Deformations of $W_{A,D,E}$ SCFTs

We discuss aspects of theories with superpotentials given by Arnold's $A,D,E$ singularities, particularly the various novelties that arise when the fields are matrices. E.g. we discuss aspects of the classical non-truncation of the chiral ring, flat directions, and the non-Abelian representations of the deformed chiral ring in the $D$ and $E$ cases. We focus on 4d ${\cal N}=1$ 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 $W_{A_k}$, $W_{D_k}$, and $W_{E_7}$ cases. As we review, the $W_{D_\text{even}}$ and $W_{E_7}$ 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 $W_{D_\text{even}}$ and $W_{E_7}$ duals, we note some challenging evidence to the contrary.