Inherited Duality and Quiver Gauge Theory

TL;DR

The work identifies a rich duality structure for ${\\cal N}=2$ and ${\\cal N}=1$ $\\hat{A}_{n-1}$ quiver gauge theories using the M5-brane picture, showing that the duality group is a faithful quotient of the mapping class group $M(1,n)$ containing the affine Weyl group $\\widehat{A}_{n-1}$, ${\\mathbb Z}_n$, and $SL(2, {\\mathbb Z})$ with $n$ non-commuting copies. It demonstrates that ${\\cal N}=1$ deformations preserve the inherited duality action on gauge and superpotential couplings, and that Seiberg duality corresponds to affine Weyl reflections on couplings and to specific transformations on the quartic superpotential coefficients. The paper also analyzes non-conformal cascades, revealing a universal structure governed by the affine Weyl group, and discusses the interplay between holographic perspectives and field-theoretic dualities. Overall, it connects brane constructions, duality symmetries, and RG flows in a unified framework with implications for broader connections to WZW and Chern-Simons theories.

Abstract

We study the duality group of $\hat{A}_{n-1}$ quiver gauge theories, primarily using their M5-brane construction. For $\mathcal{N}=2$ supersymmetry, this duality group was first noted by Witten to be the mapping class group of a torus with $n$ punctures. We find that it is a certain quotient of this group that acts faithfully on gauge couplings. This quotient group contains the affine Weyl group of $\hat{A}_{n-1}$, $\mathbb{Z}_n$ and $SL(2,\mathbb{Z})$. In fact there are $n$ non-commuting $SL(2,\mathbb{Z})$ subgroups, related to each other by conjugation using the $\mathbb{Z}_n$. When supersymmetry is broken to $\mathcal{N}=1$ by masses for the adjoint chiral superfields, an RG flow ensues which is believed to terminate at a CFT in the infrared. We find the explicit action of this duality group for small values of the adjoint masses, paying special attention to when the sum of the masses is non-zero. In the $\mathcal{N}=1$ CFT, Seiberg duality acts non-trivially on both gauge couplings and superpotential couplings and we interpret this duality as inherited from the $\mathcal{N}=2$ parent theory. We conjecture the action of S-duality in the CFT based on our results for small mass deformations. We also consider non-conformal deformations of these $\mathcal{N}=1$ theories. The cascading RG flows that ensue are a one-parameter generalization of those found by Klebanov and Strassler and by Cachazo {\it et. al.}. The universality exhibited by these flows is shown to be a simple consequence of paths generated by the action of the affine Weyl group.

Figures (1)

• Figure 1: Weyl reflections acting on fundamental weights of $\widehat{A}_1$.