arxiv: 2604.04393 · v1 · submitted 2026-04-06 · ✦ hep-th · hep-ph

Recognition: 2 theorem links

· Lean Theorem

Anomalies in family unification models from bordism classification

Tsubasa Sugeno , Hiroki Wada

Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:13 UTC · model grok-4.3

classification ✦ hep-th hep-ph

keywords bordism groupsglobal anomaliessigma modelsfamily unificationE7 cosetsAtiyah-Hirzebruch spectral sequencegauged symmetriesanomaly cancellation

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The pith

Global sigma model anomalies are absent in E7 family unification models because the torsion in their bordism groups vanishes.

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

The paper establishes that four-dimensional N=1 supersymmetric nonlinear sigma models with target spaces built from E7 and its subgroups contain no global anomalies. These models produce three generations of quarks and leptons as superpartners of the sigma-model fields. The authors compute the relevant bordism groups using the Atiyah-Hirzebruch spectral sequence and find that their torsion subgroups are trivial, which encodes the absence of global anomalies. They further show that gauging the isotropy subgroup of the coset space does not introduce new global anomalies. A reader cares because such anomalies would make the models inconsistent, while their absence supports the viability of these geometric constructions for explaining fermion families.

Core claim

In family unification models realized as supersymmetric sigma models on E7 coset spaces, global anomalies are encoded in the torsion of the associated bordism groups. Explicit computation of these groups via the Atiyah-Hirzebruch spectral sequence shows that the torsion vanishes, establishing the absence of global sigma model anomalies. When the isotropy subgroup is gauged, the corresponding bordism groups likewise have no torsion, so no additional global anomalies arise.

What carries the argument

The Atiyah-Hirzebruch spectral sequence applied to the bordism groups of E7 coset spaces, where the torsion part classifies possible global sigma model anomalies.

Load-bearing premise

The Atiyah-Hirzebruch spectral sequence converges to the correct bordism groups for these specific E7 coset spaces without errors in the differentials or missed contributions.

What would settle it

An independent computation of one of the relevant bordism groups, for instance by the Adams spectral sequence, that finds non-zero torsion in the degree corresponding to 4d global anomalies would falsify the absence of anomalies.

Figures

Figures reproduced from arXiv: 2604.04393 by Hiroki Wada, Tsubasa Sugeno.

**Figure 1.** Figure 1: The Dynkin diagram of e7. The numbering of the simple roots is the same as [78]. Lie groups that are necessary to describe these models. In Section 3.2, we review the construction of the models and the resulting quantum numbers of the fermions that arise as superpartners of the sigma model fields. We also discuss how to introduce additional fermions into the theory. To determine the possible quantum number… view at source ↗

read the original abstract

We study anomalies in family unification models within the framework of the bordism classification of invertible field theories. These models are based on four-dimensional $\mathcal{N}=1$ supersymmetric nonlinear sigma models, in which the three generations of quarks and leptons arise as superpartners of the sigma model fields. We focus on models whose target spaces are constructed from the exceptional group $E_{7}$ and its subgroups. For the consistency of the theory, sigma model anomalies must be cancelled. We show the absence of global sigma model anomalies, which are encoded in the torsion part of the relevant bordism groups, by explicitly computing these groups using the Atiyah-Hirzebruch spectral sequence. In constructing family unification models, symmetries acting on the coset spaces are gauged, which may introduce additional anomalies. We identify the relevant bordism groups in this setting and demonstrate that no global anomalies arise when the isotropy subgroup of the coset space is gauged.

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

This paper computes the bordism groups for E7 coset spaces in supersymmetric family unification sigma models and finds the torsion vanishes, implying no global anomalies even after gauging the isotropy.

read the letter

The main point is that the authors ran the Atiyah-Hirzebruch spectral sequence on the cohomology of the relevant E7 cosets and their gauged versions, and the torsion subgroups come out trivial. That directly supports their claim of no global sigma model anomalies in these family unification setups. The result is a concrete consistency check rather than a broad new principle. It applies the bordism classification framework to a specific class of N=1 supersymmetric nonlinear sigma models where the three generations come from the target space geometry. The explicit computation for these particular cosets is the new piece; prior work on bordism anomalies did not cover this E7 family unification case. The method itself is standard, and the abstract states that the differentials vanish in the relevant degrees with no leftover torsion after gauging. If the spectral sequence pages and convergence are handled correctly, this is reliable evidence that the models pass the global anomaly test. The soft spot is that the vanishing of differentials and the absence of extension problems need to be shown in detail for these specific spaces. The abstract asserts the outcome, but any missed differential or unresolved extension in the AHSS could reintroduce torsion and change the conclusion. The assumption that the sigma model target space fully encodes the global anomalies of the underlying family unification model is also taken as given. This paper is aimed at people building supersymmetric models with exceptional groups or using bordism to constrain QFT consistency. A reader already working on anomaly cancellation in sigma models or E7-based unification will find the explicit groups useful for checking their own constructions. I would send it to peer review. The calculation is falsifiable and narrow enough that referees can verify the spectral sequence steps without needing new data or broad claims.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that family unification models based on E7 coset spaces in 4d N=1 supersymmetric nonlinear sigma models have no global sigma model anomalies, as shown by explicit computation of the torsion subgroups of the relevant bordism groups via the Atiyah-Hirzebruch spectral sequence; it further claims that gauging the isotropy subgroup introduces no additional global anomalies.

