Exact Large Charge in $\mathcal{N}=4$ SYM and Semiclassical String Theory

In four-dimensional $\mathcal{N}=4$ super Yang-Mills theory with gauge group $SU(N)$, we present a closed-form solution for a family of integrated four-point functions involving stress tensor multiplet composites of arbitrary R-charge. These integrated correlators are shown to be equivalent to a one-dimensional semi-infinite lattice of harmonic oscillators with nearest-neighbor interactions, evolving over the fundamental domain of $SL(2,\mathbb{Z})$. The solution, exceptionally simple in an $SL(2,\mathbb{Z})$-invariant eigenbasis, is exact in the R-charge $p$, rank $N$, and complexified gauge coupling $\tau$. This permits a systematic and non-perturbative large charge expansion for any $N$ and $\tau$. Especially novel is a double-scaled "gravity regime" in which $p \sim N^2 \gg 1$, holographically dual to a large charge regime of semiclassical type IIB string theory in AdS$_5\, \times$ S$^5$. Our results in this limit provide a holographic computation of integrated semiclassical string amplitudes at arbitrary string coupling, including an emergent string scale with a large charge dressing factor. We compare to extremal correlators in superconformal QCD, for which we predict new genus expansions at large charge scaling with $N$.