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\section*{ADE singularity}

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - 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{ResolutionBySpheresTouchingAlongADynkinDiagram}{Resolution by spheres touching along a Dynkin diagram}\dotfill \pageref*{ResolutionBySpheresTouchingAlongADynkinDiagram} \linebreak
\noindent\hyperlink{from_coincident_kkmonopoles}{From coincident KK-monopoles}\dotfill \pageref*{from_coincident_kkmonopoles} \linebreak
\noindent\hyperlink{bridgeland_stability_conditions}{Bridgeland stability conditions}\dotfill \pageref*{bridgeland_stability_conditions} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{via_bridgeland_stability}{Via Bridgeland stability}\dotfill \pageref*{via_bridgeland_stability} \linebreak
\noindent\hyperlink{ReferencesInStringTheory}{In string theory}\dotfill \pageref*{ReferencesInStringTheory} \linebreak
\noindent\hyperlink{mtheory_on_adeorbifolds_reducing_to_d6branes_in_type_ii}{M-theory on ADE-orbifolds reducing to D6-branes in type II}\dotfill \pageref*{mtheory_on_adeorbifolds_reducing_to_d6branes_in_type_ii} \linebreak
\noindent\hyperlink{heterotic_mtheory_on_adeorbifolds}{Heterotic M-theory on ADE-orbifolds}\dotfill \pageref*{heterotic_mtheory_on_adeorbifolds} \linebreak
\noindent\hyperlink{heterotic_string_theory_on_adeorbifolds}{Heterotic string theory on ADE-orbifolds}\dotfill \pageref*{heterotic_string_theory_on_adeorbifolds} \linebreak
\noindent\hyperlink{type_string_theory_on_adeorbifolds}{Type $I'$-string theory on ADE-orbifolds}\dotfill \pageref*{type_string_theory_on_adeorbifolds} \linebreak
\noindent\hyperlink{type_string_theory_on_adeorbifolds_2}{Type $I$-string theory on ADE-orbifolds}\dotfill \pageref*{type_string_theory_on_adeorbifolds_2} \linebreak
\noindent\hyperlink{mtheory_on_orbifolds_with_adesingularities}{M-theory on $G_2$-orbifolds with ADE-singularities}\dotfill \pageref*{mtheory_on_orbifolds_with_adesingularities} \linebreak
\noindent\hyperlink{ftheory_with_adesingularities}{F-theory with ADE-singularities}\dotfill \pageref*{ftheory_with_adesingularities} \linebreak


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

An \emph{ADE singularity} is an [[orbifold]] [[fixed point]] locally of the form $\mathbb{C}^2\sslash\Gamma$ with $\Gamma \hookrightarrow SU(2)$ a [[finite subgroup of SU(2)]] given by the [[ADE classification]] (and $SU(2)$ is understood with its defining linear [[action]] on the [[complex numbers|complex]] [[vector space]] $\mathbb{C}^2$).

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

\hypertarget{ResolutionBySpheresTouchingAlongADynkinDiagram}{}\subsubsection*{{Resolution by spheres touching along a Dynkin diagram}}\label{ResolutionBySpheresTouchingAlongADynkinDiagram}

The [[blow-up]] of an ADE-singularity is given by a [[union]] of [[Riemann spheres]] that touch each other such as to form the shape of the [[Dynkin diagram]] whose A-D-E label corresponds to that of the given [[finite subgroup of SU(2)]].

This statement is originally due to (\hyperlink{duVal1934I}{duVal 1934 I, p. 1-3 (453-455)}). A description in terms of [[hyper-Kähler geometry]] is due to \hyperlink{Kronheimer89a}{Kronheimer 89a}.

Quick survey of this fact is in \hyperlink{Reid87}{Reid 87}, a textbook account is \hyperlink{Slodowy80}{Slodowy 80}.

 In [[string theory]] this fact is interpreted in terms of [[gauge enhancement]] of the [[M-theory]]-lift of coincident [[black brane|black]] [[D6-branes]] to an [[MK6]] at an ADE-singularity (\href{enhanced+gauge+symmetry#Sen97}{Sen 97}):

$\backslash$begin\{center\} $\backslash$begin\{imagefromfile\} ``file\_name'': ``ADESingularity.jpg'', ``width'': 760 $\backslash$end\{imagefromfile\} $\backslash$end\{center\}

\begin{quote}%
graphics grabbed from \hyperlink{HSS18}{HSS18}


\end{quote}
See at \emph{[[M-theory lift of gauge enhancement on D6-branes]]} for more.

