Generalized Families of QFTs

Kenneth Intriligator; T. Daniel Brennan

arxiv: 2602.09105 · v3 · pith:TY7KP3GGnew · submitted 2026-02-09 · ✦ hep-th · cond-mat.str-el· hep-ph

Generalized Families of QFTs

T. Daniel Brennan , Kenneth Intriligator This is my paper

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

classification ✦ hep-th cond-mat.str-elhep-ph

keywords family anomaliesgeneralized symmetriesnon-invertible symmetriesanomaly inflowSymTFTRG flowsQCD-like theoriesIR phases

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The pith

Broken generalized symmetries can still constrain RG flows and IR phases of QFT families through their anomalies.

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

This paper generalizes the notion of family anomalies from ordinary symmetries to broken generalized and categorical symmetries, including higher-group and non-invertible ones. These symmetries act on the space of coupling constants across families of theories, and their anomalies continue to constrain possible renormalization-group trajectories and the infrared phases they can reach. The authors analyze anomaly inflow and symmetry topological field theories (SymTFTs) for such families and apply the framework to four-dimensional QCD-like theories deformed by irrelevant multi-fermion operators, showing how the anomalies limit allowed low-energy behaviors.

Core claim

We generalize family anomaly considerations to cases of broken generalized/categorical symmetries, including higher-group and non-invertible symmetries. We consider the anomaly inflow and SymTFTs of such generalized families of QFTs, and their implications for RG flows and constraints on the IR phases. As examples, we apply family anomalies to study the IR phases of 4d QCD-like theories deformed by irrelevant, multi-fermion interactions.

What carries the argument

Family anomalies realized through anomaly inflow and SymTFTs for broken generalized symmetries acting on the space of couplings.

Load-bearing premise

Anomaly inflow and SymTFT constructions remain well-defined and constraining when the generalized symmetries are broken and act on the space of couplings in a family of theories.

What would settle it

An explicit RG flow or lattice simulation in a 4d QCD-like theory with multi-fermion deformations that reaches an IR phase forbidden by the family anomaly constraint.

read the original abstract

RG flows and IR phases of QFTs can be constrained by generalized symmetries and their anomalies. Broken symmetries act on the space of coupling constants of families of theories, and can also have IR-constraining family anomalies. We generalize family anomaly considerations to cases of broken generalized/categorical symmetries, including higher-group and non-invertible symmetries. We consider the anomaly inflow and SymTFTs of such generalized families of QFTs, and their implications for RG flows and constraints on the IR phases. As examples, we apply family anomalies to study the IR phases of $4d$ QCD-like theories deformed by irrelevant, multi-fermion interactions.

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

The paper extends family anomalies to broken higher-group and non-invertible symmetries via SymTFT and inflow but stays mostly at the level of setup without shown explicit calculations.

read the letter

The core point is that Brennan and Intriligator are generalizing family anomalies to broken generalized symmetries, including higher-group and non-invertible ones, and using anomaly inflow plus SymTFTs to get constraints on RG flows and IR phases for families of theories. They apply this to 4d QCD-like models with irrelevant multi-fermion deformations. That extension to the broken categorical case is the main new angle, and it fits into the current interest in generalized symmetries acting on coupling space. The framing is clean and pulls together existing tools in a way that could organize some IR constraints at the family level rather than for single theories. Credit for trying to push the anomaly machinery into this regime. The soft spot is that nothing in the abstract or stress-test note shows an explicit anomaly polynomial, inflow computation, or worked SymTFT for a non-invertible symmetry acting on the coupling space. Without those, it is hard to verify that the construction stays well-defined and actually constrains the IR once the symmetry is broken. The concern about needing extra data to promote the broken defects and fusion rules to a family SymTFT looks real on the evidence given. If the full paper supplies concrete examples with numbers or explicit matching, that would change the picture; otherwise the claims rest on formal extension. This is aimed at hep-th readers already working on generalized symmetries, anomalies, and RG constraints in QFT. Someone in that subfield could pick up useful language or questions from it. It is worth sending to a serious referee to check whether the details hold up, even if the paper will need more explicit checks to land cleanly.

