### Cliff Harvey

Shared publicly -**The relation between the icosahedral group and E8**in classical discussions remains a bit mysterious. N=2 super Yang-Mills theory clarifies it.

Various seemingly unrelated structures in mathematics fall into an "ADE classification" [1]. 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 in 1989 [3]. He showed that the (resolutions of) the orbifold quotients C^2/Gamma for finite subgroups Gamma of SU(2) are precisely the generic form of the gauge orbits of the direct product of U(ni)-s acting in the evident way on the direct sum of Hom(C^ni, C^nj)-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 product of U(ni) 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(C^ni, C^nj)-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 says that if the quiver is a Dynkin diagram, then this gauge orbit looks like 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, in the technical sense [4], 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 [5].

Now here is the final ingredient: it is N=2 super Yang-Mills gauge theories [6] ("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 section 2.3.4 in the thesis [7].

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 finite 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 C^2 by a discrete subgroup of SU(2).

[1] http://ncatlab.org/nlab/show/ADE+classification

[2] http://ncatlab.org/nlab/show/ADE+--+table

[3] http://ncatlab.org/nlab/show/ALE+space#Kronheimer89

[4] http://ncatlab.org/nlab/show/simple+Lie+group

[5] http://ncatlab.org/nlab/show/stabilizer+group

[6] http://ncatlab.org/nlab/show/N=D2+D=4+super+Yang-Mills+theory

[7] http://ncatlab.org/nlab/show/N=2+D=4+super+Yang-Mills+theory#Albertsson03

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