One ring to rule them all: a unified topological framework for 4D $$ \mathcal{N}=1 $$ 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 $$ \mathcal{N}=1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> 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.