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\section*{BPS state}

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

\hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{General}{General}\dotfill \pageref*{General} \linebreak
\noindent\hyperlink{in_supergravity}{In supergravity}\dotfill \pageref*{in_supergravity} \linebreak
\noindent\hyperlink{in_superstring_theory}{In superstring theory}\dotfill \pageref*{in_superstring_theory} \linebreak
\noindent\hyperlink{InTermsOfHigherDifferentialGeometry}{Formalization in higher differential geometry}\dotfill \pageref*{InTermsOfHigherDifferentialGeometry} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_11d_supergravity}{In 11d Supergravity}\dotfill \pageref*{in_11d_supergravity} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{IntroductoryReferences}{Introductions, surveys and lectures}\dotfill \pageref*{IntroductoryReferences} \linebreak
\noindent\hyperlink{in_supergravity_2}{In supergravity}\dotfill \pageref*{in_supergravity_2} \linebreak
\noindent\hyperlink{SpectralNetworksReferences}{Spectral networks}\dotfill \pageref*{SpectralNetworksReferences} \linebreak


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

\hypertarget{General}{}\subsubsection*{{General}}\label{General}

In [[supersymmetry|supersymmetric]] [[quantum field theory]] with [[extended supersymmetry]], certain extremal [[supermultiplets]] have some of the [[supersymmetry|supersymmetries]] retained (have 0-[[eigenvalue]] under some of the [[supersymmetry]] generators). These are called \textbf{Bogomol'nyi--Prasad--Sommerfield saturated solutions}.

More in detail, where in a plain [[supersymmetry]] [[super Lie algebra]] a suitable [[basis]] $\{Q_A\}$ of supersymmetry generators has odd bracket proportional to the spacetime translation and hence to an [[energy]]/[[mass]] operator $E$ (with terminology as at [[unitary representation of the Poincaré group]])

\begin{displaymath}
\{Q_A, Q_B\} = E \delta_{A B}
\end{displaymath}
for [[extended supersymmetry]] there are further bosonic super Lie algebra generators $K_{A B}$ (charges) such that

\begin{displaymath}
\{Q_A, Q_B\} = E \delta_{A B} -  K_{A B}
 \,.
\end{displaymath}
It follows from the supersymmetry algebra that $(E \delta_{A B} -  K_{A B})$ is a positive definite [[bilinear form]], which puts a lower bound on the [[energy]] given the values of these extra charges. This is called the \emph{BPS bound}. See also at \emph{[[Bridgeland stability condition]]}.

In particular when this bound is satisfied in that some of the [[eigenvalues]] of the [[matrix]] $(K_{A B})$ are actually equal to the energy/mass, then the corresponding component of the right hand side in the above equation vanishes and hence then the corresponding supersymmetry generators may annihilate the given state, then called a \emph{BPS state}. This way enhanced supersymmetry of states goes along with certain charges taken extremal values.

States with similar behaviour are also considered also in some models of [[soliton]] theory (English Wikipedia: \href{http://en.wikipedia.org/wiki/Bogomol'nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound}{Bogomol'nyi--Prasad--Sommerfield bound}).

BPS states play a central role in the investigation of [[moduli spaces]] of classical [[vacua]] as they form part of the moduli problem which is often the most tractable.

Several mathematical theories in [[geometry]] are interpreted as counting BPS-states in the sense of integration on appropriate compactification of the [[moduli space]] of BPS-states in a related physical model attached to the underlying geometry: most notably the [[Gromov-Witten invariants]], [[Donaldson-Thomas invariants]] and the [[Thomas-Pandharipande invariants]]; all the three seem to be deeply interrelated though they are defined in rather very different terms. The compactification of the moduli space involves various [[stability conditions]].

\hypertarget{in_supergravity}{}\subsubsection*{{In supergravity}}\label{in_supergravity}

In the context of [[supergravity]] BPS states correspond to [[super spacetimes]] admitting [[Killing vectors]]. These notably include extremal [[black brane]] solutions.

\hypertarget{in_superstring_theory}{}\subsubsection*{{In superstring theory}}\label{in_superstring_theory}

Specifically in [[superstring theory]] BPS states in [[target space]] correspond to string states on the worldsheet which are annihilated by the left-moving (say) half of the [[Dirac-Ramond operator]]. These are counted by the [[Witten genus]], see at \emph{\href{Witten+genus#RelationToBPSStateCounting}{Witten genus -- Relation to BPS states}}.

