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

by Lakshya Bhardwaj, Max Hubner, Sakura Schafer-Nameki

Submission summary

As Contributors: Max Hubner
Arxiv Link: (pdf)
Date submitted: 2021-09-03 17:33
Submitted by: Hubner, Max
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


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.

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Submission & Refereeing History

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Submission 2106.10265v2 on 3 September 2021

Reports on this Submission

Anonymous Report 1 on 2021-10-3 (Invited Report)


The paper introduces a method to detect confinement for some 4d N=1 theories obtained from class S theories. Since the method is based on the study of the geometry of the compactification, it could be applied also to non-Lagrangian theories.


Very technical presentation throughout the paper.


The paper addresses the problem of detecting confinement using a geometric method. More specifically, the authors consider four-dimensional N=1 deformations of N=2 class S theories, and argue that the behavior of vevs of the line operators of the resulting theory is encoded in the 1-cycles of the N=1 curve associated to the chosen vacuum. Most of the paper is devoted to checking that the geometric analysis leads to the same results as the field theory one. Following the formal definition of confinement for a gauge theory, they then conclude that this provides a method for detecting the "confining phase" of non-Lagrangian theories.

After minor revisions, I would recommend this paper for publication.

Requested changes

1) In Section 1, the authors state that their "considerations apply to general class S setup". Is it obvious that a more general setup would have the same behavior?

2) The authors propose a method for detecting confinement, and check in detail that it reproduces the results, known from a field-theoretical analysis, for the cases of $\mathfrak{su}(n)$ N=1 SYM and the theory studied by Cachazo-Seiberg-Witten. Both these theories have confining vacua. Would it be a good proof of concept for their method to consider a case where it is known from field theory that the N=1 deformation does not admit a confining vacuum?

3) Some minor additional points:
- There are a few mathematical notations/quantities that are used but then defined only later on. For instance, $\Lambda_{\mathcal{N}=2}$ is first used in p. 8, and then defined at p. 36. Similarly, the Pontryagin dual is already used at p. 15, but only defined at p. 18.
- The contents of footnote 3 p. 8 are not common knowledge for a non-expert reader. Could the authors provide a reference?
- At p. 17: "we will only study theories for which $\mathcal{L}$ is an abelian group under OPE of line operators". Is there a case in which it is not?
- The text around (3.5) p. 19 contains repetitions and colloquialisms: "is modded out by modding out... that are modded out..."
- Minor typos (understandable in a 83-page paper): p. 9 "led", p. 18 "where $\hat{\Lambda}$ is the Pontryagin dual of $\Lambda$", p. 18 footnote 9 "The Pontryagin dual group..."

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: excellent

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