Instantons and the geometry of the nilpotent variety.

*(English)*Zbl 0725.58007This paper provides a new viewpoint on the remarkable relationship between rational double points, Dynkin diagrams and subgroups of SU(2) investigated in particular by E. Brieskorn [Actes Congr. Int. Math. 1970, 2, 279-284 (1971; Zbl 0223.22012)]. The setting is that of the gradient flow of the function
\[
\phi (A_ 1,A_ 2,A_ 3)=\sum^{3}_{1}(A_ i,A_ i)+(A_ 1,[A_ 2,A_ 3])
\]
defined on \({\mathfrak g}\otimes R^ 3\). This can also be thought of as the Chern- Simons functional on a space of left-invariant G-connections on the 3- sphere, which reinterprets the space of trajectories as a space of instantons on \(S^ 4\). In this interpretation the space also has a natural hyperkähler structure, a fact which underlies many of the features in the paper, but which the author does not emphasize too strongly. The main analytical result is to parametrize the solutions of the ODE which describes the gradient flow of \(\phi\) by using the complex nilpotent orbits of the complex group \(G^ c\). This is carried out in a manner modelled on Donaldson’s parametrization of monopoles [S. K. Donaldson, Commun. Math. Phys. 96, 387-407 (1984; Zbl 0603.58042)].

There are two particularly attractive outcomes of this result. One is a new proof of Brieskorn’s theorem that the nilpotent variety has a singularity along the subregular orbit of the type \(C^ 2/\Gamma\) where \(\Gamma\) is a finite subgroup of SU(2). The other is a demonstration of two copies of SO(3) in the group \(E_ 8\) which intersect in the icosahedral group.

There are two particularly attractive outcomes of this result. One is a new proof of Brieskorn’s theorem that the nilpotent variety has a singularity along the subregular orbit of the type \(C^ 2/\Gamma\) where \(\Gamma\) is a finite subgroup of SU(2). The other is a demonstration of two copies of SO(3) in the group \(E_ 8\) which intersect in the icosahedral group.

Reviewer: N.J.Hitchin (Coventry)