Instantons and fermion condensate in adjoint two-dimensional QCD

We show that two-dimensional QCD with adjoint fermions involves instantons due to nontrivial ${\mathrm{\pi }}_1$[SU(N)/${\mathit{Z}}_{\mathit{N}}$]=${\mathit{Z}}_{\mathit{N}}$. At high temperatures, the quasiclassical approximation works and the action and the form of the effective (with account of quantum corrections) instanton solution can be evaluated. The instanton presents a localized configuration with a size ∝${\mathit{g}}^{\mathrm{-}1}$. At N=2, it involves exactly 2 zero fermion modes and gives rise to the fermion condensate 〈λ${\mathrm{¯}}^{\mathit{a}}$ ${\lambda }^{\mathit{a}}$ ${\mathrm{〉}}_{\mathit{T}}$ which falls off ∝exp{-${\mathrm{\pi }}^{3/2}$T/g} at high T but remains finite. At low temperatures, both instanton and bosonization arguments also exhibit the appearance of the fermion condensate 〈λ${\mathrm{¯}}^{\mathit{a}}$ ${\lambda }^{\mathit{a}}$ ${\mathrm{〉}}_{\mathit{T}=0\mathrm{}}$∼g. For N>2, the situation is paradoxical. There are 2(N-1) fermion zero modes in the instanton background which implies the absence of the condensate in the massless limit. On the other hand, bosonization arguments suggest the appearance of the condensate for any N. Possible ways to resolve this paradox (which occurs also in some four-dimensional gauge theories) are discussed.