Modular invariant holographic correlators for $\mathcal{N}=4$ SYM with general gauge group

Abstract

We study the stress tensor four-point function for $\mathcal{N}=4$ SYM with gauge group $G=SU(N)$, $SO(2N+1)$, $SO(2N)$ or $USp(2N)$ at large $N$. When $G=SU(N)$, the theory is dual to type IIB string theory on $Ad{S}_5\times S^5$ with complexified string coupling ${\tau }_s$, while for the other cases it is dual to the orbifold theory on $Ad{S}_5\times S^5/{\mathbb{Z}}_2$. In all cases we use the analytic bootstrap and constraints from localization to compute 1-loop and higher derivative tree level corrections to the leading supergravity approximation of the correlator. We give perturbative evidence that the localization constraint in the large $N$ and finite complexified coupling $\tau$ limit can be written for each $G$ in terms of Eisenstein series that are modular invariant in terms of ${\tau }_s\propto \tau$, which allows us to fix protected terms in the correlator in that limit. In all cases, we find that the flat space limit of the correlator precisely matches the type IIB S-matrix. We also find a closed form expression for the $SU(N)$ 1-loop Mellin amplitude with supergravity vertices. Finally, we compare our analytic predictions at large $N$ and finite $\tau$ to bounds from the numerical bootstrap in the large $N$ regime, and find that they are not saturated for any $G$ and any $\tau$, which suggests that no physical theory saturates these bootstrap bounds.

58 pages, 3 figures, v2: clarifications about USp(2N), include numerical bootstrap data, v3: clarifications about modular invariance, include 1-loop Mellin amplitude in ancillary file, v4: JHEP version, v5: add journal ref

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• The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators
• Modular-invariant large-$N$ completion of an integrated correlator in $\mathcal{N}=4$ supersymmetric Yang-Mills theory
• Generating functions and large-charge expansion of integrated correlators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory
• Bootstrapping 4d $\mathcal{N}=2$ gauge theories: the case of SQCD
• Integrated correlators in $\mathcal{N}=4$ super Yang-Mills and periods
• The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals
• Laplace-difference equation for integrated correlators of operators with general charges in $\mathcal{N}=4$ SYM
• Exact strong coupling results in $\mathcal{N}$=2 $Sp(2N)$ superconformal gauge theory from localization
• Pure anti-de Sitter supergravity and the conformal bootstrap
• Integrated correlators at strong coupling in an orbifold of $\mathcal{N}=4$ SYM
• Holographic S-fold theories at one loop
• The Kaluza-Klein AdS Virasoro-Shapiro Amplitude near Flat Space
• Integrated correlators of coincident Wilson lines in $SU(N)$ gauge theories at strong coupling
• 't Hooft expansion of $SO(N)$ and $Sp(N)$ $\mathcal{N}=4$ SYM revisited