BPS Wilson loops in Minkowski spacetime and Euclidean space

TL;DR

The paper demonstrates that spacelike BPS Wilson loops cannot exist in Minkowski spacetime due to supersymmetry constraints but can exist in Euclidean space where reality conditions are relaxed, aligning with AdS/CFT expectations.

Contribution

It provides a proof that spacelike BPS Wilson loops do not preserve supersymmetry in Minkowski space and clarifies their existence in Euclidean space.

Findings

Spacelike Wilson loops do not preserve supersymmetry in Minkowski space.

Spacelike Wilson loops exist in Euclidean space after Wick rotation.

The reality conditions of spinors explain the non-existence in Minkowski space.

Abstract

We give evidence that spacelike BPS Wilson loops do not exist in Minkowski spacetime. We show that spacelike Wilson loops in Minkowski spacetime cannot preserve any supersymmetries, in $d = 4$ $\mathcal N = 4$ super Yang-Mills theory, $d = 3$ $\mathcal N = 2$ super Chern-Simons-matter theory, and $d = 3$ $\mathcal N = 6$ Aharony-Bergman-Jafferis-Maldacena theory. We not only show this using infinite straight lines and circles as examples, but also we give proofs for general curves. We attribute this to the conflicts of reality conditions of the spinors. However, spacelike Wilson loops do exist in Euclidean space. There are both BPS Wilson loops along infinite straight lines and circular BPS Wilson loops. This is because the reality conditions of the spinors are lost after Wick rotation. The result is reasonable in view of the AdS/CFT correspondence.