Local Anomaly Descent and Wess–Zumino Consistency

A perturbative local anomaly is a nontrivial ghost-number-one class of the local BRST cohomology modulo spacetime derivatives. Wess–Zumino consistency is the closure condition, characteristic descent constructs representatives from an anomaly polynomial, and local counterterms change representatives by exact terms. This classification applies to local perturbative anomalies in a fixed dimension and field complex. It does not detect determinant-line holonomy, torsion, or other global anomalies.

Required background. Local BRST cohomology supplies $H^{1,n}(s\mid d)$; the quantum master equation supplies the loopwise defect; and Wess–Zumino consistency and descent supplies the physical anomaly calculation.

Helpful background. Consistent and covariant anomalies distinguish two current conventions, while anomaly polynomials and inflow place descent in one higher dimension.

Consistency as a BRST cocycle condition

Let $\Gamma[A]$ be a renormalized effective action and define its infinitesimal gauge variation by

$$\delta_\epsilon\Gamma=\mathcal A(\epsilon,A).$$

Because gauge transformations close, two variations must obey

$$\delta_{\epsilon_1}\mathcal A(\epsilon_2,A) -\delta_{\epsilon_2}\mathcal A(\epsilon_1,A) =\mathcal A([\epsilon_1,\epsilon_2],A).$$

This is the Wess–Zumino consistency condition derived as an integrability condition for anomalous Ward identities Wess and Zumino 1971, pp. 95–97. Replacing $\epsilon$ by the odd ghost $c$ turns it into

$$s\mathcal A=0, \qquad \operatorname{gh}\mathcal A=1.$$

If $\mathcal A=sB$ for an allowed local ghost-number-zero functional $B$, adding $-B$ to $\Gamma$ removes the anomaly. For densities, equality is modulo $d$, so the invariant object is $[\mathcal A]\in H^{1,n}(s\mid d)$. This statement assumes the total derivative integrates to zero; boundaries require their own anomaly inflow or boundary theory.

Closure is necessary, not sufficient for a genuine anomaly. A regulator can break gauge symmetry by a cocycle in the trivial class. Conversely, a nontrivial class is a possible obstruction, but its coefficient may vanish for the chosen matter representation. Cohomological classification and coefficient calculation are distinct steps.

Characteristic descent

For chiral fermions in $2n$ dimensions, the index density gives a closed invariant $(2n+2)$-form

$$I_{2n+2}=\left[\widehat A(TM)\operatorname{ch}_R(F)\right]_{2n+2}.$$

Locally choose a Chern–Simons form $I^{(0)}_{2n+1}$ and apply BRST variation:

$$I_{2n+2}=dI^{(0)}_{2n+1}, \qquad sI^{(0)}_{2n+1}+dI^{(1)}_{2n}=0, \qquad sI^{(1)}_{2n}+dI^{(2)}_{2n-1}=0.$$

The superscript is ghost number. The consistent anomaly is proportional, with convention-dependent normalization, to $\int I^{(1)}_{2n}$. Applying $s$ to the first descent equation and using $s^2=0$ gives the next equation, so consistency is built into the construction. The general local descent, its lifts, and its obstructions are analyzed in Barnich, Brandt, and Henneaux 2000, §§9–11, pp. 70–112.

In four dimensions the pure gauge term of $I_6$ is proportional to $\operatorname{tr}_R F^3$. Its coefficient is the symmetrized cubic trace of representation generators. Mixed gauge–gravitational terms and Abelian factors must be retained when present. A vanishing $\operatorname{tr}_R F^3$ removes that perturbative pure-gauge polynomial; it does not test a global anomaly associated with a large transformation.

Consistent versus covariant currents

The consistent current is defined by differentiating one effective action, so its divergence satisfies the Wess–Zumino condition. It need not transform covariantly. Adding the local Bardeen–Zumino current,

$$J_{\mathrm{cov}}^\mu=J_{\mathrm{cons}}^\mu+K_{\mathrm{BZ}}^\mu[A],$$

produces a covariant current and covariant anomaly. The shift changes the local representative and often its numerical coefficient, but does not create a second independent quantum obstruction. In general the covariant current is not the functional derivative of the same local effective action, so one must not impose the consistent integrability equation on it unchanged. Bardeen and Zumino construct the gauge and gravitational shifts in Bardeen and Zumino 1984, §§2–5, pp. 424–443.

Four-dimensional chiral application

Start from $I_6$ for the declared left-handed representation $R$, choose one normalization for the trace and curvature, and compute $I_5^{(0)}$ and $I_4^{(1)}$. The result gives the consistent anomaly. Differentiating the effective action identifies $J_{\mathrm{cons}}$; adding $K_{\mathrm{BZ}}$ gives $J_{\mathrm{cov}}$. This is the exact application passed to Standard-Model anomaly cancellation: descend the four-dimensional chiral gauge anomaly from the six-form polynomial and compare the two currents after the Bardeen–Zumino shift.

The independent checks are: $dI_6=0$ by invariant-polynomial identities; $s\int I_4^{(1)}=0$ on a closed spacetime; the representation coefficient changes sign under chirality reversal; and adding a local counterterm changes the representative by $sB+dC$. None of these checks decides a mapping-torus phase.

Failure test: a removable breaking term

Suppose a regulator produces $\mathcal A_{\mathrm{reg}}=sB+dC$ with local $B,C$. Treating it as a genuine anomaly confuses a scheme-dependent Ward-identity violation with a nontrivial class. Add the counterterm $-B$, keep the induced current improvement, and re-evaluate the identity; the breaking disappears. The strongest valid conclusion before this cohomology test is only that the chosen regulator violates the Ward identity.

The opposite error is equally serious: canceling a local polynomial and declaring the theory anomaly-free. Local descent cannot see torsion or determinant-line holonomy. Those obstructions require the global methods of the next page.

Exercises

Verify the bidegrees in the first two descent equations.

Solution

$I^{(0)}_{2n+1}$ has form degree $2n+1$ and ghost number zero. Both $sI^{(0)}_{2n+1}$ and $dI^{(1)}_{2n}$ therefore have bidegree $(1,2n+1)$. Applying $s$ gives terms of bidegree $(2,2n+1)$, matched by $dI^{(2)}_{2n-1}$ after the next descent step.

Why can a covariant anomaly fail the original Wess–Zumino integrability test?

Solution

The consistent anomaly is a variation of one effective action, which enforces integrability. The Bardeen–Zumino shift makes the current transform covariantly, but the shifted current need not be a functional derivative of that same action. Covariance and integrability are different requirements.

References

• Bardeen, William A., and Bruno Zumino. “Consistent and Covariant Anomalies in Gauge and Gravitational Theories.” Nuclear Physics B 244 (1984): 421–453. DOI.
• Barnich, Glenn, Friedemann Brandt, and Marc Henneaux. “Local BRST Cohomology in Gauge Theories.” Physics Reports 338 (2000): 439–569. DOI; Open PDF.
• Wess, Julius, and Bruno Zumino. “Consequences of Anomalous Ward Identities.” Physics Letters B 37 (1971): 95–97. DOI; CERN PDF.