Chiral gauge theories on the lattice

Putting chiral fermions on a lattice is obstructed by the Nielsen–Ninomiya theorem: a local, translation-invariant, hermitian lattice fermion action with the right symmetries necessarily produces species doublers, spoiling chirality. This is why nonperturbatively defining the electroweak (chiral) sector is hard, whereas vector-like QCD is fine.

What is known: The Ginsparg–Wilson relation and overlap/domain-wall constructions avoid the naive doubling obstruction by modifying its assumptions. Lüscher's anomaly-free Abelian construction is nonperturbative, with exact gauge invariance including finite-volume topological sectors; it is not merely an all-orders perturbative result. Symmetric mass generation and interacting mirror-fermion approaches provide further proposals, whose locality and mirror decoupling must be checked for the intended representation.

Published progress (July 2026): Thorngren, Preskill and Fidkowski propose symmetry disentanglers that convert suitable non-on-site symmetries into on-site ones. Their explicit Abelian examples and low-dimensional constructions do not establish a general non-Abelian regulator with the desired interacting continuum limit.

A general, nonperturbative construction of an arbitrary anomaly-free non-abelian chiral gauge theory with the correct continuum limit is still open. Related: QF4.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.