  $QCD_2$ with fermions in the adjoint representation is invariant under
$SU(N)/Z_N$ and thereby is endowed with a non-trivial vacuum structure
(k-sectors). The static potential between adjoint charges, in the limit of
infinite mass, can be therefore obtained by computing Wilson loops in the pure
Yang-Mills theory with the same non-trivial structure. When the (Euclidean)
space-time is compactified on a sphere $S^2$, Wilson loops can be exactly
expressed in terms of an infinite series of topological excitations
(instantons). The presence of k-sectors modifies the energy spectrum of the
theory and its instanton content. For the exact solution, in the limit in which
the sphere is decompactified, a k-sector can be mimicked by the presence of
k-fundamental charges at $\infty$, according to a Witten's suggestion. However
this property neither holds before decompactification nor for the genuine
perturbative solution which corresponds to the zero-instanton contribution on
$S^2$.
