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

TL;DR

This paper provides a geometric interpretation of the general solution to $U(1)$ anomaly equations, simplifying the derivation and revealing that permutation is the only necessary operation, with new parameterizations and solutions for even $n$.

Contribution

It introduces a geometric approach to solve $U(1)$ anomaly equations, simplifying the process and offering new parameterizations and solutions, especially for even numbers of charges.

Findings

Geometric interpretation of anomaly solutions.

Permutation is the only operation needed.

New parameterizations for the general solution.

Abstract

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