Algebraic Generalization of the Ginsparg-Wilson Relation

The generalization of the Ginsparg-Wilson relation to the form $\gamma_{5}(\gamma_{5}D) + (\gamma_{5}D)\gamma_{5} = 2a^{2k+1}(\gamma_{5}D)^{2k+2}$ is discussed, where $k$ stands for a non-negative integer and $k=0$ corresponds to the ordinary Ginsparg-Wilson relation. It is shown that the instanton-related index of all these operators is identical, but the degree of chiral symmetry breaking is different. We thus have an infinite tower of lattice Dirac operators which satisfy the index theorem, but a large enough lattice is required to accomodate such a Dirac operator with a large value of $k$. We illustrate explicitly a generalization of Neuberger's overlap Dirac operator to the case $k=1$ and show that the chiral symmetry of the Dirac operator is improved at the near continuum configurations compared to the original overlap Dirac operator. We also briefly sketch the construction of the lattice Dirac operator for any value of $k$.