Lattice Chiral Gauge Theories
Maarten Golterman
TL;DR
This work surveys non-perturbative lattice realizations of chiral gauge theories, contrasting a gauge-invariant program based on a Dirac operator $D$ satisfying the Ginsparg–Wilson relation with gauge-non-invariant strategies such as two-cutoff interpolation and gauge fixing. It highlights Lüscher’s phase construction as a path toward gauge invariance for anomaly-free theories and discusses perturbative completions, locality, and admissibility constraints, while also detailing non-perturbative open questions in the non-abelian case and global obstructions. The analysis shows how gdofs (gauge degrees of freedom) interact with chiral fermions, producing distinct infrared structures (poles versus cuts) across approaches, and emphasizes the practical and theoretical hurdles in achieving a robust, non-perturbative regulator for four-dimensional chiral gauge dynamics. Overall, these methods point toward potential four-dimensional regulators relevant for the electroweak sector, but non-abelian implementation, unitarity proofs, and full numerical exploration remain significant challenges.
Abstract
I review the substantial progress which has been made recently with the non-perturbative construction of chiral gauge theories on the lattice. In particular, I discuss three different approaches: a gauge invariant method using fermions which satisfy the Ginsparg-Wilson relation, and two gauge non-invariant methods, one using different cutoffs for the fermions and the gauge fields, and one using gauge fixing. Open problems within all three approaches are addressed.