Wilson loops in SYM theory: from weak to strong coupling

TL;DR

This work surveys Wilson loops in ${\cal N}=4$ SYM across weak and strong coupling, emphasizing exact results and AdS/CFT interpretations. It shows how minimal-area string worldsheets in ${\rm AdS}^5\times S^5$ capture strong-coupling behavior (e.g., $\langle W(C)\rangle$ scaling like $e^{\sqrt{\lambda}}$ for simple loops) and how matrix-model techniques yield exact results for the circular loop that interpolate between perturbative and strong-coupling regimes. The analysis of OPE coefficients with chiral primary operators reveals exact strong-coupling expressions involving Bessel functions, clarifying how large-$N$ correlators encode both gauge-theory and stringy physics. Together, these results provide nontrivial tests of AdS/CFT and yield precise, testable predictions for Wilson-loop observables across coupling regimes, including phase transitions and string-breaking phenomena. The work also outlines open questions and directions for extending these methods beyond conformal setups.

Abstract

We review Wilson loops in N=4 supersymmetric Yang-Mills theory with emphasis on the exact results. The implications are discussed in the context of the AdS/CFT correspondence.

Figures (5)

• Figure 1: Before the conformal transformation, the regularization cuts the slice of thickness $\varepsilon$ near the boundary. After the transformation, regularization cuts the exterior of the sphere of radius $1/\varepsilon$.
• Figure 2: (a) A correlation function of the Wilson loop with a local operator is determined by an exchange of the supergravity mode between the classical string world sheet and the point of operator insertion on the boundary of $AdS^5$. (b) At large distances the correlator factorizes, and the OPE coefficient is given by an integral of the appropriate vertex operator over the world sheet.
• Figure 3: String breaking.
• Figure 4: $\ln\left\langle W(C_1)W(C_2) \right\rangle$ vs. the distance between the loops $C_1$ and $C_2$ for concentric circles of radius $R$Olesen:2000ji. The Gross-Ooguri phase transition is of the first order and takes place at $L_c=0.91 R$Zarembo:1999bu.
• Figure 5: A typical diagram that contributes to the correlator of the circular Wilson loop with CPO.