Genus-2 Holographic Correlator on $AdS_5 \times S^5$ from Localization

TL;DR

This work computes the four-point function of the stress-tensor multiplet in $ ext{N}=4$ SYM at large $N$ and large $ au$, via its AdS$_5$/CFT$_4$ holographic dual, combining supersymmetric localization and Mellin-space bootstrap with the flat-space limit. The authors introduce an all-orders $1/N^2$ expansion for the $ ext{N}=2^*$ free energy using topological recursion and use it to fix the genus-zero and genus-one $R^4$ corrections, complete the 1-loop supergravity term, and determine the genus-two $D^4R^4$ correction, marking the first explicit holographic computation at genus two. They also extract unprotected CFT data to $O(c^{-3})$, including anomalous dimensions for the lowest-twist double-trace operators, and verify spin-analyticity properties in this regime. The results provide nontrivial checks against the IIB S-matrix in the flat-space limit and illustrate how localization constraints can fix higher-genus contributions to holographic correlators, with potential applications to broader holographic theories and numerical bootstrap approaches.

Abstract

We consider the four-point function of the stress tensor multiplet superprimary in $\mathcal{N}=4$ super-Yang-Mills (SYM) with gauge group $SU(N)$ in the large $N$ and large 't Hooft coupling $λ\equiv g_\text{YM}^2N$ limit, which is holographically dual to the genus expansion of IIB string theory on $AdS_5\times S^5$. In \cite{Binder:2019jwn} it was shown that the integral of this correlator is related to derivatives of the mass deformed $\mathcal{N}=2^*$ sphere free energy, which was computed using supersymmetric localization to leading order in $1/N^2$ for finite $λ$. We generalize this computation to any order in $1/N^2$ for finite $λ$ using topological recursion, and use this any order constraint to fix the $R^4$ correction to the holographic correlator to any order in the genus expansion. We also use it to complete the derivation of the 1-loop supergravity correction, and show that analyticity in spin fails at zero spin in the large $N$ expansion as predicted from the Lorentzian inversion formula. In the flat space limit, the $R^4$ term in the holographic correlator matches that of the IIB S-matrix in 10d, which is a precise check of AdS$_5$/CFT$_4$ for local operators at genus-one. Using the flat space limit and localization we then fix $D^4R^4$ in the holographic correlator to any order in the genus expansion, which is nontrivial at genus-two, i.e. $1/N^6$. This is the first result at two orders beyond the planar limit at strong coupling for a holographic correlator.