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\begin{document}

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\section*{brane}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{phyics}{}\paragraph*{{Phyics}}\label{phyics}

[[!include physicscontents]]

\begin{quote}%
This article is about the general concept of branes. For more specific details see also also at \emph{[[super p-brane]]}, \emph{[[black brane]]} and \emph{[[D-brane]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Dbranes}{Boundary conditions or D-branes}\dotfill \pageref*{Dbranes} \linebreak
\noindent\hyperlink{in_terms_of_the_algebraic_data_of_the_qft_on_the_worldvolume}{In terms of the algebraic data of the QFT on the worldvolume}\dotfill \pageref*{in_terms_of_the_algebraic_data_of_the_qft_on_the_worldvolume} \linebreak
\noindent\hyperlink{in__rational_cft}{In $2d$ rational CFT}\dotfill \pageref*{in__rational_cft} \linebreak
\noindent\hyperlink{in__tft}{In $2d$ TFT}\dotfill \pageref*{in__tft} \linebreak
\noindent\hyperlink{in_terms_of_geometric_data_of_the_model_background}{In terms of geometric data of the $\sigma$-model background}\dotfill \pageref*{in_terms_of_geometric_data_of_the_model_background} \linebreak
\noindent\hyperlink{fundamental_or_model_branes}{Fundamental or $\sigma$-model branes}\dotfill \pageref*{fundamental_or_model_branes} \linebreak
\noindent\hyperlink{black_branes}{Black branes}\dotfill \pageref*{black_branes} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{worldvolume_theories}{Worldvolume theories}\dotfill \pageref*{worldvolume_theories} \linebreak
\noindent\hyperlink{the_superbrane_scan}{The super-brane scan}\dotfill \pageref*{the_superbrane_scan} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{prehistory}{Prehistory}\dotfill \pageref*{prehistory} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{boundary_conditions__dbranes}{Boundary conditions / D-branes}\dotfill \pageref*{boundary_conditions__dbranes} \linebreak
\noindent\hyperlink{fundamental_branes}{Fundamental branes}\dotfill \pageref*{fundamental_branes} \linebreak
\noindent\hyperlink{branes_ending_on_branes}{Branes ending on branes}\dotfill \pageref*{branes_ending_on_branes} \linebreak


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

The term \emph{brane} in formal high energy [[physics]], and in particular in [[string theory]], refers to entities that one thinks of as physical objects that generalize the notion \emph{point particles} to higher dimensional objects.

The term derives from the word \emph{membrane} that was originally used to describe 2-dimensional ``particles''. When the need was felt to speak also about 3-, 4- and higher dimensional such ``particles'' the usage ``3-brane'', ``4-brane'' etc. was introduced. Ordinary particles would be 0-branes in this counting, the strings in [[string theory]] would be 1-branes and membranes themselves 2-branes.

Generally, there are two different incarnations of branes

\begin{enumerate}%
\item \textbf{fundamental $p$-branes} (as in ``[[fundamental particle]]''): these are given by [[sigma-models]] on $(p+1)$-dimensional [[worldvolumes]] describing propagation of single $p$-dimensional objects on certain [[target spacetimes]]. If these sigma-models are required to exhibit manifest [[target spacetime]] [[supersymmetry]], then they are [[Green-Schwarz sigma models]], which are classified by a ``[[brane scan]]'' in [[super L-infinity algebra]] [[L-infinity algebra cohomology|cohomology]];


\item \textbf{[[black branes|black p-branes]]} (as in ``[[black hole]]''): these are [[soliton|solitonic]] solutions to [[field theories]], typically [[supergravity]] theories, with [[singularities]] of [[dimension]] $(p+1)$. In analogy to how a [[charged black hole]] ($p = 0$) [[source|sources]] an [[electromagnetic field]] with [[field strength]] 2-form, so black $p$-branes source $(p+2)$-form [[higher gauge fields]] and hence appear in those [[supergravity]] theories where such exists.



