geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
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
finite subgroups of the special orthogonal group $SO(3)$ (Milnor 57, see e.g. Keenan 03, theorem 4)
simple singularities? (see ADE singularity)
singularities of elliptic fibrations (see there are classification of singular fibers)
certain 4d ALE spaces (Kronheimer 89)
certain 2d CFTs
certain 6d CFTs
intersection diagrams of vanishing 2-cycles in K3s (e.g. BBS 07, p.423)
finite subgroups of $SO(8)$ acting freely on the 7-sphere, such that the quotient has spin structure and more than 3 linearly independent Killing spinors (MFFGME 09)
and many more.
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 $\mathbb{C}^n //\Gamma$ for $\Gamma$ a finite subgroup of $SL_2(\mathbb{C})$.
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 goverened by the combinatorics of Dynkin diagrams.
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 8[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 simpe 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 $SU(2)$.
A survey is in
which in turn is summarized in
See also
Discusion of the free finite group actions on spheres goes back to
Review inclues
Discussion of ALE spaces via ADE include
Related stuff includes…
on immersions of 3-spheres into $\mathbb{R}^4$:
The original articles explaining the appearance of ADE classification from within string theory include
Michael Douglas, Gregory Moore, D-branes, Quivers, and ALE Instantons (arXiv:hep-th/9603167)
Clifford Johnson, Robert Myers, Aspects of Type IIB Theory on ALE Spaces, Phys.Rev. D55 (1997) 6382-6393 (arXiv:hep-th/9610140)
Michael Douglas, Brian Greene, David Morrison, Orbifold Resolution by D-Branes, Nucl.Phys. B506:84-106,1997 (arXiv:hep-th/9704151)
Brian Greene, Calin Lazaroiu, Mark Raugas, D-branes on Nonabelian Threefold Quotient Singularities, Nucl.Phys. B553 (1999) 711-749 (arXiv:hep-th/9811201)
Andrea Cappelli, Jean-Bernard Zuber, A-D-E Classification of Conformal Field Theories (arXiv:0911.3242)
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
Edward Witten, Some Comments On String Dynamics (arXiv:hep-th/9507121)
Jonathan Heckman, David Morrison, Cumrun Vafa, On the Classification of 6D SCFTs and Generalized ADE Orbifolds (arXiv:1312.5746)
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