$\,$

[[!include ADE -- table]]

\hypertarget{from_coincident_kkmonopoles}{}\subsubsection*{{From coincident KK-monopoles}}\label{from_coincident_kkmonopoles}

[[!include KK-monopole geometries -- table]]

\hypertarget{bridgeland_stability_conditions}{}\subsubsection*{{Bridgeland stability conditions}}\label{bridgeland_stability_conditions}

For $G_{ADE} \subset SU(2)$ a [[finite subgroup of SU(2)]], let $\tilde X$ be the [[resolution of singularities|resolution]] of the corresponding ADE-singularity as \hyperlink{ResolutionBySpheresTouchingAlongADynkinDiagram}{above}.

Then the [[connected component]] of the space of [[Bridgeland stability conditions]] on the bounded [[derived category]] of [[coherent sheaves]] over $\tilde X$ can be described explicitly (\hyperlink{Bridgeland05}{Bridgeland 05}).

Specifically for type-A singularities the space of stability conditions is in fact [[connected topological space|connected]] and [[simply-connected topological space]] (\hyperlink{IshiiUedaUehara10}{Ishii-Ueda-Uehara 10}).

Brief review is in \hyperlink{Bridgeland09}{Bridgeland 09, section 6.3}.

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

\begin{itemize}%
\item [[singularity]]


\item [[conical singularity]]


\item [[quiver gauge theory]]


\item [[M-theory on G2-manifolds]]


\item [[D7-brane]]



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

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

Original articles include

\begin{itemize}%
\item [[Patrick du Val]], \emph{On isolated singularities of surfaces which do not affect the conditions of adjunction. I}, Proceedings of the Cambridge Philosophical Society, 30 (4): 453–459 (1934a) (\href{https://doi.org/10.1017/S030500410001269X}{doi:10.1017/S030500410001269X})


\item [[Patrick du Val]], \emph{On isolated singularities of surfaces which do not affect the conditions of adjunction. II}, Proceedings of the Cambridge Philosophical Society, 30 (4): 460–465 (1934) (\href{https://doi.org/10.1017/S0305004100012706}{doi:10.1017/S0305004100012706})


\item [[Patrick du Val]], \emph{On isolated singularities of surfaces which do not affect the conditions of adjunction. III}, Proceedings of the Cambridge Philosophical Society, 30 (4): 483–491 (1934) (\href{https://doi.org/10.1017/S030500410001272X}{doi:10.1017/S030500410001272X})



\end{itemize}
Textbook accounts include

\begin{itemize}%
\item Alan H. Durfee, \emph{Fifteen characterizations of rational double points and simple critical points}, L'Enseignement Mathématique Volume: 25 (1979) (\href{http://dx.doi.org/10.5169/seals-50375}{doi:10.5169/seals-50375}, \href{http://www.maths.ed.ac.uk/~v1ranick/papers/durfee15.pdf}{pdf})


\item [[Peter Slodowy]], \emph{Simple singularities and simple algebraic groups}, in Lecture Notes in Mathematics 815, Springer, Berlin, 1980.


\item [[Klaus Lamotke]], chapter IV of \emph{Regular Solids and Isolated Singularities}, Vieweg, Braunschweig, Wiesbaden 1986.


\item [[Miles Reid]], \emph{Young persons guide to canonical singularities, in [[Spencer Bloch]] (ed.),}\href{https://www.ams.org/books/pspum/046.1/}{Algebraic geometry -- Bowdoin 1985, Part 1}\emph{, Proc. Sympos. Pure Math. 46 Part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 345-414 (\href{http://www.maths.ed.ac.uk/cheltsov/quotient/pdf/reid3.pdf}{pdf})}

(The last formula on page 409 has a typo: there should be no $r$ in the [[denominator]].)



