Large charge meets semiclassics in $\mathcal{N}=4$ super Yang-Mills

We study the large-charge sector of $\mathcal{N}=4$ super Yang-Mills theory (SYM) with $SU(N)$ gauge group by constructing a special class of half-BPS heavy operators, termed "canonical operators". Such operators exhibit remarkable simplicity in the large-charge 't Hooft limit, where the dimension of the operators $\Delta \to \infty$ with $\Delta\, g_{\text{YM}}^2$ held finite. Canonical operator insertions in this regime map $\mathcal{N}=4$ SYM onto the Coulomb branch, by assigning a classical profile to the scalar fields with non-vanishing values along the diagonals given by the roots of unity. We follow a semiclassical approach to study two-point, three-point and Heavy-Heavy-Light-Light (HHLL) correlators. In particular we show that HHLL correlators in the large-charge 't Hooft limit are computed as two-point functions in a background determined by the classical profiles. We provide consistent evidence of our findings by computing the same observables via supersymmetric localization.