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\section*{ADE classification}

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

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{ViaSuperYangMillsTheory}{Via $N=2$ super Yang-Mills theory}\dotfill \pageref*{ViaSuperYangMillsTheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general_surveys}{General Surveys}\dotfill \pageref*{general_surveys} \linebreak
\noindent\hyperlink{in_string_theory}{In string theory}\dotfill \pageref*{in_string_theory} \linebreak


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

A long list of mathematical structures happens to have a classification that is in [[bijection]] with the [[simply laced Dynkin diagrams]] of types A, D and E (but excluding type B and C), for instance

\begin{itemize}%
\item [[Platonic solids]]


\item [[finite group|finite]] [[subgroups]] of the [[special orthogonal group]] $SO(3)$ and of the [[special unitary group]] $SU(2)$ (see at [[classification of finite rotation groups]]) (\hyperlink{Milnor57}{Milnor 57}, see e.g. \hyperlink{Keenan03}{Keenan 03, theorem 4})


\item [[simply laced Dynkin diagrams]] (their [[simple Lie groups]])



\end{itemize}
$\backslash$begin\{imagefromfile\} ``file\_name'': ``ADEDynkin.jpg'', ``width'': 490 $\backslash$end\{imagefromfile\}

\begin{itemize}%
\item connected [[quivers]] with a [[finite number]] of [[indecomposable object|indecomposable]] [[quiver representations]] over an [[algebraically closed field]]

([[Gabriel's theorem]])


\item 7d [[spherical space forms]] with [[spin structure]] carrying $N \geq 4$ [[Killing vectors]] (see at \href{spherical+space+form#7DSphericalSpaceFormsWithSpinStructure}{spherical space form -- 7d with spin structure})

equivalently: [[near horizon geometries]] of smooth (i.e. non-[[orbifold]]) $\geq \tfrac{1}{2}$ [[BPS state|BPS]] [[black brane|black]] [[M2-brane]]-solutions of the [[equations of motion]] of [[11-dimensional supergravity]]

This is due to \hyperlink{MFFGME09}{MFFGME 09}.



\end{itemize}
[[!include 7d spherical space forms -- table]]

\begin{itemize}%
\item [[du Val singularities]] (see [[ADE singularity]])


\item [[singularities]] of [[elliptic fibrations]] (see there are \emph{\href{elliptic+fibration#ClassificationOfSingularFibers}{classification of singular fibers}})


\item certain 4d [[ALE spaces]] (\hyperlink{Kronheimer89}{Kronheimer 89})


\item certain [[2d CFTs]]


\item certain [[6d (2,0)-supersymmetric QFT|6d CFTs]]


\item intersection diagrams of vanishing 2-cycles in [[K3]]s (e.g. \hyperlink{BBS07}{BBS 07, p.423})



\end{itemize}
and many more.

[[!include ADE -- table]]

The obvious question for what might be the conceptual origin of this joint classification is attributed to (\hyperlink{Arnold76}{Arnold 76}).

Starting with (\hyperlink{DouglasMoore96}{Douglas-Moore 96}) is the observation that many of these structures are naturally aspects of the description of [[string theory]] [[KK-compactification|KK-compactified]] on \emph{[[orbifolds]] with [[ADE singularities]]} of the form $\mathbb{C}^n \sslash \Gamma$ for $\Gamma$ a [[finite group|finite]] [[subgroup]] of $SL_2(\mathbb{C})$.

\hypertarget{ViaSuperYangMillsTheory}{}\subsection*{{Via $N=2$ super Yang-Mills theory}}\label{ViaSuperYangMillsTheory}

Various seemingly unrelated structures in mathematics fall into an ``ADE classification''. Notably [[finite group|finite]] [[subgroups]] of [[special unitary group|SU(2)]] and [[compact Lie group|compact]] [[simple Lie groups]] do. The way this works usually is that one tries to classify these structures somehow, and ends up finding that the classification is governed by the combinatorics of [[Dynkin diagrams]] (see also \emph{[[McKay correspondence]]}).

While that does explain a bit, it seems the statement that both the [[icosahedral group]] and the Lie group [[E8]] are related to the same [[Dynkin diagram]] somehow is still more a question than an answer. Why is that so?

