Application of Noncommutative Differential Geometry on Lattice to Anomaly

The chiral anomaly in lattice abelian gauge theory is investigated by applying the geometric and topological method in noncommutative differential geometry(NCDG). A new kind of double complex and descent equation are proposed on infinite hypercubic lattice in arbitrary even dimensional Euclidean space, in the framework of NCDG. Using the general solutions to proposed descent equation, we derive the chiral anomaly in Abelian lattice gauge theory. The topological origin of anomaly is nothing but the Chern classes in NCDG.