green-Schwarz sigma model}}\label{as_a_greenschwarz_sigma_model} See at \emph{[[Green-Schwarz sigma model]]} and \emph{[[brane scan]]}. \hypertarget{AsABlackBrane}{}\subsubsection*{{As a black brane}}\label{AsABlackBrane} As a [[black brane]] solution to the [[equations of motion]] of [[11-dimensional supergravity]] the M2 is the [[spacetime]] $\mathbb{R}^{2,1} \times (\mathbb{R}^8-\{0\})$ with [[pseudo-Riemannian metric]] being \begin{displaymath} g = H^{-2/3} g_{\mathbb{R}^{2,1}} + H^{1/3}g_{\mathbb{R}^8-\{0\}} \end{displaymath} where \begin{displaymath} H = \alpha + \frac{\beta}{r^6} \end{displaymath} for $(\alpha,\beta) \in \mathbb{R}^2 \setminus \{(0,0)\}$; and the [[field strength]] of the [[supergravity C-field]] is \begin{displaymath} F = d vol_{\mathbb{R}^{2,1}} \wedge \mathbf{d} H^{-1} \,. \end{displaymath} For $\alpha \beta \neq 0$ this is a 1/2 [[BPS state]] of 11d sugra. In the above [[coordinates]] the metric is ill-defined at $r = - \beta^{1/6} \alpha$, but in fact it may be smoothly continued through this point (\hyperlink{DuffGibbonsTownsend94}{Duff-Gibbons-Townsend 94, section 3}), which is a [[event horizon]]. An actual singularity is at $r = 0$. The [[near horizon geometry]] of this spacetime is the [[Freund-Rubin compactification]] [[anti de Sitter spacetime|AdS4]]$\times$[[7-sphere|S7]]. For more on this see at \emph{[[AdS-CFT]]}. [[!include black branes in supergravity -- table]] More generally, one may classify those solutions of [[11-dimensional supergravity]] of the form $AdS_4 \times X_7$ for some [[closed manifold]] $X_7$, that are at least 1/2 [[BPS states]]. One finds (\hyperlink{MedeirosFigueroa10}{Medeiros-Figueroa 10}) that all these are of the form $AdS_4 \times S^7/G_{ADE}$, where $S^7 / G_{ADE}$ is an [[orbifold]] of the [[7-sphere]] (a [[spherical space form]] in the smooth case, see \href{spherical+space+form#7DSphericalSpaceFormsWithSpinStructure}{there}) by a [[finite subgroup of SU(2)]] $G_{ADE} \hookrightarrow SU(2)$, i.e. a finite group in the [[ADE-classification]] [[!include ADE -- table]] Here for $5 \leq \mathcal{N} \leq 8$-supersymmetry then the action of $G_{ADE}$ on $S^7$ is via the canonical action of $SU(2)$ as in the [[quaternionic Hopf fibration]] (\hyperlink{MedeirosFigueroa10}{Medeiros-Figueroa 10}), while for $\mathcal{N} = 4$ then there is an extra twist to the action (\hyperlink{MFFGME09}{MFFGME 09}). See the table \hyperlink{WorldvolumeTheory}{below}. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{QuantizationViaBFSSMatrixModel}{}\subsubsection*{{Quantization via BFSS matrix model}}\label{QuantizationViaBFSSMatrixModel} A regularized [[quantization]] of the [[Green-Schwarz sigma-model]] for the [[M2-brane]] yields the [[BFSS matrix model]] (\hyperlink{NicolaiHelling98}{Nicolai-Helling 98}, \hyperlink{DasguptaNicolaiPlefka02}{Dasgupta-Nicolai-Plefka 02}). In this correspondence, [[matrix]] blocks around the diagonal correspond to blobs of [[membrane]], while off-diagonal matrix elements correspond to thin tubes of membrane connecting these blobs. \begin{quote}% graphics grabbed from \hyperlink{DasguptaNicolaiPlefka02}{Dasgupta-Nicolai-Plefka 02} \end{quote} \hypertarget{WorldvolumeTheory}{}\subsubsection*{{Worldvolume theory -- BLG and ABJM}}\label{WorldvolumeTheory} The [[worldvolume]] [[QFT]] of [[black brane|black]] M2-branes is a [[3d superconformal gauge field theory]]: [[!include superconformal symmetry -- table]] Specifically, [[worldvolume]] [[quantum field theory]] of M2-branes sitting at [[ADE singularities]] (as \hyperlink{BlackM2AtADESingularity}{above}) is supposed to be described by [[ABJM theory]] and, for the special case of $SU(2)$ gauge group, by the [[BLG model]]. See also at \emph{[[gauge enhancement]]}. [[!include 7d spherical space forms -- table]] \hypertarget{ads4cft3_duality}{}\subsubsection*{{AdS4-CFT3 duality}}\label{ads4cft3_duality} Under [[AdS-CFT duality]] the M2-brane is given by \emph{\href{AdS-CFT#AdS4CFT3}{AdS4-CFT3 duality}}. (\hyperlink{Maldacena97}{Maldacena 97, section 3.2}, \hyperlink{KlebanovTorri10}{Klebanov-Torri 10}). $\backslash$linebreak \hypertarget{M2M5BoundStates}{}\subsubsection*{{M2/M5 bound states}}\label{M2M5BoundStates} For \emph{[[M2-M5 brane bound states]]}, i.e. [[bound states]] of [[M2-branes]] with [[M5-branes]] ([[dyon|dyonic]] M2-branes and [[giant gravitons]]), see the references \hyperlink{ReferencesDyonic}{below}. For the [[type II string theory]]-version see at \emph{[[NS5-brane]]} the sectoin \emph{\href{NS5-brane#NS5D4D2BoundStates}{NS5/D4/D2 bound states}}. $\backslash$linebreak \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[membrane instanton]], [[SM2-brane]] \item [[fractional M2-brane]] \item [[triple membrane junction]] \item [[C-field tadpole cancellation]] \item [[BLG model]], [[ABJM model]], [[membrane matrix model]] \item [[string theory]] \item [[11-dimensional supergravity]], [[M-theory]] \item [[string]], [[superstring]] \item [[D-brane]] \item [[M5-brane]], [[6d (2,0)-superconformal QFT]] \item [[M9-brane]] \item [[topological membrane]] \item [[supergravity Lie 3-algebra]], [[M-theory super Lie algebra]] \end{itemize} [[!include table of branes]] \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{as_a_fundamental_brane_gstype_model}{}\subsubsection*{{As a fundamental brane (GS-type $\sigma$-model)}}\label{as_a_fundamental_brane_gstype_model} The [[Green-Schwarz sigma-model]]-type formulation of the supermembrane (as in the [[brane scan]]) first appears in \begin{itemize}% \item [[Eric Bergshoeff]], [[Ergin Sezgin]], and [[Paul Townsend]], \emph{Supermembranes and eleven-dimensional supergravity}, Phys. Lett. B189 (1987) 75--78. (\href{http://inspirehep.net/record/248230/}{spire:248230}) \end{itemize} The [[equations of motion]] of the super membrane are derived via the [[superembedding approach]] in \begin{itemize}% \item [[Igor Bandos]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], [[Dmitry Volkov]], Chapter 3 of \emph{Superstrings and supermembranes in the doubly supersymmetric geometrical approach}, Nucl. Phys. B446:79-118, 1995 (\href{https://arxiv.org/abs/hep-th/9501113}{arXiv:hep-th/9501113}) \end{itemize} and the [[Lagrangian density]] for the super membrane is derived via the [[superembedding approach]] in \begin{itemize}% \item [[Paul Howe]], [[Ergin Sezgin]], \emph{The supermembrane revisited}, Class. Quant. Grav. 22 (2005) 2167-2200 (\href{https://arxiv.org/abs/hep-th/0412245}{arXiv:hep-th/0412245}) \end{itemize} Its [[quantization]] of the was explored in \begin{itemize}% \item [[Mike Duff]], T. Inami, [[Christopher Pope]], [[Ergin Sezgin]], [[Kellogg Stelle]], \emph{Semiclassical Quantization of the Supermembrane}, Nucl.Phys. B297 (1988) 515-538 (\href{http://inspirehep.net/record/247064}{spire:247064}) \item [[Bernard de Wit]], [[Jens Hoppe]], [[Hermann Nicolai]], \emph{On the Quantum Mechanics of Supermembranes}, Nucl. Phys. B305 (1988) 545. (\href{http://pubman.mpdl.mpg.de/pubman/item/escidoc:153408:1/component/escidoc:153407/353961.pdf}{pdf}, [[deWitHoppeNicolai88.pdf:file]], \href{http://inspirehep.net/record/261702}{spire:261702}) \item [[Bernard de Wit]], W. L\"u{}scher, [[Hermann Nicolai]], \emph{The supermembrane is unstable}, Nucl. Phys. B320 (1989) 135 (\href{http://inspirehep.net/record/266584/}{spire:266584}, ) \item Daniel Kabat, [[Washington Taylor]], section 2 of \emph{Spherical membranes in Matrix theory}, Adv.Theor.Math.Phys.2:181-206,1998 (\href{https://arxiv.org/abs/hep-th/9711078}{arXiv:hep-th/9711078}) \end{itemize} The [[double dimensional reduction]] of the M2-brane to the [[Green-Schwarz superstring]] was observed in \begin{itemize}% \item [[Michael Duff]], [[Paul Howe]], T. Inami, [[Kellogg Stelle]], \emph{Superstrings in $D=10$ from Supermembranes in $D=11$}, Phys. Lett. B191 (1987) 70 and in [[Michael Duff]] (ed.) \emph{[[The World in Eleven Dimensions]]} 205-206 (1987) (\href{http://inspirehep.net/record/245249}{spire}) \end{itemize} The interpretation of the membrane as as an object related to [[string theory]] via [[double dimensional reduction]], hence as the \emph{M2-brane} was proposed in \begin{itemize}% \item [[Paul Townsend]], \emph{The eleven-dimensional supermembrane revisited}, Phys.Lett.B350:184-187, 1995 (\href{http://arxiv.org/abs/hep-th/9501068}{arXiv:hep-th/9501068}) \end{itemize} around the time when [[M-theory]] became accepted due to \begin{itemize}% \item [[Edward Witten]], \emph{[[String Theory Dynamics In Various Dimensions]]} (\href{http://arxiv.org/abs/hep-th/9503124}{arXiv:hep-th/9503124}) \end{itemize} \hypertarget{RegularizationReferences}{}\paragraph*{{Regularization and relation to BFSS}}\label{RegularizationReferences} The proposed regularization, due to \hyperlink{deWitHoppeNicolai88}{deWit-Hoppe-Nicolai 88}, of area-preserving diffeomorphisms on the [[membrane]] [[worldvolume]] by [[SU(n)|SU(N)]]-matrices and the resulting equivalence of the [[quantization]] of the membrane to the [[BFSS matrix model]] of [[D0-branes]] is reviewed and further dicussed in the following articles: \begin{itemize}% \item [[Hermann Nicolai]], Robert Helling, \emph{Supermembranes and M(atrix) Theory}, In \emph{Trieste 1998, Nonperturbative aspects of strings, branes and supersymmetry} 29-74 (\href{http://arxiv.org/abs/hep-th/9809103}{arXiv:hep-th/9809103}, \href{http://inspirehep.net/record/476366}{spire:476366}) \item Arundhati Dasgupta, [[Hermann Nicolai]], [[Jan Plefka]], \emph{An Introduction to the Quantum Supermembrane}, Grav.Cosmol.8:1,2002; Rev.Mex.Fis.49S1:1-10, 2003 (\href{http://arxiv.org/abs/hep-th/0201182}{arXiv:hep-th/0201182}) \item [[Gijs van den Oord]], \emph{On Matrix Regularisation of Supermembranes}, 2006 (\href{http://web.science.uu.nl/itf/Teaching/2006/vandenOord.pdf}{pdf}) \end{itemize} \hypertarget{as_a_black_brane_2}{}\subsubsection*{{As