$\mathcal{N}=4$ Super-Yang-Mills Correlators at Strong Coupling from String Theory and Localization

We compute $1/\lambda$ corrections to the four-point functions of half-BPS operators in $SU(N)$ $\mathcal{N}=4$ super-Yang-Mills theory at large $N$ and large 't Hooft coupling $\lambda=g_\text{YM}^2 N$ using two methods. Firstly, we relate integrals of these correlators to derivatives of the mass deformed $S^4$ free energy, which was computed at leading order in large $N$ and to all orders in $1/\lambda$ using supersymmetric localization. Secondly, we use AdS/CFT to relate these $1/\lambda$ corrections to higher derivative corrections to supergravity for scattering amplitudes of Kaluza-Klein scalars in IIB string theory on $AdS_5\times S^5$, which in the flat space limit are known from worldsheet calculations. These two methods match at the order corresponding to the tree level $R^4$ interaction in string theory, which provides a precise check of AdS/CFT beyond supergravity, and allow us to derive the holographic correlators to tree level $D^4R^4$ order. Combined with constraints from arXiv:1809.10670, our results can be used to derive CFT data to one-loop $D^4R^4$ order. Finally, we use AdS/CFT to fix these correlators in the limit where $N$ is taken to be large while $g_{\rm YM}$ is kept fixed. In this limit, we present a conjecture for the small mass limit of the $S^4$ partition function that includes all instanton corrections and is written in terms of the same Eisenstein series that appear in the study of string theory scattering amplitudes.