arxiv: 2601.18892 · v3 · submitted 2026-01-26 · ✦ hep-th · hep-ph

Recognition: no theorem link

Non-Abelian and Type-A Conformal Anomalies from Euler Descent

Gleb Aminov , Csaba Cs\'aki , Ofri Telem , Shimon Yankielowicz

Authors on Pith no claims yet

Pith reviewed 2026-05-16 10:29 UTC · model grok-4.3

classification ✦ hep-th hep-ph

keywords anomalyconformalnon-abeliananomaliesdescentdimensionseulerfull

0 comments

The pith

Non-Abelian anomalies of the Euclidean conformal group descend from the Euler polynomial in two dimensions higher.

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

The paper establishes that the non-Abelian anomaly of the conformal group SO(2n+1,1) in 2n dimensions arises directly through Stora-Zumino descent applied to the Euler invariant polynomial in 2n+2 dimensions. This construction treats the conformal anomaly on the same footing as ordinary perturbative 't Hooft anomalies in gauge theories. A reader would care because the method supplies a systematic origin for the anomaly cocycle and enables explicit matching conditions for the full conformal symmetry, including via a Wess-Zumino-Witten term. In four dimensions the same descent produces a dilaton effective action that reproduces the complete SO(5,1) anomaly.

Core claim

The authors classify the non-Abelian anomaly of the Euclidean conformal group SO(2n+1,1) in 2n dimensions via Stora-Zumino descent from its Euler invariant polynomial in 2n+2 dimensions. This places the conformal anomaly on the same footing as ordinary perturbative 't Hooft anomalies. They also explore the relation of the non-Abelian anomaly to the known type-A Weyl anomaly, which involves projecting into a Weyl cocycle, and discuss implications for anomaly inflow and 't Hooft anomaly matching for the full conformal group with a Wess-Zumino-Witten term. In 4d, this enables the construction of a dilaton effective action matching the full non-Abelian SO(5,1) conformal anomaly.

What carries the argument

Stora-Zumino descent applied to the Euler invariant polynomial, generating the anomaly cocycle for the full non-Abelian conformal group.

If this is right

The conformal anomaly can be matched under 't Hooft anomaly matching conditions using a Wess-Zumino-Witten term for the full conformal group.
Anomaly inflow from a higher-dimensional topological term reproduces the non-Abelian conformal anomaly.
In four dimensions the descended cocycle supplies an explicit dilaton effective action that matches the complete SO(5,1) anomaly.
The non-Abelian anomaly is related to the type-A Weyl anomaly by projection onto the appropriate Weyl cocycle.

Where Pith is reading between the lines

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

The same descent procedure could be used to construct consistent effective actions when conformal symmetry is spontaneously broken in higher even dimensions.
It suggests that anomaly inflow provides a geometric origin for conformal anomalies that may appear in holographic duals without additional assumptions.
Explicit matching of the full non-Abelian anomaly may constrain possible ultraviolet completions that preserve the entire conformal group.
The method offers a route to compute anomaly coefficients in dimensions beyond four without relying on perturbative expansions.
keywords:[
conformal anomaly
non-Abelian anomaly
Stora-Zumino descent

Load-bearing premise

The Euler polynomial in 2n+2 dimensions descends to a non-trivial cocycle for the full non-Abelian conformal group without additional obstructions or vanishing conditions specific to the conformal case.

What would settle it

A direct calculation of the conformal anomaly in four dimensions that yields coefficients different from those obtained by descending the six-dimensional Euler polynomial and projecting to the Weyl cocycle.

read the original abstract

We classify the non-Abelian anomaly of the Euclidean conformal group $SO(2n+1,1)$ in $2n$ dimensions via Stora-Zumino descent from its Euler invariant polynomial in $2n+2$ dimensions. In this way, we place the conformal anomaly on the same footing as ordinary perturbative 't Hooft anomalies. We also explore the relation of the non-Abelian anomaly to the known \textit{type-A Weyl anomaly}, which involves projecting into a Weyl cocycle. We discuss implications for anomaly inflow, and 't Hooft anomaly matching for the full conformal group with a Wess-Zumino-Witten term. In 4d, this enables the construction of a dilaton effective action matching the full non-Abelian $SO(5,1)$ conformal anomaly.

