Pions and Generalized Cohomology

Daniel S. Freed

REVIEW

Pions and Generalized Cohomology

Daniel S. Freed

High Energy Physics - Theory · Submitted 19 Jul 2006 · 7 references

Pith's verdict: A careful, genuinely new construction of the WZW term in differential E-cohomology, with one tedious-but-load-bearing computation left to the reader.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 7 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-th/0607134 (version 2)

Pith IDWDGNYU4Q, Pith's permanent record of this paper and its review

CitationDaniel S. Freed. "Pions and Generalized Cohomology." arXiv:hep-th/0607134 (2006). https://pith.science/paper/WDGNYU4Q

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

This is Freed doing what Freed does: taking a physics object everyone thought was understood and rebuilding it in the right mathematical home. The real content is Theorem 2.2 — the determinant line bundle of a family of Dirac operators on spin 4-manifolds, with its covariant derivative, is the fiber integral of the differential E-class μ̌. That is a theorem with a proof, and the proof is honest: curvature from the index density, holonomy from APS plus a spin-bordism vanishing lemma (2.6) that is itself proved. The WZW definition (4.1) then falls out as the integral of ν̌, and the four advertised properties — spin dependence, Z/2-graded Hilbert space with the right soliton statistics (4.10), Nf = 2, and anomaly matching — are each checked rather than asserted. The Standard Model corollary in Example 3.4 is a genuine consequence of the machinery, not a slogan, though it is a new route to a known answer rather than a surprise; the SU(5) embedding argument is the clean way to see it, and the differential lift step the stress-test flags is fillable because the curvature vanishes for that real representation. Worth saying plainly: the paper is aware that E is a minimal choice, not a canonical one, and says so in a footnote. That is the right level of honesty about a boutique theory. The soft spot is real but narrower than it looks. The three properties of ηE in (5.16) are stated and explicitly left unchecked — the author's own line about the 'exhausting task' is an acknowledgment, not a hidden gap. Those properties are Chern-Simons-type form computations: tedious, standard, and pinned by the fiber restriction, so a wrong normalization would show up immediately as a failure of (i). The compressed Stokes argument for well-definedness of the gauged WZW factor at the end of §5 is the weaker of the two. Both affect Proposition 4.17, not Theorem 2.2 or Example 3.4. Who should read it: anyone working on anomalies, WZW terms, or differential cohomology applied to QFT. The Hopkins-Singer framework is heavy going for a physics audience, but the paper is unusually clear about what it is doing and why. This deserves referees, and it deserved them in 2007.

Referee report

Significance

If correct, this is a substantial contribution on several fronts. It gives the WZW term a definition that makes sense on an arbitrary spin 4-manifold rather than only on spheres or boundaries, and it explains structurally why the low-energy theory knows about spin structure, fermionic statistics, and the N_f = 2 case — features that spherical compactification cannot see. Theorem 2.2 is a genuine index theorem in a generalized differential cohomology theory, and it is proved rather than conjectured: the curvature computation (2.8), the grading computation (2.9), and the holonomy computation via APS (2.10) are each carried out, with the bordism lemma supplying the needed extension. The paper is parameter-free: no fitted coefficients, no data, and the normalization of νˇ is fixed once and for all by (1.15) and then checked against Witten's independent baryon-number/statistics formula in (4.10). The Standard Model corollary in Example 3.4 is a crisp, checkable statement with two independent arguments (the SU₅ embedding and the Spin(10) half-spin embedding). The anomaly-matching statement of Proposition 4.17 is exactly the kind of falsifiable structural prediction that gives the cohomol

Major objections (3)

