arxiv: 2305.18296 · v2 · submitted 2023-05-29 · ✦ hep-th · cond-mat.str-el· hep-ph· math.CT

Recognition: no theorem link

ICTP Lectures on (Non-)Invertible Generalized Symmetries

Sakura Schafer-Nameki

Authors on Pith no claims yet

Pith reviewed 2026-05-16 07:46 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elhep-phmath.CT

keywords non-invertible symmetriesgeneralized symmetriestopological defectstheta defectsSymTFTgauginghigher-form symmetriesTQFT

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The pith

Non-invertible symmetries arise when a quantum field theory is stacked with topological QFTs and a diagonal symmetry is gauged, producing topological defects without inverses.

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

The paper explains how global symmetries in QFTs have been generalized to include higher-form, higher-group, and non-invertible symmetries, all identified with topological defects. The core construction for many non-invertible symmetries involves stacking the original theory with TQFTs and gauging a shared non-anomalous symmetry, after which the TQFTs manifest as (twisted) theta defects. These defects include condensation defects and self-duality defects, and appear in gauge theories with disconnected groups. The lectures also cover how such symmetries act on extended operators through higher-representations rather than ordinary group representations. Finally, the Symmetry Topological Field Theory is introduced as a tool to organize symmetries, their gauging, and the associated charges.

Core claim

The foundational construction for non-invertible symmetries is the stacking of a QFT with topological QFTs followed by gauging a diagonal non-anomalous global symmetry; the TQFTs then become topological defects in the resulting theory known as (twisted) theta defects, which encompass condensation defects, self-duality defects, and non-invertible symmetries in gauge theories with disconnected gauge groups.

What carries the argument

The stacking-and-gauging procedure that converts TQFTs into (twisted) theta defects serving as non-invertible symmetry generators.

Load-bearing premise

Symmetries correspond to topological defects in a general QFT, and this identification continues to hold when the defects are non-invertible.

What would settle it

A calculation in a specific gauge theory showing that the predicted theta defect does not commute with all local operators in the expected way or fails to satisfy the fusion rules derived from the construction.

read the original abstract

What comprises a global symmetry of a Quantum Field Theory (QFT) has been vastly expanded in the past 10 years to include not only symmetries acting on higher-dimensional defects, but also most recently symmetries which do not have an inverse. The principle that enables this generalization is the identification of symmetries with topological defects in the QFT. In these lectures, we provide an introduction to generalized symmetries, with a focus on non-invertible symmetries. We begin with a brief overview of invertible generalized symmetries, including higher-form and higher-group symmetries, and then move on to non-invertible symmetries. The main idea that underlies many constructions of non-invertible symmetries is that of stacking a QFT with topological QFTs (TQFTs) and then gauging a diagonal non-anomalous global symmetry. The TQFTs become topological defects in the gauged theory called (twisted) theta defects and comprise a large class of non-invertible symmetries including condensation defects, self-duality defects, and non-invertible symmetries of gauge theories with disconnected gauge groups. We will explain the general principle and provide numerous concrete examples. Following this extensive characterization of symmetry generators, we then discuss their action on higher-charges, i.e. extended physical operators. As we will explain, even for invertible higher-form symmetries these are not only representations of the $p$-form symmetry group, but more generally what are called higher-representations. Finally, we give an introduction to the Symmetry Topological Field Theory (SymTFT) and its utility in characterizing symmetries, their gauging and generalized charges. Lectures prepared for the ICTP Trieste Spring School, April 2023.

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.

Desk Editor's Note private letter to a colleague

These are clear lecture notes that organize the stacking-and-gauging construction for non-invertible symmetries, but they contain no new results.

read the letter

These ICTP lectures synthesize the recent literature on generalized symmetries, with the main emphasis on non-invertible cases. The central point is that many such symmetries arise by stacking a QFT with a TQFT and gauging a diagonal non-anomalous symmetry, which produces defects including twisted theta defects, condensation defects, and self-duality defects in gauge theories with disconnected groups. The notes lay out this mechanism and then show how the resulting defects act on extended operators through higher representations, a clarification that applies even to ordinary higher-form symmetries. The closing section on the Symmetry TFT gives a compact way to track gauging and charges across examples. All of this is drawn from established papers and presented with concrete cases rather than abstract claims. The notes do this job cleanly and without circular arguments or unsubstantiated steps. Because they are lecture notes rather than a research article, the derivations stay at the level of outlines with references to the original literature instead of full proofs. That is a minor limitation for anyone who wants to work through every calculation from scratch, but it matches the stated purpose of the notes. The foundational assumption that symmetries correspond to topological defects is used throughout without re-derivation, which is standard in the field and does not create internal problems here. The paper is aimed at researchers and advanced students who already know basic QFT and want a structured entry point into the non-invertible symmetry literature. It would be useful for a reading group or as background when starting a project in this area. I would send it for peer review at a venue that accepts lecture notes or reviews, since the organization is careful and the examples are well chosen.

Referee Report

0 major / 2 minor

Summary. The manuscript consists of lecture notes introducing generalized symmetries in QFT, beginning with invertible higher-form and higher-group symmetries before focusing on non-invertible symmetries. The central construction is the stacking of a QFT with suitable TQFTs followed by gauging a diagonal non-anomalous symmetry, which produces topological defects (twisted theta defects) that realize condensation defects, self-duality defects, and non-invertible symmetries of gauge theories with disconnected groups. The notes then address the action of these symmetries on extended operators (via higher representations) and introduce the Symmetry Topological Field Theory (SymTFT) as a tool for characterizing symmetries, gauging, and charges. Numerous concrete examples are provided throughout.

Significance. These lectures consolidate recent developments in generalized symmetries into a coherent pedagogical framework. The stacking-plus-gauging mechanism is presented as a unifying principle supported by established literature, with explicit examples that illustrate fusion rules and defect actions. The inclusion of higher representations and the SymTFT perspective adds practical value for computing charges and gauging procedures. If the explanations are accurate, the notes would serve as a useful reference for researchers entering the field and for teaching advanced topics in hep-th.

minor comments (2)

In the section introducing the SymTFT, the relation between the SymTFT and the original QFT's symmetry data could be made more explicit by adding a short diagram or table that maps SymTFT operators to the corresponding defects and charges in the physical theory.
The discussion of higher representations for invertible higher-form symmetries (around the transition from invertible to non-invertible cases) would benefit from one additional low-dimensional example (e.g., 1-form symmetry in 4d) to illustrate the distinction between ordinary representations and higher representations before moving to the non-invertible setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the lecture notes and for recommending acceptance. We appreciate the recognition that the stacking-plus-gauging construction provides a unifying pedagogical framework, supported by concrete examples, and that the inclusion of higher representations and the SymTFT perspective adds practical value.

Circularity Check

0 steps flagged

No significant circularity; standard literature review of stacking-gauging constructions

full rationale

The lectures present the stacking-plus-gauging construction for non-invertible symmetries as a standard procedure drawn from prior literature, with the identification of symmetries as topological defects treated as an established starting point rather than a novel derivation internal to the notes. No equations or claims reduce by construction to parameters or definitions fitted within the paper itself; all central steps reference external results and concrete examples from established cases. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The lectures rest on standard QFT assumptions and the recent identification of symmetries with topological defects; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption Symmetries in QFT can be identified with topological defects
This is the central principle stated in the abstract that enables the generalization to non-invertible symmetries.

pith-pipeline@v0.9.0 · 5610 in / 1195 out tokens · 33847 ms · 2026-05-16T07:46:43.873408+00:00 · methodology

discussion (0)

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