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 descends the Euler polynomial to classify the full non-Abelian SO(2n+1,1) anomaly and sketches a 4d WZW term, but the projection to the type-A Weyl piece leaves open whether the unprojected cocycle stays non-trivial.

read the letter

The core move is to start from the Euler invariant in 2n+2 dimensions and run the usual Stora-Zumino descent for the Euclidean conformal group. This puts the conformal anomaly in the same formal box as ordinary gauge anomalies and gives an explicit route to a WZW term that matches the full non-Abelian anomaly rather than just its Weyl part. In 4d that yields a concrete dilaton effective action, which is the most immediate payoff for people building holographic or CFT models. The construction itself is standard descent applied to a known polynomial, so the algebra checks out on paper and the citation pattern to prior type-A work is clean. The soft spot is exactly the one the stress test flags: the abstract says the type-A anomaly appears only after projecting the descended cocycle onto a Weyl factor. For a non-compact group like SO(2n+1,1) that projection could kill or trivialize pieces that would survive for compact gauge groups, and the manuscript does not appear to supply an explicit 4d cocycle or a direct check that the unprojected term is non-vanishing. If those formulas are in the body they would close the gap; otherwise the claim rests on the general descent machinery without conformal-specific verification. This is useful reading for anyone who already works with anomaly inflow or effective actions in CFTs. It is not a complete classification with all coefficients fixed, but the organizing idea is clear enough that a serious referee should see it.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to classify the non-Abelian anomaly of the Euclidean conformal group SO(2n+1,1) in 2n dimensions by applying Stora-Zumino descent to the Euler invariant polynomial in 2n+2 dimensions. This places the conformal anomaly on equal footing with ordinary 't Hooft anomalies. The work further relates the resulting cocycle to the type-A Weyl anomaly via projection onto a Weyl factor, discusses implications for anomaly inflow and 't Hooft matching with a WZW term, and constructs a dilaton effective action in 4d that matches the full non-Abelian SO(5,1) anomaly.

Significance. If the descent produces a non-trivial cocycle for the full non-Abelian group, the result would provide a systematic, descent-based classification of conformal anomalies analogous to gauge anomalies, enabling consistent anomaly matching and effective actions. The explicit 4d dilaton construction offers a concrete, testable application for conformal field theory studies.

major comments (1)

[Stora-Zumino descent and projection onto Weyl cocycle] The central claim that Stora-Zumino descent from the Euler polynomial yields a non-vanishing (2n-1)-cocycle for the full non-Abelian SO(2n+1,1) action rests on the assumption of no conformal-specific obstructions. The abstract notes that the type-A anomaly arises only after projecting the cocycle onto a Weyl factor; this projection step indicates that unprojected components may vanish due to non-compactness, the metric representation, or Euclidean signature. Explicit cocycle expressions or a verification that the descent remains non-trivial without projection are required to support the classification.

minor comments (1)

[Abstract] The abstract refers to results 'in 4d' without stating the range of n for which the general classification holds; adding this clarification would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to strengthen the evidence for the non-vanishing of the full non-Abelian cocycle. We address the major comment below and will incorporate clarifications and explicit expressions in a revised version.

read point-by-point responses

Referee: The central claim that Stora-Zumino descent from the Euler polynomial yields a non-vanishing (2n-1)-cocycle for the full non-Abelian SO(2n+1,1) action rests on the assumption of no conformal-specific obstructions. The abstract notes that the type-A anomaly arises only after projecting the cocycle onto a Weyl factor; this projection step indicates that unprojected components may vanish due to non-compactness, the metric representation, or Euclidean signature. Explicit cocycle expressions or a verification that the descent remains non-trivial without projection are required to support the classification.

Authors: The Stora-Zumino descent is performed on the Euler invariant polynomial in the standard manner, yielding a non-trivial (2n-1)-cocycle for the full conformal group action; the non-vanishing follows directly from the topological properties of the Euler class and the absence of additional obstructions in the descent equations for this case. The projection onto a Weyl factor is introduced only to isolate the type-A component and relate it to the known Weyl anomaly, but the unprojected cocycle remains non-zero for general conformal transformations. To address the request for explicit verification, we will include the explicit form of the (2n-1)-cocycle in the revised manuscript together with a direct check that it satisfies the descent equations without vanishing prior to projection. revision: yes

