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arxiv: 2507.10459 · v2 · submitted 2025-07-14 · ✦ hep-th · hep-lat· math-ph· math.MP

Discrete p-Form Symmetry and Higher Coulomb Phases

Leron Borsten , Hyungrok Kim This is my paper

Pith reviewed 2026-05-19 04:48 UTC · model grok-4.3

classification ✦ hep-th hep-latmath-phmath.MP
keywords p-form symmetryCoulomb phasehigher-form gauge theorydiscrete symmetryinfrared phasesAbelian electrodynamicslattice gauge theory
0
0 comments X p. Extension

The pith

A field theory with a discrete Z_N p-form symmetry generically admits a Coulomb phase where the infrared theory is Abelian p-form electrodynamics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that any field theory equipped with a Z_N p-form symmetry can enter one of three phases: Higgs, confining, or Coulomb. In the Coulomb phase the long-distance physics reduces to free Abelian p-form gauge theory. This extends the familiar phase diagram of ordinary gauge theories to higher-form symmetries. The claim is supported by explicit continuum and lattice constructions that realize all three phases.

Core claim

A field theory with a Z_N p-form symmetry generically admits, in addition to a Higgs phase and a confining phase, a Coulomb phase in which the infrared theory contains Abelian p-form electrodynamics, similar to the behaviour of Yang-Mills theory coupled to adjoint or fundamental matter.

What carries the argument

The discrete Z_N p-form symmetry, which remains exact and unbroken and thereby restricts the allowed infrared phases to include a Coulomb phase with massless p-form fields.

If this is right

  • The infrared effective theory contains free massless Abelian p-form gauge fields.
  • The phase structure mirrors that of ordinary gauge theories coupled to matter.
  • Both continuum and lattice models can be built to realize the Coulomb phase explicitly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Higher-form symmetries could be used to protect Coulomb-like infrared phases in a wider range of quantum field theories.
  • Analogous phase structures may appear in condensed-matter systems that realize discrete higher-form symmetries.

Load-bearing premise

The discrete p-form symmetry remains exact and unbroken in the infrared, forcing the theory into one of three phases without other dynamical effects dominating.

What would settle it

A concrete field theory with an exact unbroken Z_N p-form symmetry whose infrared limit lacks Abelian p-form electrodynamics would falsify the generic claim.

Figures

Figures reproduced from arXiv: 2507.10459 by Hyungrok Kim, Leron Borsten.

Figure 1
Figure 1. Figure 1: An example of a tiling of ℝ3 , using rhombic triacontahedra. (Credit: Françoise Delpérée, Wikimedia Commons). • If both 𝜙 and 𝐽 acquire vacuum expectation values, then we obtain a U(1) gauge theory perturbed by monopole operators: 𝑆 = ∫ 1 2𝑔 2 𝐴 d𝐴 ∧ ★d𝐴 + 𝐴˜ ∧ 𝐽 + · · · . The term 𝐴˜ ∧ 𝐽 may be relevant [3, 4] and lead to unbroken ℤ𝑁 one-form symmetry. (Note that, in the case 𝑝 + 2 = 𝑑, then one can gener… view at source ↗
read the original abstract

We argue that a field theory with a $\mathbb Z_N$ $p$-form symmetry generically admits, in addition to a Higgs phase and a 'confining' phase, a Coulomb phase in which the infrared theory contains Abelian $p$-form electrodynamics, similar to the behaviour of Yang-Mills theory coupled to adjoint or fundamental matter. We illustrate our claim with continuum and lattice examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript argues that a field theory with a discrete ℤ_N p-form symmetry generically admits three phases—a Higgs phase, a confining phase, and a Coulomb phase whose infrared theory contains free Abelian p-form electrodynamics—analogous to the phase structure of Yang-Mills theory with adjoint or fundamental matter. The claim is illustrated through specific continuum and lattice constructions rather than derived from a general classification of infrared theories consistent with the symmetry.

