higher geometry / derived geometry
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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 $\mathbb{C}^2$).
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 via McKay correspondence | Platonic solid | finite subgroups of SO(3) | finite subgroups of SU(2) | simple Lie group |
---|---|---|---|---|
$A_n$ | cyclic group $\mathbb{Z}_{n+1}$ | cyclic group $\mathbb{Z}_{n+1}$ | special unitary group | |
$D_{n+4}$ | dihedron, hosohedron | dihedral group $D_{n+2}$ | binary dihedral group $2 D_{n+2}$ | special orthogonal group |
$E_6$ | tetrahedron | tetrahedral 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 |
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 $r$ in the denominator.)
Discussion in terms of hyper-Kähler geometry:
Peter Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. (euclid:1214443066)
Peter Kronheimer, A Torelli-type theorem for gravitational instantons, J. Differential Geom. Volume 29, Number 3 (1989), 685-697 (euclid:1214443067)
Reviews and lecture notes include
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
See also
Discussion in string theory:
Ashoke Sen, A Note on Enhanced Gauge Symmetries in M- and String Theory, JHEP 9709:001,1997 (arXiv:hep-th/9707123)
Luis Ibáñez, Angel Uranga, section 6.3.3 of String Theory and Particle Physics – An Introduction to String Phenomenology, Cambridge University Press 2012
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.