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 $\tau$ is placed on a round four-sphere and deformed by an $\mathcal{N}=2$-preserving mass parameter $m$, its free energy $F(m,\tau ,\bar{\tau })$ can be computed exactly using supersymmetric localization. In this work, we derive a new exact relation between the fourth derivative ${\partial }_m^{4}F(m,\tau ,\bar{\tau }){|}_{m=0}$ 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 ${\partial }_{\tau }{\partial }_{\bar{\tau }}{\partial }_m^{2}F(m,\tau ,\bar{\tau }){|}_{m=0}$, and type IIB superstring theory scattering amplitudes) to determine various perturbative terms in the large $N$ and large 't Hooft coupling $\lambda$ expansion of the $\mathcal{N}=4$ SYM correlator at separated points. In particular, we determine the leading large-$\lambda$ term in the $\mathcal{N}=4$ SYM correlation function at order $1/N^8$. This is three orders beyond the planar limit.

arXiv comment25 pages plus appendices, v2 submitted to JHEP, minor typos fixed

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