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

TL;DR

This paper investigates deformations of superpotentials based on Arnold's $A,D,E$ singularities in 4d ${ m extbf{N}}=1$ SQCD-like theories, examining their RG flows, IR fixed points, and dualities, and challenging some existing conjectures.

Contribution

It analyzes the effects of deformations on $W_{A,D,E}$ superpotentials, providing new insights into their RG flows and questioning the validity of certain dualities and quantum truncation assumptions.

Findings

Challenging evidence against quantum truncation of the chiral ring.

Insights into RG flows and IR fixed points of deformed $W_{A,D,E}$ theories.

Questions raised about the validity of dualities for $W_{D_ ext{even}}$ and $W_{E_7}$ cases.

Abstract

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 ${\cal N}=1$ 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.