The degeneracy of BPS states in string theory has been used to provide a microscopic interpretation of [[Bekenstein-Hawking entropy]] of [[black holes]], see at \emph{[[black holes in string theory]]}.

\hypertarget{InTermsOfHigherDifferentialGeometry}{}\subsection*{{Formalization in higher differential geometry}}\label{InTermsOfHigherDifferentialGeometry}

\begin{quote}%
The following are some observations on the formalization of BPS states from the [[nPOV]], in [[higher differential geometry]], following (\hyperlink{SatiSchreiber15}{Sati-Schreiber 15}).


\end{quote}
Let $\mathbb{R}^{d-1,1|N}$ be a [[super-Minkowski spacetime]], let $(d,N,p)$ be in the [[brane scan]] and write

\begin{displaymath}
\phi \coloneqq \bar{\psi} \wedge E^{\wedge p} \wedge \psi \in \Omega^{p+2}(\mathbb{R}^{d-1,1|N})
\end{displaymath}
for the correspoding [[super Lie algebra]] [[Lie algebra cohomology|cocycle]], as discussed at \emph{[[Green-Schwarz action functional]]}, see (\hyperlink{FSS13}{FSS 13}) for the perspective invoked here.

Consider then $X$ a [[super-spacetime]] locally modeled on $\mathbb{R}^{d-1,1|N}$ as a [[Cartan geometry]], solving the relevant [[supergravity]] [[equations of motion]] (e.g. [[11-dimensional supergravity]] for $d= 11$, [[heterotic supergravity]] for $d = 10$ and $N = (1,0)$, [[type IIA supergravity]] for $d = 10$ and $N= (1,1)$ or [[type IIB supergravity]] for $d = 10$ $N= (2,0)$).

This means in particular that $X$ carries a [[super differential form]]

\begin{displaymath}
\omega \in \Omega^{p+2}(X)
\end{displaymath}
which is [[definite form|definite]] on $\phi$. This is the [[curvature]] of the [[WZW-term]] which defines the relevant [[super p-brane sigma-model]] with [[target space]] $X$.

By (\hyperlink{AGIT89}{AGIT 89}) $X$ is a BPS state to the extent that it carries [[Killing spinors]] which form a central [[Lie algebra extension]] of a sub-algebra of the [[supersymmetry]] algebra (i.e. of the [[super translation Lie algebra]]) by $H^p_{dR}(X)$ which is classified by the [[Lie algebra cohomology|cocycle]] given by

\begin{displaymath}
(\epsilon_1, \epsilon_2) \mapsto
  \omega(\epsilon_1,\epsilon_2)
  \in 
  \Omega^p(X)_{cl}/im(\mathbf{d}_{dR})
  \,.
\end{displaymath}
Now we observe that by (\hyperlink{hgpII}{hgpII, theorem 3.3.1}) this is precisely the [[0-truncation]] of the [[super L-infinity algebra|super]]-[[Poisson bracket Lie n-algebra]] $\mathfrak{Pois}(X,\omega)$ induced by regarding $(X,\omega)$ as an [[pre-n-plectic manifold|pre-n-plectic]] [[supermanifold]] and restricting along the inclusion of the [[Killing vectors]]/[[Killing spinors]] into all the [[Hamiltonian vector fields]].

\begin{displaymath}
H^p_{dR}(X) \to \tau_0 \mathfrak{Pois}(X,\omega) \to Vect_{Ham}(X)
\end{displaymath}
(Here we are using that if an [[n-type]] is an extension of a [[0-type]], then its 0-truncation is still an extension by the 0-truncation of the original homotopy fiber.)

The elements in $H^p_{dR}(X)$ here are precisely the $p$-brane charges, as discussed in (\hyperlink{AGIT89}{AGIT 89, p. 8}).

Hence $X$ is the more BPS the more odd-graded elements there are in $\tau_0 \mathfrak{Pois}(X,\omega)$ (or its restriction to super-isometries). Hence $X$ is a 1/2 BPS state of supergravity if the odd dimension of this is half that of $\mathbb{R}^{d-1,d|N}$, it is 1/4 BPS if the odd dimension is one fourth of that of $\mathbb{R}^{d-1,d|N}$, etc.

