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\section*{BLG model}

\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{the_lagrangian}{The Lagrangian}\dotfill \pageref*{the_lagrangian} \linebreak
\noindent\hyperlink{boundary_conditions}{Boundary conditions}\dotfill \pageref*{boundary_conditions} \linebreak
\noindent\hyperlink{3AlgebraStructure}{The ``$3$-algebra'' structure}\dotfill \pageref*{3AlgebraStructure} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The BLG model is a 3-dimensional [[SCFT]] involving a [[Chern-Simons theory]] coupled to matter. It is argued to be the [[worldvolume]] theory of 2 coincident [[black brane|black]] [[M2-branes]] with 16 manifest [[supersymmetries]]. The generalization to an arbitrary number of M2-branes is supposed to be given by the [[ABJM model]].

[[!include 7d spherical space forms -- table]]

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

\hypertarget{the_lagrangian}{}\subsubsection*{{The Lagrangian}}\label{the_lagrangian}

(\ldots{})

\hypertarget{boundary_conditions}{}\subsubsection*{{Boundary conditions}}\label{boundary_conditions}

Discussion of [[boundary conditions]] of the BLG model, leading to [[brane intersection]] with [[M-wave]], [[M5-brane]] and [[MO9-brane]] is in (\hyperlink{ChuSmith09}{Chu-Smith 09}, \hyperlink{BPST09}{BPST 09}).

\hypertarget{3AlgebraStructure}{}\subsubsection*{{The ``$3$-algebra'' structure}}\label{3AlgebraStructure}

The BLG model Lagrangian involves a trilinear operation on the scalar fields $\phi \in V$

\begin{displaymath}
[-,-,-] : V^{\otimes 3} \to V
  \,.
\end{displaymath}
Moreover, the [[supersymmetry]] of the Lagrangian hinges on the fact that this map satisfies a condition that has some similarity to a [[Jacobi identity]] for the binary operation on a [[Lie algebra]].

Therefore, superficially, it looks like this might be the trinary bracket on an \emph{[[L-∞ algebra]]} structure on the space $V$.

On the one hand, indeed, by the discussion at \emph{[[supergravity C-field]]} , the M2-brane is charged under a [[circle n-bundle with connection|circle 3-bundle with connection]] whose [[higher gauge theory]] is controled by [[Lie n-algebra|Lie 3-algebra]]s in direct analogy to how the [[higher gauge theory]] of the [[string]] is controled by [[gerbes]]/[[principal 2-bundles]] and their [[Lie 2-algebra]]s and that of charged [[particles]] by ordinary [[Lie algebra]]s.

Apparantly motivated by an intuition along these lines, (\hyperlink{BaggerLambert}{BaggerLambert}) named $(V,[-,-,-])$ a \textbf{3-algebra}. This terminology was picked up by many authors In the process, it transmuted sometimes to ``3-Lie algebra'' and sometimes even to ``Lie 3-algebra''.

Unfortunately, the Bagger-Lambert ``3-algebra'' is \textbf{not} a \emph{[[Lie n-algebra|Lie 3-algebra]]} in the established sense of an [[L-∞ algebra]] structure on a [[graded vector space]] $V$ concentrated in the lowest three degrees. At least not without some modifications in the interpretation of the map $[-,-,-]$.

The reason is that for the notion of an [[L-∞ algebra]] (as discussed there) it is crucial that $V$ is a $\mathbb{N}$-graded (or $\mathbb{Z}$-graded) vector space and that the $n$-ary brackets respect the degree in a certain way. But in the \hyperlink{BaggerLamber}{BaggerLambert}-proposal, $V$ is all concentrated in a single degree (is regarded as ungraded). One immediately finds that in this case the $L_\infty$-respect of $[-,-,-]$ for the grading would imply that $V$ is taken to be in degree $1/2$. Since this is not in $\mathbb{N}$, it does not yield an $L_\infty$-algebra.

Notice that the $\mathbb{N}$-grading (or $\mathbb{Z}$-grading) of $L_\infty$-algebras is crucial for the [[homotopy theory|homotopy theoretic]] interpretation of [[L-∞ algebra]]s as higher Lie algebras. None of the good theory of $L_\infty$-algebras survives when this grading is dropped. This grading has its origin in the [[Dold-Kan correspondence]], which establishes integral graded homological structures as models for structures in [[higher category theory]]. Notably, a higher Lie algebra is supposed to have a [[Lie integration]] to a [[smooth ∞-groupoid|smooth $n$-groupoid]]. Under this process, the elements in degree $k$ of the higher Lie algebra become tangents to the space of [[k-morphisms]] of this smooth $n$-groupoid. Clearly, here only integer $k$ do make sense.

On the other hand, it is of course possible to consider the structure on ``$L_\infty$-algebras without grading'', even if these will not have a good theory. This notion has once been introduced by Filippov (Sib. Math. Zh. No 6 126--140 (195)) under the term \emph{[[n-Lie algebra]]} .

Beware, therefore, that the innocent-looking difference between the terms

\begin{itemize}%
\item [[Lie n-algebra]]


\item [[n-Lie algebra]]



\end{itemize}
corresponds, unfortunately, to a major difference in the behaviour of the concepts behind these terms.

In conclusion, it is clear that 2-brane physics is governed by Lie 3-algebraic structures, but it is not yet clear how the trinary operation highlighted by \hyperlink{BaggerLambert}{BaggerLambert} would be an example.

