Genus-2 Holographic Correlator on $Ad{S}_5\times S^5$ from Localization

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 $\lambda \equiv g_{\text{YM}}^{2}N$ limit, which is holographically dual to the genus expansion of IIB string theory on $Ad{S}_5\times S^5$. In \cite{Binder:2019jwn} it was shown that the integral of this correlator is related to derivatives of the mass deformed $\mathcal{N}=2^{\ast }$ sphere free energy, which was computed using supersymmetric localization to leading order in $1/N^2$ for finite $\lambda$. We generalize this computation to any order in $1/N^2$ for finite $\lambda$ 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 $D^4{R}^4$ 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.

arXiv comment26 pages plus appendices, 1 figure, v4 typos corrected

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