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Generalized global symmetries and holography

by Diego M. Hofman, Nabil Iqbal

Submission summary

As Contributors: Diego Hofman
Arxiv Link: http://arxiv.org/abs/1707.08577v3
Date accepted: 2018-01-18
Date submitted: 2018-01-16
Submitted by: Hofman, Diego
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: High-Energy Physics - Theory

Abstract

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.

Current status:

Ontology / Topics

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Chern-Simons theory Discrete gauge theories Goldstone modes Holography Magnetohydrodynamics

Author comments upon resubmission

We thank the referee for the kind words and for the careful reading of the draft and subsequent detailed suggestions. Below we respond to the specific suggestions made by the referee:

1. Indeed, in (1.9) b_e can be thought of as an external fixed electric charge density that sources the dynamical Maxwell fields. We agree that it makes sense to discuss this sooner, and have put an extra note below (1.11) emphasizing this when discussing the Maxwell equations that follow from the action.

2. This is a typo, we thank the referee for catching this and have put back the \star in eq (1.20).

3. Indeed, the simple form of (2.41) follows from a membrane paradigm-type result; we have added a sentence below (2.39) emphasizing this. We are reluctant however to say that this necessarily provides a “natural” explanation of the independence from the Landau pole, as it does not seem inconceivable that in a more complicated example quantities evaluated at the horizon could also depend on the boundary conditions at infinity through the background Einstein equations.

4. This is a very good point; we understand the referee’s concerns and have weakened the appropriate paragraphs on page 19 to explain that this is largely an analogy and not a precise statement about universal dynamics. Note that we have kept the “collisionless” terminology though we have modified the parenthetical comment that follows to read "(linearly dispersing)"; while we again completely agree with the referee that the system is not really collisionless, the terminology has now become standard and we feel the referee’s suggestion of “CFT” is also not really ideal in this specific case (as the system is not a CFT in the UV due to the logarithmic running). For most examples of applied holography this would have been a good notation.

5. Regarding point #5:
a. We agree with the referee that it makes more sense to place Figure 6 first; we have thus interchanged the order of Figures 5 and 6 and think that this aids the flow.
b. We now address the following question of the referee: "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”.
I. As the bulk theory is gapped, the only *universal* information that exists from the “explicit form” of \phi(x) and A(x) is really the charges of the monopole (all other data characterizing the solution falls off exponentially in space with a correlation length that has been taken to be zero in the limit in which the CS theory applies).
II. Nevertheless, given the fact that the CS theory is the IR limit of a UV complete theory (that in (3.1)) one could ask if there is any more information present in (3.1) about the monopole. In this case, the statement that \phi(x) is the monopole field then provides a complete specification of the UV core of the monopole, as \phi(x) is a fundamental field that has no internal substructure. This can be contrasted with the situation in the next section, where the “monopole” is a wrapped D5 or NS5 brane, that is not created in a simple way by an elementary field operator in the bulk and it thus possesses quite a lot of internal structure (which is nevertheless in principle fixed by the known stringy physics of these branes).
III. As these issues are mostly irrelevant for the mundane purposes of the narrative (which uses only the fact that the monopole “exists”), we chose to not go into these issues in the paper. We have however added a line in the paragraph below Figure 6 on p27 explaining that \phi \sim e^{i\theta} is the monopole field to help explain this connection.

6. Indeed a similar structure exists in all dimensions; an example is given in the earlier section, where the 4d fields B and A are together dual to only a single 1-form current j_e, but there is no electric-magnetic duality present (or necessary) on the 3d boundary.

7. We considered this revision: as this is largely a stylistic point on the order of presentation of equations, ultimately we felt that the original derivation was clearer and have chosen to not change the presentation.

Submission & Refereeing History

Resubmission 1707.08577v3 on 16 January 2018
Submission 1707.08577v2 on 3 November 2017

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