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Cited by 791Black Holes and Theoretical PhysicsRelativity and Gravitational TheoryNoncommutative and Quantum Gravity Theories

The DOI matches its Crossref record; 130 of the 130 references it cites were found in a registry or an index; none of the 45 sources its claims cite is checked for support yet.

Twenty years of the Weyl anomaly

Open access
Duff, M J·1994·Classical and Quantum Gravity
DOI 10.1088/0264-9381/11/6/004
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Abstract

In 1973 two Salam protégés (Derek Capper and the author) discovered that the conformal invariance under Weyl rescalings of the metric tensor displayed by classical massless-field systems in interaction with gravity no longer survives in the quantum theory. Since then these Weyl anomalies have found a variety of applications in black-hole physics, cosmology, string theory and statistical mechanics. We give a nostalgic review.

Overview

CoChat Summary

TL;DR

This is a retrospective review of the Weyl anomaly, twenty years after Capper and Duff found that classical Weyl invariance of massless fields coupled to gravity fails quantum mechanically. It surveys the anomaly’s form, its connection to Schwinger-DeWitt coefficients, and applications in black-hole physics, cosmology, supersymmetry, string theory, and statistical mechanics.

Key findings

  • Capper and Duff discovered in 1973 that the finite part of the two-point function violates the trace identity: \(n'_{\mu\nu}(\text{finite}) \neq 0\), so the classical Weyl invariance \(g_{\mu\nu} \to \Omega^2(x)g_{\mu\nu}\) does not survive quantization.
  • The trace anomaly takes dimension-dependent forms: for \(n=2\), \(g^{\alpha\beta}\langle T_{\alpha\beta}\rangle = aR\); for \(n=4\), \(g^{\alpha\beta}\langle T_{\alpha\beta}\rangle = \alpha R^2 + \beta R_{\mu\nu}R^{\mu\nu} + \gamma R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} + \delta \Box R + cF_{\mu\nu}F^{\mu\nu}\), with no anomaly for odd \(n\) at one loop ignoring boundary terms. These are the Schwinger-DeWitt \(b_4\) coefficients.
  • In string theory, Polyakov showed the critical dimensions correspond to absence of the two-dimensional Weyl anomaly: the bosonic-string anomaly is proportional to \(D-26\), and the fermionic-string anomaly to \(D-10\), giving critical dimensions \(D=26\) and \(D=10\).
  • The four-dimensional gravitational anomaly depends on two constants \(b\) and \(b'\); explicit calculations give \(b = \frac{1}{120(4\pi)^2}[N_s + 6N_F + 12N_V]\), and the combined anomaly cancels for the \(N=8\) and \(N=4\) supermultiplets.

Methods

  • Historical/nostalgic review of the Weyl anomaly over twenty years, synthesizing the author’s recollections and selected literature rather than presenting new primary data.
  • Discusses dimensional regularization, Ward identities, trace identities, general covariance, heat-kernel \(b\) coefficients, and applications across black holes, cosmology, supersymmetry, strings, and statistical mechanics.
  • No formal sample, dataset, or systematic review protocol is stated.

Limitations

  • The paper states uneasiness about the identity relating integrated trace anomalies to non-invariance under constant rescalings, because the Polyakov action is a counterexample; it also notes that the equivalence versus inequivalence of dual formulations remains controversial and that “the final word on this subject has still not been written.”
  • The paper states that attempts to generalize Zamolodchikov’s \(c\)-theorem to higher dimensions are “still inconclusive.”
  • Evident from the text: this is a retrospective review, not a systematic review, and some equations/names are garbled in the provided OCR, limiting independent verification of coefficients.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1088/0264-9381/11/6/004

Cites

130 of 130 references verified0 citation errors · 0 not checkable

Statistics

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Verification

Claims the paper makes

0 Valid · 1 Not valid · 33 Not checked · 32 Own results and interpretations

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Citations

Cites

Show all 130 references

Research works printed but not linked: 12

  1. Alvarez-Gaumé L, 1983, Nucl. Phys., 234, 264
  2. Weyl H, 1918, Sitz. Ber. Preuss. Ak. Wiss., 465
  3. Weyl H, 1918, Ann. Phys., 59, 101
  4. Dowker J S, 1976, Phys. Rev., 13
  5. Antoniadis I, 1992, CPTH-A214.1292, LA-UR-92-4333
  6. Antoniadis I, 1992, CPTH-C207.1192
  7. 10.1016/0550-3213(80)90423-x · DOI
  8. Alvarez-Gaumé L, 1991, Helv. Phys. Acta, 64, 361
  9. Zamolodchikov A B, 1986, JETP Lett., 43, 730
  10. 10.1016/0370-2693(91)91736-f · DOI
  11. Siegel W, 1980, Phys. Lett., 138B, 107
  12. Duff M J, 1993, CTP/TAMU-49/93, CERN-TH.7088/93

Cited by

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Metadata

Authors
Duff, M. J.
Affiliations
Texas A&M University (US)
Corresponding author
Duff, M J
ORCID
Duff, M J
Venue
Classical and Quantum Gravity
Publisher
IOP Publishing
Published
1994-06-01
Volume
11
Issue
6
Pages
1387-1403
ISSN
  • 0264-9381 (ISSN-L)
  • 1361-6382
DOI
10.1088/0264-9381/11/6/004
OpenAlex
W1970179815
Cited by
785
References
130
Citations by year
2026: 59 · 2025: 43 · 2024: 50 · 2023: 49 · 2022: 29 · 2021: 57 · 2020: 49 · 2019: 37 · 2018: 37 · 2017: 32 · 2016: 28 · 2015: 25 · 2014: 23 · 2013: 33 · 2012: 22
FWCI
3.52
Type
journal article
Language
en
Open access
Locations
  • Classical and Quantum Gravity (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • National Science Foundation (9106593)
  • Imperial College London
Topics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
Keywords
trace anomaly, conformal invariance, Weyl rescaling, massless fields, quantum gravity, black hole physics, cosmology, string theory, statistical mechanics, history of physics

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