Wilson Loops in the Adjoint Representation and Multiple Vacua in Two-Dimensional Yang–Mills Theory

Abstract

QCD2 with fermions in the adjoint representation is invariant under SU(N)/ZN and thereby is endowed with a nontrivial vacuum structure (k-sectors). The static potential between adjoint charges, in the limit of infinite mass, can be therefore obtained by computing Wilson loops in the pure Yang–Mills theory with the same nontrivial structure. When the (Euclidean) space-time is compacted on a sphere S2, Wilson loops can be exactly expressed in terms of an infinite series of topological excitations (instantons). The presence of k-sectors modifies the energy spectrum of the theory and its instanton content. For the exact solution, in the limit in which the sphere is decompacted, a k-sector can be mimicked by the presence of k-fundamental charges at ∞, according to Witten's suggestion. However, this property does not hold before decompaction or for the genuine perturbative solution which corresponds to the zero-instanton contribution on S2.