a black brane}}\label{as_a_black_brane_2} The back membrane solution of [[11-dimensional supergravity]] was found in \begin{itemize}% \item [[Mike Duff]], [[Kellogg Stelle]], \emph{Multimembrane solutions of $D = 11$ supergravity}, Phys. Lett. B 253, 113 (1991) (\href{http://inspirehep.net/record/299386?ln=en}{spire}) \end{itemize} Its regularity throught the [[event horizon]] is due to \begin{itemize}% \item [[Mike Duff]], [[Gary Gibbons]], [[Paul Townsend]], \emph{Macroscopic superstrings as interpolating solitons}, Phys.Lett.B332:321-328,1994 (\href{https://arxiv.org/abs/hep-th/9405124}{arXiv:hep-th/9405124}) \end{itemize} The [[Horava-Witten theory|Horava-Witten]]-[[orientifold]] of the black M2, supposedly yielding the [[black brane|black]] [[heterotic string]] is discussed in \begin{itemize}% \item Zygmunt Lalak, Andr\'e{} Lukas, [[Burt Ovrut]], \emph{Soliton Solutions of M-theory on an Orbifold}, Phys. Lett. B425 (1998) 59-70 (\href{https://arxiv.org/abs/hep-th/9709214}{arXiv:hep-th/9709214}) \item Ken Kashima, \emph{The M2-brane Solution of Heterotic M-theory with the Gauss-Bonnet $R^2$ terms}, Prog.Theor.Phys. 105 (2001) 301-321 (\href{https://arxiv.org/abs/hep-th/0010286}{arXiv:hep-th/0010286}) \end{itemize} Meanwhile [[AdS-CFT duality]] was recognized in \begin{itemize}% \item [[Juan Maldacena]], \emph{The Large N limit of superconformal field theories and supergravity}, Adv. Theor. Math. Phys. 2:231, 1998, \href{http://arxiv.org/abs/hep-th/9711200}{hep-th/9711200}; \end{itemize} where a dual description of the [[worldvolume]] theory of M2-brane appears in section 3.2. More on this is in \begin{itemize}% \item [[Igor Klebanov]], Giuseppe Torri, \emph{M2-branes and AdS/CFT}, Int.J.Mod.Phys.A25:332-350,2010 (\href{http://arxiv.org/abs/0909.1580}{arXiv:0909.1580}) \end{itemize} An account of the history as of 1999 is in \begin{itemize}% \item [[Mike Duff]], chapter II of \emph{[[The World in Eleven Dimensions]]: Supergravity, Supermembranes and M-theory}, IoP 1999 (\href{https://www.crcpress.com/The-World-in-Eleven-Dimensions-Supergravity-supermembranes-and-M-theory/Duff/9780750306720}{publisher}) \end{itemize} More recent review is in \begin{itemize}% \item Georgios Linardopoulos, chapter 13 of \emph{Classical Strings and Membranes in the AdS/CFT Correspondence} (\href{http://users.uoa.gr/~glinardo/Thesis.pdf}{pdf}, \href{https://inspirehep.net/record/1391031/}{spire}) \end{itemize} A detailed discussion of this [[black brane]]-realization of the M2 and its relation to [[AdS-CFT]] is in \begin{itemize}% \item [[Gianguido Dall'Agata]], Davide FabbKri, Christophe Fraser, [[Pietro Fré]], Piet Termonia, Mario Trigiante, \emph{The $Osp(8|4)$ singleton action from the supermembrane}, Nucl.Phys.B542:157-194, 1999 (\href{http://arxiv.org/abs/hep-th/9807115}{arXiv:hep-th/9807115}) \end{itemize} The generalization of this to $\geq 1/2$ BPS sugra solutions of the form $AdS_4 \times X_7$ is due to \begin{itemize}% \item Paul de Medeiros, [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], \emph{Half-BPS quotients in M-theory: ADE with a twist}, JHEP 0910:038,2009 (\href{http://arxiv.org/abs/0909.0163}{arXiv:0909.0163}, \href{http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf}{pdf slides}) \item Paul de