Significance. If the bordism computations are correct, the result supplies a topological obstruction argument confirming consistency of these specific models, extending bordism methods to concrete particle-physics constructions and providing evidence that global anomalies are absent when the isotropy subgroup is gauged.

major comments (1)

The central claim of vanishing torsion (hence no global anomalies) rests on the Atiyah-Hirzebruch spectral sequence for the E7 coset spaces and gauged versions. The manuscript asserts that all differentials vanish in the relevant degrees and that there are no extension problems producing torsion, but provides neither the input cohomology groups of the cosets, the explicit differentials, the convergence details, nor the resulting bordism groups (e.g., no tables or step-by-step pages of the spectral sequence). Without these, the vanishing cannot be verified and the claim is unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comment regarding the details of the Atiyah-Hirzebruch spectral sequence computations below. We believe that providing the requested details will strengthen the paper and confirm our results.

read point-by-point responses

Referee: The central claim of vanishing torsion (hence no global anomalies) rests on the Atiyah-Hirzebruch spectral sequence for the E7 coset spaces and gauged versions. The manuscript asserts that all differentials vanish in the relevant degrees and that there are no extension problems producing torsion, but provides neither the input cohomology groups of the cosets, the explicit differentials, the convergence details, nor the resulting bordism groups (e.g., no tables or step-by-step pages of the spectral sequence). Without these, the vanishing cannot be verified and the claim is unsupported.

Authors: We appreciate the referee's point that the explicit details of the Atiyah-Hirzebruch spectral sequence (AHSS) computations are necessary to verify the vanishing of the torsion subgroups. In the manuscript, we stated the results of these computations—namely, that the torsion parts of the bordism groups for the E7 coset spaces and their gauged versions are trivial—based on our explicit calculations using the AHSS. However, to facilitate verification, we will include in the revised manuscript an appendix that provides: (1) the input cohomology groups of the relevant coset spaces G/H, computed using the known cohomology rings of E7 and its subgroups; (2) the pages of the AHSS, including the differentials in low degrees relevant to global anomalies (up to dimension 4 or 5); (3) confirmation that all differentials vanish in the relevant range or do not affect the torsion; and (4) the converged bordism groups showing no torsion. These details were part of our internal computations but were summarized in the main text for brevity. We disagree that the claim is unsupported, as the computations were performed, but we agree they should be presented more explicitly for the reader. revision: yes

Circularity Check

0 steps flagged

No significant circularity in bordism computation via AHSS

full rationale

The paper derives the absence of global sigma model anomalies by explicitly computing the torsion subgroups of the relevant bordism groups for E7 coset spaces and their gauged isotropy subgroups, using the Atiyah-Hirzebruch spectral sequence. This is a direct, self-contained topological calculation that applies standard spectral sequence techniques to the specific spaces and does not reduce to any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The central claim follows from the computed groups rather than from any input that is restated by construction. No ansatz smuggling, uniqueness theorems from the same authors, or renaming of known results is indicated.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on established mathematical frameworks for anomaly classification without introducing new free parameters or postulated entities.

axioms (2)

domain assumption Bordism groups of invertible field theories classify global anomalies in nonlinear sigma models
This is the central framework adopted for the consistency analysis.
standard math The Atiyah-Hirzebruch spectral sequence computes the relevant bordism groups for the E7 coset spaces
Standard tool from algebraic topology invoked to obtain the torsion parts.

pith-pipeline@v0.9.0 · 5458 in / 1378 out tokens · 56115 ms · 2026-05-10T20:13:32.028823+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/AlexanderDuality.lean alexander_duality_circle_linking unclear

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

We show the absence of global sigma model anomalies, which are encoded in the torsion part of the relevant bordism groups, by explicitly computing these groups using the Atiyah-Hirzebruch spectral sequence.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

the anomaly is classified by the Anderson dual (I_Z Ω^Spin)_{d+2}(X)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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