A note on conical solutions in 3D Vasiliev theory

TL;DR

This work constructs a class of smooth conical solutions in the 3D Vasiliev higher-spin theory with gauge algebra $hs[\lambda]$, generalizing known $sl(N)$ conical defects and organizing them via diagonal $\star$-projectors labeled by Young diagrams. It provides a concrete matrix realization of $hs[\lambda]$, analyzes holonomy-based smoothness, and shows that the leading large-$c$ charges of these solutions match the primary charges of the classical $\cal W_\infty[\lambda]$ algebra, identifying each $a_{\Lambda}$ with the primary $(\Lambda,0)_\lambda$ (and $a_{\bar{\Lambda}}$ with $(\overline{\Lambda},0)_\lambda$ in related sectors). The results connect to the Gaberdiel-Gopakumar-'t Hooft duality by analytic continuation $N\to\lambda$, clarifying how conical bulk states map to a wide class of minimal-model primaries in the large central charge limit, while highlighting the distinct semiclassical vs. 't Hooft limits and the role of triality in relating different regimes. The discussion outlines future directions, including extensions to capture the unitary 't Hooft limit and the thermodynamic structure of higher-spin gravity on the torus, such as conical BTZ branches, which may reflect the absence of a Hawking-Page transition in the dual CFT.

Abstract

We construct a class of smooth solutions in three-dimensional Vasiliev higher spin theories based on the gauge algebra hs[λ]. These solutions naturally generalize the previously constructed conical defect solutions in higher spin theories with sl(N) gauge algebra, to which they reduce when λis taken to be equal to N. We provide evidence for their identification with specific primary states of the W_\infty [λ] algebra in a particular classical limit. In terms of the Gaberdiel-Gopakumar-'t Hooft limit of the W_N minimal models, this limit corresponds to a regime where the 't Hooft coupling becomes large.