Special contact Wilson loops

TL;DR

This work identifies a tractable subclass of AdS$_5$ extremal surfaces—special Legendrian submanifolds—that correspond to Wilson loops in ${\cal N}=4$ SYM. By formulating first-order differential equations that determine these surfaces from boundary contours, the paper introduces special contact Wilson loops and derives two necessary (and conjecturally sufficient near circular loops) conditions on boundary contours, linking geometry to strong-coupling functional derivatives. It provides explicit examples, analyzes infinitesimal deformations, demonstrates the general absence of supersymmetry for noncircular cases, and studies near-boundary behavior and the variation of the regularized area, revealing structural constraints on $W[C]$ at strong coupling. The results offer a concrete, geometrically controlled framework for understanding Wilson loops through special Legendrian geometry and its boundary conditions, with potential implications for exact strong-coupling computations.

Abstract

Wilson loops in ${\cal N}=4$ supersymmetric Yang-Mills theory correspond at strong coupling to extremal surfaces in $AdS_5$. We study a class of extremal surfaces known as special Legendrian submanifolds. The "hemisphere" corresponding to the circular Wilson loop is an example of a special Legendrian submanifold, and we give another example. We formulate the necessary conditions for the contour on the boundary of $AdS_5$ to be the boundary of the special Legendrian submanifold and conjecture that these conditions are in fact sufficient. We call the solutions of these conditions "special contact Wilson loops". The first order equations for the special Legendrian submanifold impose a constraint on the functional derivatives of the Wilson loop at the special contact contour which should be satisfied in the Yang-Mills theory at strong coupling.