QFT In the literature: open

Gauge-invariant lattice definition of nonabelian chiral gauge theories

In plain words

A grid version exists for chiral theories with the simplest, electromagnetism-like forces. For the non-commuting forces of the Standard Model it is still missing.

Precise statement

For a 4D anomaly-free chiral gauge theory with nonabelian gauge group (for example SU(5) with 5bar + 10, or one Standard Model generation), construct a lattice regularization with exact gauge invariance at finite lattice spacing, a local action and fermion measure, and the correct continuum limit, extending Luscher's 1999 abelian construction based on Ginsparg-Wilson fermions beyond perturbation theory.

What would settle it

An explicit construction with a nonperturbative proof of the locality and gauge invariance of the fermion measure for a nonabelian anomaly-free theory.

Status in the literature

Unverified note

The abelian case was completed in 1999 and the nonabelian case to all orders of perturbation theory in 2000; new proposals in 2024-2026 use domain walls with gauge-field flow and symmetry disentanglers.