\end{itemize}
Discussion of [[resolution of singularities|resolution]] of ADE-singularities in terms of [[hyper-Kähler geometry]]:

\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:1214443066})


\item [[Peter Kronheimer]], \emph{A Torelli-type theorem for gravitational instantons}, J. Differential Geom. Volume 29, Number 3 (1989), 685-697 (\href{https://projecteuclid.org/euclid.jdg/1214443067}{euclid:1214443067})



\end{itemize}
and in terms of preprojective algebras:

\begin{itemize}%
\item William Crawley-Boevey, Martin P. Holland, \emph{Noncommutative deformations of Kleinian singularities}, Duke Math. J. Volume 92, Number 3 (1998), 605-635 (\href{https://projecteuclid.org/euclid.dmj/1077231679}{euclid:1077231679})

\end{itemize}
Reviews and lecture notes include

\begin{itemize}%
\item \emph{\href{http://www.mathematik.uni-kl.de/~zca/Reports_on_ca/29/paper_html/node10.html}{Classification of singularities}}


\item Igor Burban, \emph{Du Val Singularities} (\href{http://www.mi.uni-koeln.de/~burban/singul.pdf}{pdf})


\item [[Miles Reid]], \emph{The Du Val Singularities $A_n$, $D_n$, $E_6$, $E_7$, $E_8$} (\href{http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf}{pdf})


\item Anda Degeratu, \emph{Crepant Resolutions of Calabi-Yau Orbifolds}, 2004 (\href{https://home.mathematik.uni-freiburg.de/degeratu/crepant.pdf}{pdf})


\item Fabio Perroni, \emph{Orbifold Cohomology of ADE-singularities} (\href{http://mathweb.uzh.ch/fileadmin/math/preprints/10-06.pdf}{pdf})


\item Kyler Siegel, section 6 of \emph{The Ubiquity of the ADE classification in Nature} , 2014 (\href{http://math.stanford.edu/~ksiegel/ADEClassifications.pdf}{pdf})


\item MathOverflow, \emph{\href{http://mathoverflow.net/q/186368/381}{Resolving ADE singularities by blowing up}}



\end{itemize}
Families of examples of [[G2 manifolds|G2 orbifolds]] with ADE singularities are constructed in

\begin{itemize}%
\item [[Frank Reidegeld]], \emph{$G_2$-orbifolds from K3 surfaces with ADE-singularities} (\href{http://arxiv.org/abs/1512.05114}{arXiv:1512.05114})


\item [[Frank Reidegeld]], \emph{$G_2$-orbifolds with ADE-singularities} (\href{https://core.ac.uk/download/pdf/159317626.pdf}{pdf})



\end{itemize}
[[Riemannian geometry]] of manifolds with ADE singularities is discussed in

\begin{itemize}%
\item Boris Botvinnik, [[Jonathan Rosenberg]], \emph{Positive scalar curvature on manifolds with fibered singularities} (\href{https://arxiv.org/abs/1808.06007}{arXiv:1808.06007})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Du_Val_singularity}{du Val singularity}}

\end{itemize}
\hypertarget{via_bridgeland_stability}{}\subsubsection*{{Via Bridgeland stability}}\label{via_bridgeland_stability}

Discussion of [[Bridgeland stability conditions]] for ([[resolution of singularities|resolutions of]]) ADE singularities includes:

\begin{itemize}%
\item [[Tom Bridgeland]], \emph{Stability conditions and Kleinian singularities}, International Mathematics Research Notices 2009.21 (2009): 4142-4157 (\href{https://arxiv.org/abs/math/0508257}{arXiv:0508257})


\item Akira Ishii, Kazushi Ueda, Hokuto Uehara, \emph{Stability conditions on $A_n$-singularities}, Journal of Differential Geometry 84 (2010) 87-126 (\href{https://arxiv.org/abs/math/0609551}{arXiv:math/0609551})



\end{itemize}
and specifically over [[Dynkin quivers]]

\begin{itemize}%
\item [[Yu Qiu]], Def. 2.1 \emph{Stability conditions and quantum dilogarithm identities for Dynkin quivers}, Adv. Math., 269 (2015), pp 220-264 (\href{https://arxiv.org/abs/1111.1010}{arXiv:1111.1010})


\item [[Tom Bridgeland]], [[Yu Qiu]], Tom Sutherland, \emph{Stability conditions and the $A_2$ quiver} (\href{https://arxiv.org/abs/1406.2566}{arXiv:1406.2566})



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

Discussion of [[ADE-singularities]] in [[string theory]] on [[orbifolds]]:

\hypertarget{mtheory_on_adeorbifolds_reducing_to_d6branes_in_type_ii}{}\paragraph*{{M-theory on ADE-orbifolds reducing to D6-branes in type II}}\label{mtheory_on_adeorbifolds_reducing_to_d6branes_in_type_ii}

