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$\mathcal{N}=4$ Super-Yang-Mills (SYM) theory is the maximally supersymmetric gauge theory in four dimensions and is an excellent playground to investigate new techniques in amplitude and Wilson loop calculations.

From what I could gather, $\mathcal{N}=4$ SYM is usually conformal, but apparently there exists a so called "Coulomb branch" of the theory on which it is not conformal and possesses very different properties. I find it very hard to find a pedagogical introduction into what exactly this "Coulomb branch" is. Perhaps someone here knows of a good learning resource to read up on this topic? Thanks for any suggestion!

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Therminology follows from vacuum expectation value of scalar fields in SUSY multiplet. In Coulomb phase gauge symmetry broken by vev: $SU(N) \to U^{N-1}(1)$. In suprconformal phase gauge group is not broken.

For more details, I recomend you chapter 10.2 in Bertolini lectures Lectures on Supersymmetry.

About N=4 you can consult chapter 3.1 Supersymmetric Gauge Theories and the AdS/CFT Correspondence.

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