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arxiv: 2606.05543 · v1 · pith:2YCDNFQXnew · submitted 2026-06-04 · ✦ hep-th

Notes on (-2)-form symmetries

Pinak Banerjee , Alonso Perez-Lona , Daniel Robbins , Subham Roy , Eric Sharpe , Xingyang Yu This is my paper

Pith reviewed 2026-06-28 00:43 UTC · model grok-4.3

classification ✦ hep-th
keywords (-2)-form symmetrySymTFTnon-invertible symmetryanomalygaugingholographyChern-Simons theoryfusion category
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0 comments X

The pith

A (-2)-form symmetry modifies the SymTFT action to relate QFTs whose ordinary global symmetries differ by anomaly data or non-invertible associator data.

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

The paper examines (-2)-form symmetries of a d-dimensional quantum field theory through a (-1)-form symmetry of its (d+1)-dimensional SymTFT. This symmetry is realized by a non-genuine codimension-one defect in the SymTFT bulk attached to a spacetime-filling topological operator. Unlike ordinary (-1)-form symmetries that only shift parameters, this construction modifies the SymTFT action itself. The modification connects theories that differ in their anomaly data or in the associator data of non-invertible symmetries. Examples appear in two-dimensional models, ABJM theories, generalized Yang-Mills theory, fusion categories, and a holographic setup with the Romans mass.

Core claim

A (-2)-form symmetry, realized via a non-genuine codimension-one defect in the SymTFT bulk attached to a spacetime-filling topological operator, modifies the SymTFT action and thereby relates theories whose ordinary global symmetries differ by anomaly data or by the associator data of a non-invertible symmetry.

What carries the argument

The non-genuine codimension-one defect in the SymTFT bulk attached to a spacetime-filling topological operator that realizes the (-2)-form symmetry and alters the SymTFT action.

If this is right

  • The construction relates two-dimensional toy models whose global symmetries differ by anomaly data.
  • It connects distinct phases of three-dimensional ABJM-type theories.
  • It applies to four-dimensional generalized Yang-Mills theory.
  • It supplies a fusion-categorical example relating the non-invertible symmetries Rep(D4) and Rep(Q8).
  • Club-sandwich and nested discrete gauging realizations interface IR phases of distinct RG flows of a common UV theory.

Where Pith is reading between the lines

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

  • The same mechanism may supply a systematic way to generate new dualities by shifting higher-form background fields that control anomaly coefficients.
  • Nested discrete gauging offers a concrete computational route to enumerate equivalent theories on the lattice.
  • The holographic identification suggests that other supergravity fluxes could be reinterpreted as (-2)-form backgrounds affecting boundary anomalies in higher-dimensional setups.

Load-bearing premise

The non-genuine codimension-one defect in the SymTFT bulk consistently realizes a (-2)-form symmetry that alters the SymTFT action without violating underlying QFT consistency conditions.

What would settle it

Explicit computation in the three-dimensional Chern-Simons-matter theory showing that a shift in the parameter identified with the (-2)-form background changes the boundary anomaly coefficients in the precise way predicted by the Romans-mass holographic realization.

Figures

Figures reproduced from arXiv: 2606.05543 by Alonso Perez-Lona, Daniel Robbins, Eric Sharpe, Pinak Banerjee, Subham Roy, Xingyang Yu.

Figure 1
Figure 1. Figure 1: The club sandwich We can summarize the situation with the following short exact sequence of groups (more generally, an exact sequence of fusion categories), 0 −→ H −→ G −→ K −→ 0, (4.1) where H is the symmetry that acts trivially in the IR. For the particular example of UV symmetry group Z4, we can construct two different such club sandwiches that effectively capture two different renormalization group flo… view at source ↗
Figure 2
Figure 2. Figure 2: The orange blob on the right hand side is a RG interface. The physical boundaries [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The orange blob is a RG interface. The IR theory in this setting is different from the [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: After fusing the interfaces Iϕ′′ and Iϕ′. The interface I in [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Starting from the 3d SymTFT of a theory with Z4 symmetry, we have gauged different 1-form symmetries. The physical boundary attached with Vec(Z2) and Vec(Z ω 2 ) are different. The interface I, represented by the blue dashed line is the product of two topological interfaces (in the ambient Drinfeld center). I := Ie 2m2 ⊗ Im2 . (4.5) There is a boundary term iπ R a ∪ aˆ term on Ie 2m2 , but we suspect upon … view at source ↗
Figure 7
Figure 7. Figure 7: By repeating the quarter gauging procedure, we have obtained the Vec( [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Boundary SymTFT 13The SymTFT for free boson can also be written using R gauge fields.: S = 1 2π Z b1 ∧ da1. (6.13) Here, both b1 and a1 are real-valued gauge fields. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Placing D vertically, produces a (−1)-form symmetry of the absolute theory (the bulk boundary coupled system). Instead, we can place the codimension one defect horizontally in the SymTFT bulk (as in [PITH_FULL_IMAGE:figures/full_fig_p040_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Placing D horizontally, produces a (−1)-form symmetry for B′ gapped. Inducing an action only on the topological corner. Fusing, B′ gapped ⊗ D = B′′ gapped. Fusing the defect D with B′ gapped will modify the lagrangian algebra on that boundary, producing a new boundary condition (or boundary theory). This action can be thought of as a (−1)-form symmetry for this boundary slab. Collapsing the whole SymTFT, … view at source ↗
read the original abstract

