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 τ . This permits a systematic and non-perturbative large charge expansion for any N and τ . Especially novel is a double-scaled “gravity regime” in which p N 2 ≫ 1, holographically dual to a large charge regime of semiclassical type IIB string theory in AdS 5 × 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 .