Type-A Conformal Anomalies from Euler Descent

We show that the type-A conformal anomaly in $2n$ dimensions follows from standard Stora-Zumino descent, starting from the Euler invariant polynomial for the Euclidean conformal group $SO(2n+1,1)$ in $6d$, thereby placing type-A anomalies on the same footing as ordinary perturbative t Hooft anomalies. 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 type-A $SO(5,1)$ conformal anomaly.

💡 Research Summary

The paper demonstrates that the type‑A conformal anomaly in even‑dimensional quantum field theories can be derived from the standard Stora‑Zumino (SZ) descent procedure, exactly as perturbative ’t Hooft anomalies are. The authors start from the Euler invariant polynomial of the Euclidean conformal group SO(2n + 1, 1) in 2n + 2 dimensions, which is a CP‑even top‑form. By applying the usual descent—first constructing the 2n + 1‑dimensional Chern‑Simons form whose exterior derivative reproduces the Euler class, then varying it under an infinitesimal gauge transformation—one obtains a 2n‑dimensional anomaly functional. When the gauge parameter is taken to be the Weyl generator, the resulting expression reproduces the familiar type‑A anomaly density (the Euler density of the background metric) up to local counterterms. The authors work out the explicit case n = 1, showing how the 2‑dimensional conformal anomaly follows from the 4‑dimensional Euler class of SO(3, 1), and then the case n = 2, reproducing the 4‑dimensional type‑A anomaly from the 6‑dimensional Euler class of SO(5, 1). They emphasize that the derivation does not require imposing the usual Riemannian constraints (vanishing torsion, zero dilatation gauge field) on the background gauge fields; the anomaly theory exists for the full non‑abelian SO(2n + 1, 1) symmetry. Consequently, ’t Hooft anomaly matching can be applied to the full conformal group, including backgrounds with torsion or non‑zero Weyl gauge fields. The paper also connects the descent construction to anomaly inflow: the bulk theory in 2n + 1 dimensions is a Chern‑Simons action built from the Euler polynomial, whose variation cancels the boundary conformal anomaly. Using the bulk‑boundary correspondence, the authors construct a dilaton effective action in four dimensions that realizes the full SO(5, 1) anomaly via a Wess‑Zumino‑Witten term for the coset SO(5, 1)/ISO(4), with the inverse‑Higgs constraint relating the dilaton to the Goldstone modes of broken conformal symmetry. The dilaton action reproduces the known non‑local part of the anomaly and provides a systematic way to match the type‑A anomaly in infrared effective field theories. Finally, the authors generalize the construction to arbitrary even dimensions, discuss the distinction between Euler‑derived (type‑A) and Pontryagin‑derived (gravitational) anomalies, and outline future directions such as exploring scattering amplitudes in the dilaton EFT, connections to holographic calculations, and extensions to supersymmetric or higher‑form symmetries. The work places type‑A conformal anomalies on the same footing as ordinary perturbative anomalies, clarifying their cohomological classification and enabling a unified treatment of anomaly inflow and matching for conformal symmetries.