We study $(-2)$-form symmetries of a $d$-dimensional quantum field theory, via a $(-1)$-form symmetry of its $(d+1)$-dimensional Symmetry Topological Field Theory (SymTFT), realized by a non-genuine codimension-one defect in the SymTFT bulk attached to a spacetime-filling topological operator. Unlike a $(-1)$-form symmetry of a $d$-dimensional theory, which merely shifts a parameter of the absolute theory, a $(-2)$-form symmetry modifies the SymTFT action, and thereby relates theories whose ordinary global symmetries differ by anomaly data or by the associator data of a non-invertible symmetry. We illustrate the construction in two-dimensional toy models, three-dimensional ABJM-type theories, four-dimensional generalized Yang--Mills theory, and in a fusion-categorical example relating the non-invertible symmetries $\operatorname{Rep}(D_4)$ and $\operatorname{Rep}(Q_8)$. We then develop a club-sandwich realization, in which a quarter-gauging operation interfaces between IR phases of distinct RG flows of a common UV theory, and an alternative realization via nested discrete gauging. Finally, we present a holographic, top-down realization in which the type IIA Romans mass plays the role of a $(-2)$-form background for a three-dimensional Chern--Simons-matter theory, with shifts of the Romans mass realizing shifts of the boundary anomaly coefficients. We also discuss related constructions for coupled bulk--boundary systems.

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

0 major / 2 minor

Summary. The manuscript introduces (-2)-form symmetries of a d-dimensional QFT, realized as a (-1)-form symmetry of its (d+1)-dimensional SymTFT via a non-genuine codimension-one defect in the SymTFT bulk attached to a spacetime-filling topological operator. This construction modifies the SymTFT action and thereby relates theories whose ordinary global symmetries differ by anomaly data or by the associator data of a non-invertible symmetry. Explicit illustrations are given in 2d toy models, 3d ABJM-type theories, 4d generalized Yang-Mills, the fusion categories Rep(D4) and Rep(Q8), club-sandwich and nested discrete gauging realizations, and a holographic top-down example in which the type IIA Romans mass acts as a (-2)-form background for 3d Chern-Simons-matter theories, with mass shifts corresponding to shifts in boundary anomaly coefficients.

Significance. If the constructions are internally consistent, the work provides a concrete extension of the SymTFT framework that unifies the treatment of anomaly shifts and non-invertible symmetry data through higher-codimension operators. The multiple low-dimensional examples, the club-sandwich and gauging constructions, and the explicit holographic realization using Romans mass constitute reproducible model-building tools that could be applied to other systems; these strengths are load-bearing for the paper's utility.

minor comments (2)
  1. [Introduction] The definition of the non-genuine codimension-one defect and its attachment to the spacetime-filling operator (abstract, paragraph 2) would benefit from an explicit local operator expression or a small diagram in the first section to make the modification of the SymTFT action fully transparent.
  2. [Holographic realization] In the holographic section, the precise dictionary between shifts of the Romans mass parameter and the resulting shifts in the boundary anomaly coefficients should be stated with the relevant Chern-Simons level or anomaly polynomial term.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the positive assessment, including the recommendation for minor revision. The referee summary accurately captures the scope of the work on (-2)-form symmetries and their realization via the SymTFT. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces (-2)-form symmetries as a conceptual extension of the SymTFT framework, realized by a non-genuine codimension-one defect attached to a spacetime-filling operator. It illustrates the idea through explicit constructions in low-dimensional models (2d toys, 3d ABJM, 4d gYM, Rep(D4)/Rep(Q8)) and a holographic Romans-mass example, all within standard SymTFT consistency. No load-bearing steps reduce by definition, fitted parameters renamed as predictions, or self-citation chains; the derivations remain self-contained extensions of prior formalism without circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard domain assumptions of quantum field theory and topological field theories; the (-2)-form symmetry itself is introduced as a new organizing entity without independent external evidence in the abstract.

axioms (2)
  • domain assumption Existence and consistency of Symmetry Topological Field Theory (SymTFT) as an auxiliary (d+1)-dimensional theory encoding d-dimensional global symmetries.
    Invoked throughout the abstract as the ambient space in which the (-2)-form symmetry is realized.
  • domain assumption Well-defined non-genuine codimension-one defects can be attached to spacetime-filling operators without spoiling topological invariance.
    Central to the realization mechanism stated in the second sentence of the abstract.
invented entities (1)
  • (-2)-form symmetry no independent evidence
    purpose: To modify the SymTFT action and thereby relate QFTs differing in anomaly or non-invertible symmetry data.
    Newly introduced concept whose only support is the constructions described in the abstract; no external falsifiable prediction is stated.

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discussion (0)

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Reference graph

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