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Generalized global symmetries and holography

by Diego M. Hofman, Nabil Iqbal

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This Submission thread is now published as SciPost Phys. 4, 005 (2018)

Submission summary

As Contributors: Diego Hofman · Nabil Iqbal
Arxiv Link: (pdf)
Date submitted: 2017-11-03 01:00
Submitted by: Hofman, Diego
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We study the holographic duals of four-dimensional field theories with 1-form global symmetries, both discrete and continuous. Such higher-form global symmetries are associated with antisymmetric tensor gauge fields in the bulk. Various different realizations are possible: we demonstrate that a Maxwell action for the bulk antisymmetric gauge field results in a non-conformal field theory with a marginally running double-trace coupling. We explore its hydrodynamic behavior at finite temperature and make contact with recent symmetry-based formulations of magnetohydrodynamics. We also argue that discrete global symmetries on the boundary are dual to discrete gauge theories in the bulk. Such gauge theories have a bulk Chern-Simons description: we clarify the conventional 0-form case and work out the 1-form case. Depending on boundary conditions, such discrete symmetries may be embedded in continuous higher-form symmetries that are spontaneously broken. We study the resulting boundary Goldstone mode, which in the 1-form case may be thought of as a boundary photon. Our results clarify how the global form of the field theory gauge group is encoded in holography. Finally, we study the interplay of Maxwell and Chern-Simons terms put together. We work out the operator content and demonstrate the existence of new backreacted anisotropic scaling solutions that carry higher-form charge.

Ontology / Topics

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Chern-Simons theory Discrete gauge theories Goldstone modes Holography Magnetohydrodynamics
Current status:
Has been resubmitted

Submission & Refereeing History

Published as SciPost Phys. 4, 005 (2018)

Resubmission 1707.08577v3 on 16 January 2018

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Submission 1707.08577v2 on 3 November 2017

Reports on this Submission

Anonymous Report 1 on 2017-12-2 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1707.08577v2, delivered 2017-12-02, doi: 10.21468/SciPost.Report.288


1) It is a nice paper discussing generalized (higher form) symmetries in holographic models. This subject might seem arcane, but I think it is a formal idea worth fleshing out since there's at least one obvious application to MHD.

2) The paper contains many helpful discussions and "warm-up" calculations in detail. Even though parts of the paper are very technical and I still noted a few parts which I found unclear I thought it was very helpful to include these parts, or sections like 3B would have been very hard to follow.


1) The only (very minor) weakness is that the paper sometimes feels a bit disjointed -- Sections 2, 3, and 4 all do rather different calculations, although I understand from the ingredients in the action why Sections 2,3,4 are separate and proceed in that order.


This is a largely well-written paper addressing generalized (higher form) symmetries in holographic models. I enjoyed reading this paper since it was a rare example of a current holography paper doing something rather different from others. While I don't know how useful this formalism will end up being, it is still interesting and it should definitely be published.

I would encourage the authors to address the points described below to improve the clarity of the paper, before publication.

Requested changes

1) In (1.9), can I think of the terms b_E F in the action as analogous to j A in a more conventional theory of electromagnetism, but written in a cleaner way? (i.e. *d * b_E ~ j) It seems this is indeed performed in a slightly different setting around (1.20), but I would mention the analogy sooner.

2) Also isn't there a Hodge star required in front of H in either (1.19) or (1.20) since j is a 1-form as defined below (1.17)?

3) Does the simple form of (2.41) follows from a generalized of membrane paradigm for resistivity? I would not be surprised and it should be easy to check. This would also seem to naturally explain the independence from the Landau pole.

4) The pole motion in Figure 1 reminds me of the story of probe brane "zero sound" in holographic models. I'm skeptical, however, that this should be interpreted as a photon, at least in the conventional sense. I'm wondering whether what's happening here is sort of analogous to how conformal sound waves in 1+1D would have c_s = 1, but c_s < 1 in higher d. Maybe this theory in higher d has a slower propagating mode. Probably a better analogy is actually to the holomorphic/anti-holomorphic decoupling in a 1+1D theory which allows for collective charge degrees of freedom to propagate at the speed of light when they would otherwise not do so.

I think resolving this question would be a separate paper as it requires thinking about a higher d bulk theory, but I might change the end of Section 2 to acknowledge this possibility. Maybe in higher d this "photon" travels slower than the speed of light because of an emergent Snell's law in the strongly coupled plasma, for example? Also, the terminology "hydrodynamic-to-collisionless" is maybe not the best -- I know it's conventionally used, but it's more like "hydrodynamic-to-CFT"...

5) I find the discussion in Section 3 to be unclear at points, but some of the topics are on things that I am unfamiliar with so I will give the authors the benefit of the doubt generally. However, I find Figure 5 a bit confusing in its placement. I think it would be easier to first disucss Figure 6, and only introduce Figure 5 afterwards in the discussion on explicitly broken symmetry. Next, I was not clear on whether there is any explicit form (in this model or any others) of the monopole on which \Psi^k can terminate from the point of view of the bulk theory. For example is the explicit form of \phi(x) and A(x) ever known in one of these monopole geometries? The authors should provide a few steps deriving the statement that the B monopole is an insertion of \phi.

6) I see how in 5d bulk holography the B and C are dual to one 2-form current J, but is that true for higher d? The electric-magnetic boundary duality is special to 4d boundary.

7) It looks to me like (4.9) follows from noting that (4.8) can be written as d[.....] = 0, and then using the fact that the resulting closed form changes by an exact form when zeta -> zeta + d\Lambda. I think the authors could re-write (4.8) and (4.9) to make this clearer by writing (4.8) as a total derivative equation, as I did, and then not moving around the i and \lambda in (4.9).

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

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