Two-dimensional instantons with bosonization and physics of adjoint two-dimensional QCD

We evaluate partition functions ZI in topologically nontrivial (instanton) gauge sectors in the bosonized version of the Schwinger model and in a gauged WZNW model corresponding to two-dimensional QCD (QCD₂) with adjoint fermions. We show that the bosonized model is equivalent to the fermion model only if a particular form of the WZNW action with a gauge-invariant integrand is chosen. For the exact correspondence, it is necessary to integrate over the ways the gauge group SU(N)ZN is embedded into the full O(N²-1) group for the bosonized matter field. For even N, one should also take into account the contributions of both disconnected components in O(N²-1). In that case, ZI∝m^n₀ for small fermion masses where 2n₀ coincides with the number of fermion zero modes in a particular instanton background. The Taylor expansion of ZIm^n₀ in mass involves only even powers of m, as it should. The physics of adjoint QCD₂ is discussed. We argue that, for odd N, the discrete chiral symmetry Z₂Z₂ present in the action is broken spontaneously down to Z₂ and the fermion condensate 〈λλ〉₀ is formed. The system undergoes a first order phase transition at Tc=0 so that the condensate is zero at an arbitrary small temperature. It is not yet quite clear what happens for even $N>~4$.