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September 23, 2025Open Access

One Ring to Rule Them All: A Unified Topological Framework for 4D Superconformal Anomalies

Key Points

Anomalies can be represented using a new topological framework focusing on polynomial rings and generalized curvatures.
It identifies Chern-type and non-Chern-type anomalies in superconformal theories, enhancing understanding of their coexistence.
The study characterizes anomalies via cohomology of a generalized BRST operator acting on the constraint ideal of the Lie algebra.
This approach bridges traditional Yang-Mills anomalies with the complex structures found in 4-dimensional superconformal gravity.

Abstract

We present a unified topological description of anomalies that generalizes the Chern-Simons formulation of Yang-Mills anomalies to encompass all 4-dimensional 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 (W₃+₂) of the generalized BRST operator δ acting on the fermion number d+2 component of the constraint ideal W₃+₂. While Yang-Mills anomalies correspond to invariant Chern curvature polynomials (where W₃+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.

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