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Goldstone modes and photonization for higher form symmetries

by Diego M. Hofman, Nabil Iqbal

Submission summary

As Contributors: Diego Hofman
Arxiv Link:
Date accepted: 2019-01-08
Date submitted: 2018-11-16
Submitted by: Hofman, Diego
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: High-Energy Physics - Theory


We discuss generalized global symmetries and their breaking. We extend Goldstone's theorem to higher form symmetries by showing that a perimeter law for an extended $p$-dimensional defect operator charged under a continuous $p$-form generalized global symmetry necessarily results in a gapless mode in the spectrum. We also show that a $p$-form symmetry in a conformal theory in $2(p+1)$ dimensions has a free realization. In four dimensions this means any 1-form symmetry in a $CFT_4$ can be realized by free Maxwell electrodynamics, i.e. the current can be photonized. The photonized theory has infinitely many conserved 0-form charges that are constructed by integrating the symmetry currents against suitable 1-forms. We study these charges by developing a twistor-based formalism that is a 4d analogue of the usual holomorphic complex analysis familiar in $CFT_2$. The charges are shown to obey an algebra with central extension, which is an analogue of the 2d Abelian Kac-Moody algebra for higher form symmetries.

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Conformal field theory (CFT) Goldstone modes

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Anonymous Report 2 on 2018-12-17 Invited Report

  • Cite as: Anonymous, Report on arXiv:1802.09512v2, delivered 2018-12-17, doi: 10.21468/SciPost.Report.749


1-The paper is written clearly, the key results are well emphasized.
2-The topic is of current interest.
3-The results suggest several potential future directions.


I could not find any.


The paper discusses the extension of Goldstone's theorem to p-form symmetries. It is shown that if charged p-dimensional objects follow a perimeter law then the theory has a Goldstone mode. For CFT's with p-form symmetry in dimension d=2(p+1), correlation functions of the higher-form current can be realized in terms of a free Goldstone mode. Moreover, there is an infinite number of conserved charges, which are shown to lead to a higher-form generalization of the Kac-Moody algebra. In 4 dimensions, the construction is formulated using twistor formalism. The results are intriguing and suggest several generalizations for future research on the topic. The paper has a clear structure, and the essential concepts are introduced in a transparent and insightful way.

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

Anonymous Report 1 on 2018-12-7 Invited Report

  • Cite as: Anonymous, Report on arXiv:1802.09512v2, delivered 2018-12-07, doi: 10.21468/SciPost.Report.717


The paper is clearly written, and it contains many important results, on topics of current interest to the community.


None that I noticed.


The paper extends Goldstone's theorem to higher-form symmetries. It moreover shows that one-form symmetries in a 4d CFT can be photonized to free Maxwell electrodynamics and, more generally, that p-form symmetries in a CFT in 2(p+1) dimensions has a free realization. In the 4d case, infinitely many conserved 0-form charges are studied by a twister-based formalism and it is shown that the charge algebra has central extension, giving an analog of 2d Kac-Moody algebra for higher form symmetries. All of these results are interesting, and give new insights into higher-form symmetry. The paper is terse in a good way - it is very clear and packed with nice results and insights. I am recommending Tier II below just because I do not know how to calibrate the level for an Editor's selection, and I think that this paper's readership might be more limited and specialized as compared with the top 10% level Editor's Select.

Requested changes


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

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