Allocate a mixed anomaly without erasing its class

Let X and Y be Abelian symmetries with background connections $a_X,a_Y$ and curvatures $f_X,f_Y$. A left-handed Weyl field with charges $(x_i,y_i)$ contributes a Chern character with exponent $(x_if_X+y_if_Y)/(2\pi)$. Background gauge transformations encode Ward identities for the associated currents. They do not add propagating gauge bosons unless the connections are integrated over. An anomalous background response is allowed as data of a global symmetry, whereas an uncanceled redundancy of a dynamical gauge field is an inconsistency.

The distinction also affects whether a continuous symmetry is exact. A current anomalous only in external probes can describe an exact global symmetry at zero sources with a 't Hooft anomaly. If Y is dynamical and X has a nonzero X–Y-squared anomaly, gauge-field configurations can violate X charge even with no external X source. The full continuous X symmetry is then generally explicitly broken by quantum effects, sometimes leaving a discrete subgroup. It should not automatically be treated as an exact symmetry in later anomaly-matching arguments.

For the displayed mixed term, a local descent representative can place the connection $a_X$ in the five-form and leave only $f_Y$ factors. Its Y gauge variation then vanishes, while its X variation produces a term proportional to $\lambda_X f_Y\wedge f_Y$. This representative is especially useful when Y is dynamical, because preserving Y gauge invariance is mandatory. Other anomaly terms must still be considered; isolating one mixed term is not a proof that a full spectrum is consistent.

A Bardeen counterterm involving $a_X\wedge a_Y\wedge f_Y$ changes the local distribution of consistent anomalies. On a closed patch calculation, its X variation has the opposite sign from its Y variation after integration by parts. One can therefore move a mixed term from one Ward identity to another. If the original anomaly polynomial is nontrivial, however, no allowed four-dimensional local counterterm makes every background symmetry exact simultaneously. The obstruction is in the shared anomaly class, not in one arbitrarily preferred current representative.

To see the relative signs directly, vary $C=b\int a_X\wedge a_Y\wedge f_Y$ in a locally trivial bundle with boundary terms absent. The result is $\delta C=b[-\int\lambda_X f_Y\wedge f_Y+\int\lambda_Y f_X\wedge f_Y]$. This shows precisely what is exchanged. Global bundle patching can restrict counterterms further, so the local expression should not be presented as a universal globally defined functional on every background. Consistent and covariant current redefinitions also require their own specified local polynomials.

An anomalous divergence fixes a source or Ward identity, not a complete constitutive equation. Turning it into a persistent current requires a state, boundary conditions, relaxation mechanisms, and a defined equilibrium or driven limit. In particular, no universal steady magnetic current follows just from reading a mixed polynomial coefficient. Background geometry organizes possible responses and constraints, while actual transport depends on additional dynamics. This separation prevents a regulator-dependent current split from being mistaken for a directly measurable charge-production rate in every physical setup.