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

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.