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\section*{McKay correspondence}

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

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

[[!include representation theory - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{McKayQuivers}{Via McKay quivers}\dotfill \pageref*{McKayQuivers} \linebreak
\noindent\hyperlink{AsAnEquivalenceOfKtheories}{As an equivalence of K-theories}\dotfill \pageref*{AsAnEquivalenceOfKtheories} \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


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

Generally, the \emph{McKay correspondence} is a subtle correspondence between the theory of [[finite subgroups of SU(2)]], the corresponding [[orbifold]] [[singularities]] ([[du Val singularities]]) and that of [[simple Lie groups]] falling into the [[ADE classification]].

[[!include ADE -- table]]

The correspondence may be understood at different levels:

\begin{enumerate}%
\item \textbf{McKay quivers.} The original correspondence due to \hyperlink{McKay80}{McKay 80} is the observation that the [[McKay quiver]] associated to the [[orbifold]] [[singularity]] $\mathbb{C}^2 \sslash G$ of a [[finite subgroup of SU(2)]] $G \subset SU(2)$ happens to be an extended [[Dynkin quiver]], hence happens to be an extended [[Dynkin diagram]] of the kind that also arises in the [[ADE-classification]] of [[simple Lie groups]].

More on this in \emph{\hyperlink{McKayQuivers}{McKay quivers}}, below.


\item \textbf{Equivariant K-theory.} A more conceptual explanation for this correspondence comes from the classical fact that the [[blow up]] $\widehat{\mathbb{C}^2\sslash G}$ of the [[ADE singularity]] $\mathbb{C}^2\sslash G$ is given by a sequence of [[spheres]] that touch along the vertices of the corresponding [[Dynkin quiver]]. In \hyperlink{GSV83}{Gonzalez-Sprinberg \& Verdier 83} it was shown that the $G$-[[equivariant K-theory]] $K_G(\mathbb{C}^2)$ of the singularity is isomorphic to the [[K-theory]] of the desingularization $\widehat {\mathbb{C}^2\sslash G}$ such that under this isomorphism the generators of $K_G(\mathbb{C}^2) \simeq R_\mathbb{C}(G)$, hence of the [[representation ring]], hence the [[irreducible representations]], map to K-theory classes supported on the [[exceptional divisors]] of this blowup. This gives a conceptual geometric/cohomological explanation for the identifications observed by \hyperlink{McKay80}{McKay 80}.

More on this in \emph{\hyperlink{AsAnEquivalenceOfKtheories}{As an equivalence of K-theories}}


\item \textbf{Seiberg-Witten theory.} Under the interpretation of [[K-theory]]-classes as [[D-brane]] charges, the K-theoretic McKay correspondence of \hyperlink{GSV83}{Gonzalez-Sprinberg \& Verdier 83} identifies [[wrapped brane]] charges of the desingularized space with [[fractional brane]] charges at the singularity.

This leads to a further (currently non-rigorous) explanation of the McKay correspondence in terms of the [[AdS-CFT duality|dual]] [[worldvolume]] [[quantum field theories]] on these branes, which are [[N=2 D=4 super Yang-Mills theory|N=2]] [[quiver gauge theories]]: The [[moduli space]] of [[scalar fields]] of these theories has two ``branches'', called the \emph{[[Higgs branch]]} and the \emph{[[Coulomb branch]]}, and the idea is that depending on which of the branches the [[vacuum state]] of the theory is in, it describes the brane as being either on the [[ADE singularity]] or on its [[resolution of singularities|resolution]], but since it's the same quantum field theory in both cases, these two situations are actually suitably equivalent.



\end{enumerate}
More on this in \emph{\hyperlink{ViaSuperYangMillsTheory}{Via N=2 SYM theory}} below.

\hypertarget{McKayQuivers}{}\subsubsection*{{Via McKay quivers}}\label{McKayQuivers}

Generally, for $G$ a [[finite group]] and $V$ a [[linear representation]] of $G$ on a [[finite dimensional vector space|finite dimensional]] [[complex vector space]], the \emph{[[McKay quiver]]} or \emph{[[McKay graph]]} associated with $V$ is the [[quiver]] whose [[vertices]] correspond to the [[irreducible representations]] $\rho_i$ of $G$ and which has $a_{i j} \in \mathbb{N}$ [[edges]] between the $i$th and the $j$th vertex, for $a_{i j}$ the [[coefficients]] in the expansion into [[irreps]] of the [[tensor product of representations]] of $V$ with these irreps:

\begin{displaymath}
V \otimes \rho_i
  \;\simeq\;
  \underset{j}{\bigoplus}
  a_{i j} \cdot \rho_j
  \,.
\end{displaymath}
Specifically this applies to the special case where $G \subset$ [[SU(2)]] a [[finite subgroup of SU(2)]] and $V$ its defining representation on $\mathbb{C}^2$.

The original \emph{McKay correspondence} (\hyperlink{McKay80}{McKay 80}) states that in this case the corresponding [[McKay quivers]] are [[Dynkin quivers]]/[[Dynkin diagrams]] in the same [[ADE classification]] as the [[ADE singularity]] $\mathbb{C}^2 \sslash G$.

More precisely: If one uses \emph{all} [[irreducible representations]] including the 1-dimensiona [[trivial representation]] $\rho_0$ then one gets the ``extended Dynkin diagram'', where the extra node corresponds to $\rho_0$. This is the vertex indicated by a cross in the following diagrams:

$\backslash$begin\{center\}  $\backslash$end\{center\}

\begin{quote}%
graphics grabbed from \hyperlink{GSV83}{GSV 83, p. 4}


\end{quote}
In particular, for $G =\mathbb{Z}_N \subset SU(2)$ a [[cyclic group]] of [[order of a group|order]] $N$, there are $N$ complex [[irreps]] and the McKay quiver, i.e. the extended Dynkin diagram, has $N$-vertices, connected by edges to form a circle.

