One ring to rule them all: a unified topological framework for 4D N=1 superconformal anomalies

A bstract We present a unified topological description of anomalies that generalizes the Chern-Simons formulation of Yang-Mills anomalies to encompass all 4-dimensional N=1 N = 1 superconformal anomalies. The key innovation is our characterization of anomalies through the constraint ideal in the polynomial ring of generalized curvatures and connections of the underlying symmetry (super) -Lie algebra. We demonstrate that anomalies in dimension d are captured by the cohomology H δ (W d +2) of the generalized BRST operator δ acting on the fermion number d + 2 component of the constraint ideal W d +2. While Yang-Mills anomalies correspond to invariant Chern curvature polynomials (where W d +2 reduces to homogeneous curvature polynomials), the constraint ideal for 4D (super) conformal gravity contains additional polynomials mixing curvatures and connections. This richer structure naturally explains the coexistence of both Chern-type (a) and non-Chern-type (c) anomalies in (super) conformal theories.