[§5, Eq. (5.16)] The three properties of the transgressing form η_E — basicness for the diagonal G-action, restriction to a fiber being cohomologous to ω_νˇ, and dη_E = π*(ω_µˇ₁ − ω_µˇ₂) — are stated and then explicitly deferred: 'We leave the reader the exhausting task of checking these properties.' These properties are precisely what make η_E a transgressing form, and they feed Theorem 4.13, Definition 4.15, and hence the anomaly-matching result of Proposition 4.17, which is advertised property (iv) of the introduction. The concern is not that the construction is wrong but that the normalization of the τ term in (5.16) is exactly what fixes the coefficient in (4.18); a sign or normalization slip there would shift the anomaly-matching coefficient, and the reader has no worked check to compare against. I would ask for at least a verification of property (iii) — the exterior derivative computation, which is the one that carries the coefficient — in an appendix or a footnote, even if the basicness check is left as an exercise. I note that this gap is localized to the gauged sector: Theorem 2.2 and the Standard Model corollary of Example 3.4 do not depend on η_E.
[§5, Eq. (5.17) and the paragraph following it] The well-definedness of the gauged WZW factor is described as a sketch: 'To conclude we sketch an argument proving that the gauged WZW factor W̃_X in Definition 4.14 is well-defined.' The argument requires the groupoid G₂ of triples (Q, Θ, γ), the morphisms given by quintuples (P, Λ, Γ, φ₀, φ₁) with the curvature condition ι(∂/∂t)Ω = 0, and a Stokes argument for (5.17) in which the integral of the ˇE-differential over [0,1] × X is asserted to vanish 'because of the constancy of Φ and Λ in the ∂/∂t direction.' Since Definition 4.14 is the definition on which the gauged theory rests, and since the morphism conditions are what make the section well-defined rather than merely a function on objects, this deserves to be written out at the same level of care as the rest of §5. In particular, the claim that the [0,1] × X integral of Φ*Γ̃*η_E equals 1 should be justified from the stated constancy conditions rather than asserted.
[Abstract and Example 3.4] The abstract states without qualification that 'there are no global anomalies in the Standard Model (in arbitrary gravitational backgrounds).' What Example 3.4 establishes is that ˇµ(ρ) = 0 for the Standard Model representation, hence that the obstruction class [Det D_{X/S}(V)] ∈ ˇE²(S) vanishes. That is the paper's own definition of anomaly, so the claim is internally consistent; but §3 is explicit that vanishing of the class 'does not provide a choice of trivialization of Det D_{X/S}(V), much less a consistent choice under gluing — a setting of the quantum integrand,' and that such a refinement of Theorem 2.2 is not available. Since the abstract's phrasing will be read by many as a stronger statement than the vanishing of a cohomology class, I would ask that the abstract and the Example 3.4 heading be tied explicitly to the definition of anomaly given in §3, e.g. 'the anomaly class vanishes' rather than 'there are no global anomalies.' This is a scope and precision request, not a claim that the computation is wrong.

Minor objections (7)