\hypertarget{AsAnEquivalenceOfKtheories}{}\subsubsection*{{As an equivalence of K-theories}}\label{AsAnEquivalenceOfKtheories}

A strong version of the McKay correspondence is obtained when the [[cohomology theory]] is taken be ([[equivariant K-theory|equivariant]]) \emph{[[K-theory]]} (\hyperlink{GSV83}{Gonzalez-Sprinberg \& Verdier 83}):

Here the McKay correspondence becomes an [[isomorphism]] between the [[equivariant K-theory]] $K_{G_{ADE}}(\ast)$ of an [[ADE-singularity]] (equivalently the [[representation ring]] $R(G_{ADE})$) to the plain [[K-theory]] $K(\tilde X)$ of its [[blow-up]] \href{ADE+singularity#ResolutionBySpheresTouchingAlongADynkinDiagram}{resolution} $\tilde X$. Much like in [[topological T-duality]], this isomorphism is given by [[integral transform]] ([[Fourier-Mukai transform]]) through the canonical [[correspondence]] that these spaces constitute:

\begin{equation}
\itexarray{
     &&
     && K( X \times_{X/G} \tilde X )
     \\
     && 
     & {}^{\mathllap{ p_1^\ast }}\nearrow && \searrow^{\mathrlap{Inv \circ (p_2)_\ast}}
     \\
     R(G) \simeq K_G(\ast) &\simeq& K_G(X) && \overset{\simeq}{\longrightarrow}  && K(\tilde X)
  }
\label{KMcKay}\end{equation}
In terms of [[physics]] ([[string theory]]), the [[K-theory]] classes appearing here may be interpreted as [[groups]] of [[fractional D-brane|fractional]] [[D-brane charges]] (see \href{fractional+D-brane#DBranesAtOrbifoldSingularities}{there}). In terms of the [[worldvolume]] [[Chan-Paton gauge field|Chan-Paton]]-[[Yang-Mills theory]] on the D-branes the McKay correspondence is then seen as passage from the [[Higgs branch]] to the [[Coulomb branch]] (see \href{fractional+D-brane#InTermsOfTheWorldvolumeGaugeTheories}{there}).

The proof of these statements generally proceeds by relating both sides of the equivalence to [[Dynkin diagrams]] of [[ADE-classification|ADE-type]].

The classical McKay correspondence, named after [[John McKay]], is a one-to-one correspondence between the [[McKay graphs]] of [[finite group|finite]] [[subgroups]] $G \subset \text{SL}_2(\mathbb{C})$ and the extended Dynkin diagrams of ADE type.

\hypertarget{ViaSuperYangMillsTheory}{}\subsubsection*{{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]].

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 quiver]]


\item [[ADE classification]]


\item [[quiver gauge theory]]


\item [[wrapped brane]]



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

The original article is

\begin{itemize}%
\item [[John McKay]], \emph{Graphs, singularities, and finite groups} Proc. Symp. Pure Math. Vol. 37. No. 183. 1980

\end{itemize}
As an isomorphism between the [[equivariant K-theory]] of [[ADE-singularity]] and the plain [[topological K-theory]] of its \href{ADE+singularity#ResolutionBySpheresTouchingAlongADynkinDiagram}{resolution}, the McKay correspondence is proven in:

\begin{itemize}%
\item [[Gérard Gonzalez-Sprinberg]], [[Jean-Louis Verdier]], \emph{Construction géométrique de la correspondance de McKay}, Ann. Sci. ́École Norm. Sup.16 (1983) 409–449. (\href{http://www.numdam.org/item?id=ASENS_1983_4_16_3_409_0}{numdam})

\end{itemize}
An analogous discussion for [[derived categories]] of coherent sheaves is in

\begin{itemize}%
\item [[Tom Bridgeland]], [[Alastair King]], [[Miles Reid]], \emph{The McKay correspondence as an equivalence of derived categories} (\href{http://www.ams.org/journals/jams/2001-14-03/S0894-0347-01-00368-X/S0894-0347-01-00368-X.pdf}{pdf})

\end{itemize}
Introductions and surveys include

\begin{itemize}%
\item Graham Leuschke, \emph{The McKay correspondence} (\href{http://www.leuschke.org/uploads/McKay-total.pdf}{pdf})


\item Bockland, \emph{Character tables and McKay quivers} (\href{https://staff.fnwi.uva.nl/r.r.j.bocklandt/notes/kleinian.pdf}{pdf})


\item Drew Armstrong, \emph{Lectures on the McKay correspondence}, 2015 (\href{http://www.math.miami.edu/~armstrong/Talks/McKay_Talca.pdf}{pdf scan of notes})


\item Max Lindh, \emph{An Introduction to the McKay Correspondence Master Thesis in Physics} (\href{http://www.diva-portal.org/smash/get/diva2:1184051/FULLTEXT01.pdf}{pdf})


\item [[John Baez]], \emph{The Geometric McKay Correspondence}, (\href{https://golem.ph.utexas.edu/category/2017/06/the_geometric_mckay_correspond.html}{Part I}, \href{https://golem.ph.utexas.edu/category/2017/07/the_geometric_mckay_correspond_1.html}{Part II})



\end{itemize}
Literature collection

\begin{itemize}%
\item [[Miles Reid]], \emph{\href{http://homepages.warwick.ac.uk/staff/Miles.Reid/McKay/}{Links to papers on McKay correspondence}}

\end{itemize}


\end{document}