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

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 $n \leftrightarrow N$. The large-N limit of our result is consistent with previous computations. Moreover we discuss the limit of small loop area $\mathcal{A}$, keeping $n^2\mathcal{A}$ 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 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 't Hooft and WML: for small area they exhibit a gap and coincide when the theory is considered on the sphere $S^2$.