Holographic Four-Point Functions in the (2, 0) Theory

TL;DR

The paper develops a symmetry-driven framework to compute holographic four-point functions for the six-dimensional (2,0) theory realized by eleven-dimensional supergravity on $AdS_7\times S^4$, extending the position-space method of Rastelli et al. to higher KK levels. By combining a position-space construction (exchange plus contact diagrams with truncation) and a Mellin-space bootstrap anchored by the superconformal Ward identity, it obtains explicit results for $k=2,3,4$ one-half BPS four-point functions and recasts them as a homogeneous holomorphic piece plus a rational inhomogeneous term, with all three cases satisfying the expected chiral algebra constraints in the Beem–Beem–Lemos twist to $W_{\infty}$. The Mellin-space formulation reveals a compact relation $\mathcal{M}_k=\widehat{\Theta}\circ\widetilde{\mathcal{M}}_k$, where $\widehat{\Theta}$ is a difference operator and $\widetilde{\mathcal{M}}_k$ encodes the dynamical data, enabling a constrained bootstrap that is conjectured to determine the full one-half BPS sector. The results robustly connect holographic correlators to the $d=2$ chiral algebra structure and lay groundwork for extending the universal Mellin-space bootstrap to all KK levels in the $AdS_7\times S^4$ background, with potential insights into the large-$n$ (2,0) theory via the $W_{n\to\infty}$ algebra.

Abstract

We revisit the calculation of holographic correlators for eleven-dimensional supergravity on $AdS_7\times S^4$. Our methods rely entirely on symmetry and eschew detailed knowledge of the supergravity effective action. By an extension of the position space approach developed in [1, 2] for the $AdS_5\times S^5$ background, we compute four-point correlators of one-half BPS operators for identical weights $k=2, 3, 4$. The $k=2$ case corresponds to the four-point function of the stress-tensor multiplet, which was already known, while the other two cases are new. We also translate the problem in Mellin space, where the solution of the superconformal Ward identity takes a surprisingly simple form. We formulate an algebraic problem, whose (conjecturally unique) solution corresponds to the general one-half BPS four-point function.

Figures (1)

• Figure 1: Connected four-point Witten diagrams at tree level.