Wilson Loops in N=4 Supersymmetric Yang--Mills Theory

TL;DR

The paper investigates Wilson loops in N=4 SYM by summing infinite classes of planar ladder diagrams and comparing with AdS/CFT predictions. It derives exact weak-coupling results up to ${\cal O}(g^4N^2)$, analyzes UV-divergence cancellations for smooth loops, and demonstrates that for the circular loop the ladder diagrams sum to $\langle W(C)\rangle_{\text{ladders}}=\frac{2}{\sqrt{g^2N}}I_1(\sqrt{g^2N})$, whose strong-coupling limit matches the expected $e^{\sqrt{g^2N}}$ scaling; vertex corrections cancel at $D=4$, hinting at an exact ladder-dominated structure. For anti-parallel lines, the ladder sum again yields a strong-coupling exponential in $\sqrt{g^2N}$ but with a coefficient that does not precisely match the AdS/CFT result, and vertex diagrams cancel at leading order as well. The work highlights how conformal symmetry and planarity constrain perturbative corrections, suggesting that certain ladder resummations may capture essential features of the Wilson loop in this theory and offers a framework to probe AdS/CFT predictions from perturbation theory.

Abstract

Perturbative computations of the expectation value of the Wilson loop in N=4 supersymmetric Yang-Mills theory are reported. For the two special cases of a circular loop and a pair of anti-parallel lines, it is shown that the sum of an infinite class of ladder-like planar diagrams, when extrapolated to strong coupling, produces an expectation value characteristic of the results of the AdS/CFT correspondence, $<W>\sim\exp((constant)\sqrt{g^2N})$. For the case of the circular loop, the sum is obtained analytically for all values of the coupling. In this case, the constant factor in front of $\sqrt{g^2N}$ also agrees with the supergravity results. We speculate that the sum of diagrams without internal vertices is exact and support this conjecture by showing that the leading corrections to the ladder diagrams cancel identically in four dimensions. We also show that, for arbitrary smooth loops, the ultraviolet divergences cancel to order $g^4N^2$.