[Footnote 1] The footnote justifies the new definition by saying that 'because the Hurewicz map π₅G → H₅G is not an isomorphism certain features were missed.' For G = SU(N) with N ≥ 3 both groups are Z and the Hurewicz map is an isomorphism; the failure is specific to N = 2, where π₅ = Z/2 and H₅ = 0. The motivating point would be sharper if the footnote distinguished the two cases, or if it pointed instead to the extra Z/2 in E⁵ coming from H³(·;Z/2), which is what actually carries the new information.
[Example 3.4] The step 'Hence c₃(V ⊕ ∧²V) = 0, and from Proposition 1.9(ii) we have µ(V ⊕ ∧²V) = 0. Thus ˇµ(V ⊕ ∧²V) = 0 as well' passes from a topological class to a differential class without comment. The differential lift is not determined by the topological class alone; the curvature form (1.16) must also vanish, which it does because the representation is real so that Tr Ω³ = 0. One sentence invoking (1.8) and the reality of the representation would close the step.
[§3, paragraph following Eq. (3.2)] The paper states 'We are unsure of the physical significance of this grading, but believe that a theory is anomalous if the grading is nonzero.' This is an honest statement of an interpretive gap, and it is worth saying what evidence would settle it — for instance, whether the grading can be nonzero in a theory whose functional integral is otherwise well defined, given the observation two sentences later that the fermionic integral vanishes when the grading is nonzero.
[§1, Eq. (1.6)] The spin orientation of E, and hence the pushforward (1.6), is asserted with the pointer 'see Proposition 4.4 for details.' Proposition 4.4 proves the dual statement, constructing a Thom class in the Anderson dual theory e and defining the integral as a product with it. A direct statement of the MSpin-orientation of E, or a forward pointer to the place where the pushforward is actually constructed, would help readers who want to use (1.6) without reconstructing the duality argument.
[§2, Eq. (2.9)] The expression 'rank V ∫ Â(X_s) − ∫ −c₂(V)' contains a double sign; the index formula gives rank V ∫Â − ∫c₂. The congruence is mod 2, so the two agree, but the displayed formula as written invites confusion and should be cleaned up.
[§4, after Theorem 4.12] There are several typographical and grammatical slips that should be fixed: 'the umker or pushforward map' at (1.6); 'does not has an equivariant extension' in the paragraph following Theorem 4.12; 'In physical term the integer degree' in the paragraph following (4.10); 'may defined via an integral' in footnote 13; and 'we simply notate' in §1.
[References] Reference [FHT] is listed as 'in preparation' and is cited only as context for twisted K-theory, a connection the text says 'plays no role here.' Since the citation is not load-bearing, either remove it or replace it with a published reference so that the bibliography contains no unpublished items.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The Wess-Zumino-Witten term for pions can be defined directly on any spin four-manifold using a generalized cohomology class, without choosing a bounding five-manifold.

The reading

The paper proposes a new geometric definition of the Wess-Zumino-Witten term in the low-energy $\sigma$ model for pions, working on arbitrary Riemannian spin four-manifolds rather than only on spacetimes that bound a five-manifold. The definition integrates a differential class $checknu$ in a generalized cohomology theory $E$ over the four-manifold; $nu$ is a class on $G=SU(N)$ that is half the generator of $H^5(G)$ in a precise sense. The author argues that this definition reproduces four features of the underlying gauge theory: it needs a spin structure, gives a $mathbb{Z}/2mathbb{Z}$-graded Hilbert space with the right soliton statistics, works for $N_f=2$, and has a gauged anomaly equal to the QCD anomaly. A companion index theorem identifies the determinant line bundle of families of Dirac operators on spin four-manifolds as an integral of a differential characteristic class $checkmu$, and this yields the claim that the Standard Model has no global anomalies in arbitrary gravitational backgrounds.

What carries the argument

The machinery is a generalized cohomology theory $E$ with exactly two nonzero homotopy groups, fitting into a fibration $Hmathbb{Z}to Eto Sigma^{-2}Hmathbb{Z}/2mathbb{Z}$ whose $k$-invariant is $betacirc Sq^2$. Its differential refinement $check E$ combines a topological class, a closed curvature form, and a cochain, and integration over spin fibers is defined. The load-bearing class is $nuin E^5(SU(N))$, characterized by $k(nu)=h_5$ and $j(nu)equiv h_3pmod 2$, together with its differential lift $checknu$ with curvature $-frac{i}{480pi^3}operatorname{Tr}theta^5$. The class $muin E^6(BSU(N))$ with $k(mu)=c_3$ and $j(mu)equiv c_2pmod 2$ is the transgressed partner; the gauged construction needs a transgressing $5$-form $eta_E$ on the universal bundle.

Extended reading notes

Core claim

The paper's central claim is that the Wess-Zumino-Witten factor for pions on a closed spin four-manifold $X$ is $exp(2pi i int_X N_c,varphi^*checknu)$, where $checknu$ is a differential $E$-theory class on $G=SU(N)$ whose underlying topological class $nu$ satisfies $k(nu)=h_5$ and $j(nu)equiv h_3 pmod 2$. The author argues that this single definition supplies the spin dependence, the $mathbb{Z}/2mathbb{Z}$ grading of the quantum Hilbert space and the soliton statistics formula, and that its gauged version has anomaly $int_{X/S}N_c(checkmu_1(Q)-checkmu_2(Q))$, matching the anomaly of gauged QCD. The supporting theorem identifies the determinant line bundle of a family of chiral Dirac operators on spin four-manifolds with $int_{X/S}checkmu(V)$ in differential $E$-theory. From this the paper concludes that the Standard Model representation has vanishing $checkmu$, hence no global anomalies in arbitrary gravitational backgrounds.

