The exact expression for Wilson loop averages winding n times on a closed contour is obtained in two dimensions for pure U (N ) Yang-Mills theory and, rather surprisingly, it displays an interesting duality in the exchange n ↔ N .The large-N limit of our result is consistent with previous computations.Moreover we discuss the limit of small loop area A, keeping n 2 A fixed, and find it coincides with the zero-instanton approximation. We deduce that small loops, both at finite and infinite "volume", are blind to instantons.Next we check the non-perturbative result by resumming 't Hooft-CPV and Wu-Mandelstam-Leibbrandt (WML)-prescribed perturbative series, the former being consistent with the exact result, the latter reproducing the zeroinstanton contribution. A curious interplay between geometry and algebraic invariants is observed. Finally we compute the spectral density of the Wilson loop operator, at large N , via its Fourier representation, both for 't Hooft and WML: for small area they exhibit a gap and coincide when the theory is considered on the sphere S 2 .

We extend the Wilson renormalization group (RG) to supersymmetric theories. As this regularization scheme preserves supersymmetry, we exploit the superspace technique. To set up the formalism we first derive the RG flow for the massless Wess-Zumino model and deduce its perturbative expansion. We then consider N=1 supersymmetric Yang-Mills and show that the local gauge symmetry -broken by the regularization-can be recovered by a suitable choice of the RG flow boundary conditions. We restrict our analysis to the first loop, the generalization to higher loops presenting no difficulty due to the iterative nature of the procedure. Furthermore, adding matter fields, we reproduce the one-loop supersymmetric chiral anomaly to the second order in the vector field.

We extend the Wilson renormalization group (RG) formulation to chiral gauge theories and show that local gauge symmetry can be implemented by a suitable choice of the RG flow boundary conditions. Since the space-time dimension is four, there is no ambiguity in handling the matrix γ 5 and left and right fermions are not coupled. As a result the ultraviolet action contains all possible globally chiral invariant interactions. Nevertheless, the correct chiral anomaly is reproduced.

We present an investigation on the invariance properties of noncommutative Yang-Mills theory in two dimensions under area preserving diffeomorphisms. Stimulated by recent remarks by Ambjorn, Dubin and Makeenko who found a breaking of such an invariance, we confirm both on a fairly general ground and by means of perturbative analytical and numerical calculations that indeed invariance under area preserving diffeomorphisms is lost. However a remnant survives, namely invariance under linear unimodular tranformations.

We study the perturbative unitarity of non-commutative quantum Yang-Mills theories, extending previous investigations on scalar field theories to the gauge case where non-locality mingles with the presence of unphysical states. We concentrate our efforts on two different aspects of the problem. We start by