Notice that if

\begin{displaymath}
\itexarray{
    &&     
     \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn}
    \\
    &  {}^{\mathllap{\mathbf{L}_{WZW}}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}}   
    \\
    X 
    &\stackrel{\omega}{\longrightarrow}&
    \mathbf{\Omega}^{p+2}_{cl}
  }
\end{displaymath}
is a [[prequantum line n-bundle|prequantization]] of $\omega$, i.e. an actual [[WZW term]] with [[curvature]] $\omega$, then $\mathfrak{Pois}(X,\omega)$ is supposed to be the [[Lie differentiation]] of the [[stabilizer group]] of $\mathbf{L}_{WZW}$, which is the [[quantomorphism n-group]] $QuantMorph(\mathbf{L}_{WZW})$. (This Lie differentiation statement is strictly shown only for $p = 0$ and $p = 1$ in [[schreiber:differential cohomology in a cohesive topos|dcct]] but clearly should hold generally.)

Hence we may regard $\mathbf{QuantMorph}(\mathbf{L}_{WZW})$ (or its restriction to super-[[isometries]]) as the [[Lie integration]] of the brane-charge extended supersymmetry algebra. By the discussion at \emph{\href{conserved+current#InHigherPrequantumGeometry}{conserved current -- In higher differential geometry}} this is indeed the [[n-group]] of conserved currents of $\mathbf{L}_{WZW}$ regraded as a [[local Lagrangian]], and so this conceptually connects back to the considerations in (\hyperlink{AGIT89}{AGIT 89}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{in_11d_supergravity}{}\subsubsection*{{In 11d Supergravity}}\label{in_11d_supergravity}

In [[11-dimensional supergravity]] ([[M-theory]]) there are four kinds of 1/2 BPS states (the [[black brane|black]] [[M-branes]]) (e.g. \hyperlink{Stelle98}{Stelle 98, section 3} \hyperlink{EHKNT07}{EHKNT 07}):

\begin{itemize}%
\item the [[M2-brane]];


\item the [[M5-brane]];


\item the [[M-wave]];


\item the [[Kaluza-Klein monopole]].



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

\begin{itemize}%
\item [[Bridgeland stability condition]]


\item [[Killing spinor]]


\item [[supermultiplet]]


\item [[wall crossing]]


\item [[protection from quantum corrections]]


\item [[positive energy theorem]]


\item [[enhanced gauge symmetry]]


\item [[geometry of physics -- BPS charges]]


\item [[intersecting branes]]

[[membrane triple junction]]



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

\hypertarget{general_2}{}\subsubsection*{{General}}\label{general_2}

The BPS bound derives its name from the discussion of [[magnetic monopoles]] in 4-dimensional [[Yang-Mills theory]] in

\begin{itemize}%
\item (E. B. Bogomolnyj) [[?. ?. ???????????]], \emph{  }, . . \textbf{24} (1976) 449--454


\item M. K. Prasad, [[Charles Sommerfield]], \emph{Exact classical solution for `t Hooft monopole and the Julia-Zee dyon}, Phys. Rev. Lett. \textbf{35} (1975) 760--762.



\end{itemize}
The extension of the term ``BPS-saturated state'' from this case to situations in [[string theory]] seems to have happened in

\begin{itemize}%
\item [[Edward Witten]], around (2.5) of \emph{[[String Theory Dynamics In Various Dimensions]]}, Nucl.Phys.B443:85-126,1995 (\href{http://arxiv.org/abs/hep-th/9503124}{arXiv:hep-th/9503124})

\end{itemize}
The original article identifying the role of BPS states in supersymmetric field theory is

\begin{itemize}%
\item [[Edward Witten]], [[David Olive]], \emph{Supersymmetry algebras that include topological charges} (2002) (\href{http://www.sciencedirect.com/science/article/pii/037026937890357X}{journal})

\end{itemize}
Exposition and review includes

\begin{itemize}%
\item [[Andrew Neitzke]], \emph{What is a BPS state?}, 2012 (\href{http://www.ma.utexas.edu/users/neitzke/expos/bps-expos.pdf}{pdf})