Medeiros, [[José Figueroa-O'Farrill]], \emph{Half-BPS M2-brane orbifolds}, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (\href{http://arxiv.org/abs/1007.4761}{arXiv:1007.4761}, \href{https://projecteuclid.org/euclid.atmp/1408561553}{Euclid}) \end{itemize} Discussion of the history includes \begin{itemize}% \item [[Mike Duff]], (\href{http://arxiv.org/abs/1501.04098}{arXiv:1501.04098}) \end{itemize} Other recent developments are discussed in \begin{itemize}% \item [[Paul Howe]], [[Ergin Sezgin]], \emph{The supermembrane revisited}, (\href{http://arxiv.org/abs/hep-th/0412245}{arXiv:hep-th/0412245}) \item [[Igor Bandos]], [[Paul Townsend]], \emph{SDiff Gauge Theory and the M2 Condensate} (\href{http://arxiv.org/abs/0808.1583}{arXiv:0808.1583}) \item [[Jonathan Bagger]], [[Neil Lambert]], Sunil Mukhi, Constantinos Papageorgakis, \emph{Multiple Membranes in M-theory} (\href{http://arxiv.org/abs/1203.3546}{arXiv:1203.3546}) \item [[Nathan Berkovits]], \emph{Towards Covariant Quantization of the Supermembrane} (\href{http://arxiv.org/abs/hep-th/0201151}{arXiv:hep-th/0201151}) \end{itemize} Formulations of multiple M2-branes on top of each other are given by the \emph{[[BLG model]]} and the \emph{[[ABJM model]]}. See there for more pointers. The relation of these to the above is discussed in section 3 of Discusson of [[boundary conditions]] in the ABJM model (for M2-branes ending on [[M5-branes]]) is in \begin{itemize}% \item [[David Berman]], Daniel Thompson, \emph{Membranes with a boundary}, Nucl.Phys.B820:503-533,2009 (\href{http://arxiv.org/abs/0904.0241}{arXiv:0904.0241}) \end{itemize} A kind of [[double dimensional reduction]] of the ABJM model to something related to [[type II superstrings]] and [[D1-branes]] is discussed in \begin{itemize}% \item [[Horatiu Nastase]], Constantinos Papageorgakis, \emph{Dimensional reduction of the ABJM model}, JHEP 1103:094,2011 (\href{http://arxiv.org/abs/1010.3808}{arXiv:1010.3808}) \end{itemize} Discussion of the ABJM model in [[Horava-Witten theory]] and reducing to [[heterotic strings]] is in \begin{itemize}% \item [[Neil Lambert]], \emph{Heterotic M2-branes} (\href{http://arxiv.org/abs/1507.07931}{arXiv:1507.07931}) \end{itemize} Discussion of general phenomena of [[M-branes]] in [[higher geometry]] and [[generalized cohomology]] is in \begin{itemize}% \item [[Hisham Sati]], \emph{[[Geometric and topological structures related to M-branes]]} (2010) \end{itemize} Discussion from the point of view of [[Green-Schwarz action functional]]-[[schreiber:∞-Wess-Zumino-Witten theory]] is in \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]} (2013) \end{itemize} \hypertarget{dualities}{}\subsubsection*{{Dualities}}\label{dualities} The role of and the relation to [[duality in string theory]] of the membrane is discussed in the following articles. Relation to [[T-duality]] is discussed in: \begin{itemize}% \item J.G. Russo, \emph{T-duality in M-theory and supermembranes} (\href{http://arxiv.org/abs/hep-th/9701188}{arXiv:hep-th/9701188}) \item M.P. Garcia del Moral, J.M. Pena, A. Restuccia, \emph{T-duality Invariance of the Supermembrane} (\href{http://arxiv.org/abs/1211.2434}{arXiv:1211.2434}) \end{itemize} Relation to [[U-duality]] is discussed in: \begin{itemize}% \item [[Martin