Load-bearing premise

The load-bearing premise is that the transgressing $5$-form in equation (5.16) really has the three properties the paper lists, since the paper asserts them and leaves the check to the reader; the gauged WZW factor's well-definedness is also only sketched at the end of §5.

Editorial extensions

If this is right

The WZW factor is well-defined on any closed spin four-manifold without requiring a bounding five-manifold, and its dependence on the spin structure is fixed by $N_c,deltasmilevarphi^*h_3[X]$.
Canonical quantization on a compact spin three-manifold produces a $mathbb{Z}/2mathbb{Z}$-graded Hilbert space whose odd states are fermionic, with the grading given by $N_c$ times the mod 2 degree of the map to $G$.
The gauged WZW anomaly equals $int_{X/S}N_c(checkmu_1(Q)-checkmu_2(Q))$, which is the same anomaly computed in gauged QCD, realizing 't Hooft anomaly matching.
Anomaly-free subgroups of the flavor symmetry can be characterized topologically by the vanishing of $(Biota)^*(mu_1-mu_2)$ in $E^6(BK)$.
The Standard Model representation has vanishing $checkmu$, so the paper concludes there are no global anomalies in the Standard Model in arbitrary gravitational backgrounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The determinant-line-bundle formula in differential $E$-theory could serve as a general anomaly calculator for other four-dimensional chiral gauge theories whose representation has determinant one.
The same construction may extend to other symmetry-breaking patterns with a five-dimensional characteristic class, provided the relevant homogeneous space carries an analogous class $nu$ and a transgressing differential cochain.
The spin-structure dependence of the WZW factor is a consistency constraint on any ultraviolet completion: the low-energy theory is only defined on spin manifolds, so the high-energy theory must supply that spin structure.
The anomaly-free subgroup criterion is a purely topological filter that could be applied to gauged flavor symmetries beyond the QCD example, separating subgroups that are safe to gauge from those that are not.

Original abstract

The Wess-Zumino-Witten term was first introduced in the low energy sigma-model which describes pions, the Goldstone bosons for the broken flavor symmetry in quantum chromodynamics. We introduce a new definition of this term in arbitrary gravitational backgrounds. It matches several features of the fundamental gauge theory, including the presence of fermionic states and the anomaly of the flavor symmetry. To achieve this matching we use a certain generalized differential cohomology theory. We also prove a formula for the determinant line bundle of special families of Dirac operators on 4-manifolds in terms of this cohomology theory. One consequence is that there are no global anomalies in the Standard Model (in arbitrary gravitational backgrounds).

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,809 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.19 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.50 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The Wess-Zumino-Witten term was first introduced in the low energy sigma-model which describes pions, the Goldstone bosons for the broken flavor symmetry in quantum chromodynamics. We introduce a new definition of this term in arbitrary gravitational backgrounds. It matches several features of the fundamental gauge theory, including the presence of fermionic states and the anomaly of the flavor symmetry. To achieve this matching we use a certain generalized differential cohomology theory. We also prove a formula for the determinant line bundle of special families of Dirac operators on 4-manifolds in terms of this cohomology theory. One consequence is that there are no global anomalies in the Standard Model (in arbitrary gravitational backgrounds).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Wess-Zumino-Witten term was first introduced in the low energy sigma-model which describes pions, the Goldstone bosons for the broken flavor symmetry in quantum chromodynamics.	background 1.00	0.04	0.02	asserted 1.00
s1	We introduce a new definition of this term in arbitrary gravitational backgrounds.	contribution 1.00	0.95	0.07	asserted 1.00
s2	It matches several features of the fundamental gauge theory, including the presence of fermionic states and the anomaly of the flavor symmetry.	result 0.89	0.77	0.09	asserted 1.00
s3	To achieve this matching we use a certain generalized differential cohomology theory.	method 0.99	0.82	0.04	asserted 1.00
s4	We also prove a formula for the determinant line bundle of special families of Dirac operators on 4-manifolds in terms of this cohomology theory.	result 0.69	0.95	0.07	asserted 1.00
s5	One consequence is that there are no global anomalies in the Standard Model (in arbitrary gravitational backgrounds).	result 0.64	0.83	0.66	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (7)