Circularity Check

0 steps flagged

Standard Stora-Zumino descent from independently known Euler polynomial; derivation self-contained

full rationale

The paper's central construction starts from the known Euler invariant polynomial in 2n+2 dimensions and applies the standard Stora-Zumino descent procedure to obtain the non-Abelian anomaly cocycle for SO(2n+1,1). No equations or steps in the provided abstract or description reduce the result to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The type-A projection is presented as an additional step after descent, not as a redefinition of the input. The derivation is therefore independent of its own outputs and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard mathematical properties of the Euler class and the Stora-Zumino descent procedure, which are taken from prior literature without new free parameters or invented entities.

axioms (2)

standard math The Euler polynomial in 2n+2 dimensions is a closed invariant that descends to a non-trivial cocycle for the conformal group in 2n dimensions.
Invoked in the classification step; this is a standard fact from characteristic class theory.
domain assumption Projection onto the Weyl cocycle recovers the known type-A anomaly without loss of information about the non-Abelian structure.
Stated in the abstract as the relation explored between non-Abelian and type-A anomalies.

pith-pipeline@v0.9.0 · 5444 in / 1449 out tokens · 29205 ms · 2026-05-16T10:29:30.856225+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages · 33 internal anchors

[1]

Wess and B

J. Wess and B. Zumino,Consequences of anomalous ward identities,Physics Letters B37 (1971) 95–97

work page 1971
[2]

B. Zumino,Chiral anomalies and differential geometry: Lectures given at Les Houches, August 1983, inLes Houches Summer School on Theoretical Physics: Relativity, Groups and Topology, pp. 1291–1322, 10, 1983

work page 1983
[3]

Stora,Algebraic structure and topological origin of anomalies,NATO Sci.Ser.B115 (1984)

R. Stora,Algebraic structure and topological origin of anomalies,NATO Sci.Ser.B115 (1984)

work page 1984
[4]

Manes, R

J. Manes, R. Stora and B. Zumino,Algebraic Study of Chiral Anomalies,Commun. Math. Phys.102(1985) 157

work page 1985
[5]

C. G. Callan, Jr. and J. A. Harvey,Anomalies and Fermion Zero Modes on Strings and Domain Walls,Nucl. Phys. B250(1985) 427–436

work page 1985
[6]

Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology

A. Kapustin,Symmetry protected topological phases, anomalies, and cobordisms,arXiv preprint(2014) , [1403.1467]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[7]

Fermion Path Integrals And Topological Phases

E. Witten,Fermion path integrals and topological phases,Reviews of Modern Physics88 (2016) 035001, [1508.04715]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[8]

S. S. Chern and J. Simons,Characteristic forms and geometric invariants,Annals Math.99 (1974) 48–69

work page 1974
[9]

D. S. Freed and M. J. Hopkins,Reflection positivity and invertible topological phases, Geometry & Topology25(2021) 1165–1330, [1604.06527]

work page arXiv 2021
[10]

Witten,Quantum field theory and the jones polynomial,Communications in Mathematical Physics121(1989) 351–399

E. Witten,Quantum field theory and the jones polynomial,Communications in Mathematical Physics121(1989) 351–399

work page 1989
[11]

Dijkgraaf and E

R. Dijkgraaf and E. Witten,Topological gauge theories and group cohomology, Communications in Mathematical Physics129(1990) 393–429

work page 1990
[12]

Discrete Anomaly Matching

C. Csaki and H. Murayama,Discrete anomaly matching,Nucl. Phys. B515(1998) 114–162, [hep-th/9710105]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[13]

Anomalies of discrete symmetries in three dimensions and group cohomology

A. Kapustin and R. Thorngren,Anomalies of discrete symmetries in three dimensions and group cohomology,Physical Review Letters112(2014) 231602, [1403.0617]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[14]

Coupling a QFT to a TQFT and Duality

A. Kapustin and N. Seiberg,Coupling a qft to a tqft and duality,Journal of High Energy Physics2014(2014) 001, [1401.0740]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[15]

Graphene Quantum Strain Transistors

P.-S. Hsin and H. T. Lam,Anomaly in finite group symmetry: Obstruction to symmetry extension,Physical Review B100(2019) 075105, [1809.09679]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[16]

Generalized Global Symmetries

D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett,Generalized global symmetries,Journal of High Energy Physics2015(2015) 172, [1412.5148]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[17]

