General solutions of the Wess-Zumino consistency condition for the Weyl anomalies

TL;DR

This paper addresses the algebraic classification of Weyl (trace) anomalies by solving the Wess-Zumino consistency conditions in arbitrary dimensions through BRST cohomology. Using a cohomological framework with the combined diffeomorphism and Weyl BRST operator $s=s_D+s_W$ and the Stora trick with $ ilde{s}=s+d$, the authors derive a general solution in terms of a local total form built from the Weyl ghost $\omega$ and a hierarchy of generalized connections, leading to a descent structure governed by the gauge covariant algebra $\mathcal{G}$. They prove that the type-A Weyl anomaly corresponds to a unique non-trivial descent, while type-B anomalies are governed by trivial descents produced by contractions of Weyl tensors; in even dimensions $n=2m$, the top component $e^n_1$ of the combined anomaly $(\alpha+\beta)$ is proportional to the Euler density, explicitly $e^n_1 = \frac{(-1)^m}{2^m} \omega \,(R_{a_1 b_1}\wedge\cdots\wedge R_{a_m b_m}) \varepsilon^{a_1 b_1\ldots a_m b_m}$. The approach is regularization-independent and dimension-agnostic, providing a purely algebraic solution that parallels the Stora-Zumino descent for chiral anomalies and offering insight into the structure of conformal anomalies across dimensions.

Abstract

The general solutions of the Wess-Zumino consistency condition for the conformal (or Weyl, or trace) anomalies are derived. The solutions are obtained, in arbitrary dimensions, by explicitly computing the cohomology of the corresponding Becchi-Rouet-Stora-Tyutin differential in the space of integrated local functions at ghost number unity. This provides a purely algebraic, regularization-independent classification of the Weyl anomalies in arbitrary dimensions. The so-called type-A anomaly is shown to satisfy a non-trivial descent of equations, similarly to the non-Abelian chiral anomaly in Yang-Mills theory.