# Seiberg-Witten geometries for Coulomb branch chiral rings which are not freely generated

## TL;DR

This paper investigates rank-1 $ ext{SCFT}$s with Coulomb branch chiral rings that are not freely generated, showing they flow to similar theories under relevant deformations, and identifying properties that could help confirm or refute their existence.

## Contribution

It systematically studies non-freely generated Coulomb branch chiral rings in rank-1 $ ext{SCFT}$s and demonstrates their RG flow behavior, highlighting their potential as a distinct subset.

## Key findings

- Rank-1 $ ext{SCFT}$s with non-freely generated CB rings flow to similar theories under deformations.
- Identified properties of these theories that could aid in their classification or exclusion.
- No known counter-examples, but evidence suggests such theories, if they exist, form a unique class.

## Abstract

Coulomb branch chiral rings of $\mathcal N=2$ SCFTs are conjectured to be freely generated. While no counter-example is known, no direct evidence for the conjecture is known either. We initiate a systematic study of SCFTs with Coulomb branch chiral rings satisfying non-trivial relations, restricting our analysis to rank 1. The main result of our study is that (rank-1) SCFTs with non-freely generated CB chiral rings when deformed by relevant deformations, always flow to theories with non-freely generated CB rings. This implies that if they exist, they must thus form a distinct subset under RG flows. We also find many interesting characteristic properties that these putative theories satisfy which may behelpful in proving or disproving their existence using other methods.