Ginsparg-Wilson relation, topological invariants, and finite noncommutative geometry

We show that the Ginsparg-Wilson (GW) relation can play an important role in defining chiral structures in finite noncommutative geometries. Employing the GW relation, we can prove the index theorem and construct topological invariants even if the system has only finite degrees of freedom. As an example, we consider a gauge theory on a fuzzy two-sphere and give an explicit construction of a noncommutative analogue of the GW relation, chirality operator, and the index theorem. The topological invariant is shown to coincide with the first Chern class in the commutative limit.