Large Charge 't Hooft Limit of $\mathcal{N}=4$ Super-Yang-Mills

The planar integrability of $\mathcal{N}=4$ super-Yang-Mills (SYM) is the cornerstone for numerous exact observables. We show that the large charge sector of the ${\rm SU}(2)$ $\mathcal{N}=4$ SYM provides another interesting solvable corner which exhibits striking similarities despite being far from the planar limit. We study non-BPS operators obtained by small deformations of half-BPS operators with $R$-charge $J$ in the limit $J\to\infty$ with $\lambda_{J}\equiv g_{\rm YM}^2 J/2$ fixed. The dynamics in this {\it large charge 't Hooft limit} is constrained by a centrally-extended $\mathfrak{psu}(2|2)^2$ symmetry that played a crucial role for the planar integrability. To the leading order in $1/J$, the spectrum is fully fixed by this symmetry, manifesting the magnon dispersion relation familiar from the planar limit, while it is constrained up to a few constants at the next order. We also determine the structure constant of two large charge operators and the Konishi operator, revealing a rich structure interpolating between the perturbative series at weak coupling and the worldline instantons at strong coupling. In addition we compute heavy-heavy-light-light (HHLL) four-point functions of half-BPS operators in terms of resummed conformal integrals and recast them into an integral form reminiscent of the hexagon formalism in the planar limit. For general ${\rm SU}(N)$ gauge groups, we study integrated HHLL correlators by supersymmetric localization and identify a dual matrix model of size $J/2$ that reproduces our large charge result at $N=2$. Finally we discuss a relation to the physics on the Coulomb branch and explain how the dilaton Ward identity emerges from a limit of the conformal block expansion. We comment on generalizations including the large spin 't Hooft limit, the combined large $N$-large $J$ limits, and applications to general $\mathcal{N}=2$ superconformal field theories.