Deformations of W A,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,D,E (X, Y) superpotentials. Many of our considerations also apply in other possible contexts for matrix-variable W A,D,E . The 4d W A,D,E SQCD-type theories RG flow to superconformal field theories, and there are proposed duals in the literature for the WAₖ , WDₖ , and WE₇ cases. As we review, the W_Dₑᵥₑₙ and WE₇ 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 WE₇ duals, we note some challenging evidence to the contrary.