\end{enumerate}
The idea is that these two concepts match where a [[condensate]] of fundamental $p$-branes turns into a black $p$-branes.

Indeed, the classical [[no-hair theorem]] matches [[fundamental particles]] (i.e. 0-branes) characterized (via the [[Wigner classification]]) by just their [[mass]], electromagnetic [[charge]] and [[angular momentum]] to [[black hole]] solutions of pure vacuum gravity. Accordingly, it is an old suggestion (\hyperlink{EinsteinInfeldHoffmann39}{Einstein-Infeld-Hoffmann 39}) that fundamental particles could be identified with singular solutions of vacuum gravity.

This matching generalizes to higher dimensional $p$-branes in higher dimensional [[supergravity]] and there is an exact correspondence between fundamental [[Green-Schwarz super p-branes]] and extremal [[BPS solutions|BPS]] black brane solutions.

In [[string theory]] there is a third incarnation of branes, known as

\begin{itemize}%
\item \textbf{[[D-branes]]}: these are the admissible [[boundary conditions]] for the 2-dimensional [[sigma models]] describing [[open strings]].

\end{itemize}
One envisions that as one passes from [[perturbative string theory]] to the [[non-perturbative effect|non-perturbative]] version of the theory (``[[M-theory]]'') these [[D-branes]] show [[back-reaction]] and turn into the [[UV-completion]] of the [[black branes]] seen in the [[supergravity]] [[effective field theory]]. This relation is key in the microscopic computation of [[black hole entropy]] for [[black holes in string theory]].

\hypertarget{Dbranes}{}\subsubsection*{{Boundary conditions or D-branes}}\label{Dbranes}

Some words on [[D-branes]]

\hypertarget{in_terms_of_the_algebraic_data_of_the_qft_on_the_worldvolume}{}\paragraph*{{In terms of the algebraic data of the QFT on the worldvolume}}\label{in_terms_of_the_algebraic_data_of_the_qft_on_the_worldvolume}

An abstractly defined $n$-dimensional [[quantum field theory]] is a consistent assignment of [[state]]-space and correlators to $n$-dimensional [[cobordisms]] with certain structure (topological structure, conformal structure, Riemannian structure, etc. see [[FQFT]]/[[AQFT]]). In an \emph{open-closed QFT} the cobordisms are allowed to have boundaries. See at \emph{[[boundary field theory]]} for more on this.

In this abstract formulation of QFT a \textbf{brane} is a type of data assigned by the QFT to boundaries of cobordisms.

\hypertarget{in__rational_cft}{}\paragraph*{{In $2d$ rational CFT}}\label{in__rational_cft}

A well understood class of examples is this one: among all 2-dimensional [[conformal field theory]], the case of \emph{full rational 2d CFT} has been understood completely, using [[FFRS-formalism]]. It is then a theorem that full 2-rational CFTs are classified by

\begin{enumerate}%
\item a [[modular tensor category]] $\mathcal{C}$ (to be thought of as being the category of representations of the [[vertex operator algebra]] of the 2d CFT);


\item a special symmetric [[Frobenius algebra]] object $A$ [[internalization|internal]] to $\mathcal{C}$.



\end{enumerate}
In this formulation a type of \textbf{brane} of the theory is precisely an $A$-[[module]] in $\mathcal{C}$ (an $A$-[[bimodule]] is a [[bi-brane]] or \emph{defect line} ):

the 2d cobordisms with boundary on which the theory defined by $A \in \mathcal{C}$ carry as extra structure on their connected boundary pieces a label given by an equivalence class of an $A$-module in $\mathcal{C}$. The assignment of the CFT to such a cobordism with boundary is obtained by

\begin{itemize}%
\item triangulating the cobordism,


\item labeling all internal edges by $A$


\item labelling all boundary pieces by the $A$-module


\item all vertices where three internal edges meet by the multiplication operation


\item and all points where an internal edge hits a moundary by the corresponding [[action]] morphism


\item and finally evaluating the resulting [[string diagram]] in $\mathcal{C}$.



