  Recently, motivated by supersymmetric gauge theory, Cachazo, Douglas,
Seiberg, and Witten proposed a conjecture about finite dimensional simple Lie
algebras, and checked it in the classical cases. Later V. Kac and the author
proposed a uniform approach to this conjecture, based on the theory of abelian
ideals in the Borel subalgebra; this allowed them to check the conjecture for
type G_2. In this note we further develop this approach, and propose three
natural conjectures which imply the three parts of the CDSW conjecture. In a
sense, these conjectures explain why the CDSW conjecture should be true. We
show that our conjectures hold for classical Lie algebras and for G_2; this, in
particular, gives a purely algebraic proof of the CDSW conjecture for SO(N)
(the original proof, due to Witten, uses the theory of instantons).
