Supersymmetric Wilson loops on S^3

TL;DR

This work classifies and analyzes a broad family of supersymmetric Wilson loops on ${ m S}^3$ in ${ m N}=4$ SYM, showing that their string duals are governed by a novel almost complex structure on ${ m AdS}_4 imes{ m S}^2$ and that their expectation values admit a generalized calibration interpretation. It provides explicit examples with varying amounts of preserved supersymmetry, derives a first-order pseudo-holomorphic system for the dual strings, and reveals a deep connection to two-dimensional Yang–Mills theory for loops restricted to ${ m S}^2$, supported by perturbative and strong-coupling checks. The results offer a coherent holographic and field-theoretic framework for exact or highly constrained calculations of nontrivial Wilson loop observables in this sector, with potential ties to matrix models and twisted gauge theories. Overall, the paper uncovers a rich structure linking SUSY Wilson loops, calibrated string geometry, and lower-dimensional gauge dynamics, suggesting avenues for exact results within ${ m AdS}/{ m CFT}$ and beyond.

Abstract

This paper studies in great detail a family of supersymmetric Wilson loop operators in N=4 supersymmetric Yang-Mills theory we have recently found. For a generic curve on an S^3 in space-time the loops preserve two supercharges but we will also study special cases which preserve 4, 8 and 16 supercharges. For certain loops we find the string theory dual explicitly and for the general case we show that string solutions satisfy a first order differential equation. This equation expresses the fact that the strings are pseudo-holomorphic with respect to a novel almost complex structure we construct on AdS_4 x S^2. We then discuss loops restricted to S^2 and provide evidence that they can be calculated in terms of similar observables in purely bosonic YM in two dimensions on the sphere.

Figures (4)

• Figure 1: Quarter-BPS Wilson loop along a latitude. In a. we show the Wilson loop along a latitude at angle $\theta_0$ on an $S^2 \subset \mathbb{R}^4$. b. depicts the scalar couplings which follow a dual latitude on $S^2 \subset S^5$. Notice that if we took b. to be the path of the loop in space, then a. would describe the associated scalar couplings. This is an explicit example of the duality between scalar and gauge field couplings discussed in the text.
• Figure 2: Quarter-BPS Wilson loop made of two longitudes. In a. we show the loop on $S^2 \subset \mathbb{R}^4$ obtained by taking two half circles, or longitudes, with opening angle $\delta$. The corresponding scalar couplings in b. turn out to be two points on the equator of $S^2 \subset S^5$ separated by an angle $\pi-\delta$.
• Figure 3: An arbitrary curve on $S^2$ divides it into two surfaces, one with area ${\mathcal{A}}_1$ and the other with area ${\mathcal{A}}_2$. In all the calculations that we did the expectation value of the Wilson loop turns out to be a function only of the product of those two areas.
• Figure 4: The quarter-BPS Wilson loop made of two longitudes (a.) can be mapped to a stereographic projection the a cusp on the plane (b.). The scalar couplings (see figure \ref{['longi-fig']} b.) are not altered and are the natural coupling for a supersymmetric cusp in the plane.