The first key insight is due to \href{ADE+singularity#Kronheimer89a}{Kronheimer 89}. He showed that the (resolutions of) the [[orbifold]] quotients $\mathbb{C}^2/\Gamma$ for finite subgroups $\Gamma$ of $SU(2)$ are precisely the generic form of the [[gauge group|gauge]] [[orbits]] of the [[direct product group]] of $U(n_i)$s acting in the evident way on the [[direct sum]] of $Hom(\mathbb{C}^{n_i}, \mathbb{C}^{n_j})$-s, where $i$ and $j$ range over the vertices of the [[Dynkin diagram]], and $(i,j)$ over its edges.

This becomes more illuminating when interpreted in terms of [[gauge theory]]: in a [[quiver gauge theory]] the [[gauge group]] is a [[direct product group]] of $U(n_i)$ factors associated with vertices of a [[quiver]], and the [[particles]] which are [[charged particle|charged]] under this gauge group arrange, as a [[linear representation]], into a [[direct sum]] of $Hom(\mathbb{C}^{n_i}, \mathbb{C}^{n_j})$-s, for each edge of the quiver.

Pick one such particle, and follow it around as the gauge group transforms it. The space swept out is its gauge [[orbit]], and \hyperlink{Kronheimer89}{Kronheimer 89} says that if the quiver is a Dynkin diagram, then this gauge orbit looks like $\mathbb{C}^2/\Gamma$.

On the other extreme, gauge theories are of interest whose gauge group is not a big direct product, but is a [[simple Lie group]], such as [[special unitary group|SU(N)]] or [[E8]]. The mechanism that relates the two classes of examples is [[spontaneous symmetry breaking]] (``[[Higgs field|Higgsing]]''): the ground state energy of the field theory may happen to be achieved by putting the fields at any one point in a higher dimensional space of field configurations, acted on by the gauge group, and fixing any one such point ``spontaneously'' singles out the corresponding [[stabilizer subgroup]].

Now here is the final ingredient: it is [[N=2 D=4 super Yang-Mills theory]] (``[[Seiberg-Witten theory]]'') which have a potential that is such that its [[vacua]] break a simple gauge group such as $SU(N)$ down to a Dynkin diagram [[quiver gauge theory]]. One place where this is reviewed, physics style, is in \href{N=2+D=4+super+Yang-Mills+theory#Albertsson03}{Albertsson 03, section 2.3.4}.

More precisely, these theories have two different kinds of vacua, those on the ``[[Coulomb branch]]'' and those on the ``[[Higgs branch]]'' depending on whether the scalars of the ``[[vector multiplets]]'' (the gauge field sector) or of the ``[[hypermultiplet]]'' (the matter field sector) vanish. The statement above is for the Higgs branch, but the Coulomb branch is supposed to behave ``dually''.

So that then finally is the relation, in the ADE classification, between the simple Lie groups and the finite subgroups of SU(2): start with an N=2 super Yang Mills theory with gauge group a simple Lie group. Let it spontaneously find its vacuum and consider the orbit space of the remaining spontaneously broken symmetry group. That is (a resolution of) the orbifold quotient of $\mathbb{C}^2$ by a discrete subgroup of $SU(2)$.

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

\begin{itemize}%
\item [[McKay correspondence]]

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

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

\begin{itemize}%
\item [[Vladimir Arnold]], \emph{Problems in present day mathematics}, (1976) in Felix E. Browder, \emph{Mathematical developments arising from Hilbert problems}, Proceedings of symposia in pure mathematics 28, American Mathematical Society, p. 46, \emph{Problem VIII. The A-D-E classifications} (V. Arnold).

\end{itemize}
A survey is in

\begin{itemize}%
\item Michael Hazewinkel, W Hesseling, Dirk Siersma, and Ferdinand Veldkamp, \emph{The ubiquity of Coxeter Dynkin diagrams (an introduction to the ADE problem)}, Nieuw Archief voor Wiskunde 25 (1977), 257-307. (\href{http://oai.cwi.nl/oai/asset/10039/10039A.pdf}{pdf})

\end{itemize}
which in turn is summarized in

\begin{itemize}%
\item Kyler Siegel, \emph{The Ubiquity of the ADE classification in Nature} , 2014 (\href{http://math.stanford.edu/~ksiegel/ADEClassifications.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/ADE_classification}{ADE classification}}

