Symmetries of Holographic Minimal Models

TL;DR

The paper establishes a precise bulk–boundary dictionary for holographic minimal models by showing that the asymptotic symmetry algebra of the bulk higher-spin theory based on $hs[\lambda]$ is the nonlinear $\mathcal{W}_\infty[\lambda]$, which at $\lambda=1$ linearizes to $\mathcal{W}_\infty^{\rm PRS}$. It then demonstrates that the wedge algebra of this $\mathcal{W}$-algebra coincides with $hs[\lambda]$ and that the large-$N$ limit of the minimal model cosets carries representations of $hs[\lambda]$, with matching characters and spin-3 zero-mode eigenvalues. Together, these results provide a nontrivial consistency check of minimal-model holography, clarify the bulk/boundary map, and situate the symmetry structure within the KP hierarchy-inspired family $\mathcal{W}_\infty[\lambda]$. The work thus links AdS$_3$ higher-spin gravity, $W$-algebras from integrable systems, and the representation theory of large-$N$ minimal models in a coherent holographic framework.

Abstract

It was recently proposed that a large N limit of a family of minimal model CFTs is dual to a certain higher spin gravity theory in AdS_3, where the 't Hooft coupling constant of the CFT is related to a deformation parameter of the higher spin algebra. We identify the asymptotic symmetry algebra of the higher spin theory for generic 't Hooft parameter, and show that it coincides with a family of W-algebras previously discovered in the context of the KP hierarchy. We furthermore demonstrate that this family of W-algebras controls the representation theory of the minimal model CFTs in the 't Hooft limit. This provides a non-trivial consistency check of the proposal and explains part of the underlying mechanism.