Multiple vacua in two-dimensional Yang-Mills theory

Two-dimensional SU(N) Yang-Mills theory is endowed with a non-trivial vacuum structure (k-sectors). The presence of k-sectors modifies the energy spectrum of the theory and its instanton content, the (Euclidean) space-time being compactified on a sphere. 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 infinity, 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 the sphere.