Chiral Rings, Singularity Theory and Electric-Magnetic Duality

TL;DR

This paper explores the duality between two supersymmetric theories using singularity theory to understand their chiral rings and quantum relations, revealing how classical differences are reconciled quantum mechanically.

Contribution

It introduces a detailed analysis of chiral ring perturbations in dual supersymmetric theories, highlighting the role of singularity theory in establishing their quantum equivalence.

Findings

Quantum relations unify dual theories' chiral rings.

Duality swaps classical and quantum relations.

Singularity theory underpins the duality mechanism.

Abstract

We study in detail the space of perturbations of a pair of dual $N=1$ supersymmetric theories based on an $SU(N_c)$ gauge theory with an adjoint $X$ and fundamentals with a superpotential which is polynomial in $X$. The equivalence between them depends on non-trivial facts about polynomial equations, i.e.\ singularity theory. The classical chiral rings of the two theories are different. Quantum mechanically there are new relations in the chiral rings which ensure their equivalence. Duality interchanges ``trivial'' classical relations in one theory with quantum relations in the other and vice versa. We also speculate about the behavior of the theory without the superpotential.