Carry an exact global-symmetry anomaly into the infrared

Start with the actual quantum global symmetry G and its allowed background fields. An axial U(1) explicitly violated by dynamical gauge configurations is not automatically part of G, even if the massless classical Lagrangian has that rotation. By contrast, a 't Hooft anomaly is an obstruction to gauging an otherwise exact global symmetry, revealed by its background generating functional. Its currents can be conserved at zero background while their sourced Ward identities have a nontrivial variation.

The symmetry's global form, charge lattice, and spacetime structures must agree in ultraviolet and infrared comparisons. Rescaling one side's charges without rescaling its background connection changes the problem. Matching one polynomial coefficient is also insufficient when mixed or gravitational terms survive. Local counterterms can shift representatives, so equality means the same anomaly class after allowed local choices, not necessarily identical-looking formulas for every current in two unrelated subtraction schemes. Global anomaly phases require matching beyond the perturbative polynomial as well.

A useful argument weakly gauges the global symmetry as a diagnostic and adds spectator fermions whose anomaly cancels the ultraviolet matter contribution. Choose the probe coupling weak enough that it does not rearrange the original dynamics over the scales of interest. The spectators retain their anomaly while the interacting sector flows toward lower energies. If the symmetry remains unbroken, the complete probe-gauged theory must still be consistent, forcing the infrared sector to cancel the same spectator contribution.

This reasoning compares responses rather than particle identities. The infrared charged excitations may be composite and have entirely different quantum numbers from the ultraviolet elementary fields, yet their combined anomaly must agree. Integrating out massive modes can leave Wess–Zumino terms or other background-dependent contributions that must be retained in the effective description. Counting only the visible low-energy particles while omitting those terms can falsely suggest that matching has failed. Conversely, declaring omitted heavy fields to cancel everything is not an infrared explanation unless their residual action is specified.

A nonzero anomaly rules out a completely trivial, unique, symmetry-preserving gapped infrared vacuum with no additional anomaly-carrying sector or ambient bulk. It can require gapless degrees of freedom, spontaneous symmetry breaking with an appropriate Goldstone effective action, or another nontrivial realization allowed by the particular anomaly and dimension. These possibilities are not interchangeable universal solutions. Some anomaly types forbid proposed gapped topological realizations too, so the precise class must be analyzed instead of invoking topological order as an automatic escape.

Even a successful set of charge sums proves only that the proposed spectrum passes those necessary tests. Stability, interactions, confinement, and whether the required composites exist remain dynamical questions. For a broken symmetry, a Wess–Zumino–Witten term can carry information absent from an ordinary two-derivative Goldstone action. The practical workflow is to specify a candidate infrared realization, compute all relevant anomaly data, and reject mismatches before spending effort on a detailed dynamical construction.