Duality in N=2 minimal model holography

TL;DR

This paper resolves the quantum structure of the ${\\cal N}=2$ ${\\cal W}_\\infty$ algebra as the DS reduction of $\\mathrm{shs}[\\mu]$, and connects it to the Kazama-Suzuki cosets underlying the duality with ${\\cal N}=2$ higher-spin gravity. It demonstrates that the algebra is fixed by the central charge $c$ and a single coupling $\\gamma$, and that multiple $\\mu$ values yield the same algebra, establishing a fourfold degeneracy and ensuring quantum symmetry equivalence at finite $N,k$, not just in the 't Hooft limit. The precise relation $\\gamma(\\mu,c) = ...$ links the bulk higher-spin data to boundary CFT data, and the wedge subalgebra check confirms the consistency with the $\\mathrm{shs}[\\mu]$ DS reduction. The analytic continuation to large $c$ clarifies the perturbative vs non-perturbative nature of dual scalar fields and aligns with alternative quantization and level-rank dualities in KS models, offering concrete predictions for quantum corrections in the higher-spin theory.

Abstract

Recently a duality between a family of \mathcal{N}=2 supersymmetric higher spin theories on AdS3, and the 't Hooft like limit of a class of Kazama-Suzuki models (that are parametrised by N and k) was proposed. The higher spin theories can be described by a Chern-Simons theory based on the infinite-dimensional Lie algebra shs[μ], and under the duality, μis to be identified with λ=N/(N+k+1). Here we elucidate the structure of the (quantum) asymptotic symmetry algebra sW_{\infty}[μ] for arbitrary μand central charge c. In particular, we show that for each value of the central charge, there are generically four different values of μthat describe the same sW_{\infty} algebra. Among other things this proves that the quantum symmetries on both sides of the duality agree; this equivalence does not just hold in the 't Hooft limit, but even at finite N and k.