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

We evaluate partition functions Z(I) 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 (QCD(2)) 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)/Z(N) is embedded into the full O(N-2-1) group for the bosonized matter field. For even N, one should also take into account the contributions of both disconnected components in O(N-2-1). In that case, Z(I) proportional to m(n0) for small fermion masses where 2n(0) coincides with the number of fermion zero modes in a particular instanton background. The Taylor expansion of Z(l)Im(n0) in mass involves only even powers of m, as it should. The physics of adjoint QCD(2) is discussed. We argue that, for odd N, the discrete chiral symmetry Z(2)xZ(2) present in the action is broken spontaneously down to Z(2) and the fermion condensate (<(lambda)over bar>lambda)(0) is formed. The system undergoes a first order phase transition at T-c=0 so that the condensate is zero at an arbitrary small temperature. It is not yet quite clear what happens for even N greater than or equal to 4.