Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

A long list of mathematical structures happens to have a classification that is in bijection with the simply laced Dynkin diagrams of types A, D and E (but excluding type B and C), for instance $N$ Killing spinors on
spherical space form $S^7/\widehat{G}$
$\phantom{AA}\widehat{G} =$spin-lift of subgroup of
isometry group of 7-sphere
3d superconformal gauge field theory
on back M2-branes
with near horizon geometry $AdS_4 \times S^7/\widehat{G}$
$\phantom{AA}N = 8\phantom{AA}$$\phantom{AA}\mathbb{Z}_2$cyclic group of order 2BLG model
$\phantom{AA}N = 7\phantom{AA}$
$\phantom{AA}N = 6\phantom{AA}$$\phantom{AA}\mathbb{Z}_{k\gt 2}$cyclic groupABJM model
$\phantom{AA}N = 5\phantom{AA}$$\phantom{AA}2 D_{k+2}$
$2 T$, $2 O$, $2 I$
binary dihedral group,
binary tetrahedral group,
binary octahedral group,
binary icosahedral group
$\phantom{AA}N = 4\phantom{AA}$$\phantom{A}2 D_{k+2}$
$2 O$, $2 I$
binary dihedral group,
binary octahedral group,
binary icosahedral group
(HLLLP 08b, Chen-Wu 10)

and many more.

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
$SU(n+1)$
A1cyclic group of order 2
$\mathbb{Z}_2$
cyclic group of order 2
$\mathbb{Z}_2$
SU(2)
A2cyclic 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)
D4dihedron 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)
D5dihedron on
triangle
dihedral group of order 6
$D_6$
binary dihedral group of order 12
$2 D_6$
SO(10), Spin(10)
D6dihedron 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
$SO(2n)$, $Spin(2n)$
$E_6$tetrahedrontetrahedral 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

The obvious question for what might be the conceptual origin of this joint classification is attributed to (Arnold 76).

Starting with (Douglas-Moore 96) is the observation that many of these structures are naturally aspects of the description of string theory KK-compactified on orbifolds with ADE singularities of the form $\mathbb{C}^n \sslash \Gamma$ for $\Gamma$ a finite subgroup of $SL_2(\mathbb{C})$.

## Via $N=2$ super Yang-Mills theory

Various seemingly unrelated structures in mathematics fall into an “ADE classification”. Notably finite subgroups of SU(2) and compact simple Lie groups do. The way this works usually is that one tries to classify these structures somehow, and ends up finding that the classification is governed by the combinatorics of Dynkin diagrams (see also McKay correspondence).

While that does explain a bit, it seems the statement that both the icosahedral group and the Lie group E8 are related to the same Dynkin diagram somehow is still more a question than an answer. Why is that so?

The first key insight is due to Kronheimer 89. He showed that the (resolutions of) the orbifold quotients $\mathbb{C}^2/\Gamma$ for finite subgroups $\Gamma$ of $SU(2)$ are precisely the generic form of the gauge orbits of the direct product group of $U(n_i)$s acting in the evident way on the direct sum of $Hom(\mathbb{C}^{n_i}, \mathbb{C}^{n_j})$-s, where $i$ and $j$ range over the vertices of the Dynkin diagram, and $(i,j)$ over its edges.

This becomes more illuminating when interpreted in terms of gauge theory: in a quiver gauge theory the gauge group is a direct product group of $U(n_i)$ factors associated with vertices of a quiver, and the particles which are charged under this gauge group arrange, as a linear representation, into a direct sum of $Hom(\mathbb{C}^{n_i}, \mathbb{C}^{n_j})$-s, for each edge of the quiver.

Pick one such particle, and follow it around as the gauge group transforms it. The space swept out is its gauge orbit, and Kronheimer 89 says that if the quiver is a Dynkin diagram, then this gauge orbit looks like $\mathbb{C}^2/\Gamma$.

On the other extreme, gauge theories are of interest whose gauge group is not a big direct product, but is a simple Lie group, such as SU(N) or E8. The mechanism that relates the two classes of examples is spontaneous symmetry breaking (“Higgsing”): the ground state energy of the field theory may happen to be achieved by putting the fields at any one point in a higher dimensional space of field configurations, acted on by the gauge group, and fixing any one such point “spontaneously” singles out the corresponding stabilizer subgroup.

Now here is the final ingredient: it is N=2 D=4 super Yang-Mills theory (“Seiberg-Witten theory”) which have a potential that is such that its vacua break a simple gauge group such as $SU(N)$ down to a Dynkin diagram quiver gauge theory. One place where this is reviewed, physics style, is in Albertsson 03, section 2.3.4.

More precisely, these theories have two different kinds of vacua, those on the “Coulomb branch” and those on the “Higgs branch” depending on whether the scalars of the “vector multiplets” (the gauge field sector) or of the “hypermultiplet” (the matter field sector) vanish. The statement above is for the Higgs branch, but the Coulomb branch is supposed to behave “dually”.

So that then finally is the relation, in the ADE classification, between the simple Lie groups and the finite subgroups of SU(2): start with an N=2 super Yang Mills theory with gauge group a simple Lie group. Let it spontaneously find its vacuum and consider the orbit space of the remaining spontaneously broken symmetry group. That is (a resolution of) the orbifold quotient of $\mathbb{C}^2$ by a discrete subgroup of $SU(2)$.

### General Surveys

• Vladimir Arnold, Problems in present day mathematics, (1976) in Felix E. Browder, Mathematical developments arising from Hilbert problems, Proceedings of symposia in pure mathematics 28, American Mathematical Society, p. 46, Problem VIII. The A-D-E classifications (V. Arnold).

A survey is in

• Michael Hazewinkel, W Hesseling, Dirk Siersma, and Ferdinand Veldkamp, The ubiquity of Coxeter Dynkin diagrams (an introduction to the ADE problem), Nieuw Archief voor Wiskunde 25 (1977), 257-307. (pdf)

which in turn is summarized in

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

Discussion of the free finite group actions on spheres goes back to

• John Milnor, Groups which act on $S^n$ without fixed points, American Journal of Mathematics Vol. 79, No. 3 (Jul., 1957), pp. 623-630 (JSTOR)

Review inclues

Discussion of ALE spaces via ADE include

• Peter Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. (Euclid)

Related stuff includes…

on immersions of 3-spheres into $\mathbb{R}^4$:

• Shumi Kinjo, Immersions of 3-sphere into 4-space associated with Dynkin diagrams of types A and D (arXiv:1309.6526)

### In string theory

The original articles explaining the appearance of ADE classification from within string theory include

Surveys include

Discussion of an ADE-classification of BPS Freund-Rubin compactifications is in

Specifically the ADE classfication involved in the 6d (2,0)-supersymmetric QFT on the M5-brane is discussed in

Discussion in the context of M-theory on G2-manifolds includes

• Bobby Acharya, section 3.1.1 of M Theory, $G_2$-manifolds and Four Dimensional Physics, Classical and Quantum Gravity Volume 19 Number 22, 2002 (pdf)

• Katrin Becker, Melanie Becker, John Schwarz, p. 423 of String Theory and M-Theory: A Modern Introduction, 2007

### In solid state physics

On ADE-classifications in/of topological phases of matter/topological order:

• Mayukh Nilay Khan, Jeffrey C. Y. Teo, Taylor L. Hughes, Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the ADE Classification, Phys. Rev. B 90 235149 (2014) $[$arXiv:1403.6478$]$