Lattice chiral non-Abelian gauge symmetry via bosonization

Abstract

A central issue in lattice formulations of chiral gauge theories is how the anomaly cancellation mechanism of the continuum theory can be realized at finite lattice spacing. In the present paper, based on non-Abelian bosonization, we propose a lattice formulation of the bosonic theory corresponding to a two-dimensional non-Abelian chiral gauge theory. In the continuum theory, the gauge anomaly of chiral fermions is represented, in the bosonized description, as anomaly inflow from a three-dimensional Chern--Simons-type bulk contribution contained in a gauged Wess--Zumino--Witten model. Motivated by this structure, we introduce gauge-neutral spectator fermions and use the resulting bosonized description. We then construct a lattice counterpart of the gauged Wess--Zumino--Witten model with a three-dimensional bulk extension under appropriate smoothness conditions. A salient feature of this lattice formulation is the cancellation of the left and right bulk contributions in the exponentiated action. This cancellation occurs even before taking the continuum limit when the anomaly-free condition is satisfied, namely when the left and right representations have identical quadratic indices. Thus, the present construction realizes the anomaly-cancellation mechanism at finite lattice spacing via the bosonized description of two-dimensional anomaly-free chiral gauge theories. Establishing the desired continuum limit remains an important open problem.