Symmetries of Ginsparg-Wilson Chiral Fermions

TL;DR

This paper analyzes the complex symmetry structure of Ginsparg-Wilson lattice chiral fermions, revealing an infinite group with universal features across various formulations and discussing implications for physical properties like reflection positivity.

Contribution

It explicitly characterizes the infinite symmetry group, its Lie algebra, and CP automorphism, highlighting universal features in lattice chiral fermion formulations.

Findings

The symmetry group contains infinitely many generators.

Some generators lead to the same Noether current.

Non-canonical symmetries relate to complex energy singularities.

Abstract

The group structure of the variant chiral symmetry discovered by Luscher in the Ginsparg-Wilson description of lattice chiral fermions is analyzed. It is shown that the group contains an infinite number of linearly independent symmetry generators, and the Lie algebra is given explicitly. CP is an automorphism of this extended chiral group, and the CP transformation properties of the symmetry generators are found. The group has an infinite-parameter invariant subgroup, and the factor group, whose elements are its cosets, is isomorphic to the continuum chiral symmetry group. Features of the currents associated with these symmetries are discussed, including the fact that some different, non-commuting symmetry generators lead to the same Noether current. These are universal features of lattice chiral fermions based on the Ginsparg-Wilson relation; they occur in the overlap, domain-wall, and perfect-action formulations. In a solvable example, free overlap fermions, these non-canonical elements of lattice chiral symmetry are related to complex energy singularities that violate reflection positivity and impede continuation to Minkowski space.