Correlation functions / Coulomb branch amplitudes duality in $\mathcal{N} = 4$ SYM

We review infrared properties of scattering amplitudes in gauge theories, with a particular emphasis on planar $\mathcal{N} = 4$ super Yang–Mills (SYM) theory. We then discuss a web of dualities that relate amplitudes to seemingly different observables in planar $\mathcal{N} = 4$ SYM: light-like polygonal Wilson loops, correlation functions of local half-BPS operators, and their strong-coupling counterparts in AdS/CFT. We then focus on a recently uncovered duality between scattering amplitudes on the Coulomb branch, in a “pseudo off-shell’’ kinematic regime with small $W$-boson masses, and correlation functions of operators with large $R$-charge. This Coulomb amplitudes/heavy-correlator duality will be illustrated in detail for the case of a five-point correlator and the corresponding five-$W$-boson amplitude evaluated to two loops in the pseudo off-shell limit. The example provides non-trivial evidence for yet another duality that connects Coulomb-branch amplitudes and heavy correlators in planar $\mathcal{N} = 4$ SYM.