  The use of adjoint (quasi) zero-modes of the Dirac operator to probe the
Yangs-Mills vacuum has been recently advocated by Gonzalez-Arroyo and Kirchner.
The construction relies on the use of the super-symmetric zero mode which, for
classical configurations, provides a direct estimate of the gauge action
density. In the lattice implementation of this idea, we show how the results
improve considerably if the overlap operator is used instead of the
Wilson-Dirac one. Before proceeding to the detailed study of Monte Carlo
ensembles, we studied here a series of potentially complicated situations which
can be encountered. In particular, we study the case of instanton
anti-instanton pairs and analyse how the results depend upon separation. The
effect of lattice artifacts is also of concern. Indeed, a statistical analysis
of zero modes of thermalised SU(2) configurations at beta=2.57 shows a
significant fraction having 4N+2 adjoint zero modes, in contradiction with the
index theorem. This violation must be associated to the roughness of the
lattice configurations. Indeed, we show that this situation occurs for
instantons of size of the order of the lattice spacing.
