nLab ADE singularity

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.

$\,$

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
$A_{n \geq 1}$		cyclic group $\mathbb{Z}_{n+1}$	cyclic group $\mathbb{Z}_{n+1}$	special unitary group $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic group of order 3 $\mathbb{Z}_3$	cyclic group of order 3 $\mathbb{Z}_3$	SU(3)
A3 = D3		cyclic group of order 4 $\mathbb{Z}_4$	cyclic group of order 4 $2 D_2 \simeq \mathbb{Z}_4$	SU(4) $\simeq$ Spin(6)
D4	dihedron on bigon	Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$	quaternion group $2 D_4 \simeq$ Q8	SO(8), Spin(8)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron on square	dihedral group of order 8 $D_8$	binary dihedral group of order 16 $2 D_{8}$	SO(12), Spin(12)
$D_{n \geq 4}$	dihedron, hosohedron	dihedral group $D_{2(n-2)}$	binary dihedral group $2 D_{2(n-2)}$	special orthogonal group, spin group $SO(2n)$, $Spin(2n)$
$E_6$	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

From coincident KK-monopoles

geometry transverse to KK-monopoles	Riemannian metric	remarks
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.

Related concepts

• singularity

• conical singularity

• quiver gauge theory

• M-theory on G₂-manifolds

• D7-brane

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:

• 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)

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:

• 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)

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

• MathOverflow, Resolving ADE singularities by blowing up

On the Chen-Ruan orbifold cohomology of ADE-singularities:

• Fabio Perroni, Orbifold Cohomology of ADE-singularities (arXiv:0510528)

• Fabio Perroni, Chen-Ruan cohomology of ADE-singularities, International Journal of Mathematics, Vol. 18, No. 9 (2007) 1009-1059 (arXiv:math/0605207, doi:10.1142/S0129167X07004436)

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

• Frank Reidegeld, $G_2$-orbifolds from K3 surfaces with ADE-singularities (arXiv:1512.05114)

• Frank Reidegeld, $G_2$-orbifolds with ADE-singularities (pdf)

Riemannian geometry of manifolds with ADE singularities is discussed in

• Boris Botvinnik, Jonathan Rosenberg, Positive scalar curvature on manifolds with fibered singularities (arXiv:1808.06007)

See also

• Wikipedia, du Val singularity

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 $A_n$-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)

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:

• 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

Heterotic M-theory on ADE-orbifolds

heterotic M-theory on ADE-orbifolds:

• Ashoke Sen, A Note on Enhanced Gauge Symmetries in M- and String Theory, JHEP 9709:001,1997 (arXiv:hep-th/9707123)

• Michael Faux, Dieter Lüst, Burt Ovrut, Intersecting Orbifold Planes and Local Anomaly Cancellation in M-Theory, Nucl. Phys. B554: 437-483, 1999 (arXiv:hep-th/9903028)

• Michael Faux, Dieter Lüst, Burt Ovrut, Local Anomaly Cancellation, M-Theory Orbifolds and Phase-Transitions, Nucl. Phys. B589: 269-291, 2000 (arXiv:hep-th/0005251)

• Michael Faux, Dieter Lüst, Burt Ovrut, An M-Theory Perspective on Heterotic K3 Orbifold Compactifications, Int. J. Mod. Phys. A18:3273-3314, 2003 (arXiv:hep-th/0010087)

• Michael Faux, Dieter Lüst, Burt Ovrut, Twisted Sectors and Chern-Simons Terms in M-Theory Orbifolds, Int. J. Mod. Phys. A18: 2995-3014, 2003 (arXiv:hep-th/0011031)

• Vadim Kaplunovsky, J. Sonnenschein, Stefan Theisen, S. Yankielowicz, On the Duality between Perturbative Heterotic Orbifolds and M-Theory on $T^4/Z_N$, Nuclear Physics B Volume 590, Issues 1–2, 4 December 2000, Pages 123-160 Nuclear Physics B (arXiv:hep-th/9912144, doi:10.1016/S0550-3213(00)00460-0)

• E. Gorbatov, Vadim Kaplunovsky, J. Sonnenschein, Stefan Theisen, S. Yankielowicz, On Heterotic Orbifolds, M Theory and Type I’ Brane Engineering, JHEP 0205:015, 2002 (arXiv:hep-th/0108135)

• John Huerta, Hisham Sati, Urs Schreiber, Real ADE-equivariant (co)homotopy and Super M-branes, CMP (2019) (arXiv:1805.05987, doi:10.1007/s00220-019-03442-3)