\end{itemize}
Discusion of the free finite group actions on spheres goes back to

\begin{itemize}%
\item [[John Milnor]], \emph{Groups which act on $S^n$ without fixed points}, American Journal of Mathematics Vol. 79, No. 3 (Jul., 1957), pp. 623-630 (\href{http://www.jstor.org/stable/2372566}{JSTOR})

\end{itemize}
Review inclues

\begin{itemize}%
\item Adam Keenan, \emph{Which finite groups act freely on spheres?}, 2003 (\href{http://www.math.utah.edu/~keenan/actions.pdf}{pdf})

\end{itemize}
Discussion of [[ALE spaces]] via ADE include

\begin{itemize}%
\item [[Peter Kronheimer]], \emph{The construction of ALE spaces as hyper-K\"a{}hler quotients}, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. (\href{https://projecteuclid.org/euclid.jdg/1214443066}{Euclid})

\end{itemize}
Related stuff includes\ldots{}

on [[immersions]] of [[3-spheres]] into $\mathbb{R}^4$:

\begin{itemize}%
\item Shumi Kinjo, \emph{Immersions of 3-sphere into 4-space associated with Dynkin diagrams of types A and D} (\href{http://arxiv.org/abs/1309.6526}{arXiv:1309.6526})

\end{itemize}
\hypertarget{in_string_theory}{}\subsubsection*{{In string theory}}\label{in_string_theory}

The original articles explaining the appearance of ADE classification from within [[string theory]] include

\begin{itemize}%
\item [[Michael Douglas]], [[Gregory Moore]], \emph{D-branes, Quivers, and ALE Instantons} (\href{http://arxiv.org/abs/hep-th/9603167}{arXiv:hep-th/9603167})


\item [[Clifford Johnson]], [[Robert Myers]], \emph{Aspects of Type IIB Theory on ALE Spaces}, Phys.Rev. D55 (1997) 6382-6393 (\href{http://arxiv.org/abs/hep-th/9610140}{arXiv:hep-th/9610140})


\item [[Michael Douglas]], [[Brian Greene]], [[David Morrison]], \emph{Orbifold Resolution by D-Branes}, Nucl.Phys. B506:84-106,1997 (\href{http://arxiv.org/abs/hep-th/9704151}{arXiv:hep-th/9704151})


\item [[Brian Greene]], [[Calin Lazaroiu]], Mark Raugas, \emph{D-branes on Nonabelian Threefold Quotient Singularities}, Nucl.Phys. B553 (1999) 711-749 (\href{http://arxiv.org/abs/hep-th/9811201}{arXiv:hep-th/9811201})


\item Andrea Cappelli, Jean-Bernard Zuber, \emph{A-D-E Classification of Conformal Field Theories} (\href{http://arxiv.org/abs/0911.3242}{arXiv:0911.3242})



\end{itemize}
Surveys include

\begin{itemize}%
\item MO discussion, \emph{\href{http://mathoverflow.net/a/34680/381}{ADE classification from string theory}}

\end{itemize}
Discussion of an ADE-classification of BPS [[Freund-Rubin compactifications]] is in

\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})

\end{itemize}
Specifically the ADE classfication involved in the [[6d (2,0)-supersymmetric QFT]] on the [[M5-brane]] is discussed in

\begin{itemize}%
\item [[Edward Witten]], \emph{Some Comments On String Dynamics} (\href{http://arxiv.org/abs/hep-th/9507121}{arXiv:hep-th/9507121})


\item [[Jonathan Heckman]], [[David Morrison]], [[Cumrun Vafa]], \emph{On the Classification of 6D SCFTs and Generalized ADE Orbifolds} (\href{http://arxiv.org/abs/1312.5746}{arXiv:1312.5746})



\end{itemize}
Discussion in the context of [[M-theory on G2-manifolds]] includes

\begin{itemize}%
\item [[Bobby Acharya]], section 3.1.1 of \emph{M Theory, $G_2$-manifolds and Four Dimensional Physics}, Classical and Quantum Gravity Volume 19 Number 22, 2002 (\href{http://users.ictp.it/~pub_off/lectures/lns013/Acharya/Acharya_Final.pdf}{pdf})


\item [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], p. 423 of \emph{String Theory and M-Theory: A Modern Introduction}, 2007



\end{itemize}
[[!redirects ADE classifications]]

[[!redirects ADE-classification]] [[!redirects ADE-classifications]]

[[!redirects A-D-E classification]] [[!redirects A-D-E classifications]]



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