Twenty years of the Weyl anomaly

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.