\item [[Tudor Dimofte]], \emph{Refined BPS invariants, Chern-Simons theory, and the quantum dilogarithm}, 2010 (\href{http://thesis.library.caltech.edu/5808/1/DimofteTDofficial.pdf}{pdf}, \href{http://thesis.library.caltech.edu/5808/}{web})



\end{itemize}
Further developments are in

\begin{itemize}%
\item [[Jeffrey Harvey]], [[Greg Moore]], \emph{Algebras, BPS states, and strings}, Nucl.Phys.B463:315--368 (1996) (\href{http://dx.doi.org/10.1016/0550-3213%2895%2900605-2}{doi}); \href{http://arxiv.org/abs/hep-th/9510182}{hep-th/9510182};


\item [[Jeffrey Harvey]], [[Greg Moore]], \emph{On the algebras of BPS states}, Comm. Math. Phys. \textbf{197} (1998), 489---519, \href{http://dx.doi.org/10.1007/s002200050461}{doi}, \href{http://arxiv.org/abs/hep-th/9609017}{hep-th/9609017}.


\item [[Ali Chamseddine]], M. S. Volkov, \emph{Non-abelian BPS monopoles in $N=4$ gauged supergravity}, Physical Review Letters 79: 3343--3346 (1997) \href{http://arxiv.org/abs/hep-th/9707176}{hep-th/9707176}.


\item [[Steven Weinberg]], \emph{The quantum theory of fields}, vol. II


\item [[Tudor Dimofte]], [[Sergei Gukov]], \emph{Refined, Motivic, and Quantum}, \href{http://arXiv.org/abs/0904.1420}{arXiv:0904.1420}


\item [[Davide Gaiotto]], [[Gregory Moore]], [[Andrew Neitzke]], \emph{Wall-crossing, Hitchin systems, and the WKB approximation}, \href{http://arxiv.org/abs/0907.3987}{arxiv:0907.3987}


\item R. Pandharipande, R.P. Thomas, \emph{Stable pairs and BPS invariants}, \href{http://arxiv.org/abs/0711.3899}{arXiv:0711.3899}


\item Markus Reineke, \emph{Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants}, \href{http://arXiv.org/abs/0903.0261}{arXiv:0903.0261}


\item Duiliu-Emanuel Diaconescu, \emph{Moduli of ADHM sheaves and local Donaldson-Thomas theory}, \href{http://arXiv.org/abs/0801.0820}{arXiv:0801.0820}


\item [[Tom Bridgeland]], \emph{Stability conditions on triangulated categories}, Ann. of Math. 166 (2007) 317--345,\href{http://arxiv.org/abs/math/0212237}{math.AG/0212237}


\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Stability structures, motivic Donaldson-Thomas invariants and cluster transformations}, \href{http://arxiv.org/abs/0811.2435}{arXiv:0811.2435}


\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Motivic Donaldson-Thomas invariants: summary of results}, \href{http://arxiv.org/abs/0910.4315}{arxiv/0910.4315}


\item [[Dominic Joyce]], Y. Song, \emph{A theory of generalized Donaldson-Thomas invariants}, \href{http://arxiv.org/abs/0810.5645}{arxiv/0810.5645}



\end{itemize}
\hypertarget{IntroductoryReferences}{}\subsubsection*{{Introductions, surveys and lectures}}\label{IntroductoryReferences}

An introduction that starts at the beginning and then covers much of the ground in some detail is

\begin{itemize}%
\item [[Greg Moore]], \emph{PiTP Lectures on BPS states and wall-crossing in $d = 4$, $\mathcal{N} = 2$ theories} (\href{http://www.physics.rutgers.edu/~gmoore/PiTP_July26_2010.pdf}{pdf})

\end{itemize}
A survey of progress on the most general picture is in

\begin{itemize}%
\item Katzutoshi Ohta, \emph{BPS state counting and related physics} (2005) (\href{http://www2.yukawa.kyoto-u.ac.jp/~qft/2005/slides/ohta.pdf}{pdf})

\end{itemize}
\hypertarget{in_supergravity_2}{}\subsubsection*{{In supergravity}}\label{in_supergravity_2}

