Admissibility Condition and Nontrivial Indices on a Noncommutative Torus

摘要

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