Anomaly free U(1) chiral gauge theories on a two dimensional torus

TL;DR

This paper investigates anomaly-free chiral U(1) gauge theories on a 2D torus, analyzing their behavior in continuum and lattice formulations, and explores methods to restore gauge invariance and obtain correct continuum limits.

Contribution

It provides a detailed comparison of continuum and lattice formulations of anomaly-free chiral U(1) theories on a 2D torus, and proposes gauge averaging of overlap phases to recover gauge invariance.

Findings

Discontinuities in fermion-induced actions under large gauge transformations.

Special boundary conditions can restore gauge invariance in the continuum.

Gauge averaging of overlap phases yields correct continuum results.

Abstract

We consider anomaly free combinations of chiral fermions coupled to $U(1)$ gauge fields on a 2D torus first in the continuum and then on the lattice in the overlap formulation. Both in the continuum and on the lattice, when the background consists of sufficiently large constant gauge potentials the action induced by the fermions varies significantly under certain singular gauge transformations. ``Ruling away'' such discontinuities cannot be justified in the continuum framework and does not naturally fit on the lattice. Complete gauge invariance in the continuum can be restored in some models by choosing special boundary conditions for the fermions. Evidence is presented that gauge averaging the overlap phases in these models produces correct continuum results.