Triality in Minimal Model Holography

TL;DR

This work determines the quantum deformation of the classical ${\cal W}_\infty[\mu]$ algebra at finite central charge by enforcing Jacobi identities, yielding a complete description in terms of $(c,\gamma)$ with $\mu$ fixed by a cubic relation. A striking triality emerges: three distinct $\mu$ values give isomorphic ${\cal W}^{\rm qu}_{\infty}[\mu]$ algebras, including the $\mu=N$ truncation to ${\cal W}_N$ and the bulk hs$[\lambda]$ theory at finite $N,k$. The authors then analyze minimal representations and their fusion, showing how analytic continuation in $c$ connects light states in the boundary CFT to conical defect configurations in the bulk, and revealing a non-perturbative role for one bulk scalar. A refined holographic picture is proposed in which the bulk theory is hs$[\lambda]$ with a single complex scalar, while certain states are interpreted as non-perturbative excitations of semiclassical defects, providing a consistent finite-$N$ duality and guiding future bulk calculations and supersymmetric extensions.

Abstract

The non-linear W_{\infty}[μ] symmetry algebra underlies the duality between the W_N minimal model CFTs and the hs[μ] higher spin theory on AdS_3. It is shown how the structure of this symmetry algebra at the quantum level, i.e. for finite central charge, can be determined completely. The resulting algebra exhibits an exact equivalence (a`triality') between three (generically) distinct values of the parameter μ. This explains, among other things, the agreement of symmetries between the W_N minimal models and the bulk higher spin theory. We also study the consequences of this triality for some of the simplest W_{\infty}[μ] representations, thereby clarifying the analytic continuation between the`light states' of the minimal models and conical defect solutions in the bulk. These considerations also lead us to propose that one of the two scalar fields in the bulk actually has a non-perturbative origin.