nLab
ADE singularity

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

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 May 17, 2018 at 04:06:41. See the history of this page for a list of all contributions to it.