nLab
ADE singularity

Contents

Contents

Idea

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

\,

ADE classification and McKay correspondence

Dynkin diagram/
Dynkin quiver
dihedron,
Platonic solid
finite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A n1A_{n \geq 1}cyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
SU(n+1)SU(n+1)
A1cyclic group of order 2
2\mathbb{Z}_2
cyclic group of order 2
2\mathbb{Z}_2
SU(2)
A2cyclic group of order 3
3\mathbb{Z}_3
cyclic group of order 3
3\mathbb{Z}_3
SU(3)
A3
=
D3
cyclic group of order 4
4\mathbb{Z}_4
cyclic group of order 4
2D 2 42 D_2 \simeq \mathbb{Z}_4
SU(4)
\simeq
Spin(6)
D4dihedron on
bigon
Klein four-group
D 4 2× 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2
quaternion group
2D 42 D_4 \simeq Q8
SO(8), Spin(8)
D5dihedron on
triangle
dihedral group of order 6
D 6D_6
binary dihedral group of order 12
2D 62 D_6
SO(10), Spin(10)
D6dihedron on
square
dihedral group of order 8
D 8D_8
binary dihedral group of order 16
2D 82 D_{8}
SO(12), Spin(12)
D n4D_{n \geq 4}dihedron,
hosohedron
dihedral group
D 2(n2)D_{2(n-2)}
binary dihedral group
2D 2(n2)2 D_{2(n-2)}
special orthogonal group, spin group
SO(2n)SO(2n), Spin(2n)Spin(2n)
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

From coincident KK-monopoles

geometry transverse to KK-monopolesRiemannian metricremarks
Taub-NUT space:
geometry transverse to
N+1N+1 distinct KK-monopoles
at r i 3i{1,,N+1}\vec r_i \in \mathbb{R}^3 \;\; i \in \{1, \cdots, N+1\}
ds TaubNUT 2U 1(dx 4+ωdr) 2+U(dr) 2, r 3,x 4/(2πR) U1+i=1N+1U i,AAωi=1N+1ω i U iR/2|rr i|,AA×ω=U i\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 NN close-by KK-monopoles
e.g. close to r=0\vec r = 0: |r i|R/2,|r|R/21\frac{{\vert \vec r_i\vert}}{R/2}, \frac{{\vert \vec r\vert}}{R/2} \ll 1
ds ALE 2U 1(dx 4+ωdr) 2+U(dr) 2, r 3,x 4/(2πR) Ui=1N+1U i,AAωi=1N+1ω i U iR/2|rr i|,AA×ω=U i\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: ω=(N+1)R/2(cos(θ)1)dψ\vec \omega = (N+1)R/2(\cos(\theta)-1) d\psi
(e.g. Asano 00, Sect. 2)
A NA_N-type ADE singularity:
ALE space in the limit
where all N+1N+1 KK-monopoles coincide at vecr i=0vec r_i = 0
ds A NSing 2|r|(N+1)R/2(dx 4+ωdr) 2+(N+1)R/2|r|(dr) 2, r 3,x 4/(2πR)\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 ADESU(2)G_{ADE} \subset SU(2) a finite subgroup of SU(2), let X˜\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 X˜\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 rr 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

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

Riemannian geometry of manifolds with ADE singularities is discussed in

See also

Via Bridgeland stability

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 nA_n-singularities, Journal of Differential Geometry 84 (2010) 87-126 (arXiv:math/0609551)

and specifically over Dynkin quivers

In string theory

Discussion of ADE-singularities in string theory on orbifolds:

M-theory on ADE-orbifolds reducing to D6-branes in type II

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

Heterotic M-theory on ADE-orbifolds

heterotic M-theory on ADE-orbifolds:

Heterotic string theory on ADE-orbifolds

heterotic string theory on ADE-orbifolds:

Type II'-string theory on ADE-orbifolds

type I' string theory on ADE-orbifolds

Type II-string theory on ADE-orbifolds

type I string theory on ADE-orbifolds

M-theory on G 2G_2-orbifolds with ADE-singularities

M-theory on G2-manifolds\, with ADE-singularities:

F-theory with ADE-singularities

F-theory with ADE-singularities

See also at F-branes – table

Last revised on September 12, 2019 at 06:54:18. See the history of this page for a list of all contributions to it.