\end{itemize}
So in this abstract algebraic formulation of QFT on the worldvolume, a brane is just the datum assigned by the QFT to the boundary of a cobordism. But abstractly defined QFTs may arise from [[quantization]] of [[sigma model]]s. This gives these boundary data a geometric interpretation in some space. This we discuss in the next section.

\hypertarget{in__tft}{}\paragraph*{{In $2d$ TFT}}\label{in__tft}

Another case where the branes of a QFT are under good mathematical control is [[TCFT]]: the [[(infinity,1)-category]]-version of a 2d [[TQFT]].

Particularly the [[A-model]] and the [[B-model]] are well understood.

\begin{itemize}%
\item the branes of the B-model (``B-branes'') form the the [[stable (infinity,1)-category]] of [[chain complex]]es of [[quasicoherent sheaves]] on the target space (often considered just in terms of its [[homotopy category of an (infinity,1)-category]], the [[derived category]] of quasicoherent sheaves);


\item the branes of the A-model form the [[Fukaya category]] of the target space.



\end{itemize}
(\ldots{})

\hypertarget{in_terms_of_geometric_data_of_the_model_background}{}\paragraph*{{In terms of geometric data of the $\sigma$-model background}}\label{in_terms_of_geometric_data_of_the_model_background}

An abstractly defined QFT (as a consistent assignment of state spaces and propagators to cobordisms as in [[FQFT]]) may be obtained by [[quantization]] from \emph{geometric data} :

Sich a \emph{[[sigma-model]] QFT} is the [[quantization]] of an [[action functional]] on a space of maps $\Sigma \to X$ from a cobordims (``worldvolume'') $\Sigma$ to some target space $X$ that may carry further geoemtric data such as a [[Riemannian metric]], or other background [[gauge field]]s.

One may therefore try to match the geometric data on $X$ that encodes the $\sigma$-model with the algebraic data of the [[FQFT]] that results after quantization. This gives a geometric interpretation to many of the otherwise purely abstract algebraic properties of the worldvolume QFT.

It turns out that if one checks which geometric data corresponds to the $A$-modules in the above discussion, one finds that these tend to come from structures that look at least roughly like \emph{submanifolds} of the target space $X$. And typically these submanifolds themselves carry their own background [[gauge field]] data.

A well-understood case is the [[Wess-Zumino-Witten model]]: for this the target space $X$ is a simple [[Lie group]] $X = G$ and the background field is a [[circle n-bundle with connection|circle 2-bundle with connection]] (a [[bundle gerbe]]) on $G$, representing the background field that is known as the [[Kalb-Ramond field]].

In this case it turns out that branes for the sigma model on $X$ are given in the smplest case by conjugacy classes $D \subset G$ inside the group, and that these carry [[twisted bundle|twisted vector bundle]] with the twist given by the Kalb-Ramond background bundle. These vector bundles are known in the [[string theory]] literature as \emph{Chan-Paton vector bundles} . The geometric intuition is that a QFT with certain boundary condition comes form a quantization of spaces of maps $\Sigma \to G$ that are restricted to take the boundary of $\Sigma$ to these submanifolds.

More generally, one finds that the geometric data that corresponds to the branes in the algebraically defined 2d QFT is given by cocycles in the twisted [[differential K-theory]] of $G$. These may be quite far from having a direct interpretation as submanifolds of $G$.

The case of rational 2d CFT considered so far is only the best understood of a long sequence of other examples. Here the collection of all [[D-branes]] -- identified with the colleciton of all internal modules over an internal frobenius algebra, forms an ordinary [[category]].

More generally, at least for 2-dimensional [[TQFT]]s analogous considerations yield not just categories but [[stable (∞,1)-categories]] of boundary condition objects. For instance for what is called the [[B-model]] 2-d [[TQFT]] the category of [[D-branes]] is the [[derived category]] of [[coherent sheaves]] on some Calabi-Yau space.

