Exact local fermionic zero modes
  We introduce a simple method to find localized exact fermionic zero modes for any local fermionic action. The zero modes are attached to specific local gauge configurations. Examples are provided for staggered and Wilson fermion actions in 2-6 dimensions, at finite and infinite lattice volumes, and for abelian and non-abelian gauge groups. One of our concrete results is that a finite density of almost zero modes must occur in quenched four dimensional lattice gauge theory simulations that use traditional methods. This density is exponentially suppressed in the gauge coupling constant. 
F. Berruto, R. Narayanan, H. Neuberger