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

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