Topological phases in the Neuberger-Dirac operator

The response of the Neubeger-Dirac fermion operator D = 1 + V in the topologically nontrivial background gauge field depends on the negative mass parameter m(0) in the Wilson-Dirac fermion operator D-w, which enters D through the unitary operator V = D-w(D(w)dagger D-w)(-1/2). We classify the topological phases of D by comparing its index to the topological charge of the smooth background gauge field. An exact discrete symmetry in the topological phase diagram is proved for any gauge configurations. A formula for the index of D in each topological phase is derived by obtaining the total chiral charge of the zero modes in the exact solution of the free fermion propagator. [S0556-2821(99)06521-2].