Far beyond the planar limit in strongly-coupled $$ \mathcal{N} $$ = 4 SYM

Abstract

When the SU(N) \( \mathcal{N} \) = 4 super-Yang-Mills (SYM) theory with complexified gauge coupling τ is placed on a round four-sphere and deformed by an \( \mathcal{N} \) = 2-preserving mass parameter m, its free energy F (m, τ, \( \overline{\tau} \)) can be computed exactly using supersymmetric localization. In this work, we derive a new exact relation between the fourth derivative \( {\partial}_m^4F\left(m,\tau, \overline{\tau}\right)\left|{{}_m}_{=0}\right. \) of the sphere free energy and the integrated stress-tensor multiplet four-point function in the \( \mathcal{N} \) = 4 SYM theory. We then apply this exact relation, along with various other constraints derived in previous work (coming from analytic bootstrap, the mixed derivative \( {\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2F\left(m,\tau, \overline{\tau}\right)\left|{{}_m}_{=0}\right. \), and type IIB superstring theory scattering amplitudes) to determine various perturbative terms in the large N and large ’t Hooft coupling λ expansion of the \( \mathcal{N} \) = 4 SYM correlator at separated points. In particular, we determine the leading large-λ term in the \( \mathcal{N} \) = 4 SYM correlation function at order 1/N⁸. This is three orders beyond the planar limit. A preprint version of the article is available at ArXiv.