Geometric general solution to the U(1) anomaly equations

Abstract

Costa et al. [Phys. Rev. Lett. 123 (2019) 151601] 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.

A ν supersymmetric anomaly-free atlas

“…[15], the gravitational and gauge anomalies of a U(1) gauge symmetry (i.e. with no SM gauge group but with charged chiral fermionic fields) were solved analytically for the charges of a priori fixed numbers of chiral fermions via an ingenious algebraic method; 2 this was soon understood from a geometric perspective [17] by using a theorem due to Mordell [14]. Similar geometric methods were employed to find an analytic solution to the more difficult problem of sm ⊕ u(1) anomaly-free charge assignments in the specific case of SM fermion content, plus three right-handed (RH) neutrinos (i.e.…”

Composite dark matter from strongly-interacting chiral dynamics

“…The cancellation of gauge anomalies, for example, is a non-trivial requirement that significantly restricts the possible models. Interesting studies in this direction include techniques for finding anomaly-free sets of fermions in theories with a U(1) chiral factor [31][32][33], and a method to construct chiral theories starting from irreducible representations of a simple, anomaly-free gauge group [34]. Chiral theories are also notoriously difficult to simulate on the lattice [35,36], and their IR behavior is still not known for simple gauge groups.…”

Anomaly cancellation with an extra gauge boson

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