Liberating confinement from Lagrangians: 1-form symmetries and lines in 4d N=1 from 6d N=(2,0)

Abstract

We study confinement in 4d \mathcal{N}=1𝒩=1 theories obtained by deforming 4d \mathcal{N}=2𝒩=2 theories of Class S. We argue that confinement in a vacuum of the \mathcal{N}=1𝒩=1 theory is encoded in the 1-cycles of the associated \mathcal{N}=1𝒩=1 curve. This curve is the spectral cover associated to a generalized Hitchin system describing the profiles of two Higgs fields over the Riemann surface upon which the 6d (2,0)(2,0) theory is compactified. Using our method, we reproduce the expected properties of confinement in various classic examples, such as 4d \mathcal{N}=1𝒩=1 pure Super-Yang-Mills theory and the Cachazo-Seiberg-Witten setup. More generally, this work can be viewed as providing tools for probing confinement in non-Lagrangian \mathcal{N}=1𝒩=1 theories, which we illustrate by constructing an infinite class of non-Lagrangian \mathcal{N}=1𝒩=1 theories that contain confining vacua. The simplest model in this class is an \mathcal{N}=1𝒩=1 deformation of the \mathcal{N}=2𝒩=2 theory obtained by gauging SU(3)^3SU(3)3 flavor symmetry of the E_6E6 Minahan-Nemeschansky theory.