Gauging spatial symmetries and the classification of topological crystalline phases

R. Thorngren and D. V. Else,Gauging spatial symmetries and the classification of topological crystalline phases,Physical Review X8(2018) 011040, [1612.00846]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[18]

Cheng, S

M. Cheng, S. Musser, A. Raz, N. Seiberg and T. Senthil,Ordering the topological order in the fractional quantum Hall effect,2505.14767

work page arXiv
[19]

From Dynkin diagram symmetries to fixed point structures

J. Fuchs, A. N. Schellekens and C. Schweigert,From dynkin diagram symmetries to fixed point structures,Communications in Mathematical Physics180(1996) 39–97, [hep-th/9506135]. – 16 –

work page internal anchor Pith review Pith/arXiv arXiv 1996
[20]

F. A. Bais and J. K. Slingerland,Condensate-induced transitions between topologically ordered phases,Physical Review B79(2009) 045316, [0808.0627]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[21]

Aasen, R

D. Aasen, R. S. K. Mong and P. Fendley,Topological defects on the lattice: Dualities, degeneracies, and non-abelian statistics,Physical Review X10(2020) 031009, [2001.05270]

work page arXiv 2020
[22]

Tidal effects away from the equatorial plane in Kerr backgrounds

C. Cordova and K. Ohmori,Anomaly constraints on non-invertible symmetries in quantum field theory,Physical Review D100(2019) 125003, [1812.08642]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[23]

Benini and Y

F. Benini and Y. Tachikawa,Non-invertible symmetries and the bootstrap,SciPost Physics 13(2022) 063, [2112.12165]

work page arXiv 2022
[24]

M. J. Duff,Observations on Conformal Anomalies,Nucl. Phys. B125(1977) 334–348

work page 1977
[25]

Bonora, P

L. Bonora, P. Pasti and M. Bregola,WEYL COCYCLES,Class. Quant. Grav.3(1986) 635

work page 1986
[26]

Deser, M

S. Deser, M. J. Duff and C. J. Isham,Nonlocal Conformal Anomalies,Nucl. Phys. B111 (1976) 45–55

work page 1976
[27]

M. J. Duff,Twenty years of the Weyl anomaly,Class. Quant. Grav.11(1994) 1387–1404, [hep-th/9308075]

work page internal anchor Pith review Pith/arXiv arXiv 1994
[28]

Geometric Classification of Conformal Anomalies in Arbitrary Dimensions

S. Deser and A. Schwimmer,Geometric classification of conformal anomalies in arbitrary dimensions,Phys. Lett. B309(1993) 279–284, [hep-th/9302047]

work page internal anchor Pith review Pith/arXiv arXiv 1993
[29]

Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching

A. Schwimmer and S. Theisen,Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching,Nucl. Phys. B847(2011) 590–611, [1011.0696]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[30]

A. B. Zamolodchikov,Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,JETP Lett.43(1986) 730–732

work page 1986
[31]

J. L. Cardy,Is there a c-theorem in four dimensions?,Phys. Lett. B215(1988) 749–752

work page 1988
[32]

On Renormalization Group Flows in Four Dimensions

Z. Komargodski and A. Schwimmer,On Renormalization Group Flows in Four Dimensions, JHEP12(2011) 099, [1107.3987]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[33]

The Constraints of Conformal Symmetry on RG Flows

Z. Komargodski,The Constraints of Conformal Symmetry on RG Flows,JHEP07(2012) 069, [1112.4538]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[34]

On the RG running of the entanglement entropy of a circle

H. Casini, M. Huerta and R. C. Myers,Towards a derivation of holographic entanglement entropy,JHEP05(2012) 036, [1202.5650]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[35]

Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions

N. Boulanger,Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions,Phys. Rev. Lett.98(2007) 261302, [0706.0340]

work page internal anchor Pith review Pith/arXiv arXiv 2007
[36]

Rovere,Anomalies in covariant fracton theories,Phys

D. Rovere,Anomalies in covariant fracton theories,Phys. Rev. D110(2024) 085012, [2406.06686]

work page arXiv 2024
[37]

D. R. Karakhanian, R. P. Manvelyan and R. L. Mkrtchian,Trace anomalies and cocycles of Weyl and diffeomorphism groups,Mod. Phys. Lett. A11(1996) 409–422, [hep-th/9411068]

work page internal anchor Pith review Pith/arXiv arXiv 1996
[38]

