Descend from a characteristic polynomial to a Ward variation

Use natural units and a compact Abelian connection a normalized so unit charge has covariant derivative $d-ia$. Its curvature is $f=da$ locally, and $x=f/(2\pi)$ represents the first Chern class. Matter has integer charges q in this normalization. For bookkeeping, assign the displayed anomaly polynomial to a left-handed complex Weyl fermion; opposite chirality contributes its negative. This declaration fixes the overall anomaly sign, which otherwise changes between chirality, orientation, and effective-action conventions in different references.

The polynomial follows from the degree-six part of $\widehat A(T)\operatorname{ch}(qf)$. The Chern character expands as $e^{qx}$ and the gravitational factor begins with $1-p_1/24$. Multiplying and retaining total form degree six gives a cubic gauge term and a mixed gauge-gravitational term. Here $p_1$ is a four-form, so its product with x has the required degree. A charge sum that cancels the cubic term need not cancel the gravitational term.

Set gravitational curvature to zero to isolate the Abelian example. Locally, the closed six-form is the exterior derivative of a Chern–Simons five-form proportional to $a\wedge f\wedge f$. Under $\delta a=d\lambda$, its variation is the exterior derivative of a four-form proportional to $\lambda f\wedge f$. Each step lowers the spacetime form degree by one while increasing the dependence on the transformation parameter. The construction is local on connection patches; a globally defined potential is not assumed on every nontrivial bundle.

Define the generating-functional phase by $\delta\log Z=2\pi i\int I_4^{(1)}$. This fixes the otherwise easy-to-miss factor connecting the normalized characteristic polynomial to the physical Ward variation. Changing a local counterterm can change the descent representative by a trivial variation, but cannot remove a nontrivial anomaly class. Statements about finite large transformations require additional global information; exponentiating an infinitesimal local formula without that information is not a general global-anomaly calculation.

The consistent current is obtained by functional differentiation of the effective action. Its anomaly satisfies the Wess–Zumino consistency condition expressing the gauge-transformation algebra. Adding a Bardeen–Zumino local polynomial to the current produces a covariant current whose transformation properties are simpler. That current generally is not the derivative of the same local effective action. For a single four-dimensional Abelian chiral current, the covariant anomaly coefficient is three times its consistent coefficient; the distinction cannot be discarded during descent.

This factor does not mean that a physical obstruction has tripled. The two currents are different local representatives of the same anomaly data and include different contact terms. Nor should the single-Weyl formula be compared directly with a Dirac axial identity after imposing exact vector conservation, which allocates anomalies differently among currents. The polynomial is best used as a compact invariant organizing cancellation and matching, while current-level calculations state separately which representative and background symmetries they preserve.