Note on the Lattice Fermion Chiral Symmetry Group

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 the chiral group, and the CP transformation properties of the symmetry generators is found. 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 strange features occur in all implementations of lattice fermions based on the Ginsparg-Wilson relation, including overlap, domain-wall, and perfect-action chiral fermions. The conclusions are illustrated in a solvable example, free overlap fermions.