In view of this it might be noteworthy that the equivalent reformulation and generalization of the BLG model by the [[ABJM model]] does not involve any ``3-algebras'' at all. In fact at least most of the ``3-algebras'' appearing in the membrane literature may be understood as being data of a plain [[Lie algebra]] with an invariant product and a representation (\hyperlink{MFMR}{MFMR 08}). These authors summarize the state of affiars on p. 3 as

\begin{quote}%
All this prompts one to question whether the 3-algebras appearing in the constructions 1--3,10,11 play a fundamental role in M-theory or, at least insofar as the effective field theory is concerned, are largely superfluous. The equivalence of 10 and 6 and the abundance of new theories (dual to known M-theory backgrounds) which seem not to involve a 3-algebra might suggest the latter. Nonetheless, given our lack of understanding of how to incorporate in Lie-algebraic terms the expected properties of M-theoretic degrees of freedom, like the entropy scaling laws for M2- and M5-brane condensates, it may be useful to understand the precise relation between the 3-algebras appearing in the recent literature on superconformal Chern--Simons theory and Lie algebras.


\end{quote}
The article (\hyperlink{MFMR}{MFMR 08}) provides this relation and under this relation the ``3-algebraic'' BLG model has then been understood as a special case of the ordinary Lie algebraic [[ABJM theory]]. A review is in (\hyperlink{BaggerLambert12}{Bagger-Lambert 12}).

It has also been suggested that ``3-algebras'' are to be interpreted in \textbf{[[n-plectic geometry|2-plectic structure]]} (\hyperlink{SaemannSzabo}{Saemann-Szabo}).

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

\begin{itemize}%
\item [[ABJM theory]]


\item [[membrane matrix model]]



\end{itemize}
[[!include superconformal symmetry -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Precursors:

\begin{itemize}%
\item [[John Schwarz]], \emph{Superconformal Chern-Simons Theories} (\href{https://arxiv.org/abs/hep-th/0411077}{arXiv:arXiv:hep-th/0411077}

\end{itemize}
The original articles are

\begin{itemize}%
\item [[Jonathan Bagger]], [[Neil Lambert]], \emph{Modeling Multiple M2's}, Phys. Rev. D75, 045020 (2007). (\href{http://arxiv.org/abs/hep-th/0611108}{hep-th/0611108}).


\item [[Jonathan Bagger]], [[Neil Lambert]], Phys. Rev. D77, 065008 (2008). (\href{http://arXiv.org/abs/0711.0955}{arXiv:0711.0955}).



\end{itemize}
and concerning the ``3-algebra''-structure also

\begin{itemize}%
\item [[Andreas Gustavsson]], Nucl. Phys. B811, 66-76 (2009). arXiv:0709.1260.

\end{itemize}
The interpretation in terms of branes at a $\mathbb{Z}/2$-[[ADE-singularities]] with [[discrete torsion]] in the [[supergravity C-field]] is due to

\begin{itemize}%
\item [[Neil Lambert]], [[David Tong]], \emph{Membranes on an Orbifold}, Phys.Rev.Lett.101:041602, 2008 (\href{https://arxiv.org/abs/0804.1114}{arXiv:0804.1114})


\item [[Jacques Distler]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], [[Mark Van Raamsdonk]], \emph{M2-branes on M-folds}, JHEP 0805:038,2008 (\href{https://arxiv.org/abs/0804.1256}{arXiv:0804.1256})



\end{itemize}
A comprehensive review is in

\begin{itemize}%
\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})

\end{itemize}
and a review talk is

\begin{itemize}%
\item [[Neil Lambert]], \emph{M-Branes: Lessons from M2’s and Hopes for M5’s}, talk at \emph{\href{http://www.maths.dur.ac.uk/lms/109/index.html}{Higher Structures in M-Theory, Durham, August 2018}} (\href{http://www.maths.dur.ac.uk/lms/109/talks/1877lambert.pdf}{pdf slides}, \href{http://www.maths.dur.ac.uk/lms/109/movies/1877lamb.mp4}{video recording})

\end{itemize}
Discussion of [[boundary conditions]] leading to [[brane intersection]] with [[M-wave]], [[M5-brane]] and [[MO9-brane]] is in

\begin{itemize}%
\item [[Chong-Sun Chu]], Douglas J. Smith, \emph{Multiple Self-Dual Strings on M5-Branes}, JHEP 1001:001, 2010 (\href{https://arxiv.org/abs/0909.2333}{arXiv:0909.2333})


\item [[David Berman]], Malcolm J. Perry, [[Ergin Sezgin]], Daniel C. Thompson, \emph{Boundary Conditions for Interacting Membranes}, JHEP 1004:025, 2010 (\href{https://arxiv.org/abs/0912.3504}{arXiv:0912.3504})



\end{itemize}
Discussion in [[Horava-Witten theory]] reducing M2-branes 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}
The interpretation of at least most of the ``3-algebra'' appearing in the membrane literature in terms of plain [[Lie algebras]] is due to

\begin{itemize}%
\item Paul de Medeiros, [[José Figueroa-O'Farrill]], Elena M\'e{}ndez-Escobar, Patricia Ritter, \emph{On the Lie-algebraic origin of metric 3-algebras}, Commun.Math.Phys.290:871-902,2009 (\href{http://arxiv.org/abs/0809.1086}{arXiv:0809.1086})

\end{itemize}
See also

\begin{itemize}%
\item [[José Figueroa-O'Farrill]], section \emph{Triple systems and Lie superalgebras} in \emph{M2-branes, ADE and Lie superalgebras}, talk at IPMU 2009 (\href{http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Hongo.pdf}{pdf})

\end{itemize}
The suggestion that BGL ``3-algebras'' are to be interpreted in [[2-plectic geometry]] appears in

\begin{itemize}%
\item [[Christian Saemann]], [[Richard Szabo]], \emph{Quantization of 2-Plectic Manifolds} (\href{http://arxiv.org/abs/1106.1890}{arXiv:1106.1890})

\end{itemize}
[[!redirects BLG-model]]



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