Chiral gauge theory
In quantum field theory, a chiral gauge theory is a quantum field theory with charged chiral (i.e. Weyl) fermions. For instance, the Standard Model is a chiral gauge theory. For topological reasons, chiral charged fermions cannot be given a mass without breaking the gauge symmetry, which will lead to inconsistencies unlike a global symmetry. It is notoriously difficult to construct a chiral gauge theory from a theory which does not already contain chiral fields at the fundamental level.[1] A consistent chiral gauge theory must have no gauge anomaly (or global anomaly). Almost by necessity, regulators will have to break the gauge symmetry. This is responsible for gauge anomalies in the first place.
Fermion doubling on a lattice
Lattice regularizations suffer from fermion doublings leading to many tastes, and a loss of chirality.
3-4-5 model
The simplest model has 1 spatial and 1 time dimension. We have a U(1) gauge symmetry. There are left-mover chiral fermions with charges ± 3 and ±; 4. We have a right-mover with charge ± 5. This is a Pythagorean triple. This theory is free from gauge anomalies.