Connes’ spectral triple and U(1) gauge theory on finite sets

We first apply Connes’ noncommutative geometry to a finite point set. The explicit form of the action functional of U(1) gauge field on this n-point set is obtained. We then construct the U(1) gauge theory on a disconnected manifold consisting of n copies of a given manifold. In this case, the explicit action functional of U(1) gauge field is also obtained.