Exact large charge in N = 4 SYM and semiclassical string theory

In four-dimensional 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, Z). The solution, exceptionally simple in an SL(2, 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 similar to N-2 >> 1, holographically dual to a large charge regime of semiclassical type IIB string theory in AdS(5) x 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.