One-loop amplitudes in $AdS_5\times S^5$ supergravity from $\mathcal{N}=4$ SYM at strong coupling

TL;DR

This work presents a general algorithm to construct one-loop AdS$_5\times$S$^5$ supergravity amplitudes for four arbitrary KK states, recasting the problem in terms of dual N=4 SYM four-point functions at strong coupling and large $N$. The approach separates the dynamical one-loop function into a generalised tree-level piece $\mathcal{T}^{(2)}$ and a minimal loop piece $\mathcal{H}^{(2)}$, with the latter encoding all nontrivial OPE data and crossing-consistent structures; the former is uniquely fixed by free-theory data and tree-level constraints. A detailed bootstrap is performed, using leading log resummations, window and below-window OPE data, and semishort multiplet recombination to determine $\mathcal{H}^{(2)}$ for several multi-channel correlators, including $\langle O_3 O_3 O_3 O_3\rangle$, $\langle O_4 O_4 O_4 O_4\rangle$, and next-to-next-to-extremal cases, with results cross-checked via Mellin-space representations and the 10d conformal symmetry encoded by $\Delta^{(8)}$ and $\widehat{\mathcal{D}}_{pqrs}$. The paper demonstrates how 1/N^4 corrections are controlled by a mix of long and protected sectors, with degeneracies lifted at one loop, and provides a unified framework that generalises prior results and paves the way for higher-loop explorations and connections to 10d symmetry structures.

Abstract

We explore the structure of maximally supersymmetric Yang-Mills correlators in the supergravity regime. We develop an algorithm to construct one-loop supergravity amplitudes of four arbitrary Kaluza-Klein supergravity states, properly dualised into single-particle operators. We illustrate this algorithm by constructing new explicit results for multi-channel correlation functions, and we show that correlators which are degenerate at tree level become distinguishable at one-loop. The algorithm contains a number of subtle features which have not appeared until now. In particular, we address the presence of non-trivial low twist protected operators in the OPE that are crucial for obtaining the correct one-loop results. Finally, we outline how the differential operators $\widehat{\mathcal{D}}_{pqrs}$ and $Δ^{(8)}$, which play a role in the context of the hidden 10d conformal symmetry at tree level, can be used to reorganise our one-loop correlators.

Figures (2)

• Figure 1: The large $N$ structure of $C_{p_1p_2,\vec{\tau}}\,C_{p_3p_4,\vec{\tau}}$ for two particle operators $\mathcal{O}_{\tau}$ in an $su(4)$ representation $[aba]$, and varying twist.
• Figure 2: Illustration of the possible $su(4)$ representations exchanged in the overlap of the $(\mathcal{O}_{p_1} \times \mathcal{O}_{p_2})$ and $(\mathcal{O}_{p_3} \times \mathcal{O}_{p_4})$ OPEs. The vertical axis represents the possible values of $b+2a$.