Generalized Ginsparg-Wilson Algebra and Index Theorem on the Lattice

Recent studies of the topological properties of a general class of lattice Dirac operators are reported. This is based on a specific algebraic realization of the Ginsparg-Wilson relation in the form _γ5(_γ5D) + (_γ5D)_γ5 = 2a^2k+2 where k stands for a non-negative integer. The choice k = 0 corresponds to the commonly discussed Ginsparg-Wilson relation and thus to the overlap operator. It is shown that local chiral anomaly and the instanton-related index of all these operators are identical. The locality of all these Dirac operators for vanishing gauge fields is proved on the basis of explicit construction, but the locality with dynamical gauge fields has not been fully established yet.