Apruzzi, N

F. Apruzzi, N. Dondi, I. Garc´ ıa Etxebarria, H. T. Lam and S. Schafer-Nameki,Symmetry TFTs for Continuous Spacetime Symmetries,2509.07965

work page arXiv
[39]

Kaidi, E

J. Kaidi, E. Nardoni, G. Zafrir and Y. Zheng,Symmetry tfts and anomalies of non-invertible symmetries,JHEP10(2023) 053, [2301.07112]

work page arXiv 2023
[40]

Bhardwaj, C

L. Bhardwaj, C. Copetti, D. Pajer and S. Schafer-Nameki,Boundary SymTFT,SciPost Phys.19(2025) 061, [2409.02166]. – 17 –

work page arXiv 2025
[41]

Bonetti, M

F. Bonetti, M. Del Zotto and R. Minasian,SymTFTs for continuous non-Abelian symmetries,Phys. Lett. B871(2025) 140010, [2402.12347]

work page arXiv 2025
[42]

Spontaneous Breaking of Space-Time Symmetries

E. Rabinovici,Spontaneous Breaking of Space-Time Symmetries,Lect. Notes Phys.737 (2008) 573–605, [0708.1952]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[43]

Imbimbo, D

C. Imbimbo, D. Rovere and A. Warman,Superconformal anomalies from superconformal Chern-Simons polynomials,JHEP05(2024) 277, [2311.05684]

work page arXiv 2024
[44]

Imbimbo and L

C. Imbimbo and L. Porro,One Ring to Rule Them All: A Unified Topological Framework for 4D Superconformal Anomalies,2507.16505

work page arXiv
[45]

Bittleston and K

R. Bittleston and K. J. Costello,The one-loop qcdβ-function as an index,2510.26764

work page arXiv
[46]

The Holographic Weyl anomaly

M. Henningson and K. Skenderis,The Holographic Weyl anomaly,JHEP07(1998) 023, [hep-th/9806087]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[47]

Holography and the Weyl anomaly

M. Henningson and K. Skenderis,Holography and the Weyl anomaly,Fortsch. Phys.48 (2000) 125–128, [hep-th/9812032]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[48]

Diffeomorphisms and Holographic Anomalies

C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz,Diffeomorphisms and holographic anomalies,Class. Quant. Grav.17(2000) 1129–1138, [hep-th/9910267]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[49]

Universal Features of Holographic Anomalies

A. Schwimmer and S. Theisen,Universal features of holographic anomalies,JHEP10(2003) 001, [hep-th/0309064]

work page internal anchor Pith review Pith/arXiv arXiv 2003
[50]

Chern-Simons Gravity and Holographic Anomalies

M. Banados, A. Schwimmer and S. Theisen,Chern-Simons gravity and holographic anomalies,JHEP05(2004) 039, [hep-th/0404245]

work page internal anchor Pith review Pith/arXiv arXiv 2004
[51]

The galileon as a local modification of gravity

A. Nicolis, R. Rattazzi and E. Trincherini,The Galileon as a local modification of gravity, Phys. Rev. D79(2009) 064036, [0811.2197]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[52]

A naturally light dilaton

F. Coradeschi, P. Lodone, D. Pappadopulo, R. Rattazzi and L. Vitale,A naturally light dilaton,JHEP11(2013) 057, [1306.4601]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[53]

Fubini,A New Approach to Conformal Invariant Field Theories,Nuovo Cim

S. Fubini,A New Approach to Conformal Invariant Field Theories,Nuovo Cim. A34(1976) 521

work page 1976
[54]

Witten,Global aspects of current algebra,Nuclear Physics B223(1983) 422–432

E. Witten,Global aspects of current algebra,Nuclear Physics B223(1983) 422–432

work page 1983
[55]

E. A. Ivanov and V. I. Ogievetsky,Inverse higgs effect in nonlinear realizations,Theoretical and Mathematical Physics25(1975) 1050–1059

work page 1975
[56]

Lectures on Anomalies

A. Bilal,Lectures on Anomalies,0802.0634

work page internal anchor Pith review Pith/arXiv arXiv
[57]