[1]

[AS] M. F. Atiyah, G. Segal, Twisted K-theory, Ukr. Mat. Visn. 1 (2004), 287–330, arXiv:math.KT/0407054. [BF] J. M. Bismut, D. S. Freed, The analysis of elliptic families: Dirac operators, eta inva ri- ants, and the holonomy theorem of Witten , Commun. Math. Phys. 107 (1986), 103–16. [BT] R. Bott, L. Tu, Diﬀerential forms in algebraic topology, Springer-V...

View on Pith arXiv 2004
[135]

[HS] M. J. Hopkins, I. M. Singer, Quadratic functions in geometry, topology,and M-theory , J. Diﬀ. Geom. 70 (2005), 329–452, arXiv:math.AT/0211216. [J] J. A. Jenquin, Classical Chern-Simons on manifolds with spin structure , arXiv:math.DG/0504524. [Jo] D. L. Johnson, Chern-Simons forms on associated bundles, and boundary term s, arXiv:math/0601182. [MMN] ...

View on Pith arXiv 2005
[324]

[W4] E. Witten, On holomorphic factorization of WZW and coset models , Commun. Math. Phys. 144 (1992), 189–212. Department of Mathematics, University of Texas, 1 Universi ty Station C1200, Austin, TX 78712-0257 E-mail address : dafr@math.utexas.edu 32

View on Pith 1992
[1979]

(G. ’t Hooft et. al., eds.), Plenum, New York, 1980, p

View on Pith 1980
[1982]

[CS] S. S. Chern, J. Simons, Characteristic forms and geometric invariants , Ann. Math. 99 (1974), 48–69. [DK] J. F. Davis, P. Kirk, Lecture notes in algebraic topology , Graduate Studies in Mathemat- ics, Volume 35, American Mathematical Society, Providence , RI,

View on Pith 1974
[1996]

[WZ] J. Wess, B. Zumino, Consequences of anomalous Ward identities , Phys Lett. B 37 (1971), 95–97. [W1] E. Witten, Global aspects of current algebra , Nucl. Phys. B 223 (1983), 422–432. [W2] E. Witten, Current algebra, baryons, and quark conﬁnement , Nuc. Phys. B 223 (1983), 433–444. [W3] E. Witten, An SU (2) anomaly, Phys. Lett. B117 (1982),

View on Pith 1971
[2001]

[DZ] J. Distler, F. Zamora, Chiral symmetry breaking in the AdS/CFT correspondence , J. High Energy Physics (2000), no. 5, Paper 5, 25 pp., arXiv:hep-th/9911040. [DHZ] J. Dupont, R. Hain, S. Zucker, Regulators and characteristic classes of ﬂat bundles , The arithmetic and geometry of algebraic cycles, (Banﬀ, AB, 1 998), Amer. Math. Soc., Providence, RI, 2...

View on Pith Pith review arXiv 2000

Forward citations

Cited by 24 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The absence of global anomalies of CP symmetry
hep-ph 2026-02 conditional novelty 7.0 of 10

Gauged CP symmetry in four dimensions introduces no new global anomalies for connected, simply-connected gauge groups; the standard model matter content is anomaly-free under a gauged CP.
The Classification of 3+1d Symmetry Enriched Topological Order
math-ph 2025-09 conditional novelty 7.0 of 10

Finite-group-enriched 3+1d topological orders are classified by 2SVect-enriched G-crossed braided fusion 2-categories, with gauging obstructions living in SW^5(BG).
Symmetry theta angles and topological Witten effects
hep-th 2025-06 conditional novelty 7.0 of 10

