BPS Wilson Loops on S^2 at Higher Loops

TL;DR

This paper computes higher-loop vacuum expectation values of supersymmetric Wilson loops on S^2 in N=4 SYM, comparing results with 2D Yang-Mills theory, and explores their potential correspondence.

Contribution

It provides second and third order perturbative calculations of Wilson loops on S^2 in N=4 SYM and compares them with 2D Yang-Mills results, extending understanding of their relation.

Findings

Results consistent with 2D Yang-Mills in certain limits

Numerical evaluation supports theoretical predictions

Identifies limitations in matching higher-order terms

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.