The hermitian Wilson-Dirac operator in smooth SU(2) instanton   backgrounds
  We study the spectral flow of the hermitian Wilson-Dirac operator $\ham(m)$ as a function of $m$ in smooth SU(2) instanton backgrounds on the lattice. For a single instanton background with Dirichlet boundary conditions on $\ham(m)$, we find a level crossing in the spectral flow of $\ham(m)$, and we find the shape of the crossing mode at the crossing point to be in good agreement with the zero mode associated with the single instanton background. With anti-periodic boundary conditions on $\ham(m)$, we find that the instanton background in the singular gauge has the correct spectral flow but the one in regular gauge does not. We also investigate the spectral flows of two instanton and instanton-anti-instanton backgrounds. 
Robert G. Edwards, Urs M. Heller and Rajamani Narayanan