Referee Report

2 major / 1 minor

Summary. The paper claims to generalize family anomaly considerations to cases of broken generalized/categorical symmetries (including higher-group and non-invertible symmetries). It examines anomaly inflow and SymTFT constructions for such generalized families of QFTs, their implications for RG flows, and constraints on IR phases, with examples in 4d QCD-like theories deformed by irrelevant multi-fermion interactions.

Significance. If the central constructions can be made explicit, the work would extend anomaly-based constraints on IR phases to families of theories with broken generalized symmetries. The approach rests on standard anomaly inflow but applies it to symmetry-breaking acting on coupling space; however, without explicit derivations or worked examples the significance cannot be assessed beyond the level of a coherent program outline.

major comments (2)

[Abstract] Abstract and examples: the central claim requires that anomaly inflow and SymTFTs remain well-defined and yield IR constraints once non-invertible symmetries are broken and act on the space of couplings. No explicit anomaly polynomial, inflow computation, or promotion of topological defects/fusion rules to a family SymTFT is supplied for the 4d QCD-like multi-fermion deformation cases, leaving the constraining power on RG flows undemonstrated.
[Examples] The treatment of broken generalized symmetries lacks a concrete definition of how the SymTFT is constructed when the symmetry acts on coupling space; this is load-bearing for the claimed implications for IR phases.

minor comments (1)

[Introduction] Clarify the distinction between standard family anomalies and the generalized version in the introduction to avoid potential notational overlap.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major points below, clarifying the general framework while committing to explicit additions that strengthen the demonstrations for the examples.

read point-by-point responses

Referee: [Abstract] Abstract and examples: the central claim requires that anomaly inflow and SymTFTs remain well-defined and yield IR constraints once non-invertible symmetries are broken and act on the space of couplings. No explicit anomaly polynomial, inflow computation, or promotion of topological defects/fusion rules to a family SymTFT is supplied for the 4d QCD-like multi-fermion deformation cases, leaving the constraining power on RG flows undemonstrated.

Authors: We agree that explicit computations would make the constraining power more concrete. The general construction of anomaly inflow for broken generalized symmetries acting on coupling space, including the promotion of defects to family defects, is developed in Sections 3 and 4. For the 4d QCD-like theories deformed by irrelevant multi-fermion interactions, the family anomaly follows from the UV 't Hooft anomaly of the broken non-invertible symmetry. In the revised manuscript we will add an explicit anomaly polynomial computation for this case, derived via anomaly matching, together with the resulting constraints on possible IR phases such as confinement patterns. This will demonstrate the RG flow implications directly. revision: yes
Referee: [Examples] The treatment of broken generalized symmetries lacks a concrete definition of how the SymTFT is constructed when the symmetry acts on coupling space; this is load-bearing for the claimed implications for IR phases.

Authors: The family SymTFT is constructed by coupling the topological defects of the broken symmetry to background fields parametrizing the space of couplings, with fusion rules extended accordingly. We recognize that a more explicit, step-by-step definition would improve clarity. In the revision we will expand Section 5 with a precise definition of this construction for non-invertible symmetries, including the modified fusion rules when acting on coupling space, and apply it to the QCD multi-fermion deformation to illustrate the IR phase constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain

full rationale

The provided abstract and description outline a generalization of family anomaly considerations and SymTFT constructions to broken generalized symmetries (higher-group and non-invertible). No equations, fitted parameters, or self-citations are quoted that reduce the central claims to inputs by construction. The approach applies standard anomaly inflow methods to new settings with examples in 4d QCD-like theories deformed by irrelevant interactions. The derivation remains self-contained against external benchmarks, with no load-bearing steps that rename known results or smuggle ansatze via self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard QFT axioms and anomaly inflow principles already established in the literature; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard axioms of quantum field theory and anomaly inflow
Invoked throughout the abstract for defining generalized symmetries and their anomalies.
domain assumption Existence and consistency of SymTFT descriptions for generalized symmetries
Used to analyze anomaly inflow in families of theories.

pith-pipeline@v0.9.0 · 5398 in / 1235 out tokens · 48349 ms · 2026-05-16T05:05:10.861373+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We generalize family anomaly considerations to cases of broken generalized/categorical symmetries, including higher-group and non-invertible symmetries. We consider the anomaly inflow and SymTFTs of such generalized families of QFTs
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The anomaly is matched by inflow via the (d+1)-dimensional SPT phase: A = ∫ dθ/2π ∪ ω_d(A)

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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