Fermion determinant and chiral anomaly on a finite lattice

The fermion determinant and the chiral anomaly of lattice Dirac operator D on a finite lattice are investigated. The condition for D to reproduce correct chiral anomaly at each site of a finite lattice for smooth background gauge fields is that D possesses exact zero modes satisfying the Atiyah-Singer index theorem. This is also the necessary condition for D to have correct fermion determinant (ratio) which plays the important role of incorporating dynamical fermions in the functional integral. We outline a scheme for dynamical fermion simulation of lattice QCD with topologically proper D.