Supersymmetric Wilson loops on S ³

This paper studies in great detail a family of supersymmetric Wilson loop operators in = 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 × 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.