Graph Rings and Integrable Perturbations of $N=2$ Superconformal Theories

TL;DR

This paper explores the relationship between integrable perturbations of $N=2$ superconformal theories and graph structures, extending previous results and providing a new way to test integrability through ground state patterns.

Contribution

It generalizes the connection between integrable perturbations and graphs to a broader class, introducing a dual ring framework labeled by ground states and graphs.

Findings

Perturbations allow diagonalization of chiral ring generators.

A dual ring is defined, labeled by ground states and encoded in graphs.

Known $ADE$ perturbations satisfy the proposed criterion.

Abstract

We show that the connection between certain integrable perturbations of $N=2$ superconformal theories and graphs found by Lerche and Warner extends to a broader class. These perturbations are such that the generators of the perturbed chiral ring may be diagonalized in an orthonormal basis. This allows to define a dual ring, whose generators are labelled by the ground states of the theory and are encoded in a graph or set of graphs, that reproduce the pattern of the ground states and interpolating solitons. All known perturbations of the $ADE$ potentials and some others are shown to satisfy this criterion. This suggests a test of integrability.