Starting with Kontsevich's [[homological algebra]] reformulation of [[homological mirror symmetry|mirror symmetry]] the study of (derived) D-brane categories has become a field in its own right in pure mathematics.

\ldots{} lots of further things to say \ldots{}

\hypertarget{fundamental_or_model_branes}{}\subsubsection*{{Fundamental or $\sigma$-model branes}}\label{fundamental_or_model_branes}

In [[string theory]] one speaks apart from the [[D-branes]] also about \textbf{fundamental branes} . These are the objects $\Sigma$ in the $n$-dimensional [[sigma model]] themselves.

\begin{itemize}%
\item For $n=0$ this describes the ordinary quantum mechanics of a point particles on $X$. And such point particles are the \emph{fundamental particles} for instance of the [[standard model of particle physics]].


\item For $n=1$ this describes the quantum propagation of a [[string theory|string]], and accordingly one speaks of the \emph{fundamental string} or F1-brane (fundamental 1-brane).


\item For $n=2$ this describes the quantum propagation of a membrane.


\item There are good indications that there is a way to describe heterotic [[string theory]] not in terms of fundamental 1-branes but in terms of the [[sigma-model]] of a fundamental 5-brane -- the [[magnetic charge|magnetic dual]] of the 1-brane in 10-dimensions.


\item etc.



\end{itemize}
[[!include brane scan]]

\hypertarget{black_branes}{}\subsubsection*{{Black branes}}\label{black_branes}

See \emph{[[black brane]]} .

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

\hypertarget{worldvolume_theories}{}\subsubsection*{{Worldvolume theories}}\label{worldvolume_theories}

[[!include table of branes]]

\hypertarget{the_superbrane_scan}{}\subsubsection*{{The super-brane scan}}\label{the_superbrane_scan}

If the worldvolume QFT of the fundamental branes (for instance the worlsheet 2d[[CFT]] of the string) is required to be a [[supersymmetric QFT]], specifically if the [[Green-Schwarz action functional]] is used only particular combinations of the dimenion $dim \Sigma = p + 1$ of the worldvolume and $D = dim X$ of [[spacetime]] are possible.

The corresponding table has been called the \textbf{[[brane scan]]}

[[branescan.gif:pic]]

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

\begin{itemize}%
\item [[wrapped brane]]


\item [[brane intersection]]


\item [[D-brane]], [[fractional D-brane]]


\item [[sigma-model]]


\item [[double dimensional reduction]]


\item [[particle]], [[superparticle]]


\item [[string]], [[superstring]]


\item [[membrane]], [[M2-brane]], [[M5-brane]]


\item [[NS5-brane]]


\item [[D-brane]]


\item [[electric eigenbrane]]. [[magnetic eigenbrane]]



\end{itemize}
[[!include field theory with boundaries and defects - table]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

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

\hypertarget{prehistory}{}\subsubsection*{{Prehistory}}\label{prehistory}

\begin{itemize}%
\item [[Albert Einstein]], [[Leopold Infeld]], B. Hoffmann, \emph{The gravitational equations and the problem of motion}, Annals of Mathematics, Vol 39, No. 1, 1938

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Black_hole_electron}{Black hole electron}}

\end{itemize}
The terminology ``$p$-brane'' originates 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})

\end{itemize}
For more history see

\begin{itemize}%
\item [[Mike Duff]], \emph{Thirty years of Erice on the brane}, Based on lectures at the International Schools of Subnuclear Physics 1987-2017 and the International Symposium ``60 Years of Subnuclear Physics at Bologna'', University of Bologna, November 2018 (\href{https://arxiv.org/abs/1812.11658}{arXiv:1812.11658})

\end{itemize}
\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Pietro Fre]], \emph{The Branes: Three Viewpoints}, In: \emph{Gravity, a Geometrical Course} Springer 2013 (\href{http://inspirehep.net/record/1242195}{spire:1242195}, \href{https://doi.org/10.1007/978-94-007-5443-0_7}{doi:10.1007/978-94-007-5443-0\_7})