[[M-theory lift of gauge enhancement on D6-branes]]:

\begin{itemize}%
\item [[Ashoke Sen]], \emph{A Note on Enhanced Gauge Symmetries in M- and String Theory}, JHEP 9709:001,1997 (\href{http://arxiv.org/abs/hep-th/9707123}{arXiv:hep-th/9707123})


\item [[Luis Ibáñez]], [[Angel Uranga]], Section 6.3.3 of: \emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]}, Cambridge University Press 2012



\end{itemize}
\hypertarget{heterotic_mtheory_on_adeorbifolds}{}\paragraph*{{Heterotic M-theory on ADE-orbifolds}}\label{heterotic_mtheory_on_adeorbifolds}

[[heterotic M-theory on ADE-orbifolds]]:

\begin{itemize}%
\item [[Ashoke Sen]], \emph{A Note on Enhanced Gauge Symmetries in M- and String Theory}, JHEP 9709:001,1997 (\href{http://arxiv.org/abs/hep-th/9707123}{arXiv:hep-th/9707123})


\item Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], \emph{Intersecting Orbifold Planes and Local Anomaly Cancellation in M-Theory}, Nucl. Phys. B554: 437-483, 1999 (\href{https://arxiv.org/abs/hep-th/9903028}{arXiv:hep-th/9903028})


\item Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], \emph{Local Anomaly Cancellation, M-Theory Orbifolds and Phase-Transitions}, Nucl. Phys. B589: 269-291, 2000 (\href{https://arxiv.org/abs/hep-th/0005251}{arXiv:hep-th/0005251})


\item Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], \emph{An M-Theory Perspective on Heterotic K3 Orbifold Compactifications}, Int. J. Mod. Phys. A18:3273-3314, 2003 (\href{https://arxiv.org/abs/hep-th/0010087}{arXiv:hep-th/0010087})


\item Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], \emph{Twisted Sectors and Chern-Simons Terms in M-Theory Orbifolds}, Int. J. Mod. Phys. A18: 2995-3014, 2003 (\href{https://arxiv.org/abs/hep-th/0011031}{arXiv:hep-th/0011031})


\item [[Vadim Kaplunovsky]], J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, \emph{On the Duality between Perturbative Heterotic Orbifolds and M-Theory on $T^4/Z_N$}, Nuclear Physics B Volume 590, Issues 1–2, 4 December 2000, Pages 123-160 Nuclear Physics B (\href{https://arxiv.org/abs/hep-th/9912144}{arXiv:hep-th/9912144}, )


\item E. Gorbatov, [[Vadim Kaplunovsky]], J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, \emph{On Heterotic Orbifolds, M Theory and Type I' Brane Engineering}, JHEP 0205:015, 2002 (\href{https://arxiv.org/abs/hep-th/0108135}{arXiv:hep-th/0108135})


\item [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]}, CMP (2019) (\href{https://arxiv.org/abs/1805.05987}{arXiv:1805.05987}, \href{http://link.springer.com/article/10.1007/s00220-019-03442-3}{doi:10.1007/s00220-019-03442-3})



\end{itemize}
\hypertarget{heterotic_string_theory_on_adeorbifolds}{}\paragraph*{{Heterotic string theory on ADE-orbifolds}}\label{heterotic_string_theory_on_adeorbifolds}

[[heterotic string theory]] on ADE-orbifolds:

\begin{itemize}%
\item [[Paul Aspinwall]], [[David Morrison]], \emph{Point-like Instantons on K3 Orbifolds}, Nucl. Phys. B503 (1997) 533-564 (\href{https://arxiv.org/abs/hep-th/9705104}{arXiv:hep-th/9705104})


\item [[Edward Witten]], \emph{Heterotic String Conformal Field Theory And A-D-E Singularities}, JHEP 0002:025, 2000 (\href{https://arxiv.org/abs/hep-th/9909229}{arXiv:hep-th/9909229})



\end{itemize}
\hypertarget{type_string_theory_on_adeorbifolds}{}\paragraph*{{Type $I'$-string theory on ADE-orbifolds}}\label{type_string_theory_on_adeorbifolds}

