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The adjoint 2-dimensional $QCD$ with the gauge group $SU(N)/Z_N$ admitstopologically 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 topologyis not associated with an integral invariant like the magnetic flux orPontryagin index. These instantons may admit fermion zero modes, but there isalways an equal number of left-handed and right-handed modes, so that theAtiyah-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 gaugefield configurations indicates the presence of $k(N-k)$ zero mode doublets inthe topological sector $k$. In this note, we demonstrate in an explicit waythat these modes are not robust under a generic enough deformation of the gaugebackground and confirm thereby the mod. 2 conjecture.  The implications for the physics of this theory (screening vs. confinementissue) are briefly discussed.

A comment on instantons and their fermion zero modes in adjoint QCD_2