The topological characteristics of lattice Dirac operators

We show that even if a lattice Dirac operator satisfies the conditions consisting of locality, free of species doublings, correct continuum behavior, $\text{gm}5$-hermiticity and the Ginsparg-Wilson relation, it does not necessarily have exact zero modes in nontrivial gauge backgrounds. This implies that each lattice Dirac operator has its own topological characteristics which cannot be fixed by these conditions. The role of topological characteristics in the axial anomaly is derived explicitly.