Discussion of extremal/BPS [[black branes]] in [[supergravity]] (especially in [[11-dimensional supergravity]] and 10d [[type II supergravity]]) includes

\begin{itemize}%
\item [[Kellogg Stelle]], \emph{BPS Branes in Supergravity} (\href{http://arxiv.org/abs/hep-th/9803116}{arXiv:hep-th/9803116})


\item [[Jerome Gauntlett]], [[Chris Hull]], \emph{BPS States with Extra Supersymmetry}, JHEP 0001 (2000) 004 (\href{http://arxiv.org/abs/hep-th/9909098}{arXiv:hep-th/9909098})


\item [[Francois Englert]], Laurent Houart, [[Axel Kleinschmidt]], [[Hermann Nicolai]], Nassiba Tabti, \emph{An $E_9$ multiplet of BPS states}, JHEP 0705:065,2007 (\href{http://arxiv.org/abs/hep-th/0703285}{arXiv:hep-th/0703285})


\item Andrew Callister, [[Douglas Smith]], \emph{Topological BPS charges in 10 and 11-dimensional supergravity}, Phys. Rev. D78:065042,2008 (\href{http://arxiv.org/abs/0712.3235}{arXiv:0712.3235})


\item Andrew Callister, [[Douglas Smith]], \emph{Topological charges in $SL(2,\mathbb{R})$ covariant massive 11-dimensional and Type IIB SUGRA}, Phys.Rev.D80:125035,2009 (\href{http://arxiv.org/abs/0907.3614}{arXiv:0907.3614})


\item Andrew Callister, \emph{Topological BPS charges in 10- and 11-dimensional supergravity}, thesis 2010 (\href{http://inspirehep.net/record/1221591?ln=en}{spire})


\item Cristine N. Ferreira, \emph{BPS solution for eleven-dimensional supergravity with a conical defect configuration} (\href{http://arxiv.org/abs/1312.0578}{arXiv:1312.0578})



\end{itemize}
Specifically for $1/2^n$-BPS states of intersecting [[M-branes]] in 11d there is discussion in

\begin{itemize}%
\item [[Arkady Tseytlin]], \emph{Harmonic superpositions of M-branes}, Nucl.Phys.B475:149-163,1996 (\href{https://arxiv.org/abs/hep-th/9604035}{arXiv:hep-th/9604035}) also in [[Mike Duff]] (ed.) chapter 5 of \emph{[[The World in Eleven Dimensions]]}

\end{itemize}
see also

\begin{itemize}%
\item [[Jerome Gauntlett]], \emph{Intersecting Branes} (\href{https://arxiv.org/abs/hep-th/9705011}{hep-th/9705011})


\item [[Igor Bandos]], [[Jose de Azcarraga]], Jose Izquierdo, Jerzy Lukierski, \emph{BPS states in M-theory and twistorial constituents}, Phys.Rev.Lett.86:4451-4454,2001 (\href{https://arxiv.org/abs/hep-th/0101113}{arXiv:hep-th/0101113})


\item U. Gran, [[Jan Gutowski]], [[George Papadopoulos]], \emph{Classification, geometry and applications of supersymmetric backgrounds} (\href{https://arxiv.org/abs/1808.07879}{arXiv:1808.07879})



\end{itemize}
Discussion in the context of multiple [[M2-branes]] in the [[BLG model]] is in

\begin{itemize}%
\item [[Jonathan Bagger]], [[Neil Lambert]], Sunil Mukhi, Constantinos Papageorgakis, section 1.6 of \emph{Multiple Membranes in M-theory} (\href{http://arxiv.org/abs/1203.3546}{arXiv:1203.3546})

\end{itemize}
Discussion for [[4d supergravity]], hence in [[KK-compactification]] of [[type II supergravity]] on a [[Calabi-Yau manifold]] is due to

\begin{itemize}%
\item [[Frederik Denef]], \emph{Supergravity flows and D-brane stability}, JHEP 0008:050,2000 (\href{http://arxiv.org/abs/hep-th/0005049}{arXiv:hep-th/0005049})


\item [[Frederik Denef]], \emph{Quantum Quivers and Hall/Hole Halos}, JHEP 0210:023,2002 (\href{http://arxiv.org/abs/hep-th/0206072}{arXiv:hep-th/0206072})



