Contents

# Contents

## Idea

An 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 vector space $\mathbb{C}^2$).

## Properties

### Resolution by spheres touching along a Dynkin diagram

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 (duVal 1934 I, p. 1-3 (453-455)). A description in terms of hyper-Kähler geometry is due to Kronheimer 89a.

Quick survey of this fact is in Reid 87, a textbook account is Slodowy 80.

In string theory this fact is interpreted in terms of gauge enhancement of the M-theory-lift of coincident black D6-branes to an MK6 at an ADE-singularity (Sen 97):

graphics grabbed from HSS18

See at M-theory lift of gauge enhancement on D6-branes for more.

$\,$

Dynkin diagram/
Dynkin quiver
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
$A_{n \geq 1}$cyclic group
$\mathbb{Z}_{n+1}$
cyclic group
$\mathbb{Z}_{n+1}$
special unitary group
$SU(n+1)$
D4Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group
$SO(2n)$
$E_6$tetrahedrontetrahedral group
$T$
binary tetrahedral group
$2T$
E6
$E_7$cube,
octahedron
octahedral group
$O$
binary octahedral group
$2O$
E7
$E_8$dodecahedron,
icosahedron
icosahedral group
$I$
binary icosahedral group
$2I$
E8

### From coincident KK-monopoles

geometry transverse to KK-monopolesRiemannian metricremarks
Taub-NUT space:
geometry transverse to
$N+1$ distinct KK-monopoles
at $\vec r_i \in \mathbb{R}^3 \;\; i \in \{1, \cdots, N+1\}$
$\array{d s^2_{TaubNUT} \coloneqq U^{-1}(d x^4 + \vec \omega \cdot d \vec r)^2 + U (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) \\ U \coloneqq 1 + \underoverset{i = 1}{N+1}{\sum} U_i\,, \phantom{AA} \vec \omega \coloneqq \underoverset{i = 1}{N+1}{\sum} \vec \omega_i \\ U_i \coloneqq \frac{R/2}{ {\vert \vec r - \vec r_i\vert} }\,, \phantom{AA} \vec \nabla \times \vec \omega= \vec \nabla U_i}$(e.g. Sen 97b, Sect. 2)
ALE space
Taub-NUT close to $N$ close-by KK-monopoles
e.g. close to $\vec r = 0$: $\frac{{\vert \vec r_i\vert}}{R/2}, \frac{{\vert \vec r\vert}}{R/2} \ll 1$
$\array{d s^2_{ALE} \coloneqq U'^{-1}(d x^4 + \vec \omega \cdot d \vec r)^2 + U' (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) \\ U' \coloneqq \underoverset{i = 1}{N+1}{\sum} U'_i\,, \phantom{AA} \vec \omega \coloneqq \underoverset{i = 1}{N+1}{\sum} \vec \omega_i \\ U'_i \coloneqq \frac{R/2}{ {\vert \vec r - \vec r_i\vert} }\,, \phantom{AA} \vec \nabla \times \vec \omega= \vec \nabla U_i}$e.g. via Euler angles: $\vec \omega = (N+1)R/2(\cos(\theta)-1) d\psi$
(e.g. Asano 00, Sect. 2)
$A_N$-type ADE singularity:
ALE space in the limit
where all $N+1$ KK-monopoles coincide at $vec r_i = 0$
$\array{d s^2_{A_N Sing} \coloneqq \frac{\vert\vec r\vert }{(N+1)R/2}(d x^4 + \vec \omega \cdot d \vec r)^2 + \frac{ (N+1)R/2}{\vert \vec r\vert} (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) }$(e.g. Asano 00, Sect. 3)

### Bridgeland stability conditions

For $G_{ADE} \subset SU(2)$ a finite subgroup of SU(2), let $\tilde X$ be the resolution of the corresponding ADE-singularity as 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 (Bridgeland 05).

Specifically for type-A singularities the space of stability conditions is in fact connected and simply-connected topological space (Ishii-Ueda-Uehara 10).

Brief review is in Bridgeland 09, section 6.3.

## References

### General

Original articles include

• Patrick du Val, 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) (doi:10.1017/S030500410001269X)

• Patrick du Val, 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) (doi:10.1017/S0305004100012706)

• Patrick du Val, 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) (doi:10.1017/S030500410001272X)

Textbook accounts include

• Alan H. Durfee, Fifteen characterizations of rational double points and simple critical points, L’Enseignement Mathématique Volume: 25 (1979) (doi:10.5169/seals-50375, pdf)

• Peter Slodowy, Simple singularities and simple algebraic groups, in Lecture Notes in Mathematics 815, Springer, Berlin, 1980.

• Klaus Lamotke, chapter IV of Regular Solids and Isolated Singularities, Vieweg, Braunschweig, Wiesbaden 1986.

• Miles Reid, Young persons guide to canonical singularities, in Spencer Bloch (ed.),Algebraic geometry – Bowdoin 1985, Part 1, Proc. Sympos. Pure Math. 46 Part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 345-414 (pdf)

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

Discussion of resolution of ADE-singularities in terms of hyper-Kähler geometry:

and in terms of preprojective algebras:

• William Crawley-Boevey, Martin P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. Volume 92, Number 3 (1998), 605-635 (euclid:1077231679)

Reviews and lecture notes include

• Classification of singularities

• Igor Burban, Du Val Singularities (pdf)

• Miles Reid, The Du Val Singularities $A_n$, $D_n$, $E_6$, $E_7$, $E_8$ (pdf)

• Anda Degeratu, Crepant Resolutions of Calabi-Yau Orbifolds, 2004 (pdf)

• Fabio Perroni, Orbifold Cohomology of ADE-singularities (pdf)

• Kyler Siegel, section 6 of The Ubiquity of the ADE classification in Nature , 2014 (pdf)

• MathOverflow, Resolving ADE singularities by blowing up

Families of examples of G2 orbifolds with ADE singularities are constructed in

Riemannian geometry of manifolds with ADE singularities is discussed in

### In the context of string theory and stability conditions

Discussion in string theory:

For more seet at M-theory on G2-manifolds the section Orbifold singularities

Discussion of Bridgeland stability conditions for (resolutions of) ADE singularities includes:

• Tom Bridgeland, Stability conditions and Kleinian singularities, International Mathematics Research Notices 2009.21 (2009): 4142-4157 (arXiv:0508257)

• Akira Ishii, Kazushi Ueda, Hokuto Uehara, Stability conditions on $A_n$-singularities, Journal of Differential Geometry 84 (2010) 87-126 (arXiv:math/0609551)

and specifically over Dynkin quivers

• Yu Qiu, Def. 2.1 Stability conditions and quantum dilogarithm identities for Dynkin quivers, Adv. Math., 269 (2015), pp 220-264 (arXiv:1111.1010)

• Tom Bridgeland, Yu Qiu, Tom Sutherland, Stability conditions and the $A_2$ quiver (arXiv:1406.2566)

Last revised on March 13, 2019 at 05:57:00. See the history of this page for a list of all contributions to it.