Liberating Confinement from Lagrangians: 1-form Symmetries and Lines in 4d N=1 from 6d N=(2,0)

TL;DR

This work develops a geometric framework to study confinement in 4d $\mathcal{N}=1$ theories obtained from deformations of Class S theories, by associating an $\mathcal{N}=1$ curve (a spectral cover of a generalized Hitchin system) to each vacuum. Confinement is read from 1-cycles on this curve through the 1-form symmetry data $\mathcal{O}=\widehat{\Lambda}$ and the vacuum-preserved subgroup $\mathcal{O}_r=\widehat{(\Lambda/\Lambda_r)}$, with the curve’s topology encoding line operators and their screening properties. The paper tests the framework on familiar Lagrangian cases such as pure $\mathcal{N}=1$ SYM and the Cachazo-Seiberg-Witten setup, and extends it to non-Lagrangian theories, including an infinite class built from $E_6$ MN and higher $A_{n-1}$ constructions, showing these too can exhibit confinement. The methods unify UV data from six-dimensional (2,0) theory, Hitchin systems, and M-theory realizations to diagnose confinement, offering a powerful tool to probe phases in both Lagrangian and non-Lagrangian $\mathcal{N}=1$ theories with potential extensions to anomalies and DV-type dualities.

Abstract

We study confinement in 4d N=1 theories obtained by deforming 4d N=2 theories of Class S. We argue that confinement in a vacuum of the N=1 theory is encoded in the 1-cycles of the associated N=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) theory is compactified. Using our method, we reproduce the expected properties of confinement in various classic examples, such as 4d N=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 N=1 theories, which we illustrate by constructing an infinite class of non-Lagrangian N=1 theories that contain confining vacua. The simplest model in this class is an N=1 deformation of the N=2 theory obtained by gauging $SU(3)^3$ flavor symmetry of the $E_6$ Minahan-Nemeschansky theory.

Figures (19)

• Figure 1: The Hanany-Witten setup realizing in Type IIA string theory the pure $\mathcal{N}=2$$\mathfrak{su}(2)$ SYM theories.
• Figure 2: Rotation of the branes in the Hanany-Witten setup, which results in breaking $\mathcal{N}=2$ to $\mathcal{N}=1$ supersymmetry. Again we show the construction for $\mathfrak{su}(2)$ pure SYM.
• Figure 3: The figure displays the sheet structures of $v$-curve and $w$-curve over the Gaiotto curve $\mathcal{C}$ with the coordinate $t$. A star denotes a singular point where the corresponding curve escapes to infinity, $\times$ denotes a non-singular branch-point, and the dashed lines denote branch cuts.
• Figure 4: A possible movement and collision of branch points that ensures that the branch structures of $v$ and $w$ curves coincide. However, as explained in the text, this branch structure is not possible.
• Figure 5: A possible movement and collision of branch points that ensures that the branch structures of $v$ and $w$ curves coincide. This configuration gives rise to a consistent $\mathcal{N}=1$ curve.