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

TL;DR

This paper derives an exact formula for Wilson loop averages with arbitrary winding numbers in two-dimensional pure U(N) Yang-Mills theory, revealing dualities, instanton effects, and spectral density properties, with implications for non-perturbative phenomena.

Contribution

It provides the first exact expression for Wilson loops with general winding numbers and explores instanton effects, dualities, and spectral properties in 2D Yang-Mills theory.

Findings

Exact Wilson loop averages with arbitrary winding numbers obtained.

Small loops are insensitive to instantons, matching zero-instanton approximation.

Spectral density exhibits a gap at small areas and matches on the sphere.

Abstract

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 ${\cal A}$, keeping $n^2 {\cal A}$ 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 $S^2$.