Symmetry theta angles are intrinsic, Lagrangian-independent parameters of any non-anomalous Abelian invertible symmetry, and their universal effect is a topological Witten effect that shuffles twisted sectors while pr...
Anomaly Matching in 6d $\mathcal{N}=(2,0)$ SCFTs from M5 Cobordism
hep-th 2025-01 conditional novelty 7.0 of 10

M5 bordism classifies the invertible phases of 6d (2,0) anomaly theories, and the Hopf-Wess-Zumino term is the phase that transgresses to the second Pontryagin class.
SymTFTs and Non-Invertible Symmetries of 6d (2,0) SCFTs of Type $D$ from M-theory
hep-th 2024-12 conditional novelty 7.0 of 10

The 7d SymTFT for 6d (2,0) D_N SCFTs is derived from M-theory on AdS7 × RP4, including the outer-automorphism Z2 sector, and is used to derive non-invertible symmetries and anomaly polynomials.

Show 19 more citing papers

Symmetry TFTs from String Theory
hep-th 2021-12 unverdicted novelty 7.0 of 10

Constructs Symmetry TFTs for M-theory compactifications by reducing the topological sector of 11d supergravity on the boundary of X using differential cohomology, with applications to 7d SYM and 5d SCFTs confirmed via...
Remarks on the Green-Schwarz terms of six-dimensional supergravity theories
hep-th 2018-08 conditional novelty 7.0 of 10

Six-dimensional supergravity anomalies cancel globally for gauge groups U(n), SU(n), Sp(n), and E8, and consistency forces the gravitational anomaly coefficient to be a characteristic element of the string charge lattice.
A Symmetry Breaking Scenario for QCD$_3$
hep-th 2017-06 conditional novelty 7.0 of 10

For 2k < N_f < N_*(N,k), SU(N)_k with N_f fundamental fermions is conjectured to break U(N_f) spontaneously to U(N_f/2+k) x U(N_f/2-k), with a Wess-Zumino term fixed by N.
Categorical Symmetry of the Standard Model from Gravitational Anomaly
hep-th 2023-02 conditional novelty 6.0 of 10

The Standard Model's gravitationally anomalous B-L symmetry survives as a noninvertible categorical symmetry, built by dressing the naive charge operator with a 3d topological field theory whose framing anomaly cancel...
Symmetry TFTs for 3d QFTs from M-theory
hep-th 2022-10 conditional novelty 6.0 of 10

Reducing M-theory on a 7-manifold with differential cohomology yields the Symmetry TFT for 3d QFTs, giving BF-couplings, global-form constraints, and a geometrically derived 1-form anomaly for ABJ(M) and Y^{p,k}(CP^2)...
Gauge Enhanced Quantum Criticality Between Grand Unifications: Categorical Higher Symmetry Retraction
hep-th 2021-11 conditional novelty 6.0 of 10

The Standard Model may sit near a quantum critical region shared by four grand-unified models, and a proposed non-invertible symmetry of that region is retracted by the full Spin(10) group.
Anomaly and Cobordism Constraints Beyond the Standard Model: Topological Force
hep-th 2020-06 conditional novelty 6.0 of 10

The Standard Model with a preserved discrete X symmetry and 15 Weyl fermions per generation carries a Z16 global anomaly that requires either a right-handed neutrino or a new hidden gapped topological sector.
Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena
cond-mat.str-el 2020-04 accept novelty 6.0 of 10

Quantized topological terms in the parameter-space effective action protect gapless diabolical points in phase diagrams and force boundary gapless points whose invariants sum to the bulk value.
Global anomalies in the Standard Model(s) and Beyond
hep-th 2019-10 accept novelty 6.0 of 10

Direct bordism computations show that all four Standard Model gauge groups, and the BSM extensions considered, are free of new global anomalies for arbitrary fermion content.
Anomaly Inflow and the $\eta$-Invariant
hep-th 2019-09 accept novelty 6.0 of 10