\item [[Greg Moore]], \emph{What is\ldots{} a brane?}, Notices of the AMS vol 52, no. 2 (\href{http://www.ams.org/notices/200502/what-is.pdf}{pdf})


\item Joan Simon, \emph{Brane Effective Actions, Kappa-Symmetry and Applications} (\href{http://arxiv.org/abs/1110.2422}{arXiv:1110.2422})


\item [[Luis Ibáñez]], [[Angel Uranga]], section 6.2 of \emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]}, Cambridge University Press 2012



\end{itemize}
\hypertarget{boundary_conditions__dbranes}{}\subsubsection*{{Boundary conditions / D-branes}}\label{boundary_conditions__dbranes}

(\ldots{})

See [[D-brane]].

For exhaustive details on D-branes in 2-dimensional rational [[CFT]] see the references given at

\begin{itemize}%
\item [[FFRS-formalism]].

\end{itemize}
A classical text describing how the physics way to think of D-branes for the [[topological string]] leads to seeing that they are objects in [[derived categories]] (of [[coherent sheaves]] for the [[B-model]]) is reviewed in

\begin{itemize}%
\item [[Paul Aspinwall]], \emph{D-Branes on Calabi-Yau Manifolds} (\href{http://arxiv.org/abs/hep-th/0403166}{arXiv})

\end{itemize}
based on

\begin{itemize}%
\item [[Michael Douglas]], \emph{D-branes, Categories and $N=1$ Supersymmetry}, J.Math.Phys. 42 (2001) 2818-2843 (\href{https://arxiv.org/abs/hep-th/0011017}{arXiv:hep-th/0011017})


\item [[Paul Aspinwall]], Albion Lawrence, \emph{Derived Categories and Zero-Brane Stability} (\href{https://arxiv.org/abs/hep-th/0104147}{arXiv:hep-th/0104147})



\end{itemize}
This can to a large extent be read as a dictionary from [[homological algebra]] terminology to that of D-brane physics.

More recent similar material with the emphasis on the [[K-theory]] aspects is

\begin{itemize}%
\item [[Richard Szabo]], \emph{[[Szabo09.pdf:file]]}

\end{itemize}
\hypertarget{fundamental_branes}{}\subsubsection*{{Fundamental branes}}\label{fundamental_branes}

The ``[[brane scan]]'' table showing the consistent dimension pairs for the [[Green-Schwarz action functional]] was depicted in

\begin{itemize}%
\item [[Michael Duff]], \emph{Supermembranes: the first fifteen weeks} CERN-TH.4797/87 (\href{http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198708425}{scan})

\end{itemize}
going back to

\begin{itemize}%
\item A. Ach\'u{}carro, J. M. Evans, [[Paul Townsend]] and D. L. Wiltshire, \emph{Super $p$-branes} Physics Letters B Volume 198, Issue 4, 3 (1987)

\end{itemize}
Further developments are in

More along these lines is in

\begin{itemize}%
\item [[Michael Duff]], S. Ferrara, \emph{Four curious supergravities} (\href{http://arxiv.org/abs/1010.3173}{arXiv})

\end{itemize}
See also [[division algebras and supersymmetry]].

\hypertarget{branes_ending_on_branes}{}\subsubsection*{{Branes ending on branes}}\label{branes_ending_on_branes}

\begin{itemize}%
\item [[Andrew Strominger]], \emph{Open P-Branes}, Phys. Lett. B383:44-47,1996 (\href{http://arxiv.org/abs/hep-th/9512059}{arXiv:hep-th/9512059})

\end{itemize}
[[!redirects branes]]

[[!redirects branes]]

[[!redirects fundamental brane]] [[!redirects fundamental branes]]

[[!redirects NS-fivebrane]] [[!redirects NS-fivebranes]]

[[!redirects p-brane]] [[!redirects p-branes]]



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