The Operator Product Expansion for Wilson Loops and Surfaces in the Large N Limit

TL;DR

This work uses the AdS/CFT correspondence at large $N$ and large $\lambda$ to compute the OPE of small Wilson loops in ${\cal N}=4$ SYM and Wilson surfaces in the 6D $(0,2)$ theory, by mapping loops to minimal-area string worldsheets and membranes. It identifies the allowed operator content by symmetry and computes OPE coefficients via two complementary approaches: perturbative field theory and supergravity, with explicit results for protected operators such as symmetric-traceless scalars and for dilaton and tachyonic KK scalars, showing distinct $\lambda$- and $N$-dependences. The analysis yields a leading scalar-exchange picture for inter-loop interactions and a UV logarithmic divergence in the Wilson surface corresponding to a rigid-string action, signaling subtle conformal-scale structure in higher dimensions. Together, these results illuminate how bulk field exchanges determine OPE data at strong coupling and how Wilson-valued observables behave in AdS/CFT beyond perturbation theory.

Abstract

The operator product expansion for ``small'' Wilson loops in {\cal N}=4, d=4 SYM is studied. The OPE coefficients are calculated in the large N and g_{YM}^2 N limit by exploiting the AdS/CFT correspondence. We also consider Wilson surfaces in the (0,2), d=6 superconformal theory. In this case, we find that the UV divergent terms include a term proportional to the rigid string action.

Figures (3)

• Figure 1: The emission of a particle from a circular Wilson loop of size $a$ onto the AdS boundary at a distance $r$ from the loop.
• Figure 2: The correlation between two circular Wilson loops.
• Figure 3: The interaction between two rectangular Wilson loops of size $a$, separated by a distance $L\gg a$.