On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

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. We prove the conjecture for type G_2, and also verify a consequence of the conjecture in the general case.