Cederwall]], \emph{M-branes on U-folds} (\href{http://arxiv.org/abs/0712.4287}{arXiv:0712.4287}) \item M.P. Garcia del Moral, \emph{Dualities as symmetries of the Supermembrane Theory} (\href{http://arxiv.org/abs/1211.6265}{arXiv}) \item \href{http://streaming.ictp.trieste.it/preprints/P/97/044.pdf}{pdf} \end{itemize} Discussion from the point of view of [[E11]]-[[U-duality]] and [[current algebra]] is in \begin{itemize}% \item [[Hirotaka Sugawara]], \emph{Current Algebra Formulation of M-theory based on E11 Kac-Moody Algebra}, International Journal of Modern Physics A, Volume 32, Issue 05, 20 February 2017 (\href{https://arxiv.org/abs/1701.06894}{arXiv:1701.06894}) \item [[Shotaro Shiba]], [[Hirotaka Sugawara]], \emph{M2- and M5-branes in E11 Current Algebra Formulation of M-theory} (\href{https://arxiv.org/abs/1709.07169}{arXiv:1709.07169}) \end{itemize} \hypertarget{ReferencesDyonic}{}\subsubsection*{{M2-M5 bound states}}\label{ReferencesDyonic} Discussion of [[M2-M5 brane bound state|M2-M5 brane bound states]], i.e. [[dyon|dyonic]]$\,$[[black brane|black]] [[M2-branes]] ([[M5-branes]] [[wrapped brane|wrapped]] on a [[3-manifold]], see also at \emph{\href{NS5-brane#ReferencesNS5D4D2BoundStates}{NS5-branes -- D2/D4/NS5-bound states}}): \begin{itemize}% \item J.M. Izquierdo, [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], \emph{Dyonic Membranes}, Nucl. Phys. B460:560-578, 1996 (\href{https://arxiv.org/abs/hep-th/9508177}{arXiv:hep-th/9508177}) \item [[Michael Green]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], \emph{Dyonic $p$-branes from self-dual $(p+1)$-branes}, Phys.Lett.B384:86-92, 1996 (\href{https://arxiv.org/abs/hep-th/9605146}{arXiv:hep-th/9605146}) \item [[Troels Harmark]], Section 3.1 of \emph{Open Branes in Space-Time Non-Commutative Little String Theory}, Nucl.Phys. B593 (2001) 76-98 (\href{https://arxiv.org/abs/hep-th/0007147}{arXiv:hep-th/0007147}) \item [[Troels Harmark]], N.A. Obers, Section 5.1 of \emph{Phase Structure of Non-Commutative Field Theories and Spinning Brane Bound States}, JHEP 0003 (2000) 024 (\href{https://arxiv.org/abs/hep-th/9911169}{arXiv:hep-th/9911169}) \item [[George Papadopoulos]], [[Dimitrios Tsimpis]], \emph{The holonomy of the supercovariant connection and Killing spinors}, JHEP 0307:018, 2003 (\href{https://arxiv.org/abs/hep-th/0306117}{arXiv:hep-th/0306117}) \item Nicolò Petri, slide 14 of \emph{Surface defects in massive IIA}, talk at \href{http://physics.ipm.ac.ir/conferences/stringtheory3/}{Recent Trends in String Theory and Related Topics} 2018 (\href{http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf}{pdf}) \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}) \item Jay Armas, Vasilis Niarchos, Niels A. Obers, \emph{Thermal transitions of metastable M-branes} (\href{https://arxiv.org/abs/1904.13283}{arXiv:1904.13283}) \end{itemize} Further [[bound states]] of M2/[[M5-branes]] to [[giant gravitons]]: \begin{itemize}% \item J. M. Camino, A. V. Ramallo, \emph{M-Theory Giant Gravitons with C field}, Phys.Lett.B525:337-346,2002 (\href{https://arxiv.org/abs/hep-th/0110096}{arXiv:hep-th/0110096}) \end{itemize} [[!redirects M2-branes]] [[!redirects M2 brane]] [[!redirects M2 branes]] [[!redirects M-theory membrane]] [[!redirects M-theory membranes]] \end{document}