Admissibility Condition and Nontrivial Indices on a Noncommutative Torus

We study the index of the Ginsparg-Wilson Dirac operator on a noncommutative torus numerically. To do this, we first formulate an admissibility condition which suppresses the fluctuation of gauge fields sufficiently small. Assuming this condition, we generate gauge configurations randomly, and find various configurations with nontrivial indices. We show one example of configurations with index 1 explicitly. This result provides the first evidence that nontrivial indices can be naturally defined on the noncommutative torus by utilizing the Ginsparg-Wilson relation and the admissibility condition.