[[type I' string theory]] on ADE-orbifolds

\begin{itemize}%
\item [[Oren Bergman]], Diego Rodriguez-Gomez, Section 3 of: \emph{5d quivers and their $AdS_6$ duals}, JHEP07 (2012) 171 (\href{https://arxiv.org/abs/1206.3503}{arxiv:1206.3503})

\end{itemize}
\hypertarget{type_string_theory_on_adeorbifolds_2}{}\paragraph*{{Type $I$-string theory on ADE-orbifolds}}\label{type_string_theory_on_adeorbifolds_2}

[[type I string theory]] on ADE-orbifolds

\begin{itemize}%
\item [[Kenneth Intriligator]], \emph{New String Theories in Six Dimensions via Branes at Orbifold Singularities}, Adv. Theor. Math. Phys.1:271-282, 1998 (\href{https://arxiv.org/abs/hep-th/9708117}{arXiv:hep-th/9708117})

\end{itemize}
\hypertarget{mtheory_on_orbifolds_with_adesingularities}{}\paragraph*{{M-theory on $G_2$-orbifolds with ADE-singularities}}\label{mtheory_on_orbifolds_with_adesingularities}

[[M-theory on G2-manifolds]] $\,$ \href{M-theory+on%20G2-manifolds#EnhancedGaugeGroups}{with ADE-singularities}:

\begin{itemize}%
\item [[Bobby Acharya]], \emph{M theory, Joyce Orbifolds and Super Yang-Mills}, Adv. Theor. Math. Phys. 3 (1999) 227-248 (\href{http://arxiv.org/abs/hep-th/9812205}{arXiv:hep-th/9812205})


\item [[Bobby Acharya]], \emph{On Realising $N=1$ Super Yang-Mills in M theory} (\href{http://arxiv.org/abs/hep-th/0011089}{arXiv:hep-th/0011089})


\item [[Bobby Acharya]], B. Spence, \emph{Flux, Supersymmetry and M theory on 7-manifolds} (\href{http://arxiv.org/abs/hep-th/0007213}{arXiv:hep-th/0007213})


\item [[Bobby Acharya]], \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 [[Michael Atiyah]], [[Juan Maldacena]], [[Cumrun Vafa]], \emph{An M-theory Flop as a Large N Duality}, J. Math. Phys.42:3209-3220, 2001 (\href{https://arxiv.org/abs/hep-th/0011256}{arXiv:hep-th/0011256})


\item [[Chris Beasley]], [[Edward Witten]], \emph{A Note on Fluxes and Superpotentials in M-theory Compactifications on Manifolds of $G_2$ Holonomy}, JHEP 0207:046,2002 (\href{http://arxiv.org/abs/hep-th/0203061}{arXiv:hep-th/0203061})


\item [[Michael Atiyah]], [[Edward Witten]] \emph{$M$-Theory dynamics on a manifold of $G_2$-holonomy}, Adv. Theor. Math. Phys. 6 (2001) (\href{http://arxiv.org/abs/hep-th/0107177}{arXiv:hep-th/0107177})


\item [[Edward Witten]], \emph{Anomaly Cancellation On Manifolds Of $G_2$ Holonomy} (\href{http://arxiv.org/abs/hep-th/0108165}{arXiv:hep-th/0108165})


\item [[Bobby Acharya]], [[Edward Witten]], \emph{Chiral Fermions from Manifolds of $G_2$ Holonomy} (\href{http://arxiv.org/abs/hep-th/0109152}{arXiv:hep-th/0109152})



\end{itemize}
\hypertarget{ftheory_with_adesingularities}{}\paragraph*{{F-theory with ADE-singularities}}\label{ftheory_with_adesingularities}

[[F-theory]] with ADE-singularities

\begin{itemize}%
\item [[Paul Aspinwall]], [[David Morrison]], \emph{Point-like Instantons on K3 Orbifolds}, Nucl. Phys. B503 (1997) 533-564 (\href{https://arxiv.org/abs/hep-th/9705104}{arXiv:hep-th/9705104})

\end{itemize}
See also at \emph{[[F-branes -- table]]}

[[!redirects ADE singularities]]

[[!redirects ADE-singularity]] [[!redirects ADE-singularities]]

[[!redirects ADE orbifold]] [[!redirects ADE orbifolds]]

[[!redirects ADE-orbifold]] [[!redirects ADE-orbifolds]]

[[!redirects du Val singularity]] [[!redirects du Val singularities]]



\end{document}