On two dimensional non-abelian chiral lattice gauge theories in Ginsparg-Wilson formalism

TL;DR

This paper proves the existence of smooth, gauge-invariant fermion measures in 2D non-abelian chiral lattice gauge theories within the Ginsparg-Wilson formalism, under anomaly-free conditions, addressing a key non-perturbative challenge.

Contribution

It provides a non-perturbative proof for the existence of fermion measures in 2D non-abelian theories with zero field strength, extending previous abelian and perturbative results.

Findings

Existence of smooth fermion measure proven for zero field strength configurations

Gauge invariance maintained under anomaly-free conditions

Supports conjecture for full non-perturbative proof in 2D non-abelian theories

Abstract

Defining chiral lattice gauge theories in the Ginsparg-Wilson formalism is complicated by the so-called fermion measure problem. It has been proven for the abelian theories that smooth well-behaved fermion measure exists if and only if the anomaly-free condition is granted, and the same was shown to hold in perturbative theories for non-abelian gauge groups, but the non-perturbative proof is absent. In this paper, we consider a simpler problem in 2-d and present a proof for the existence of smooth and gauge invariant fermion measure on the gauge field configuration space with zero field strengths for arbitrary compact Lie groups, provided the anomaly-free conditions are satisfied. It is conjectured that such consideration is sufficient for the unknown full proof.