\end{itemize}
Discussion of more general classification of solutions to [[supergravity]] preserving some [[supersymmetry]], i.e. admitting some [[Killing spinors]] includes

\begin{itemize}%
\item [[Jerome Gauntlett]], Stathis Pakis, \emph{The Geometry of $D=11$ Killing Spinors}, JHEP 0304 (2003) 039 (\href{http://arxiv.org/abs/hep-th/0212008}{arXiv:hep-th/0212008})


\item [[Eric D'Hoker]], John Estes, Michael Gutperle, Darya Krym, Paul Sorba, \emph{Half-BPS supergravity solutions and superalgebras}, JHEP0812:047,2008 (\href{http://arxiv.org/abs/0810.1484}{arXiv:0810.1484})



\end{itemize}
The conceptual identification of the relevant brane-charge extension of the [[supersymmetry]] algebra as that of the [[conserved currents]] of the [[Green-Schwarz super p-brane sigma models]] for branes is due to

\begin{itemize}%
\item [[José de Azcárraga]], [[Jerome Gauntlett]], J.M. Izquierdo, [[Paul Townsend]], \emph{Topological Extensions of the Supersymmetry Algebra for Extended Objects}, Phys. Rev. Lett. 63 (1989) 2443 (\href{http://inspirehep.net/record/26393?ln=en}{spire})

\end{itemize}
reviewed in

\begin{itemize}%
\item [[José de Azcárraga]], Jos\'e{} M. Izquierdo, section 8.8. of \emph{Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics} , Cambridge monographs of mathematical physics, (1995)

\end{itemize}
This is for branes in the old [[brane scan]] ([[strings]], [[membranes]], [[NS5-branes]]), excluding [[D-branes]] and [[M5-brane]].

The generalization oft this perspective to the [[M5-brane]] is discussed in

\begin{itemize}%
\item [[Dmitri Sorokin]], [[Paul Townsend]], \emph{M-theory superalgebra from the M-5-brane}, Phys.Lett. B412 (1997) 265-273 (\href{http://arxiv.org/abs/hep-th/9708003}{arXiv:hep-th/9708003})

\end{itemize}
and the generalization to [[D-branes]] is discussed in

\begin{itemize}%
\item Hanno Hammer, \emph{Topological Extensions of Noether Charge Algebras carried by D-p-branes}, Nucl.Phys. B521 (1998) 503-546 (\href{http://arxiv.org/abs/hep-th/9711009}{arXiv:hep-th/9711009})

\end{itemize}
Detailed discussion of examples for various backgrounds is in

\begin{itemize}%
\item Takeshi Sato, \emph{Superalgebras in Many Types of M-Brane Backgrounds and Various Supersymmetric Brane Configurations}, Nucl.Phys. B548 (1999) 231-257 (\href{http://arxiv.org/abs/hep-th/9812014}{arXiv:hep-th/9812014})

\end{itemize}
Discussion of this in [[higher differential geometry]] via the [[Poisson bracket Lie n-algebra]] is in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Lie n-algebras of BPS charges]]} (\href{http://arxiv.org/abs/1507.08692}{arXiv:1507.08692})

\end{itemize}
Discussion of relation of [[M5-brane]] BPS states to [[knot invariants]] includes

\begin{itemize}%
\item [[Edward Witten]], \emph{Fivebranes and Knots} (\href{http://arxiv.org/abs/1101.3216}{arXiv:1101.3216})


\item [[Sergei Gukov]], Marko Sto\v{s}i, \emph{Homological algebra of knots and BPS states} (\href{http://arxiv.org/abs/1112.0030}{arXiv:1112.0030})


\item Ross Elliot, [[Sergei Gukov]], \emph{Exceptional knot homology} (\href{http://arxiv.org/abs/1505.01635}{arXiv:1505.01635})



\end{itemize}
\hypertarget{SpectralNetworksReferences}{}\subsubsection*{{Spectral networks}}\label{SpectralNetworksReferences}

\begin{itemize}%
\item [[Davide Gaiotto]], [[Greg Moore]], [[Andrew Neitzke]], \emph{Spectral networks} (\href{http://arxiv.org/abs/1204.4824}{arXiv:1204.4824}, \href{http://www.ma.utexas.edu/users/neitzke/spectral-network-movies/}{illustrating animations})

\end{itemize}
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