Supersymmetric Wilson loops at two loops

TL;DR

This work tests a conjecture that certain 1/8–BPS supersymmetric Wilson loops on S^2 in N=4 SYM are exactly captured by the zero-instanton sector of YM$_2$ on the sphere, i.e., by a Gaussian matrix model. The authors develop a finite two-loop framework for these loops, revealing a nontrivial cancellation between ladder-type and interacting diagrams and deriving a compact finite expression for the order-$g^4$ contribution. They apply this to a wedge-shaped cusped loop on S^2, performing analytical and numerical evaluations that yield $W_1$ and $W_2^{\mathrm{mnb}}$ in agreement with the YM$_2$ matrix-model expansion, thus supporting the conjectured localization-like description. The results point to a richer localization structure for these loops and motivate further checks at higher orders and for other contours, with potential implications for exact Wilson loop computations in four dimensions.

Abstract

We study the quantum properties of certain BPS Wilson loops in ${\cal N}=4$ supersymmetric Yang-Mills theory. They belong to a general family, introduced recently, in which the addition of particular scalar couplings endows generic loops on $S^3$ with a fraction of supersymmetry. When restricted to $S^2$, their quantum average has been further conjectured to be exactly computed by the matrix model governing the zero-instanton sector of YM$_2$ on the sphere. We perform a complete two-loop analysis on a class of cusped Wilson loops lying on a two-dimensional sphere, finding perfect agreement with the conjecture. The perturbative computation reproduces the matrix-model expectation through a highly non-trivial interplay between ladder diagrams and self-energies/vertex contributions, suggesting the existence of a localization procedure.