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):

ADE 2Cycle

graphics grabbed from HSS18

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

\,

ADE classification

Dynkin diagram via
McKay correspondence
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A nA_ncyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
D n+4D_{n+4}dihedron,
hosohedron
dihedral group
D n+2D_{n+2}
binary dihedral group
2D n+22 D_{n+2}
special orthogonal group
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

References

General

Original articles include

  • Patrick du Val, (1934a), “On isolated singularities of surfaces which do not affect the conditions of adjunction. I”, Proceedings of the Cambridge Philosophical Society, 30 (4): 453–459, doi:10.1017/S030500410001269X

  • Patrick du Val, (1934b), “On isolated singularities of surfaces which do not affect the conditions of adjunction. II”, Proceedings of the Cambridge Philosophical Society, 30 (4): 460–465, doi:10.1017/S0305004100012706

  • Patrick du Val, (1934c), “On isolated singularities of surfaces which do not affect the conditions of adjunction. III”, Proceedings of the Cambridge Philosophical Society, 30 (4): 483–491, 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 in terms of hyper-Kähler geometry:

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

In the context of string compactifications

Discussion in string theory:

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

See also at F-branes – table

Last revised on September 13, 2018 at 15:11:48. See the history of this page for a list of all contributions to it.