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

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