  We study a number of different ingredients related to $\theta$ dependence,
the non-dispersive contribution in topological susceptibility with the "wrong"
sign, topological sectors in gauge theories, and related subjects using a
simple "deformed QCD". This model is a weakly coupled gauge theory, which
however has all the relevant essential elements allowing us to study difficult
and nontrivial questions which are known to be present in real strongly coupled
QCD. Essentially we want to test the ideas related to the $U(1)_A$ problem in a
theoretically controllable manner using the "deformed QCD" as a toy model. One
can explicitly see microscopically how the crucial elements work.
