Localization of the four-dimensional N=4 SYM to a two-sphere and 1/8 BPS Wilson loops

TL;DR

The authors demonstrate that four-dimensional $ ext{N}=4$ SYM on $S^4$ localizes to a two-dimensional constrained Hitchin–Yang–Mills theory on $S^2$, with the interesting 1/8 BPS Wilson loops on the equatorial $S^2$ captured by this 2d theory. They implement localization via a Hermitian supercharge $Q$, reduce the problem to a three-dimensional setup on a ball, and show that the physical action becomes a boundary term that defines a 2d boundary theory, identified as a constrained complexified Yang–Mills theory. Perturbatively, the 2d boundary theory coincides with ordinary bosonic 2d YM in the zero-instanton sector, supporting the Drukker–Giombi–Ricci–Trancanelli conjecture about 1/8 BPS loops on $S^2$; non-perturbative sectors are discussed but shown to be suppressed or constrained by fermionic zero modes. The work connects the 4d Wilson loop problem to Hitchin–HYM (cHYM) and, under a hyper-Kähler rotation viewpoint, to aYM, outlining how exact results in 2d YM could inform the understanding of 4d observables, and highlighting open issues related to determinants and non-perturbative contributions.

Abstract

We localize the four-dimensional N=4 super Yang-Mills theory on a four-sphere to the two-dimensional constrained Hitchin/Higgs-Yang-Mills (cHYM) theory on a two-sphere S^2. We show that expectation values of certain 1/8 BPS supersymmetric Wilson loops on S^2 in the 4d N=4 SYM is captured by the 2d cHYM theory. We further argue that expectation values of Wilson loops in the cHYM theory agree with the prescription "two-dimensional bosonic Yang-Mills excluding instanton contributions". Hence, we support the recent conjecture by Drukker, Giombi, Ricci and Trancanelli on the 1/8 BPS Wilson loops on S^2 in the 4d N=4 SYM.