A Criterium for the Index Theorem on the Lattice with the Ginsparg Wilson Relation

The Ginsparg Wilson Relation and the Neuberger action recently improved the understanding of the Index Theorem on the lattice, however this problem has not been solved yet. With the presently available criteria, we develop a technique based on the position of roots and poles to build a new simple solution of the Ginsparg Wilson Relation. In the free fermion case, we study the roots, the poles, and the locality of the Wilson action, the simple action and the Neuberger action. Then we explore the effects of topological gauge configurations on the eigenvalues of the different actions. In the discrete version of the Schwinger model we examine how the GWR solutions project the eigenvalues of the Wilson action to a circle in the complex Argand plane. Finally we propose a new criterium for solutions of the Ginsparg Wilson Relation constructed with the Wilson action, and we conclude that the Neuberger action is the only one that complies with the Index Theorem.