Heterotic string theory on ADE-orbifolds

heterotic string theory on ADE-orbifolds:

• Paul Aspinwall, David Morrison, Point-like Instantons on K3 Orbifolds, Nucl. Phys. B503 (1997) 533-564 (arXiv:hep-th/9705104)

• Edward Witten, Heterotic String Conformal Field Theory And A-D-E Singularities, JHEP 0002:025, 2000 (arXiv:hep-th/9909229)

Type II-string theory on ADE-orbifolds

The observation that the worldsheet 2d CFT correspoding to a string probing (a D-brane on) an $A_{\kappa-1}$-type singularity $\mathbb{H}/_{C_{\kappa + 2}}$ is the chiral WZW model for the affine Lie algebra su(2) at level $\kappa - 2$ (plus some trivial summands):

• Hirosi OoguriCumrun Vafa, p. 10-12 of: Two-Dimensional Black Hole and Singularities of CY Manifolds, Nucl. Phys. B 463 (1996) 55-72 (arXiv:hep-th/9511164, doi:10.1016/0550-3213%2896%2900008-9)

• Wolfgang Lerche, Carsten Andrew Lütken, Christoph Schweigert, p. 4 of: D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories, Nucl.Phys. B 622 (2002) 269-278 (doi:10.1016/S0550-3213%2801%2900613-7, arXiv:hep-th/0006247)

On how this $\widehat{\mathfrak{su}(2)}^{\kappa-2}$-CFT encodes the BPS states of $SU(\kappa)$-SYM on D3-branes transverse to the singularity:

• Wolfgang Lerche, On a Boundary CFT Description of Nonperturbative $\mathcal{N} = 2$ Yang-Mills Theory (arXiv:hep-th/0006100)

An interpretation of this phenomenon, under the expected K-theory classification of D-brane charge, as due to the (somewhat neglected) sector of twisted equivariant K-theory where the twist is by an inner local system which may appear inside an A-type singularity:

• Hisham Sati, Urs Schreiber, (around p. 23 of) Anyonic defect branes in TED-K-theory (arXiv:2203.11838)

Type $I'$-string theory on ADE-orbifolds

type I' string theory on ADE-orbifolds

• Oren Bergman, Diego Rodriguez-Gomez, Section 3 of: 5d quivers and their $AdS_6$ duals, JHEP07 (2012) 171 (arxiv:1206.3503)

Type $I$-string theory on ADE-orbifolds

type I string theory on ADE-orbifolds

• Kenneth Intriligator, New String Theories in Six Dimensions via Branes at Orbifold Singularities, Adv. Theor. Math. Phys.1:271-282, 1998 (arXiv:hep-th/9708117)

M-theory on $G_2$-orbifolds with ADE-singularities

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

• Bobby Acharya, M theory, Joyce Orbifolds and Super Yang-Mills, Adv. Theor. Math. Phys. 3 (1999) 227-248 (arXiv:hep-th/9812205)

• Bobby Acharya, On Realising $N=1$ Super Yang-Mills in M theory (arXiv:hep-th/0011089)

• Bobby Acharya, B. Spence, Flux, Supersymmetry and M theory on 7-manifolds (arXiv:hep-th/0007213)

• Bobby Acharya, M Theory, $G_2$-manifolds and Four Dimensional Physics, Classical and Quantum Gravity Volume 19 Number 22, 2002 (pdf)

• Michael Atiyah, Juan Maldacena, Cumrun Vafa, An M-theory Flop as a Large N Duality, J. Math. Phys.42:3209-3220, 2001 (arXiv:hep-th/0011256)

• Chris Beasley, Edward Witten, A Note on Fluxes and Superpotentials in M-theory Compactifications on Manifolds of $G_2$ Holonomy, JHEP 0207:046 (2002) (arXiv:hep-th/0203061)

• Michael Atiyah, Edward Witten M-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)

• Edward Witten, Anomaly Cancellation On Manifolds Of $G_2$ Holonomy (arXiv:hep-th/0108165)

• Bobby Acharya, Edward Witten, Chiral Fermions from Manifolds of $G_2$ Holonomy (arXiv:hep-th/0109152)

• Ron Donagi, Martin Wijnholt: The M-Theory Three-Form and Singular Geometries [arXiv:2310.05838]

F-theory with ADE-singularities

F-theory with ADE-singularities

• Paul Aspinwall, David Morrison, Point-like Instantons on K3 Orbifolds, Nucl. Phys. B503 (1997) 533-564 (arXiv:hep-th/9705104)

See also at F-branes – table