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

Abstract

A bstract The planar integrability of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 super-Yang-Mills (SYM) is the cornerstone for numerous exact observables. We show that the large charge sector of the SU(2) $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ {\lambda}_J\equiv {g}_{\textrm{YM}}^2J/2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>J</mml:mi> </mml:msub> <mml:mo>≡</mml:mo> <mml:msubsup> <mml:mi>g</mml:mi> <mml:mi>YM</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mi>J</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> fixed. The dynamics in this large charge ’t Hooft limit is constrained by a centrally-extended $$ \mathfrak{psu} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>psu</mml:mi> </mml:math> (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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 superconformal field theories.

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