Two-Dimensional Instantons with Bosonization and Physics of Adjoint QCD2

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 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 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(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(N2 − 1). In that case, ZI ∝ mn0 for small fermion masses where 2n0 coincides with the number of fermion zero modes in a particular instanton background. The Taylor expansion of ZI/m 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 Z2⊗Z2 present in the action is broken spontaneously down to Z2 and the fermion condensate < λ̄λ >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.