The paper derives the nonperturbative anomaly inflow formula Z = |Pf D+| exp(-iπη/2) from the Euclidean path integral, justifying a result previously expected from the Dai-Freed theorem.
A Non-Perturbative Definition of the Standard Models
hep-th 2018-09 conditional novelty 6.0 of 10

Cobordism classification of Spin(10) and SU(5) anomalies shows the mirror sector of the corresponding chiral fermion theories can be gapped, giving the Standard Model a non-perturbative lattice definition.
Dai-Freed anomalies in particle physics
hep-th 2018-07 conditional novelty 6.0 of 10

Bordism-based Dai-Freed anomaly checks leave the Standard Model and its unified extensions anomaly-free, but impose a mod-9 constraint on baryon triality that only three generations satisfy.
Quantization of anomaly coefficients in 6D $\mathcal{N}=(1,0)$ supergravity
hep-th 2017-11 conditional novelty 6.0 of 10

The anomaly coefficients of 6D N=(1,0) supergravity must lie in 2H^4(BG;Z) ⊗ Λ_S, a constraint realized in F-theory and used to identify the global structure of the gauge group from geometry.
M2-branes on M-folds
hep-th 2008-04 conditional novelty 6.0 of 10

The Bagger-Lambert A4 theory at level k has moduli space (R^8 x R^8)/D_{2k}, matching two M2-branes on a Z_{2k} orbifold.
Perturbiner methods in scattering amplitudes
hep-th 2026-07 accept novelty 5.5 of 10

Perturbiner multi-particle solutions of classical field equations generate Berends–Giele currents and tree-level amplitudes across scalars, gauge theory, gravity, NLSM, AdS, and one-loop integrands, including several ...
A Note On Lattice Definition of Gauge Theories With Chiral Fermions
hep-lat 2026-11 conditional novelty 5.0 of 10

A lattice regularization of chiral gauge theories is defined by coupling continuum chiral fermions to interpolated admissible lattice gauge fields; the resulting non-positive measure makes it implementable in principl...
Topological symmetry in quantum field theory
hep-th 2022-09 unverdicted novelty 5.0 of 10

Authors introduce a TFT-based framework for finite topological symmetries in QFT, including gauging, condensation defects, and duality defects, with an appendix on finite homotopy theories.
Revisiting Wess-Zumino-Witten terms
hep-th 2020-08 accept novelty 5.0 of 10

The 4d Wess-Zumino-Witten terms for SU and SO QCD require a spin structure to be well-defined, and the mixed anomaly of the Z2 1-form symmetry of SO QCD is reproduced in the low-energy sigma model coupled to a Z2 gaug...
Notes on generalized global symmetries in QFT
hep-th 2015-08 conditional novelty 4.0 of 10

Higher-form symmetries of quantum field theories are recast as special cases of 2-groups and higher groups, with examples and a conjecture that anomalies transmute ordinary symmetries into higher-group symmetries.

How the literature uses it

45 arXiv papers cite this in 90 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 67 sentences without a confident label

Contests it 1 paper

Moreover, there exists a more modern diagnosis of WZW terms by cobordism groups [ref] , but this classification also fails for the usual choices of cobordism, as we discuss in [sec42]Section 4.2.

Anomaly Matching in 6d $\mathcal{N}=(2,0)$ SCFTs from M5 Cobordism · Introduction · contest 0.85

Builds on it 1 paper

Additionally, this complements the results of the WZW term in the cases of $SU$ QCD and $SO$ QCD [ref] .

Wess-Zumino-Witten Terms of $Sp$ QCD by Bordism Theory · Introduction · 0.85

Compares with it 2 papers

We will find that both the $SU(5)$ and $Spin(10)$ GUT's, as well as the Standard Model itself, are free from Dai-Freed anomalies.[This result was previously obtained for $$ manifolds in [ref] using different techniques.

Dai-Freed anomalies in particle physics · Introduction · 0.80
This is completely parallel to the appearance of generalized cohomology theories in the Wess-Zumino terms of the non-linear sigma model describing pions in QCD [ref] .

