Wilson loops on the Coulomb branch of $N=4$ super-Yang-Mills

TL;DR

The paper studies Wilson loops on the Coulomb branch of N=4 SYM by solving for minimal surfaces in AdS5×S5 that end on a boundary contour and attach to a floating D3-brane. It maps the phase diagram of a circular Wilson loop, revealing a Gross-Ooguri-type transition between connected and hemisphere saddles as functions of radius, brane position z0, and misalignment angle phi, and provides evidence that the straight-line loop remains tree-level exact. At large circle size the results reproduce a perimeter-law behavior, connecting strong-coupling holography with perturbative expectations. The workHighlights the richness of Coulomb-branch holography, defect/CFT interfaces, and the interplay between integrability hints and non-perturbative Wilson-loop physics.

Abstract

We study Wilson loops on the Coulomb branch of $N = 4$ super-Yang-Mills theory, by solving for minimal surfaces that connect the contour on the boundary with the D3-brane in the bulk of AdS$_5 \times S^5$. The circular loop undergoes the Gross-Ooguri transition as a function of the radius and angular separation, and we fully map its phase diagram. As a byproduct we find evidence that the expectation value of the straight line is tree-level exact.

Figures (6)

• Figure 1: This figure depicts the string configuration of the Wilson loop where the contour is a straight line. The bottom gray plane represents the $AdS_5$-boundary, the green plane represents the D3-brane, the yellow shaded surface represents the worldsheet traced out by the string.
• Figure 2: Two types of minimal surfaces conrtributing to the string path integral: (a) the hemisphere, connected to the brane by an infinitely thin tube, and (b) cylindrical minimal surface with a neck.
• Figure 3: The relation between the physical parameters of the string configuration and the energy $\epsilon$: (a) the worldsheet area; (b) the ratio of the bulk position of the brane $z_0$ and the radius $R$ of the circle. The results for the connected surface smoothly match to the hemisphere solution zero energy.
• Figure 4: The area of the minimal surface. The green line is the true minimum of the string action that switches between the connected and disconnected saddle-points at $z_0=z_{\rm c}$. The lower branch of the connected solution is stable, the upper is unstable, they meet at $z_0=z_{\rm max}$, the maximal hight the minimal surface can reach.
• Figure 5: The area of the connected minimal surface for various values of $j$. The dashed line is the area of the disconnected surface: in the shaded region above this line the true minimum of the string action is the hemisphere solution.