Ginsparg-Wilson Relation and Admissibility Condition in Noncommutative Geometry

Noncommutative (NC) geometry1) had attracted much attention recently from various motivations. Topologically nontrivial configurations in finite NC geometries or matrix models2)3) have been constructed based on algebraic K-theory and projective modules in many papers, but it would be better if we could obtain an index operator which takes an integer even at a finite cutoff, since we need to perform the Kaluza-Klein compactification of extra dimensions with nontrivial indices to construct four dimensional chiral gauge theories. This can be realized if we utilize the Ginsparg-Wilson (GW) relation4) and the admissibility condition5)6),7) which were developed in lattice gauge theory (LGT) to construct chiral gauge theories.8)