Significance. If substantiated, the result would extend the known phase taxonomy for theories with higher-form symmetries and clarify how discrete p-form symmetries can protect a massless p-form photon in the infrared. Concrete continuum and lattice realizations are provided, which could serve as starting points for further studies of duality and confinement in generalized-symmetry settings.

major comments (2)
  1. [Abstract] Abstract: the central claim that the ℤ_N p-form symmetry 'generically admits' a Coulomb phase is load-bearing for the paper's contribution, yet it rests on explicit examples; the manuscript does not supply a general argument showing why other symmetry-preserving relevant operators or strong-coupling fixed points cannot gap the would-be massless p-form gauge field while preserving the symmetry.
  2. [Lattice examples] Lattice constructions: the emergence of the Coulomb phase must be shown to survive generic deformations that preserve the exact ℤ_N p-form symmetry; without an explicit check that the lattice action does not introduce symmetry-allowed mass terms or confine the p-form photon, the step from tuned examples to the generic statement remains incomplete.
minor comments (1)
  1. [Terminology] The quotation marks around 'confining' phase suggest a non-standard usage; a short clarification of the precise infrared behavior (e.g., area-law vs. perimeter-law for appropriate operators) would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised help clarify the scope of our claims. We have revised the abstract and added discussion in the relevant sections to address the concerns about generality and robustness. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the ℤ_N p-form symmetry 'generically admits' a Coulomb phase is load-bearing for the paper's contribution, yet it rests on explicit examples; the manuscript does not supply a general argument showing why other symmetry-preserving relevant operators or strong-coupling fixed points cannot gap the would-be massless p-form gauge field while preserving the symmetry.

    Authors: We thank the referee for highlighting this distinction. Our use of 'generically' is meant to convey that the symmetry permits a Coulomb phase and that this phase arises in standard constructions without additional fine-tuning, in analogy with the Coulomb phase of Yang-Mills theory with matter. We do not claim to have performed a complete classification of all possible infrared theories consistent with the symmetry, nor do we assert that the Coulomb phase is the only possible phase. In the revised manuscript we have updated the abstract to state that the Coulomb phase is realized in explicit continuum and lattice examples. We have also added a paragraph in the introduction that discusses leading symmetry-preserving relevant operators and explains why they do not gap the p-form photon within the models we consider. A general no-go result or exhaustive classification of all symmetry-compatible fixed points lies beyond the scope of the present work. revision: partial

  2. Referee: [Lattice examples] Lattice constructions: the emergence of the Coulomb phase must be shown to survive generic deformations that preserve the exact ℤ_N p-form symmetry; without an explicit check that the lattice action does not introduce symmetry-allowed mass terms or confine the p-form photon, the step from tuned examples to the generic statement remains incomplete.

    Authors: We agree that demonstrating stability under generic symmetry-preserving deformations strengthens the claim. In the lattice models the exact ℤ_N p-form symmetry forbids local mass terms for the p-form gauge field. We have added an explicit discussion in the lattice section of the revised manuscript showing that the leading symmetry-allowed operators are either irrelevant in the infrared or do not produce confinement of the p-form photon. For the concrete actions we study, the Coulomb phase persists under small deformations that preserve the symmetry. While we have not performed an exhaustive scan of all possible higher-order terms, the symmetry itself protects the massless photon at long distances. This limitation is now noted in the text. revision: yes

Circularity Check

0 steps flagged

Symmetry-based phase classification with explicit examples shows no circularity

full rationale

The manuscript argues that a discrete ℤ_N p-form symmetry permits three IR phases (Higgs, confining, Coulomb with free Abelian p-form electrodynamics) by direct appeal to symmetry constraints and then illustrates the claim with concrete continuum and lattice constructions. No step equates a derived quantity to its own input by definition, renames a fitted parameter as a prediction, or reduces the central claim to a self-citation chain whose content is unverified. The generalization from examples to the generic case rests on standard field-theory reasoning about allowed relevant operators rather than on any self-referential construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the existence of an exact discrete p-form symmetry and on the standard assumption that the infrared phase structure is dictated by that symmetry in the manner of ordinary gauge theories.

axioms (1)
  • domain assumption The theory possesses an exact ℤ_N p-form symmetry that constrains the possible infrared phases.
    Invoked when stating that the symmetry generically admits the three phases.

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