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<p>We present a unified topological description of anomalies that generalizes the Chern-Simons formulation of Yang-Mills anomalies to encompass all 4-dimensional <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> <mo>=</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N}=1 \)</EquationSource> </InlineEquation> 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.</p><p>We demonstrate that anomalies in dimension <i>d</i> are captured by the cohomology <i>H</i><sub><i>δ</i></sub>(<i>W</i><sub><i>d</i>+2</sub>) of the generalized BRST operator <i>δ</i> acting on the fermion number <i>d</i> + 2 component of the constraint ideal <i>W</i><sub><i>d</i>+2</sub>. While Yang-Mills anomalies correspond to invariant Chern curvature polynomials (where <i>W</i><sub><i>d</i>+2</sub> 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 (<i>a</i>) and non-Chern-type (<i>c</i>) anomalies in (super)conformal theories.</p>

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One ring to rule them all: a unified topological framework for 4D \( \mathcal{N}=1 \) superconformal anomalies

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  • 发表日期：

    2026年2月13日

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  • Camillo Imbimbo,
  • Ludovico Porro

摘要

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 \( \mathcal{N}=1 \) 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 Hδ(Wd+2) of the generalized BRST operator δ acting on the fermion number d + 2 component of the constraint ideal Wd+2. While Yang-Mills anomalies correspond to invariant Chern curvature polynomials (where Wd+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.