BPS Wilson Loops on S^2 at Higher Loops

TL;DR

This work investigates supersymmetric Wilson loops on the two-sphere in planar ${\cal N}=4$ SYM and tests their conjectured reduction to a Wu–Mandelstam–Leibbrandt-regulated 2-d Yang–Mills theory on $S^2$. By computing two-loop VEVs for loops formed by two longitudes and by a wavy latitude, and a three-loop connected correlator of two latitudes, the authors compare to zero-instanton sector results and to the reduced 2-d model in light-cone gauge. They find precise agreement for single-loop observables at ${\cal O}(\lambda^2)$, with cancellations between interacting diagrams and ladder diagrams reproducing 2-d YM predictions; for the connected correlator, the coincident-limit analysis reveals a leading ${|h|}$-dependence with controlled finite remnants, compatible with the reduced model but requiring further higher-order checks. These results strengthen the case that a reduced 2-d description captures key features of certain Wilson loops in ${\cal N}=4$ SYM and provide detailed benchmarks for future studies at higher perturbative orders and in the strong-coupling regime. The work also clarifies the structure of ladder versus interaction diagrams and highlights the intricate cancellations that emerge in this AdS/CFT-relevant sector.

Abstract

We consider supersymmetric Wilson loops of the variety constructed by Drukker, Giombi, Ricci, and Trancanelli, whose spatial contours lie on a two-sphere. Working to second order in the 't Hooft coupling in planar N=4 Supersymmetric Yang-Mills Theory (SYM), we compute the vacuum expectation value of a wavy-latitude and of a loop composed of two longitudes. We evaluate the resulting integrals numerically and find that the results are consistent with the zero-instanton sector calculation of Wilson loops in 2-d Yang-Mills on S^2 performed by Bassetto and Griguolo. We also consider the connected correlator of two distinct latitudes to third order in the 't Hooft coupling in planar N=4 SYM. We compare the result in the limit where the latitudes become coincident to a perturbative calculation in 2-d Yang-Mills on S^2 using a light-cone Wu-Mandelstam-Leibbrandt prescription. We are not able to calculate the SYM result at the required order in the separation between the latitudes necessary for a match with 2-d Yang-Mills; the result, however, does not preclude such a match.

Figures (15)

• Figure 1: The three geometries of Wilson loops on $S^2$ we consider: two longitudes, wavy-latitude, and two latitudes.
• Figure 2: The two-loop, non-ladder/rainbow diagrams contributing to $\langle W \rangle$. The Wilson loop is indicated by the outer circle. Internal solid lines refer to scalar and gauge fields, while the greyed-in bubble represents the one-loop correction to the propagator.
• Figure 3: A Wilson loop composed of two longitudes.
• Figure 4: A subset of the two-loop diagrams.
• Figure 5: A (different) subset of the two-loop diagrams.