Correlation Functions in Holographic Minimal Models
Kyriakos Papadodimas, Suvrat Raju
TL;DR
This work probes the Gaberdiel–Gopakumar duality between ${\cal W}_N$ minimal models and a higher-spin AdS$_3$ theory by computing exact four-point functions of the boundary operators dual to bulk scalars and extracting three-point couplings. It introduces two computational frameworks—the Coulomb-gas free-field representation and a coset-to-WZW reduction—to obtain explicit correlators, including mixed and double-trace channels, and demonstrates that the boundary theory exhibits a controlled $1/N$ expansion despite a dense additional light sector. A central finding is that, at leading order in $1/N$, the ordinary (bulk-like) sector decouples, but at finite $N$ the ordinary and additional sectors couple via nontrivial three-point functions such as $C^{\omega}_{\phi_+\phi_-} \sim 1/N$, and descendants of $\omega$ contribute to tree-level four-point functions. This implies that reproducing the boundary correlators within the bulk requires augmenting the bulk theory with additional light fields (e.g., $\omega$) and interactions, potentially modifying the naive higher-spin dual; the paper also outlines a concrete bulk-interaction ansatz and discusses broader implications for bulk reconstruction and fusion-rule consistency.
Abstract
We compute exact three and four point functions in the W_N minimal models that were recently conjectured to be dual to a higher spin theory in AdS_3. The boundary theory has a large number of light operators that are not only invisible in the bulk but grow exponentially with N even at small conformal dimensions. Nevertheless, we provide evidence that this theory can be understood in a 1/N expansion since our correlators look like free-field correlators corrected by a power series in 1/N . However, on examining these corrections we find that the four point function of the two bulk scalar fields is corrected at leading order in 1/N through the contribution of one of the additional light operators in an OPE channel. This suggests that, to correctly reproduce even tree-level correlators on the boundary, the bulk theory needs to be modified by the inclusion of additional fields. As a technical by-product of our analysis, we describe two separate methods -- including a Coulomb gas type free-field formalism -- that may be used to compute correlation functions in this theory.