The semiclassical limit of W_N CFTs and Vasiliev theory

TL;DR

The paper studies the semiclassical limit of large-$c$ $W_N$ CFTs and their holographic dual via 3d Vasiliev theory at $\lambda=-N$, arguing that bulk configurations are organized by an $sl(N)_{\Lambda^-}$ twisted wedge algebra and that the conical surplus saddles correspond to the CFT primary $|0,\Lambda^-\rangle$. It then shows that light states in the 't Hooft limit emerge as bound states of perturbative scalars with these surpluses, not as defects alone, and provides a detailed bulk–boundary matching through 1-loop partition functions and scalar fluctuations. The work develops the scalar sector at $\lambda=-N$, demonstrates non-unitary finite-dimensional representations of the bulk isometries, and constructs a consistent dictionary between bulk multiparticle states and CFT primaries $(\Lambda^+,\Lambda^-)$. Overall, it offers a concrete, calculable realization of a non-unitary semiclassical higher-spin holography, with explicit matches between bulk determinants and boundary characters and clear directions for extending the duality. The results illuminate how higher-spin symmetries organize nonperturbative sectors and how scalar matter encodes the light states in a controlled AdS$_3$/CFT$_2$ setting.

Abstract

We propose a refinement of the Gaberdiel-Gopakumar duality conjecture between W_N conformal field theories and 2+1-dimensional higher spin gravity. We make an identification of generic representations of the W_N CFT in the semiclassical limit with bulk configurations. By studying the spectrum of the semiclassical limit of the W_N theories and mapping to solutions of Euclidean Vasiliev gravity at λ=-N, we propose that the `light states' of the W_N minimal models in the 't Hooft limit map not to the conical defects of the Vasiliev theory, but rather to bound states of perturbative scalar fields with these defects. Evidence for this identification comes from comparing charges and from holographic relations between CFT null states and bulk symmetries. We also make progress in understanding the coupling of scalar matter to sl(N) gauge fields.

Figures (1)

• Figure 1: Large $c$ behavior of Virasoro ($N=2$) maximally degenerate representations. The plot shows $h/c$ as a function of $c$. The red lines (top in each band) are the first three $(0, \Lambda^-)$ representations (corresponding to Young diagrams with $0,1,2$ boxes), while the blue lines denote the states $(\Lambda^+, \Lambda^-)$ for the first four representations $\Lambda^+$ (Young diagrams with $0,1,2,3$ boxes).