  The adjoint 2-dimensional $QCD$ with the gauge group $SU(N)/Z_N$ admits
topologically nontrivial gauge field configurations associated with nontrivial
$\pi_1[SU(N)/Z_N] = Z_N$. The topological sectors are labelled by an integer
$k=0,\ldots, N-1$. However, in contrast to $QED_2$ and $QCD_4$, this topology
is not associated with an integral invariant like the magnetic flux or
Pontryagin index. These instantons may admit fermion zero modes, but there is
always an equal number of left-handed and right-handed modes, so that the
Atiyah-Singer theorem, which determines in other cases the number of the modes,
does not apply.
  The mod. 2 argument suggests that, for a generic gauge field configuration,
there is either a single doublet of such zero modes or no modes whatsoever.
However, the known solution of the Dirac problem for a wide class of gauge
field configurations indicates the presence of $k(N-k)$ zero mode doublets in
the topological sector $k$. In this note, we demonstrate in an explicit way
that these modes are not robust under a generic enough deformation of the gauge
background and confirm thereby the mod. 2 conjecture.
  The implications for the physics of this theory (screening vs. confinement
issue) are briefly discussed.
