Aspects of domain-wall fermions on the lattice

I derive the generating functional for n-point Green's function of the effective fermion (quark) fields in domain-wall fermions, and clarify the issues pertainent to finite N_s and the lattice spacing a_5 in the fifth dimension. These issues can be examined by computing the index, the axial anomaly, and the fermion determinant of the effective lattice Dirac operator, in 2D abelian background gauge fields. For the optimal domain-wall fermion, the fermion determinant, the index and the axial anomaly of the effective lattice Dirac operator are in good agreement with the exact solutions in the continuum. However, for the conventional domain-wall fermion, the effective lattice Dirac operator cannot reproduce the correct fermion determinant while its index and axial anomaly agree with the topological charge and density of the background gauge field. This demonstrates that even though an effective lattice Dirac operator (N_s = infinite) satisfies the Ginsparg-Wilson relation, gamma_5-hermiticity, doubler-free, exponential-locality, and correct continuum (free fermion) behavior, it does not necessarily reproduce the (complex conjugate pairs) spectrum correctly for topologically nontrivial gauge backgrounds, due to its dependence on a_5