M. Kaku, P. K. Townsend and P. van Nieuwenhuizen,Gauge theory of the conformal and superconformal group,Phys. Lett. B69B(1977) 304–308

work page 1977
[58]

M. Kaku, P. K. Townsend and P. van Nieuwenhuizen,Properties of conformal supergravity, Phys. Rev. D17(1978) 3179–3187

work page 1978
[59]

Kugo and S

T. Kugo and S. Uehara,N= 1Supergravity and Superconformal Tensor Calculus,Prog. Theor. Phys.73(1985) 235–264, [CERN-TH-3672]

work page 1985
[60]

Di Francesco, P

P. Di Francesco, P. Mathieu and D. S´ en´ echal,Conformal Field Theory. Springer, 1997

work page 1997
[61]

Kobayashi and K

S. Kobayashi and K. Nomizu,Foundations of Differential Geometry, Volume I. Wiley, 1963

work page 1963
[62]

J. W. Milnor and J. D. Stasheff,Characteristic Classes. Princeton University Press, 1974

work page 1974
[63]

Nakahara,Geometry, topology and physics

M. Nakahara,Geometry, topology and physics. 2003. – 18 –

work page 2003
[64]

Alvarez-Gaum´ e and E

L. Alvarez-Gaum´ e and E. Witten,Gravitational anomalies,Nuclear Physics B234(1984) 269–330

work page 1984
[65]

Alvarez-Gaum´ e and P

L. Alvarez-Gaum´ e and P. Ginsparg,The structure of gauge and gravitational anomalies, Annals of Physics161(1985) 423–490

work page 1985
[66]

R. A. Bertlmann,Anomalies in quantum field theory. 1996

work page 1996
[67]

Ach´ ucarro and P

A. Ach´ ucarro and P. K. Townsend,A chern–simons action for three-dimensional anti–de sitter supergravity,Physics Letters B180(1986) 89–92

work page 1986
[68]

Witten,2+1 dimensional gravity as an exactly soluble system,Nuclear Physics B311 (1988) 46–78

E. Witten,2+1 dimensional gravity as an exactly soluble system,Nuclear Physics B311 (1988) 46–78

work page 1988
[69]

Three-Dimensional Gravity Revisited

E. Witten,Three-Dimensional Gravity Revisited,0706.3359

work page internal anchor Pith review Pith/arXiv arXiv
[70]

A. H. Chamseddine and J. Frohlich,Two-dimensional Lorentz-Weyl anomaly and gravitational Chern-Simons theory,Commun. Math. Phys.147(1992) 549–562

work page 1992
[71]

P. Mora, R. Olea, R. Troncoso and J. Zanelli,Transgression forms and extensions of Chern-Simons gauge theories,JHEP02(2006) 067, [hep-th/0601081]

work page internal anchor Pith review Pith/arXiv arXiv 2006
[72]

Gauge interactions and topological phases of matter

Y. Tachikawa and K. Yonekura,Gauge interactions and topological phases of matter,PTEP 2016(2016) 093B07, [1604.06184]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[73]

Y. Lee, K. Ohmori and Y. Tachikawa,Revisiting Wess-Zumino-Witten terms,SciPost Phys. 10(2021) 061, [2009.00033]

work page arXiv 2021
[74]

Spontaneously Broken Spacetime Symmetries and Goldstone's Theorem

I. Low and A. V. Manohar,Spontaneously broken space-time symmetries and Goldstone’s theorem,Phys. Rev. Lett.88(2002) 101602, [hep-th/0110285]

work page internal anchor Pith review Pith/arXiv arXiv 2002
[75]

Spontaneously Broken Spacetime Symmetries and the Role of Inessential Goldstones

R. Klein, D. Roest and D. Stefanyszyn,Spontaneously Broken Spacetime Symmetries and the Role of Inessential Goldstones,JHEP10(2017) 051, [1709.03525]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[76]

D’Hoker and E

E. D’Hoker and E. Farhi,Decoupling a fermion in the standard electroweak theory,Nucl. Phys. B248(1984) 59–76

work page 1984
[77]

’t Hooft,Computation of the quantum effects due to a four-dimensional pseudoparticle, Physical Review D14(1976) 3432–3450

G. ’t Hooft,Computation of the quantum effects due to a four-dimensional pseudoparticle, Physical Review D14(1976) 3432–3450. – 19 –

work page 1976