Large charge ’t Hooft limit of $$ \mathcal{N} $$ = 4 super-Yang-Mills

Abstract

The planar integrability of $$ \mathcal{N} $$ N = 4 super-Yang-Mills (SYM) is the cornerstone for numerous exact observables. We show that the large charge sector of the SU(2) $$ \mathcal{N} $$ 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 → ∞ with $$ {\lambda}_J\equiv {g}_{\textrm{YM}}^2J/2 $$ λ J ≡ g YM 2 J / 2 fixed. The dynamics in this large charge ’t Hooft limit is constrained by a centrally-extended $$ \mathfrak{psu} $$ 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 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 $$ \mathcal{N} $$ N = 2 superconformal field theories.