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Costa et al. [Phys. Rev. Lett. 123, 151601 (2019)] recently gave a generalsolution to the anomaly equations for $n$ charges in a $U(1)$ gauge theory.`Primitive' solutions of chiral fermion charges were parameterised and it wasshown how operations performed upon them (concatenation with other primitivesolutions and with vector-like solutions) yield the general solution. We showthat the ingenious methods used there have a simple geometric interpretation,corresponding to elementary constructions in number theory. Viewing them inthis context allows the fully general solution to be written down directly,without the need for further operations. Our geometric method also allows us toshow that the only operation Costa et al. require is permutation. It also givesa variety of other, qualitatively similar, parameterisations of the generalsolution, as well as a qualitatively different (and arguably simpler) form ofthe general solution for $n$ even.

Geometric General Solution to the $U(1)$ Anomaly Equations