Deformations of WA,D,E SCFTs

We discuss aspects of theories with superpotentials given by Arnold's A, D, E singularities, particularly the novelties that arise when the fields are matrices. We focus on 4d N = 1 variants of susy QCD, with U(N-c) or SU(N-c) gauge group, N-f fundamental flavors, and adjoint matter fields X and Y appearing in W-A,W- D,W- E(X, Y) superpotentials. Many of our considerations also apply in other possible contexts for matrix-variable W-A,W- D,W- E. The 4d W-A,W- D,W- E SQCD-type theories RG flow to superconformal field theories, and there are proposed duals in the literature for the W-Ak, W-Dk, and W-E7 cases. As we review, the W-Deven and W-E7 duals rely on a conjectural, quantum truncation of the chiral ring. We explore these issues by considering various deformations of the W-A, (D), (E) superpotentials, and the resulting RG flows and IR theories. Rather than finding supporting evidence for the quantum truncation and W-Deven and W-E7 duals, we note some challenging evidence to the contrary.