Instanton contributions to Wilson loops with general winding number in two dimensions and the spectral density Original Research Article

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 n2A 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 ʹt Hooft-CPV and Wu-Mandelstam-Leibbrandt (WML)-prescribed perturbative series, the former being consistent with the exact result, the latter reproducing the zero-instanton 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 ʹt Hooft and WML: for small area they exhibit a gap and coincide when the theory is considered on the sphere S2.