The SU ( N) Wilson loop average in two dimensions

Abstract

We solve explicitly a closed, linear loop equation for the SU(2) Wilson loop average on a two-dimensional plane and generalize the solution to the case of the SU( N) Wilson loop average with an arbitrary closed contour. Furthermore, the flat space solution is generalized to any two-dimensional manifold for the SU(2) Wilson loop average and to any two-dimensional manifold of genus O for the SU( N) Wilson loop average. The SU( N) Wilson loop average folows an area law W( C) = Σ r P' r exp[ - Σ i j ri 2 S i ], where j ri 2 is the quadratic Casimir operator for the window with area S i . Only certain combinations of the Casimir operators are allowed in the sum over i. We give a physical interpretation of the constants P r 1 in the case of a non self-intersecting composed path C and of the constraints determining in which combinations the Casimir operators occur.