Instanton corrections to circular Wilson loops in N=4 Supersymmetric Yang-Mills

TL;DR

This work computes the one-instanton contribution to a circular Wilson loop in ${\cal N}=4$ SU(2) Yang–Mills in the semi-classical regime, showing a finite, regulator-independent result after subtracting a perimeter divergence. By leveraging residual $SO(2,2)$ symmetry, a six-dimensional conformal embedding, and a superspace construction via a supercoset $SU(2,2|4)/H$, the authors integrate over both bosonic and fermionic instanton moduli to obtain a finite density for the loop. A nonzero finite contribution arises for the circle, in contrast to the straight line where the instanton effect vanishes after subtraction; the perimeter divergence is identified as a regulator artifact tied to small instantons touching the loop. The results are extended to $SU(N)$ in the small 't Hooft coupling limit and discussed in the context of AdS/CFT and S-duality, including implications for instanton correlators with local operators and potential strong-coupling pictures via D-instantons in $AdS_5\times S^5$.

Abstract

It is argued that whereas supersymmetry requires the instanton contribution to the expectation value of a straight Wilson line in the N=4 supersymmetric SU(2) Yang-Mills theory to vanish, it is not required to vanish in the case of a circular Wilson loop. A non-vanishing value can arise from a subtle interplay between a divergent integral over bosonic moduli and a vanishing integral over fermionic moduli. The one-instanton contribution to such Wilson loops is explicitly evaluated in semi-classical approximation. The method utilizes the symmetries of the problem to perform the integration over the bosonic and fermionic collective coordinates of the instanton. The integral is singular for small instantons touching the loop and is regularized by introducing a cutoff at the boundary of the (euclidean) AdS_5 moduli space. In the case of a circular loop a nonzero finite result is obtained when the cutoff is removed and a perimeter divergence subtracted. This is contrasted with the case of the straight line where the result is zero after subtraction of an identical divergence per unit length. The linear divergence is an artifact of our non supersymmetric regulator that deserves further consideration. The generalization to gauge group SU(N) with arbitrary N is straightforward in the limit of small 't Hooft coupling. The extension to strong 't Hooft coupling is more challenging and only a qualitative discussion is given of the AdS/CFT correspondence