Quantization of anomaly coefficients in 6D $\mathcal{N}=(1,0)$ supergravity · Self-duality condition and the modulus · 0.90

Cites it as background 16 papers

topological terms are classified up to continuous ones by a group $Inv_^d(X)$ [ref] , which sits in the middle of the short exact sequence [equation] In addition, ordinary theta angles invisible in $Inv_^d(X)$ are classified by $Hom(Ω_d^(X),U(1))$ along with discrete theta angles [ref] .

Topological terms of (2+1)d flag-manifold sigma models · Introduction and Summary · 0.85
In Sec. [ref], we review the case of $SU$ QCD as given by [ref] .

Wess-Zumino-Witten Terms of $Sp$ QCD by Bordism Theory · Introduction · 0.80
But there are two more common choices for fermionic phases: restricted supercohomology $_1$ as introduced by Freed [ref] and Gu-Wen [ref] , and extended supercohomology $_2$ as defined by Kapustin-Thorngren [ref] and Wang-Gu [ref] .

Weak topological phases in the presence of interactions · The free-to-interacting map constrains the spectrum of SPT phases · 0.95

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

[AS] M. F. Atiyah, G. Segal, Twisted K-theory, Ukr. Mat. Visn. 1 (2004), 287–330, arXiv:math.KT/0407054. [BF] J. M. Bismut, D. S. Freed, The analysis of elliptic families: Dirac operators, eta inva ri- ants, and the holonomy theorem of Witten , Commun. Math. Phys. 107 (1986), 103–16. [BT] R. Bott, L. Tu, Diﬀerential forms in algebraic topology, Springer-V...

View on Pith arXiv 2004

[2] [135]

[HS] M. J. Hopkins, I. M. Singer, Quadratic functions in geometry, topology,and M-theory , J. Diﬀ. Geom. 70 (2005), 329–452, arXiv:math.AT/0211216. [J] J. A. Jenquin, Classical Chern-Simons on manifolds with spin structure , arXiv:math.DG/0504524. [Jo] D. L. Johnson, Chern-Simons forms on associated bundles, and boundary term s, arXiv:math/0601182. [MMN] ...

View on Pith arXiv 2005

[3] [324]

[W4] E. Witten, On holomorphic factorization of WZW and coset models , Commun. Math. Phys. 144 (1992), 189–212. Department of Mathematics, University of Texas, 1 Universi ty Station C1200, Austin, TX 78712-0257 E-mail address : dafr@math.utexas.edu 32

View on Pith 1992

[4] [1979]

(G. ’t Hooft et. al., eds.), Plenum, New York, 1980, p

View on Pith 1980

[5] [1982]

[CS] S. S. Chern, J. Simons, Characteristic forms and geometric invariants , Ann. Math. 99 (1974), 48–69. [DK] J. F. Davis, P. Kirk, Lecture notes in algebraic topology , Graduate Studies in Mathemat- ics, Volume 35, American Mathematical Society, Providence , RI,

View on Pith 1974

[6] [1996]

[WZ] J. Wess, B. Zumino, Consequences of anomalous Ward identities , Phys Lett. B 37 (1971), 95–97. [W1] E. Witten, Global aspects of current algebra , Nucl. Phys. B 223 (1983), 422–432. [W2] E. Witten, Current algebra, baryons, and quark conﬁnement , Nuc. Phys. B 223 (1983), 433–444. [W3] E. Witten, An SU (2) anomaly, Phys. Lett. B117 (1982),

View on Pith 1971

[7] [2001]

[DZ] J. Distler, F. Zamora, Chiral symmetry breaking in the AdS/CFT correspondence , J. High Energy Physics (2000), no. 5, Paper 5, 25 pp., arXiv:hep-th/9911040. [DHZ] J. Dupont, R. Hain, S. Zucker, Regulators and characteristic classes of ﬂat bundles , The arithmetic and geometry of algebraic cycles, (Banﬀ, AB, 1 998), Amer. Math. Soc., Providence, RI, 2...

View on Pith Pith review arXiv 2000